Systems and Control Theory An Introduction A. Astolfi First draft – September 2006
Contents Preface
vii
1 Introduction
1
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2.1
Growth of a family of rabbits . . . . . . . . . . . . . . . . . . . . . .
2
1.2.2
Model of an infectious disease . . . . . . . . . . . . . . . . . . . . . .
2
1.2.3
A scholastic population . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.4
A tractortrailer system . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2.5
A simplified atmospheric model . . . . . . . . . . . . . . . . . . . . .
5
1.2.6
A perspective vision system . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.7
The ABS system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.8
A simplified guitar string . . . . . . . . . . . . . . . . . . . . . . . .
8
1.2.9
Approximate discretetime models . . . . . . . . . . . . . . . . . . .
9
1.2.10 Google page rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3
1.4
The notion of system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.1
Parametric representations . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2
Causal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.3
The notion of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.4
Definition of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.5
Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.6
Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Examples revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 iii
iv
CONTENTS
2 Stability
25
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2
Existence and unicity of solutions . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3
Trajectory, motion and equilibrium . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.1
Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4
Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5
Lyapunov stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5.2
Stability of linear systems . . . . . . . . . . . . . . . . . . . . . . . . 46
2.6
Coordinates transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.7
Stability in the first approximation . . . . . . . . . . . . . . . . . . . . . . . 53
3 The structural properties
57
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2
Reachability and controllability . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3
3.2.1
Reachability of discretetime systems . . . . . . . . . . . . . . . . . . 59
3.2.2
Controllability of discretetime systems . . . . . . . . . . . . . . . . 62
3.2.3
Construction of input signals . . . . . . . . . . . . . . . . . . . . . . 63
3.2.4
Reachability and controllability of continuoustime systems . . . . . 64
3.2.5
A canonical form for reachable systems . . . . . . . . . . . . . . . . 68
3.2.6
Description of nonreachable systems . . . . . . . . . . . . . . . . . . 71
3.2.7
PBH reachability test . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Observability and reconstructability . . . . . . . . . . . . . . . . . . . . . . 74 3.3.1
Observability of discretetime systems . . . . . . . . . . . . . . . . . 75
3.3.2
Reconstructability of discretetime systems . . . . . . . . . . . . . . 77
3.3.3
Computation of the state . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3.4
Observability and reconstructability for continuoustime systems . . 80
3.3.5
Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4 Design tools
89
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2
The notion of feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3
State feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
CONTENTS
v
4.3.1
Stabilizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4
The notion of filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.5
State observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.5.1
Detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5.2
Reduced order observer . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.6
The separation principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.7
Tracking and regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.7.1
The full information regulator problem . . . . . . . . . . . . . . . . . 104
4.7.2
The FBI equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.7.3
The error feedback regulator problem . . . . . . . . . . . . . . . . . 107
4.7.4
The internal model principle . . . . . . . . . . . . . . . . . . . . . . 110
Exercises
I
vi
CONTENTS
Preface These lecture notes are intended as a reference for a 3rd year, 20 hours ”Control Engineering” course at the EEE Department of Imperial College London. In addition, they can be partly used as support material for a 50 hours ”Automatic Control” course at the University of Rome Tor Vergata. (The reason why similar topics are taught in such a diverse time range is, and will probably remain, a mistery to me, even after various years of experience with students and courses from both institutions!) In both cases, this is the second controllike course taken by the students. The main goal of these notes is to provide a selfcontained and rigorous background on systems theory and an introduction to state space analysis and design methods for linear systems. In preparing these notes I was deeply influenced by the approach pursued in the book ”Teoria dei sistemi”, by A. Ruberti and A. Isidori (Boringheri, 1985) and by my research experience on nonlinear control theory. Different approaches can be pursued, and the socalled behavioural approach, developed by J. Willems, is (in my opinion) the best alternative. These notes are organized as follows. I start giving an abstract mathematical description of the notion of system, illustrated by several pratical examples (Chapter 1). I then move to the study of properties of autonomous systems (Chapter 2), and then to the study of the inputtostate, the statetooutput, and the inputtooutput interactions (Chapter 3). Finally, I discuss a few basic design tools (Chapter 4). I try to deal at the same time with continuous and discretetime systems, pointing out similarities and differences. The reader should be familiar with standard calculus and linear algebra. I hope the reader will find these notes a valuable starting point to proceed in more advanced areas of systems and control theory. In particular, these notes should provide the necessary tools for the 4th year control courses and the Control M.Sc. course at Imperial College London. I am aware that there are several excellent books where the same topics are dealt with in detail. The idea of these notes is to provide a condensed, yet precise, introduction to systems and control theory amenable for a short second (even first) undergraduates systems and control course, and not to be a substitute for more indepth study.
vii
viii
PREFACE
Chapter 1
Introduction
2
CHAPTER 1. INTRODUCTION
1.1
Introduction
Aim of this chapter is to introduce the notion of system. In the (abstract) definition of system, we follow a simple and natural approach, the socalled inputoutput approach, which is motivated by the study of simple systems. Among the various mathematical representations we give particular emphasys to the one based on the introduction of an auxiliary variable, the state variable, which is denoted as state space representation. This representation plays a fundamental role in systems and control theory, hence we discuss in detail its properties and the conditions under which it can be introduced.
1.2
Examples
In this section we consider a few motivating examples, which are instrumental to introduce the notion of (abstract) system.
1.2.1
Growth of a family of rabbits
The number of pairs of rabbits n months after a single pair begins breeding (and newly born bunnies are assumed to begin breeding when they are two months old) is given by the socalled Fibonacci numbers, which are recursively defined as F0 = 0
F1 = 1
Fn = Fn−1 + Fn−2 .
(1.1)
Interestingly • the Fibonacci number Fn+1 gives the number of ways for 2 × 1 dominoes to cover a 2 × n checkerboard; • the Fibonacci number Fn+2 gives the number of ways of picking a set (including the empty set) from the numbers 1, 2, ... n, without picking two consecutive numbers; • the probability of not getting two heads in a row in n tosses of a coin is
Fn+2 2n .
Finally, given a resistor network of 1Ω resistors, each incrementally connected in series or parallel to the preceeding resistors, the net resistance is a rational number having maximum possible denominator equal to Fn+1 . This example shows that the same mathematical object models several physical situations or properties. This justifies therefore the study of the feature of the abstract object (1.1) without reference to the real situation that it represents.
1.2.2
Model of an infectious disease
There are several mathematical models which describe the interactions between HIV and immunocytes in human body. The most commonly used model for longterm excitement
3
1.2. EXAMPLES
of the immune response, and hence for medication purposes, is described by the equations x˙ = λ − dx − ηβxy
y˙ = ηβxy − ay − yI,
(1.2)
where x denotes the population of uninfected CD4 Thelper cell (in a unit volume of blood), y denotes the population of infected CD4 Thelper cell (in a unit volume of blood), I denotes the action of the immune system, and λ, d, β, η and a are positive parameters. The population of healthy (uninfected) CD4 Thelper cells (produced by thymus) increases at a rate λ, and decreases at a rate dx (since a cell dies naturally). The healthy CD4 Thelper cells are a target of HIV, hence its population decreases proportionally to x and y, because infected CD4 Thelper cells produce the virus, i.e. when a cell is infected it generates new virus. The infected cells die out at a rate ay, increase at a rate proportional to x and y and are affected by the immune system. In general, and without any medication, the model (1.2) has three main operating conditions. One which corresponds to a healthy patient, one which represents a patient with HIV but not with AIDS, and one which represents a patient in which AIDS dominates. It can be (rigorously) shown that the first two operating conditions are unstable, whereas the third one is stable (formal definitions of stability and instability will be given in Chapter 2). This justifies the difficulty in treating HIV infected patients.
1.2.3
A scholastic population
Consider a threeyear course and the problem of modelling the number of students in each year. Let • u(k) be the number of incoming first year students at time k; • y(k) be the number of graduated students at time k; • xi (k) be the number of students in the ith year at time k; • αi (k) ∈ [0, 1] be the rate of promotion in the ith year at time k. The behaviour of the students’ population can be described by the equations x1 (k + 1) = (1 − α1 (k))x1 (k) + u(k)
x2 (k + 1) = (1 − α2 (k))x2 (k) + α1 (k)x1 (k) x3 (k + 1) = (1 − α3 (k))x3 (k) + α2 (k)x2 (k) y(k)
(1.3)
= α3 (k)x3 (k).
In the ideal situation in which αi (k) = 1 for all k we have that y(k) = u(k − 3), which clearly shows that all incoming students are graduated after three years. Finally, in the extreme situation in which, αi (k) = 0 for all k and for some i, we have lim y(k) = 0.
k→∞
4
1.2.4
CHAPTER 1. INTRODUCTION
A tractortrailer system
Consider a vehicle (see Figure 1.1) consisting of a wheeled tractor with two reardrive wheels and a frontsteering wheel, towing a trailer, possibly with offaxle hitching. The offaxle length c has to be regarded as a variable with sign, being negative when the joint is in front of the wheel axle, and positive otherwise. L1 and L2 are constants depending on the geometry of the vehicle. The longitudinal speed v1 and the steering angle δ of the tractor can be (independently) manipulated so that the guidepoint P1 follows a desired path with an assigned velocity. Suppose that the vehicle has to follow, at a given speed, a circular path of radius R1 .
L1 Jo c
d
s
j J
v1 H
P1
L2
Jp lo
s
P0
P2
Figure 1.1: Vehicle’s geometry and pathtracking offsets los and ϑos . To describe the motion of the vehicle, define los and ϑos as the tractor lateral offset and its orientation offset, respectively. They are measured with reference to the projection of the point P1 of the tractor onto the path. Moreover, let ϕos = ϕ − ϕp be the difference between the current angle ϕ between tractor and trailer and its steady state value ϕp along the prescribed path. Pathtracking can be viewed as the task of driving these offsets asymptotically to zero. The offsets are such that l˙os
= −σ v1  sin ϑos
cos ϑos u − σ v1  ϑ˙ os = v1 L1 R1 + los v1 v1 ϕ˙ os = − sin (ϕos + ϕp ) − (c cos (ϕos + ϕp ) + L2 ) u, L2 L1 L2
5
1.2. EXAMPLES
where u = tan δ is the manipulated variable, and the parameter σ is used to distinguish between counterclockwise (σ = 1) or clockwise (σ = −1) directions.
In many applications, the absolute value of the steering angle δ is bounded by a saturation value δM < π/2.
1.2.5
A simplified atmospheric model
Consider a rectangular slice of air heated from below and cooled from above by edges kept at constant temperatures. This is our atmosphere in its simplest description. The bottom is heated by the earth and the top is cooled by the void of outer space. Within this slice, warm air rises and cool air sinks. In the model, as in the atmosphere, convection cells develop, transferring heat from bottom to top. The state of the atmosphere in this model can be completely described by three variables, namely the convective flow x, the horizontal temperature distribution y, and the vertical temperature distribution z; by three parameters, namely the ratio of viscosity to thermal conductivity σ, the temperature difference between the top and bottom of the slice ρ, and the width to height ratio of the slice β, and by three differential equations describing the appropriate laws of fluid dynamics, namely x˙ = σ(y − x)
y˙ = ρx − y − xz
z˙ = xy − βz.
(1.4)
These equations were introduced by E.N. Lorenz in 1963, to model the strange behaviour of the atmosphere and to justify why weather forecast can be erroneous, and have been recently shown to play an important role on models of lasers and electrical generators. Note that the Lorenz equations are still at the basis of modern weather forecast algorithms.
1.2.6
A perspective vision system
A classical problem in machine vision is to determine the position of an object moving in the threedimensional space by observing the motion of its projected feature on the twodimensional image space of a chargecoupled device camera. In this case, the problem of determining the object space coordinates reduces to the problem of estimating the depth (or range) of the object. The motion of an object undergoing rotation, translation and linear deformation can be described by the equation 2 6 4
x˙ 1 x˙ 2 x˙ 3
3
2
7 5
=4
6
a11 a12 a13 a21 a22 a23 a31 a32 a33
32 76 54
x1 x2 x3
3
2
7 6 5+4
b1 b2 b3
3 7 5,
(1.5)
where (x1 , x2 , x3 ) ∈ IR3 are the coordinates of the object in an inertial reference frame, with x3 being perpendicular to the camera image space, as shown in Figure 1.2. The parameters aij , bi , known as motion parameters, are timevarying and known.
6
CHAPTER 1. INTRODUCTION x2
y2
y1
x3
x1
Figure 1.2: Diagram of the perspective vision system.
Using the perspective (or “pinhole”) model for the camera, the measurable coordinates on the image space are given by y=
y1 , y2
′
=ǫ
x1 , x3
x2 x3
′
,
(1.6)
where ǫ is the focal length of the camera, i.e. the distance between the camera and the origin of the imagespace axes. The perspective estimation problem consists in reconstructing the coordinates x1 , x2 , x3 from measurements of the imagespace coordinates y1 , y2 .
1.2.7
The ABS system
Electronic Antilock Braking Systems (ABS) have recently become a standard for all modern cars. ABS can greatly improve the safety of a vehicle in extreme circumstances, as it maximizes the longitudinal tireroad friction while keeping large lateral forces, which guarantee vehicle steerability. For the preliminary modelling of braking systems, the socalled quartercar model is used (see Figure 1.3). The model is described by J ω˙ = rFx − Tb
mv˙ = −Fx
where • ω is the angular speed of the wheel; • v is the longitudinal speed of the vehicle body;
(1.7)
7
1.2. EXAMPLES
Figure 1.3: Quarter car vehicle model. • Tb is the braking torque; • Fx is the longitudinal tireroad contact force; • J, m and r are the moment of inertia of the wheel, the quartercar mass, and the wheel radius, respectively. The dynamic behavior is hidden in the expression of Fx , which depends on the variables v and ω, and can be approximated as follows Fx = Fz µ(λ, βt , θr ), where • Fz is the vertical force at the tireroad contact point; • λ is the longitudinal slip, defined as1 λ=
v − ωr ; max{ωr, v}
• βt is the wheel sideslip angle; • θr is a set of parameters which characterize the shape of the static function µ(λ, βt ; θr ) and which depend upon the road conditions. 1 By definition, λ ∈ [−1, 1]; during braking, though, as ωr ≤ v, the wheel slip is defined as λ = and λ ∈ [0, 1].
v−ωr v
8
1.2.8
CHAPTER 1. INTRODUCTION
A simplified guitar string
Consider a guitar string and assume that it can be modelled by n identical segments (linear lumped springs of unity mass) which interact by means of elastic forces (depending on a tension parameter k). Let xi , x˙ i and x ¨i be the position, velocity, and acceleration, respectively, of the ith segment. The string can be described by x ¨1 = −k(x1 − x2 ) x ¨2 = −k(x2 − x1 ) − k(x2 − x3 ) .. . x ¨i
= −k(xi − xi−1 ) − k(xi − xi+1 ) .. .
(1.8)
x ¨n = −k(xn − xn−1 ). This can be rewritten in compact form as 2
x ¨=
6 6 6 6 6 6 6 6 4
−k k 0 0 ... 0 k −2k k 0 ... 0 0 k −2k k ... 0 .. .. .. .. .. .. . . . . . . 0 ... 0 k −2k k 0 ... 0 0 k −k
3 7 7 7 7 7x 7 7 7 5
= Ax,
where x = [x1 , x2 , . . . , xn ]′ . The main frequency of oscillation of the string, hence the tune of the string, is a function of the square root of the largest nonzero eigenvalue of A, and this depends on k, hence on the tension on the string. Remark. A more precise model of a guitar string of length L is given by the socalled onedimensional wave equation ∂ 2 x(y, t) ρ ∂ 2 x(y, t) = , ∂y 2 T ∂t2
(1.9)
where y ∈ [0, L] denotes the position of a point on the string, x(y, t) the deformation of the point y at time t with respect to the rest position, ρ the mass per unit of length, and T the tension of the string. From this equation it is possible to obtain the given approximate model by considering the finite difference approximation of the derivative, namely x(y + h, t) − 2x(y, t) + x(y − h, t) ∂ 2 x(y, t) ≈ ∂y 2 h2 defining the variables x1 = x(0, t)
x2 = x(h, t)
···
xn = x(L, t),
and using the constraints x(L + h) − x(L) = 0
x(0) − x(−h) = 0.
9
1.2. EXAMPLES Note finally, that the wave equation admits a close solution given by nπy , x(y, t) = A sin ωn t sin L where ωn2 =
n2 π 2 T . L2 ρ ⋄
1.2.9
Approximate discretetime models
Consider the differential equation x˙ = f (x),
(1.10)
with x ∈ IRn , together with the initial condition x(0) = x0 , for some given x0 , and the problem of obtaining a solution x(t), for t ≥ 0, of such equation. With the exception of very specific examples, it is in general not possible to compute a closed form solution x(t). This implies that x(t) has to be computed numerically, i.e. the idea is to select a sequence of time instants 0 < t1 < t2 · · · and to construct a numerical algorithm yielding values x1 , x2 , · · · which approximate x(t1 ), x(t2 ), · · · . To this end, the simplest possible approach is to consider an equally spaced sequence of time instants, namely {0, τ, 2τ, · · · , kτ, · · · }, where τ > 0 is the socalled samplingtime, and approximate the time derivative with the first difference, namely x(kτ + τ ) − x(kτ ) . x(kτ ˙ )≈ τ The differential equation can thus be approximated by x(kτ + τ ) − x(kτ ) = f (x(kτ )) τ yielding the integration algorithm xk+1 = xk + τ f (xk ),
(1.11)
with k ≥ 0, and with x0 given. The equation (1.11) is known as the Euler discretetime approximation of the differential equation (1.10). The sequence {xk } obtained from the Euler approximate model is such that kxk − x(kτ )k ≤ τ ψ(k) where ψ(k) is a function of k which is not, in general, bounded. This implies that the use of the Euler approximation yields an error that can be reduced (under certain technical conditions) reducing the sampling interval τ but may become unbounded as k → ∞, i.e. the Euler approximate model cannot be used for long term prediction of the solution of differential equations.
10
1.2.10
CHAPTER 1. INTRODUCTION
Google page rank
The speed, and success, of Google can be attributed in large part to the efficiency of the search algorithm which, linked with a good hardware architecture, creates an excellent search engine. The main part of the search engine is PageRankT M , a system for ranking web pages developed by Google’s founders Larry Page and Sergey Brin at Stanford University. The main idea of the algorithm is the following. The web can be represented as an oriented (and sparse) graph in which the nodes are the web pages and the oriented paths between nodes are the hyperlinks. The basic idea of PageRank is to walk randomly on the graph assigning to each node a vote proportional to the frequency of return to the node. If xi (k) denotes the vote of the node i at time k, one has xi (k + 1) =
X
xj (k) , nj j:j→i
where nj is the number of nodes connected to the node i, for i = 1, . . . , N , where N ≈ 3.000.000.000. The graph, representing the web, is not strongly connected, therefore to improve the algorithm for computing the vote, one considers a random jump, with probability p (typically 0.15) to another node (i.e. another page). As a result xi (k + 1) = (1 − p)
N X xj (k) xj (k) +p . nj N j:j→i j=1
X
(1.12)
Collecting the variables xi in a vector x we can rewrite the equations for the votes in the form x(k + 1) = Ax(k), for some matrix A ∈ IRN ×N . It is not difficult to prove that the matrix A has one eigenvalue equal to one and all other eigenvalues λi are such that λi  < 1. This implies that (we will discuss this issue in detail, after introducing the notion of stability in Chapter 2) ¯, lim x(k) = x
k→∞
where all elements x ¯i of x ¯ are nonnegative and bounded. The vector x ¯ (after a certain normalization) is Google Page Rank. The computation of x ¯ is numerically very difficult, and it is performed once a month.
1.3
The notion of system
The discussion in Section 1.2 highlights the fact that it is possible to describe the behaviour of several objects, natural or artificial, by means of mathematical expressions (differential or difference equations) of diverse forms, and with diverse properties.
11
1.3. THE NOTION OF SYSTEM
The notion of system is thus introduced to provide tools to study such a wide variety of objects on the basis of their mathematical, hence abstract, description. Therefore, by definition, an abstract system is an entity which does not depend upon the physical properties of the associated object. This implies that it is possible to associate the same system to several different objects, and at the same time several systems can be associated to the same object (depending upon the properties that have to be investigated). We stress that the definition of an abstract notion of system has the advantage that it allows to interprete and study, within a unified framework, diverse phenomena and processes, and provides a unique language for several different areas of applications. However, because of its generality, it raises several difficult issues, which can be solved or addressed from several perspectives. In this notes we give a definition of system which is based on the consideration of the input and output signals. With this in mind, note that the simplest way of associating a system to an object is to consider all possible behaviours (as a function of time) of the input signals and of the corresponding output signals. This approach does not depend upon the physical properties of the signals and upon the mechanisms which determine such signals. Remark. Throughout these notes we assume that the objects under study are deterministic. Similar considerations can be performed in a probabilistic setting. These however require somewhat more sophisticated mathematical tools. ⋄ The process of association of a system to an object can be regarded as the collection of data from experiments performed on the object thought of as a black box. The experiments can be carried out as follows: fix an initial time instant t0 , consider a possible input signal for all t ≥ t0 and the corresponding output signals. In this way we collect one or more pairs of functions, denoted as inputoutput pairs, which are defined for all t ≥ t0 . Collecting together all such pairs we have a set of inputoutput pairs, which is used to obtain the definition of system. In particular, if we consider the set U of all input signals and the set Y of all output signals, we have that all inputoutput pairs determine a relation S which is such that S ⊂ U × Y. This implies that the natural way of giving a formal definition of system is to define an abstract system as a set of relations, where each relation describes all inputoutput pairs obtained from experiments performed starting from a given time instant. In particular, consider an ordered subset T of the set IR, which is the set of time instants of interest for the system, and define the subset of future time instants2 F (t0 ) = {t ∈ T  t ≥ t0 }; 2
F stands for “future”.
12
CHAPTER 1. INTRODUCTION
the set U F (t0 ) of all input functions defined for t ≥ t0 , and the set Y F (t0 ) of all output functions defined for t ≥ t0 . Then, a relation St0 ⊂ U F (t0 ) × Y F (t0 ) can be used to describe all experiments, hence all inputouput pairs, starting at t0 . From the above discussion we conclude that an abstract system can be defined as the set of all relations St0 for all t0 ∈ T . Note however that the sets St0 and St1 , for t1 > t0 are not independent, because we can consider some of the pairs in St1 as obtained from experiments started at t0 and disregarding all data for t < t1 . A formal definition of system must therefore take this issue into consideration. Example 1.1 Consider a synchronous Dtype flipflop with input u and output y. The set T is the set of all time instants in which the clock goes high, the set of all input values is U = [0, 1], the set of all output values is Y = [0, 1], and the set of all input and output functions is such that U F (t0 ) = Y F (t0 ) and this is the set of all sequences of ’1’s and ’0’s. To understand the structure of the relation St0 we have to consider the behaviour of the flipflop. For example, for some given t0 , the pair of input and output sequences {′ 0011′ ,′ 0010′ } does not belong to St0 , whereas the pairs {′ 0011′ ,′ 0001′ }
{′ 0011′ ,′ 1001′ }
do. This is consistent with the fact that the output lags the input of one clock cycle, that the initial value of y cannot be determined from the current input and that the value of u at some time instant does not affect y at the same time instant. This implies that all pairs of input and output sequence of the form {′ 001˜ x′ ,′ x ˆ001′ } with x ˜ and x ˆ either ’1’ or ’0’ belong to St0 . Note finally that, for a given input sequence we can generate several (two in this example) output sequences. Definition 1.1 Consider an ordered subset T of IR and two (nonempty) sets U and Y . An abstract system is a set of relations S = {St0 ⊂ U F (t0 ) × Y F (t0 )  t0 ∈ T } such that3 for all t0 ∈ T and for all t1 ∈ F (t0 ) (u0 , y0 ) ∈ St0 ⇒ (u0 F (t1 ) , y0 F (t1 ) ) ∈ St1 .
(1.13)
For this system T is the set of time instants, U the set of values of the input signal, and Y the set of values of the output signal. 3
uF (t1 ) denotes the restriction of u to t ∈ F (t1 ).
13
1.3. THE NOTION OF SYSTEM
Condition (1.13) implies that the relation St1 contains all inputoutput pairs which are obtained truncating any other inputoutput pair originated at a time instant t0 ≤ t1 . Note however, that the relation St1 may contain inputoutput pairs which cannot be obtained by truncation of other pairs. There is, however, a class of systems, of special interest in applications, for which every relation contains only pairs obtained by truncation. This class is characterized as follows. Definition 1.2 A system is uniform if there exists a relation S ⊂ UT × Y T
(1.14)
such that for all t0 ∈ T (u, y) ∈ S ⇒ (uF (t0 ) , yF (t0 ) ) ∈ St0 and (u0 , y0 ) ∈ St0 ⇒ ∃(u, y) ∈ S : (uF (t0 ) , yF (t0 ) ) = (u0 , y0 ). A uniform system can therefore be assigned by means of the relation S which, roughly speaking, allows to generate all relations St0 for t0 ∈ T . Example 1.2 Consider an ideal and instantaneous quantizer, i.e. a device which receives at its input a signal u and delivers at its output a signal y, which is obtained via quantization, with a certain quantization interval q. For such a system T = IR, U = IR and Y = {· · · , −2q, −q, 0, q, 2q, · · · }. The relation St0 is given in Figure 1.4, and it is not hard to argue that the system is uniform.
1.3.1
Parametric representations
The use of relations to represent a system provides a very powerful point of view which is applicable to a very general set of objects. It is however interesting to study if it is possible to determine all inputoutput pairs by means of functions. To this end, we recall the following basic result. Lemma 1.1 Consider two nonempty sets A and B and a relation R ⊂ A × B. Then it is always possible to define a set C and a function4 f : C × DR → RR 4
DR denotes the domain of R, and RR denotes the range of R.
14
CHAPTER 1. INTRODUCTION
y 3q 2q q
u q 2q 3q
Figure 1.4: The relation St0 for an ideal quantizer.
such that and
(a, b) ∈ R ⇒ ∃c ∈ C : b = f (c, a) c ∈ C, a ∈ DR ⇒ (a, f (c, a)) ∈ R.
This lemma shows that the function f can be used to specify all, and only, the pairs in the relation R. The set C is the set of parameters and the function f is a parametric representation of R. Finally, the association of f and C to R is said parameterization. The result expressed in Lemma 1.1 can be used to obtain parametric representations for a system S. To this end, for any relation St0 it is possible to perform a parameterization, i.e. it is possible to define a set of parameters Xt0 and a function ft0 : Xt0 × DSt0 → RSt0 . It is then possible to define a parametric representation of the system S by means of a set of functions F = {ft0 : Xt0 × DSt0 → RSt0  t0 ∈ T }. Note that, for uniform systems, described by the relation (1.14), it is enough to consider a single function f : X × DS → RS.
Remark. For given x0 ∈ Xt0 and given u0 ∈ DSt0 , the function ft0 can be used to compute the output of the system as y = ft0 (x0 , u0 ). ⋄
1.3. THE NOTION OF SYSTEM
1.3.2
15
Causal systems
The class of systems considered so far are socalled oriented, i.e. there is a natural flow of information from the input to the output. This implies that we regard the input as a cause and the output as a consequence. However, these quantities are functions of time. It is therefore natural to study the relationship between the observed effect at time t¯ and the time evolution of the causes. Because abstract systems are often used to describe the behaviour of physical objects or processes, it is natural to consider that the above relation be causal, i.e. the output at time t¯ should depend only upon the input at times t < t¯, or possibly upon the input at times t ≤ t¯. It is not easy to formalize this idea to inputoutput relations. The simplest way is to resort to the parametric representation of the system and to consider the following definition. Definition 1.3 A system S is causal if it possesses at least one parametric representation F which is such that for all t0 ∈ T , for all x0 ∈ Xt0 and for all t¯ ∈ F (t0 ) u[t0 ,t¯] = u′ [t0 ,t¯] ⇒ ft0 (x0 , u)(t¯) = ft0 (x0 , u′ )(t¯). A system S is strictly causal if it possesses at least one parametric representation F which is such that for all t0 ∈ T , for all x0 ∈ Xt0 and for all t¯ ∈ F (t0 ) u[t0 ,t¯) = u′ [t0 ,t¯) ⇒ ft0 (x0 , u)(t¯) = ft0 (x0 , u′ )(t¯). We stress that the difference between the notions of causality and strict causality is only on the constraint on u. In the former case u and u′ have to be identical for all t ∈ [t0 , t¯], in the latter for all t ∈ [t0 , t¯).
1.3.3
The notion of state
The crucial point in the definition of abstract system by means of a relation is the fact that, in the relation St0 , to any given input signal we can associate several output signals. To single out one output signal it is thus necessary to specify, besides the input function, further information. This infomation is associated to the notion of state. To understand this notion, recall that we have associated, via the process of parameterization, to each relation St0 a function which associates to an input signal and a parameter x0 ∈ Xt0 a single output signal. The parameter x0 may be therefore regarded as the additional piece of information needed to specify the output signal in a unique way. However, to give a formal and precise, hence useful, definition, we have to make sure that the parameterizations performed at each time instant be related in some way, i.e. they cannot be independent but must satisfy a socalled consistency condition. This consistency property will be discussed in the framework of causal systems. To begin with, note that the set Xt0 , associated with the parameterization St0 , may be different from the set Xt1 , associated with the parameterization St1 , and so on for all
16
CHAPTER 1. INTRODUCTION
t ∈ T . It is therefore convenient to define a unique set X such that all sets Xt¯, with t¯ ∈ T , are subsets of X. Consider now an element x0 ∈ X, thought of as an element of Xt0 , which is the set of parameters associated to the parameterization of St0 , and a second element x1 ∈ X, thought of as an element of Xt1 , which is the set of parameters associated to the parameterization of St1 , with t1 > t0 . If the system is causal there should be a relation between x1 and x0 . In particular, if we assume that x0 depends only upon the input values for t < t0 and x1 depends only upon the input values for t < t1 , then, recalling that t1 > t0 , it is natural to assume that x1 depends upon the input values for t ∈ [t0 , t1 ) and upon x0 , and this can be written as x1 = φ(t1 , t0 , x0 , u[t0 ,t1 ) ) for some function φ5 . The discussion above can be extended to any pair of elements in X, hence it is possible to define the function φ for all pairs of elements of X, for all pairs {t0 , t1 } such that t0 < t1 , and once t0 is fixed to all u ∈ DSt0 . Finally, for any t1 ≥ t0 the output is computed as
y(t1 ) = ft0 (x0 , u[t0 ,t1 ] )(t1 ). Setting t0 = t1 in the above relation allows to obtain the relation y(t1 ) = η(t1 , x1 , u(t1 )), which shows that the output at time t1 depends (only) upon t1 , the parameter x1 (which represents the memory of the system) and the value of the input signal at time t1 . We conclude this discussion noting that, to a causal system, we have associated a space of parameters and two functions which provide an alternative representation of inputoutput pairs. In fact, for all t1 ≥ t0 , we have y(t1 ) = η(t1 , φ(t1 , t0 , x0 , u[t0 ,t1 ) , u(t1 )) and this yields, varying x0 in X and u in DSt0 , all inputoutput pairs in St0 .
This representation is characterized by the appearance, aside the input and output signals, of an auxiliary quantity which takes values in the space X. The role of this quantity is to summarize the effect of past input values, hence to render unique the determination of the present and future output. This quantity is denominated state, or state variable, and the space X is denominated state space.
1.3.4
Definition of state
In the previous section we have informally introduced the notion of state, and we have highlighted its main properties. We now provide a formal definition of state. 5
With an alternative convention we could have obtained x1 = φ(t1 , t0 , x0 , u[t0 ,t1 ] ).
17
1.3. THE NOTION OF SYSTEM
Definition 1.4 Given a system S and a space of input functions U. A set X is a state space for the system S if there exist two functions φ:T ×T ×X ×U →X η :T ×X ×U →Y such that the following conditions hold. • For all t0 ∈ T , u ∈ U and x0 ∈ X {(u0 , y0 ) ∈ U F (t0 ) × Y F (t0 ) : u0 (t) = u(t), y0 (t) = η(t, φ(t, t0 , x0 , u[t0 ,t) , u(t))} = St0 . • (Causality) For all t0 ∈ T , for all t ≥ t0 and for all x0 ∈ X u[t0 ,t) = u′ [t0 ,t) ⇒ φ(t, t0 , x0 , u[t0 ,t) ) = φ(t, t0 , x0 , u′ [t0 ,t) ). • (Consistency) For all t ∈ T , and for all u ∈ U φ(t, t, x0 , u) = x0 . • (Separation)6 For all t0 ∈ T , for all t ≥ t0 , for all x0 ∈ X and for all u ∈ U t > t1 > t0 ⇒ φ(t, t0 , x0 , u[t0 ,t) ) = φ(t, t1 , φ(t1 , t0 , x0 , u[t0 ,t1 ) ), u[t1 ,t) ). The function φ is called state transition function, and the function η output transformation. The triple {X, φ, η} is called state space representation, or inputstateoutput representation, of S. Remark. In what follows, and to simplify the equations, we use the notation φ(t, t0 , x0 , u) in place of φ(t, t0 , x0 , u[t0 ,t) ). ⋄ Remark. For the special class of systems in which U , Y and X are composed of a finite number of elements, for examples all digital electronics systems, the above definition is equivalent to the definition of a Mealytype finite state machine. To obtain the definition of a Mooretype finite state machine it is necessary to alter the definition of state, requiring that the state represent the effect of all past and current input values (see Footnote 5). ⋄ At this stage, one may wonder if the given definition of state space representation is sufficiently general to develop a systematic theory. To this end, we need to address the issues of existence, and then of unicity of state space representations. While a complete treatment of such topics is outside the scope of these notes, we give a few important results and facts. 6
This property is sometimes called semigroup property.
18
CHAPTER 1. INTRODUCTION
Theorem 1.1 Given a system S and a space of input functions7 U. The system has a state space representation if and only if it is causal. The above statement implies that, under mild technical assumptions, a causal system admits at least one state space representation. However, such representation need not be unique. In fact, given a system S and a state space representation {X, φ, η} it is possible to obtain other state space representations, for example by means of one of the following procedures. • (State space transformation) Consider a system S, with state space representation {X, φ, η}. Let ψ : X → Z be an invertible map, i.e. there exists ψ −1 : Z → X such that z = ψ(ψ −1 (z)) and x = ψ −1 (ψ(x)), for all z ∈ Z and x ∈ X. Define the function φz : T × T × Z × U → Z˜ such that φz (t, t0 , z, u[t0 ,t) ) = ψ(φ(t, t0 , ψ −1 (z), u[t0 ,t) ) and the function ηz : T × Z × U → Y˜ such that ηz (t, z, u) = η(t, ψ −1 (z), u). Then {Z, φz , ηz } is a state space representation of S. • (State augmentation) Consider a system S, with state space representation {X, φ, η}. ˜ where X ˜ is a nonempty set, Let Xa = X × X,
φa =
φ φ˜
,
for some function ˜ φ˜ : T × T × Xa × U → X, and ηa : T × T × Xa × U → Y is such that ηa − η = 0. Then {Xa , φa , ηa } is a state space representation of S. We conclude that if a system S admits a state space representation, it admits an infinite number of representations. Thus, it makes sense to distinguish between the various representations of a system, and to determine (if possible) a representation which is more convenient, or adequate, for a certain goal. To this end, we conclude this section introducing a few new concepts, which will be useful in the study a state space representations of particular classes of systems. 7 To be precise, we should assume that the space U be complete, i.e. that it is closed with respect to concatenation and that, for all t ∈ T , {u(t) ∈ U : u ∈ U} = U .
1.3. THE NOTION OF SYSTEM
19
Definition 1.5 (Equivalent representations) Two state space representations {X, φ, η} and {X ′ , φ′ , η ′ } of a system S are equivalent if • for all t0 ∈ T and for all x0 ∈ X there exists x′0 ∈ X ′ such that for all u ∈ U and t ≥ t0 η(t, φ(t, t0 , x0 , u), u(t)) = η ′ (t, φ′ (t, t0 , x′0 , u), u(t)); • for all t0 ∈ T and for all x′0 ∈ X ′ there exists x0 ∈ X such that for all u ∈ U and t ≥ t0 η ′ (t, φ′ (t, t0 , x′0 , u), u(t)) = η(t, φ(t, t0 , x0 , u), u(t)). Definition 1.6 (Equivalent states) Consider a system S and a state space representation {X, φ, η}. Two elements xa and xb of X are equivalent at t0 if for all u ∈ U and for all t ≥ t0 η(t, φ(t, t0 , xa , u), u(t)) = η(t, φ(t, t0 , xb , u), u(t)). Remark. Equivalent states are sometimes referred to as nondistinguishable states, because it is not possible to distingh between them by measurements of the output. ⋄ Definition 1.7 (Reduced state space) Consider a system S and a state space representation {X, φ, η}. The state space X is said to be reduced at t0 if there are no pairs of states equivalent at t0 . If X is reduced at t0 the representation {X, φ, η} is said reduced at t0 . Remark. A state space can be reduced at some time t0 but not reduced at some other time t1 . It is possible to give a notion of reduction independent of time requiring that X be reduced at least at one time instant. ⋄
1.3.5
Classification
The notion of system introduced is very general. In applications, it is often possible to consider special classes of systems, i.e. to restrict our interest to systems with special properties. To clarify this issue we introduce a classification of systems on the basis of some of the key ingredients discussed. Definition 1.8 A system S is a continuoustime system if T = IR. A system S is a discretetime system if T = Z. Remark. There is a class of system, increasingly studied and used in applications, in which for some state variables T = IR and for some other state variables T = Z, i.e.
20
CHAPTER 1. INTRODUCTION
some variables vary continuously with time, and other variables vary only at discrete time instants. This is the case in systems where a physical component, for example a robot, is connected with a supervisor, for example a machine that decides which operation the robot has to perform. These systems are denominated hybrid systems. ⋄ Definition 1.9 A system S is said timeinvariant if for all t0 ∈ T and for all δ such that t0 + δ ∈ T (u0 (t), y0 (t)) ∈ St0 ⇒ (u0 (t − δ), y0 (t − δ)) ∈ St0 +δ . Definition 1.10 A state space representation is said timeinvariant if for all t0 ∈ T , all x0 ∈ X, all u ∈ U and all t¯ ∈ T φ(t, t0 , x0 , u) = φ(t − t0 , 0, x0 , u)
η(t, x, u(t)) = η(t¯, x, u(t)).
Remark. For a timeinvariant representation we can always select t0 = 0.
⋄
Definition 1.11 A system S is said linear if X, U and Y are linear spaces and if, for all t0 ∈ T , St0 is a linear subspace of U F (t0 ) × Y F (t0 ) . A system S which is not linear is called nonlinear. Definition 1.12 A state space representation is linear if • the sets U , Y and X are linear spaces; • the set U is a linear subspace of U T ; • for all t0 ∈ T and t ∈ T such that t ≥ t0 the function φ is linear on X × U; • for all t ∈ T the function η is linear on X × U . Definition 1.13 A state space representation is a finite state representation if the sets U , Y and X have a finite number of elements. Definition 1.14 A state space representation is a finitedimensional representation if the sets U , Y and X are linear, finitedimensional, spaces.
1.3.6
Generating functions
In this section we show that, under certain regularity assumptions, it is possible to obtain an alternative description of a system. To begin with, consider discretetime systems and rewrite the state transition function for t − t0 = 1, i.e. x(t + 1) = φ(t + 1, t, x(t), u[t,t+1) ) = φ(t + 1, t, x(t), u(t)).
21
1.3. THE NOTION OF SYSTEM
This equation shows that for a discretetime system the value of the state at time t + 1 depends upon t, x(t) and u(t). We can therefore write x(t + 1) = f (t, x(t), u(t)), where the function f is called generating function, or onestep state update function. Note that, from the function f it is possible to reconstruct (uniquely) the state transition function φ. Thus, the triple {X, f, η} can be regarded as the state space representation of a discretetime system. It is now natural to wonder if a similar representation can be obtained for continuoustime systems. To this end, consider the generating function of a discretetime system and note that (if X is a linear space) x(t + 1) − x(t) = f (t, x(t), u(t)) − x(t), which shows that the variation of the state in a time unit is a function of t, x(t) and u(t). This means that, for continuoustime systems, we are looking at the class of systems for which the rate of change of the state x(t) can be written as a function of t, x(t) and u(t). These systems have special importance in applications, where they arise naturally whenever first principles are used to derive their representation (see some of the examples in Section 1.2). Motivated by these considerations we say that the state space representation {X, φ, η} is regular, or differentiable, if there exists a function f : IR × X × U → X such that, for any t0 , for any x0 ∈ X and for any u ∈ U the function φ(t, t0 , x0 , u) is, for all t ∈ F (t0 ) the (unique) solution of the differential equation dφ(t, t0 , x0 , u) = f (t, φ(t, t0 , x0 , u), u(t)), dt
(1.15)
with the initial condition φ(t0 , t0 , x0 , u) = x0 . Note that equation (1.15) can be rewritten as x(t) ˙ =
dx(t) = f (t, x(t), u(t)). dt
We conclude noting that a regular representation {X, φ, η} can be alternatively described by the triple {X, f, η}. Remark. A state space representation {X, f, η} is timeinvariant in f does not depend explicitely on t and η is as in Definition 1.10. ⋄
22
CHAPTER 1. INTRODUCTION
1.4
Examples revisited
We conclude this chapter by revisiting the examples discussed in Section 1.2 in terms of the concepts, and notions introduced. • The system (1.1) is a discretetime, timeinvariant, linear, finitedimensional system without input. To obtain a state space representation {X, f, η} consider the state variables x1 = Fn−2 and x2 = Fn−1 and note that X = {(x1 , x2 ) ∈ IR2 }
f (x) =
x2 x1 + x2
and η(x) = x1 + x2 . • The system (1.2) is a continuoustime, nonlinear, timeinvariant, finite dimensional system, with input I ∈ IR+ , and state (x, y) ∈ IR+ × IR+ , described by a generating function. For such a system we have not defined an output transformation. • The system (1.3) is a discretetime, nonlinear, finitedimensional system, with input u ∈ IR+ , state (x1 , x2 , x3 ) ∈ IR+ × IR+ × IR+ , and output y ∈ IR+ , described by means of a generating function. The system is nonlinear because the input, state and output spaces are not linear spaces. • The system (1.4) is a continuoustime, nonlinear, timeinvariant, finitedimensional system with input u ∈ [− tan δM , tan δM ] and state (los , ϑos , ϕos ) ∈ IR × (−π, π] × (−π, π], and described by means of a generating function. For such a system we have not defined an output transformation. • The system (1.4) is a continuoustime, nonlinear, timeinvariant, finitedimensional system, without input, and with state (x, y, z) ∈ IR3 , described by means of a generating function. For such a system we have not defined an output transformation. • The system (1.5)(1.6) is a continuoustime, nonlinear, finitedimensional system, without input, with state (x1 , x2 , x3 ) ∈ IR3 , and output y ∈ IR2 , described by a generating function. The system is nonlinear because the output transformation is not linear. • The system (1.7) is a continuoustime, nonlinear, timeinvariant, finitedimensional system, with input Tb ∈ IR+ , and state (ω, v) ∈ IR2 , described by a generating function. The output variable can be selected as the longitudinal slip λ ∈ [−1, 1]. • The system (1.8) is a continuoustime, linear, timeinvariant, finitedimensional system, without input, and with state (x1 , · · · , xn ) ∈ IRn , described by a generating function. For such a system we have not defined an output transformation. The system (1.9) is a continuoustime, timeinvariant, infinitedimensional, linear
1.4. EXAMPLES REVISITED
23
system without input of with state the set of all functions x(y, t) defined and twice differentiable in [0, L] × IR. For such a system we have not defined an output transformation. • The system (1.10) (resp. (1.11)) is a continuous (resp. discrete) time, timeinvariant, nonlinear (in general), finitedimensional system, with state x ∈ X and without input. For such a system we have not defined an output transformation. • The system (1.12) is a discretetime, linear, timeinvariant, finitedimensional system, without input, and with state x ∈ IRN . For such a system the output transformation can be regarded as the identity map.
24
CHAPTER 1. INTRODUCTION
Chapter 2
Stability
26
CHAPTER 2. STABILITY
2.1
Introduction
In Chapter 1 we have seen that, under some regularity conditions, continuous and discretetime causal systems, with state space X, can be described by means of a generating function and an output transformation, namely x˙ = f (t, x, u)
y = η(t, x, u)
(2.1)
y = η(t, x, u),
(2.2)
and1 x+ = f (t, x, u)
where all signals have to be understood as evaluated at time t, and t ∈ IR if the system is continuoustime, whereas t ∈ Z if the system is discretetime. In what follows, whenever convenient and for compactness, we also use the notation σx = f (t, x, u)
y = η(t, x, u),
(2.3)
where σx stands for x˙ if the system is continuoustime, and σx stands for x+ if the system is discretetime.
2.2
Existence and unicity of solutions
The simplest question that can be posed in the study of the equations (2.1) and (2.2) is the following. Given an initial time t0 , an initial value of the state x(t0 ) = x0 and an input signal u ∈ U F (t0 ) , is it possible to obtain a solution of the equation (2.3)? By a solution we mean a function x(t), defined for all t ≥ t0 , and such that σx(t) = f (t, x(t), u(t)) for all2 t ∈ F (t0 ), or for all t ∈ [t0 , t¯), for some t¯ > t0 . 1
To simplify notation we replace x(t + 1) with x+ and x(t) with x. It is enough to require that the equality holds for almost all t, i.e. the condition may be violated for some t ∈ Ts ⊂ T , provided that Ts has zero Lebesgue measure. To illustrate this point consider the differential equation x˙ = −sign(x), (2.4) 2
where the signum function is defined as (
sign(x) = For a given x(0) > 0 we have
§
x(t) =
1 0 −1
x(0) − t 0
if x > 0 if x = 0 if x > 0. for t ≤ x(0) for t ≥ x(0),
which shows that equation (2.4) does not hold for all t, in fact x(t) is not differentiable at t = x(0).
2.2. EXISTENCE AND UNICITY OF SOLUTIONS
27
The answer to this question is trivial in the case of discretetime systems, provided the state space X coincides with IRn and the generating function is continuous. In fact, if the function f is continuous and the input signal u(t) is bounded for all finite t ∈ T , then equation (2.2) describes a continuous mapping from T × IRn × U to IRn , hence the solution x(t) is unique and it is such that x(t) ⊂ X = IRn for all finite t ≥ t0 .
Note however that, if the function f is not continuous, or if the state space X is not IRn , then solutions of the equation (2.2) may not be defined for all t ≥ t0 . In the case of continuoustime systems the situation is much more involved, and continuity of f is not enough to guarantee existence and uniqueness of solutions of equation (2.1). To understand the issues involved in this problem we start considering two examples. Example 2.1 Consider the nonlinear system, without input, described by x˙ = x3 ,
(2.5)
with x ∈ IR, and the initial condition x(0) = x0 . A simple integration by parts yields x(t) =
x0
È
1 − 2x20 t
,
which shows that if x0 6= 0 then x(t) is defined only for t ∈ [0,
1 ). 2x20
This situation is often referred to as the existence of a finite escape time for the solutions of the differential equation. Note that the time of escape depends upon the initial condition. Example 2.2 Consider the nonlinear system, without input, described by x˙ = x1/3 ,
(2.6)
with x ∈ IR, and the initial condition x(0) = 0. Clearly x(t) = 0 is a solution of the differential equation for the given initial condition. However
x(t) =
2 t 3
3 2
is also a solution of the differential equation for the given initial condition. Remark. The finite escape time phenomenon is not only a mathematical curiosity, but it may naturally arise in applications. For example, consider a mass m, sliding without friction along a line, and acted upon by an external force F . Let x be the position of the mass, and assume that the mass is constrained to stay in the set I = (−1, 1), to avoid
28
CHAPTER 2. STABILITY
contacts with hard physical constraints (this may be the case in a car suspension). By Newton’s law, the differential equation of the mass is m¨ x = F, which holds provided x ∈ I. The differential equation can be rewritten in state space representation (see Example 2.8 for a general discussion on high order differential/difference equations) as F x˙ 1 = x2 x˙ 2 = , m where x1 = x ∈ [−1, 1] and x2 = x. ˙ To take into account the presence of the constraints we can define a new variable z = ψ(x) = − log(1 − x) + log(1 + x) and note that the function ψ maps the interval [−1, 1] into IR. Note now that the system can be rewritten as F (ez + 1)2 x2 x˙ 2 = . z˙ = z 2e m Suppose that F = 0 for all t, that x1 (0) = 0 and that x2 (0) 6= 0. Then x2 (t) = x2 (0) for all t and 1 + x2 (0)t z(t) = log 1 − x2 (0)t
which shows that z(t) has finite escape time at t = x21(0) . This is the mathematical description of the obvious fact that the mass, with a nonzero initial velocity, and without any external action, hits one of the constraints in finite time, i.e. at the time of escape of z(t). Note that, even in the presence of a nonzero force, the problem of determining F to avoid hitting the constraints, i.e. to avoid the occurence of finite escape time, is nontrivial. ⋄ Example 2.3 Consider the Euler approximate models of systems (2.5) and (2.6), with sampling time τ , namely x+ = x + τ x3 and x+ = x + τ x1/3 . Note that these systems do not present the finite escape time phenomenon, neither the existence of multiple solutions. This shows that the numerical integration of nonlinear continuoustime systems may be very difficult and requires, in general, the use of dedicated algorithms. From the above examples we infer that the issues of existence and unicity of solutions for continuoustime systems may be very delicate. It is possible to show that if the function
2.2. EXISTENCE AND UNICITY OF SOLUTIONS
29
f is continuous then, for any x0 , the differential equation (2.1) has at least one solution, defined for all t ≥ t0 and such that t − t0 is sufficiently small (local existence). The critical issue is therefore the existence of trajectories for t → ∞ (global existence). It is very hard to characterize such a property, and we therefore discuss only sufficient conditions. Theorem 2.1 (Existence of solutions for Lipschitz differential equations) Consider the differential equation x˙ = f (t, x), (2.7) with x ∈ IRn , and the initial condition x(t0 ) = x0 . Suppose that f is piecewise continuous in t and it is such that the (global) Lipschitz condition kf (t, x) − f (t, y)k ≤ Lkx − yk holds for all x ∈ IRn and y ∈ IRn , and for some constant L > 0. Then for any x0 the differential equation (2.7) has a unique solution defined for all t ≥ t0 . As a direct consequence of the above statement we have the following result. Corollary 2.1 (Existence of solutions for linear differential equations) Consider the differential equation x˙ = A(t)x + g(t), with x ∈ IRn , and the initial condition x(t0 ) = x0 . Assume that A(t) and g(t) are piecewise continuous. Then for any x0 ∈ IRn the differential equation has a unique solution defined for all t ≥ t0 . The Lipschitz condition of Theorem 2.1 is very restrictive, and there are several differential equations for which it does not hold, but which have unique solutions defined for all t ≥ t0 . Note that existence conditions for nonLipschitz differential equations may be very involved. We thus restrict our discussion to a simple sufficient condition which is however very important in applications. Theorem 2.2 Consider the differential equation (2.7), with x ∈ IRn , and the initial condition x(t0 ) = x0 . Suppose that f is piecewise continuous in t and it is differentiable3 in x. Suppose, in addition, that there exists a compact set W such that, for all x0 ∈ W , the solutions of the differential equation remain in W for all t ≥ t0 . Then for any x0 ∈ W the differential equation (2.7) has a unique solution defined for all t ≥ t0 . Example 2.4 Consider the differential equation x˙ = a(t)x − x3 + u, 3
Locally Lipschitz is enough.
30
CHAPTER 2. STABILITY
with x ∈ IR, a(t) continuous and such that a(t) ≤ 1, and u ∈ [−1, 1]. We now show that for any x0 ∈ IR the differential equation admits a unique solutions for all t ≥ t0 . Note first that, for any fixed x ¯ > 2, x>x ¯ ⇒ x˙ < 0
x < −¯ x ⇒ x˙ > 0.
This implies that, for all x0 ∈ [−¯ x, x ¯], the corresponding solution x(t) is such that x(t) ∈ [−¯ x, x ¯], for all t ≥ t0 . Hence, by Theorem 2.2 we infer that, for any x0 ∈ IR the differential equation has a unique solution for all t ≥ t0 .
2.3
Trajectory, motion and equilibrium
We now consider again a system described by means of an inputstateoutput representation {X, φ, η} and define a few typical dynamic behaviours of the system. Definition 2.1 (Trajectory) Consider a system {X, φ, η}. A trajectory is the set T = {x ∈ X : x = φ(t, t0 , x0 , u)} ⊂ X, i.e. is the set of points in X reached by the state x(t), for t ≥ t0 , and for a specific initial state x0 and input signal u. Definition 2.2 (Motion) Consider a system {X, φ, η}. A motion is the set M = {(t, x(t)) ∈ T × X : t ∈ F (t0 ), x(t) = φ(t, t0 , x0 , u)} ⊂ T × X, i.e. is the set of points in T × X taken by the pairs (t, x(t)), for t ≥ t0 , and for a specific initial state x0 and input signal u. The main differences between a trajectory and a motion are that they leave in different spaces, and the motion is parameterized by t, whereas the trajectory does not contain any information on t. This means that the trajectory provides solely information on the points of the state space X visited by the system during his evolution, whereas the motion specifies in addition when each point has been visited. Note, however, that the (natural) projection of a motion along T yields a trajectory. Figure 2.1 gives an example of a motion, and of the corresponding trajectory, for a continuoustime system, with X = IR2 and t0 = 0. Definition 2.3 (Equilibrium) Consider a system {X, φ, η}. Assume the input u is constant, i.e. u(t) = u0 for all t and for some constant u0 . A state xe is an equilibrium of the system associated to the input u0 if xe = φ(t, t0 , xe , u0 ), for all t ≥ t0 , i.e. an equilibrium is a trajectory composed of a single point.
31
2.3. TRAJECTORY, MOTION AND EQUILIBRIUM
10
8
6
t 4
2
0 2
1
0
x
−1
−2
−2
−1
1
0
2
x
1
2
Figure 2.1: A motion (dashed line) and the corresponding trajectory (solid line).
If the system {X, φ, η} possesses a generating function, hence can be described by means of the triple {X, f, η}, then the computation of equilibria requires the solution of the system of equations 0 = f (t, xe , u0 ), for continuoustime systems, and of the systems of equations xe = f (t, xe , u0 ), for discretetime systems.
2.3.1
Linear systems
In this section we discuss the notions introduced for the special class of linear, timeinvariant, finitedimensional systems, i.e. systems described by equations of the form σx = Ax + Bu
y = Cx + Du,
(2.8)
with x ∈ X = IRn , u(t) ∈ IRm , y(t) ∈ IRp and A ∈ IRn×n , B ∈ IRn×m, C ∈ IRp×n , and D ∈ IRp×m.
32
CHAPTER 2. STABILITY
Proposition 2.1 (Equilibria of linear systems) Consider a linear, timeinvariant, system σx = Ax + Bu, with x ∈ IRn and u(t) ∈ IRm . The set of equilibria is a linear subspace4 . Moreover, the following hold. • For u(t) = u0 = 0, the origin is always an equilibrium. • For continuoustime systems, if A is invertible, for any u0 there is a unique equilibrium xe = −A−1 Bu0 . If A is not invertible the system has either infinitely many equilibria (spanning a linear subspace) or it has no equilibria. • For discretetime systems, if I − A is invertible, for any u0 there is a unique equilibrium xe = (I − A)−1 Bu0 . If I − A is not invertible the system has either infinitely many equilibria (spanning a linear subspace) or it has no equilibria. Proposition 2.2 (Trajectories of linear, continuoustime, systems) Consider the continuoustime, timeinvariant, linear system x˙ = Ax + Bu
y = Cx + Du,
with x ∈ X = IRn , u(t) ∈ IRm , y(t) ∈ IRp and the initial condition5 x(0) = x0 . Then6 x(t) = eAt x0 +
Z t 0
eA(t−τ ) Bu(τ )dτ
(2.9)
4
This property holds also for general linear systems, i.e. not necessarily timeinvariant. Without loss of generality we set t0 = 0. 6 Given a (square) matrix F , the matrix exponential eF t is formally defined as 5
eF t = I + F t +
(F t)2 (F t)3 + + ··· . 2! 3!
The matrix exponential has the following properties, which can be derived from its definition. • For every t1 and t2 , eF t1 eF t2 = eF (t1 +t2 ) . ˜ ˜ ˜ • eF t eF t = eF t eF t = e(F +F )t if and only F F˜ = F˜ F , i.e. if and only if F and F˜ commute. ′
• (eF t )−1 = e−F t and (eF t )′ = eF t . • If v is an eigenvector of F with eigenvalue λ then v is also an eigenvector of eF t with eigenvalue eλt . •
d Ft e dt
= F eF t = eF t F .
• T −1 eF t T = e(T
−1
F T )t
.
• If F is diagonal, i.e. F diag(eλ1 t , eλ2 t , · · · , eλn t ).
=
diag(λ1 , λ2 , · · · , λn ) then eF t is also diagonal and eF t
Finally, eF t =
mi r X X i=1 k=1
Rik
tk−1 λi t e , (k − 1)!
for some mi ≥ 1 (see Footnote 9), where r is the number of distinct eigenvalues of F .
=
33
2.3. TRAJECTORY, MOTION AND EQUILIBRIUM and
Z t
At
y(t) = Ce x0 +
0
CeA(t−τ ) Bu(τ )dτ + Du(t).
(2.10)
Proof. To begin with consider the differential equation x(t) ˙ = Bu(t), with x(0) = x0 and note that, by a simple integration, Zt
x(t) = x0 +
Bu(τ )dτ. 0
Consider now the variable z(t) = e−At x(t) and note that z(0) = x(0), x(t) = eAt z(t) and z˙ = e−At Bu. Hence
Zt
z(t) = z0 +
e−Aτ Bu(τ )dτ,
0
from which we obtain directly equation (2.9). Equation (2.10) is trivially obtained replacing x(t) in the output transformation. ⊳
Example 2.5 Let Fλ ∈ IRn×n be defined as 2
Fλ =
then
6 6 6 6 6 6 4
2
eFλ t =
6 6 6 6 λt 6 e 6 6 6 4
1 λ .. .
0 1 .. .
··· ··· .. .
0 0 .. .
0 ··· 0 ···
0 ···
λ 0
1 λ
1
t
t2 2
···
0
1
t
.. .. . . 0 ··· 0 ···
..
···
λ 0 .. .
.
..
0 ···
1 0
3 7 7 7 7 7 7 5
tn−1 (n−1)!
3 7
tn−2 7 (n−2)! 7 7
.
.. . t 1
7. 7 7 7 5
To establish the claim note that Fλ = λI + F0 and that I and F0 commutes, hence eFλ t = e(λI+F0 )t = eλt eF0 t ,
34
CHAPTER 2. STABILITY
where, by definition of the matrix exponential, 2
eF0 t =
6 6 6 6 6 6 6 6 4
1
t
t2 2
···
0
1
t
.. .. . . 0 ··· 0 ···
..
···
Example 2.6 Let
Fλ t
e
..
0 ···
1 0
=e
7. 7 7 7 5
.. . t 1
cos ωt sin ωt − sin ωt cos ωt
λt
7
tn−2 7 (n−2)! 7 7
.
λ ω −ω λ
Fλ = then
.
3
tn−1 (n−1)!
.
To establish the claim note that Fλ = λI + F0 , and that I and F0 commutes, hence eFλ t = e(λI+F0 )t = eλt eF0 t , where, by definition of the matrix exponential, and recalling the series expansions of sin ωt and cos ωt, cos ωt sin ωt F0 t . e = − sin ωt cos ωt Example 2.7 Consider a continuoustime, timeinvariant, linear system with x ∈ IR2 , u(t) ∈ IR and y(t) ∈ IR. Suppose that x(0) = [1, 1]′ , that u(t) = 0 for all t, and that
A= Note that
A=L
−1
˜ = AL
0 1 −2 −3 1 1 −1 −2
C=
−1 0 0 −2
1 0
.
2 1 −1 −1
−1
hence At
e
˜ (L−1 AL)t
=e
=L
−1
e−t 0 0 e−2t
Therefore
x(t) =
−t
L=e
2 1 −2 −1
−2e−2t + 3e−t 4e−2t − 3e−t
−2t
+e
−1 −1 2 2
.
and y(t) = −2e−2t + 3e−t . Note that, the state and the output are linear combinations of exponential functions with exponents given by t times the eigenvalues of A.
35
2.3. TRAJECTORY, MOTION AND EQUILIBRIUM
Remark. Continuoustime systems are reversible, i.e. the knowledge of x(t) and of the input in the interval [0, t) allows to compute x0 . In fact, from equation (2.9) we obtain x0 = e−At x(t) −
Z t 0
e−Aτ Bu(τ )dτ. ⋄
Proposition 2.3 (Trajectories of linear, discretetime, systems) Consider the discretetime, timeinvariant, linear system x(k + 1) = Ax(k) + Bu(k)
y(k) = Cx(k) + Du(k),
with x ∈ X = IRn , u(k) ∈ IRm , y(k) ∈ IRp and the initial condition x(0) = x0 . Then x(k) = Ak x0 +
k−1 X
Ak−1−i Bu(i)
(2.11)
i=0
and y(k) = CAk x0 +
k−1 X
CAk−1−i Bu(i) + Du(k).
(2.12)
i=0
Proof. Using the state space representation of the system we have that x(1) = Ax0 + Bu(0) x(2) = Ax(1) + Bu(1) = A2 x0 + ABu(0) + Bu(1) x(3) = Ax(2) + Bu(2) = A3 x0 + A2 Bu(0) + ABu(1) + Bu(2) .. . from which we obtain the expression of x(k). Finally, y(k) is obtained replacing x(k) in the output transformation. ⊳ Remark. The expression of x(k) can be rewritten as 2 k
x(k) = A x0 +
Ak−1 B
B AB · · ·
6 6 6 6 4
3
u(k − 1) u(k − 2) .. .
7 7 7. 7 5
u(0)
This expression highlights the role of the matrix B, AB, · · · , Ak−1 B in the computation of x(k). ⋄
36
CHAPTER 2. STABILITY
Remark. If the matrix A is invertible then the system is reversible, i.e. the knowledge of x(k) and of the input sequence in the interval [0, k) allows to compute x0 . In fact, from equation (2.11) we obtain x0 = A−k x(k) −
k−1 X
A−i−1 Bu(i).
i=0
⋄ Remark. Reversibility of continuoustime systems does not require any assumption on the matrix A. This is because, for any A, the matrix eAt is invertible for any t. ⋄ Remark. The matrices CAh B, known as Markov parameters, appearing in equation (2.12) have a simple and interesting interpretation for singleinput singleoutput discretetime systems. Suppose that x(0) = 0, that u(0) = 1 and that u(i) = 0 for i ≥ 1. Then y(0) = D
y(1) = CB
y(2) = CAB
···
y(h) = CAh−1 B,
i.e. the output yields directly information on the matrices A, B, C and D. The problem of determining such matrices, hence a state space representation for the system, from the above output sequence is the socalled realization problem. ⋄ It is interesting to interprete the results of the Propositions 2.2 and 2.3 in the light of the general discussion in Chapter 1. Equations (2.9) and (2.11) show that the state of the system at time t is the linear combination of two contributions, the former depends only upon the initial condition x0 , and is denoted free response of the state of the system, the latter depends only upon the input signal u and is denoted forced response of the state of the system. Note that the initial condition and the input can be regarded as two independent causes acting on the system, hence equations (2.9) and (2.11) show that the principle of superposition holds for such systems. Analogously, equations (2.10) and (2.12) show that the output of the system at time t is the linear combination of two contributions, the former depends only upon the initial condition x0 , and is denoted free response of the output of the system, the latter depends only upon the input signal u and is denoted forced response of the output of the system. Finally, note that equations (2.9) and (2.10) ((2.11) and (2.12), resp.) yield directly the functions φ and η of a state space representation of the system. Remark. Consider the continuoustime, timeinvariant, linear system x˙ = Ax + Bu
y = Cx
with x ∈ X = IRn , u(t) ∈ IRm and y(t) ∈ IRp . Suppose the system is between a zeroorder hold and a sampler with sampling period T , i.e. the input is constant over each
37
2.3. TRAJECTORY, MOTION AND EQUILIBRIUM
time interval [kT, (k + 1)T ), for k ≥ 0, and the output is measured at t = kT , for k ≥ 0. The system viewed from outside the zeroorder hold and the sampler is a discretetime, timeinvariant, linear system. To obtain a state space representation of this discretetime system let xk = x(kT ), uk = u(kT ) and yk = y(kT ). Integrating the above differential equation for t ∈ [kT, (k + 1)T ) yields (recall equation (2.9)) xk+1 = eAT xk +
Z T 0
eA(T −τ ) Bdτ uk = Ad xk + Bd uk ,
whereas the output transformation is given by yk = Cxk . These equations provide a state space representation of the discretetime system seen from outside the zeroorder hold and the sampler. Note that, unlike the approximate discretetime models described in Section 1.2.9, the obtained discretetime model is exact, i.e. under the stated operating conditions xk = x(kT ), for all k ≥ 0. ⋄ Example 2.8 (Input/output models) Linear, timeinvariant, systems can be also described by means of highorder differential or difference equations involving only the external signals, i.e. the input and the output. Consider for simplicity singleinput single outputsystems and the equation7 σ n y + an−1 σ n−1 y + · · · + a1 σy + a0 y = bn−1 σ n−1 u + · · · + b1 σu + b0 u,
(2.15)
where the ai and bi are constant coefficients. We now show that this equation defines a linear, timeinvariant, system as discussed in Chapter 1. To this end we derive a state space representation with a generating function by means of the following procedure (known as realization). Assume, for simplicity, that bn−1 = bn−2 = · · · = b1 = 0 and that b0 6= 0. Let x ∈ X = IRn , and define x1 = y x2 = σy · · · xn−1 = σ n−2 y xn = σ n−1 y.
Note that
2
σx =
6 6 6 6 6 6 4
3
2
7 7 7 7 7 7 σxn−1 5
6 6 6 6 6 6 4
σx1 σx2 .. .
=
σxn 7
x2 x3 .. . xn −a0 x1 − a1 x2 − · · · − an−1 xn + b0 u
3 7 7 7 7 7 7 5
The operator σ i is such that ¨ i
σ α(t) = for continuoustime systems, and such that i
σ α(t) =
¨
α(t)
i=0
di α(t) dti
i>1
(2.13)
α(t)
i=0
α(t + i)
i>1
for discretetime systems. To simplify notation we write α instead of σ 0 α and σα instead of σ 1 α.
(2.14)
38
CHAPTER 2. STABILITY
hence
2
σx =
and
6 6 6 6 6 6 4
0 0 .. .
1 0 .. .
0 1 .. .
0 0 ··· −a0 −a1 · · ·
··· ··· .. .
0 0 .. .
0 1 −an−1 −an
3
2
7 6 7 6 7 6 7x + 6 7 6 7 6 5 4
0 0 .. . 0 b0
3 7 7 7 7 u, 7 7 5
y = [1, 0, · · · , 0]x. The above equations provide the state space representation of the system defined by equation (2.15), under the considered simplifying assumptions. Note that, a similar procedure can be used in the general case, i.e. in the case of nonzero bi ’s.
2.4
Linearization
Linear systems can be often used to approximate the behaviour of nonlinear systems around given operating conditions. The procedure that allows to associate a linear system to a nonlinear system together with one of its operating conditions is called linearization. The advantage in using a linear system to approximate a nonlinear system resides clearly in the fact that linear systems are simpler to study, and it is possible to assess several of their properties with simple tests. In addition, it is also often possible to determine local properties of the original nonlinear system, i.e. properties around a certain operating condition, by properties of the linearization. The process of linearization is performed on a nonlinear system and for a specific initial condition and input signal, i.e. it is assumed that a system, described by equations of the form (2.3), an initial (nominal) state x(t0 ) = x0 and a (nominal) input function uN are assigned. Suppose, in addition, that the motion of the system, for the given initial state and input signal, is welldefined for all t ≥ t0 . Let xN (t) be such that xN (t0 ) = x0 and, for t ≥ t0 , σxN = f (t, xN , uN ), and let yN (t) = η(t, xN , uN ). xN (t) and yN (t) are referred to as the nominal state and the nominal output trajectories. Consider now a perturbed initial condition and a perturbed input signal, i.e. assume that the system (2.3) has been initialized at time t0 with x(t0 ) = xN (0) + δx (t0 ) and that the input signal is uP = uN + δu . Let xP (t) be such that xP (t0 ) = xN (0) + δx (t0 ) and, for t ≥ t0 , σxP = f (t, xP , uP ),
39
2.4. LINEARIZATION and let yP (t) = η(t, xP , uP ).
xP (t) and yP (t) are referred to as the perturbed state and the perturbed output trajectories. Note now that δx (t) = xP (t) − xN (t) is such that σδx = σxP − σxN = f (t, xP , uP ) − f (t, xN , uN ) = f (t, xN + δx , uN + δu ) − f (t, xN , uN ), and define δy = yP (t) − yN (t). If the generating function is differentiable, using Taylor series expansion around xN and uN , we obtain8 σδx = f (t, xN + δx , uN + δu ) − f (t, xN , uN )
∂f (t, x, u) ∂f (t, x, u) (t, xN , uN )δx + (t, xN , uN )δu + ∂x ∂u O(kδx k2 + kδu k2 ) − f (t, xN , uN ).
= f (t, xN , uN ) +
If the perturbations δx (0) and δu are sufficiently small, and as long as δx (t) remains small, it is possible to approximate σδx with the linear terms of the Taylor series expansion, i.e. σδx ≈
∂f (t, x, u) ∂f (t, x, u) (t, xN , uN )δx + (t, xN , uN )δu = A(t)δx + B(t)δu . ∂x ∂u
(2.16)
Similarly, if η is differentiable, δy = η(t, xP , uP ) − η(t, xN , uN ) = η(t, xN , uN ) +
∂η(t,x,u) (t, xN , uN )δx ∂x
+
∂η(t,x,u) (t, xN , uN )δu + ∂u
O(kδx k2 + kδu k2 ) − η(t, xN , uN ), hence δy can be approximated, provided δu and δx are small, with the linear terms of the Taylor series expansion, i.e. δy ≈
∂η(t, x, u) ∂η(t, x, u) (t, xN , uN )δx + (t, xN , uN )δu = C(t)δx + D(t)δu . ∂x ∂u
(2.17)
In summary, we have shown that, under suitable differentiability assumptions, the behaviour of a nonlinear system around a nominal operating condition can be approximated 8
The Jacobian of a funcion g : IRn → IRm at a point x ¯ is defined as 2
∂g(x) 6 (¯ x) = 4 ∂x
∂g1 (x) (¯ x) ∂x1
.. . ∂gm (x) (¯ x) ∂x1
··· .. . ···
∂g1 (x) (¯ x) ∂xn
.. . ∂gm (x) (¯ x) ∂xn
3 7 5.
40
CHAPTER 2. STABILITY
considering input, state and output perturbations around the nominal behaviour and such perturbations are the input, state and output variables of a linear, timevarying system. We stress, once more, that the approximation makes sense only if δx and δu are small. While it is possible to select δu small by selecting uP close to uN , selecting δx (t0 ) small does not guarantee that δx (t) be small for all t ≥ t0 . We will see in Section 2.7 that there is an important practical situation in which a small δx (t0 ) implies that δx (t) remains small for all t ≥ t0 . Remark. If we consider, as nominal operating condition, a constant input and an equilibrium point, and if in addition the functions f and η do not depend explicitely on time, then the linearized system is timeinvariant, i.e. the matrices A(t), B(t), C(t) and D(t) have constant entries. ⋄ Example 2.9 Consider the nonlinear, timeinvariant, system x˙ = sin x + u
y = sin 2x,
with x ∈ IR, u(t) ∈ IR and y(t) ∈ IR, and the nominal operating condition uN = 0 and xN (0) = 0, yielding xN (t) = 0 and yN (t) = 0 for all t ≥ 0. The linearization of the system around the given nominal behaviour is given by ∂(sin x + u) ∂(sin x + u) (0, 0)δx + (0, 0)δu = δx + δu δ˙x = ∂x ∂u and
∂ sin 2x ∂ sin 2x (0, 0)δx + (0, 0)δu = 2δx . ∂x ∂u Note that, as discussed above, the linearized system is timeinvariant. δy =
Consider again the same system but with operating condition uN = −1 and xN (0) = π/2, yielding xN (t) = π/2 and yN (t) = 0 for all t ≥ 0. The linearized system is now δ˙x = δu
δy = −2δx .
Note the (obvious) fact that linearizing a nonlinear system around different operating conditions yields different linearized systems! Example 2.10 The problem of minimizing a function f : IRn → IR can be dealt with, if the function is twice differentiable, considering the sequence of points generated, starting from some initial guess x0 of the minimum, by the socalled Newton’s iteration, namely
−1
∂ 2 f (x) (x) x =x− ∂x2 +
∂f (x) (x). ∂x
(2.18)
This is in general a nonlinear, discretetime system, and any stationary point of the function f is an equilibrium of this system. Let x ¯ be a stationary point of f such that the matrix ∂ 2 f (x) (¯ x) (2.19) ∂x2
41
2.5. LYAPUNOV STABILITY
is nonsingular. The linearization of system (2.18) around such stationary point is given by
δx+ =
h
I−
∂
i−1 ∂f (x) ∂ 2 f (x) (x) ∂x (x) ∂x2
∂x
Ǒ
δx = 0.
(¯ x)
Example 2.11 Consider the system x˙ 1 = g(x1 x2 )x1 x˙ 2 = −2x2 + x3 x˙ 3 = −x3 ,
(2.20)
with x = (x1 , x2 , x3 ) ∈ IR3 , where g(x1 x2 ) is a differentiable function such that g(1) = 1. Consider an initial condition (x1 (0), x2 (0), x3 (0)) such that x1 (0) 6= 0, x2 (0) = 1/x1 (0) and x3 (0) = x2 (0) and note that the corresponding trajectory is x1 (t) = et x1 (0)
x2 (t) =
e−t x1 (0)
x3 (t) =
e−t . x1 (0)
(2.21)
The system linearized around this nominal trajectory is described by
δ˙x1
=
δ˙x2
= −2δx2 + δx3
δ˙x3
e−t ∂g(x1 x2 ) t ∂g(x1 x2 ) ) + 1 δx1 + (e x1 (0), ∂x1 x1 (0) ∂x2
e−t e x1 (0), x1 (0) t
δx2 (2.22)
= −δx3 ,
and this is not, in general, timeinvariant.
2.5
Lyapunov stability
To study the qualitative behaviour of a system, hence to describe the properties of its trajectories for all t ∈ T and for t → ∞, we introduce the notion of stability. In addition, this notion allows to study the behaviour of trajectories close to an equilibrium point or to a certain motion. Remark. The notion of stability that we discuss has been introduced in 1882 by the Russian mathematician A.M. Lyapunov, in its Doctoral Thesis (hence it is often referred to as Lyapunov stability). There are other notions of stability due for example to Lagrange (Lagrange stability) or introduced in the past 20 years. Nevertheless, the concept of Lyapunov stability is the most commonly used in applications. ⋄
42
2.5.1
CHAPTER 2. STABILITY
Definition
Consider a system described via a state space representation {X, φ, η} and denote with x(t) the value of the function φ(t, t0 , x0 , 0), i.e. the value of the state at time t when the input is identically zero and x(t0 ) = x0 . Recall that x(t) describes the socalled free evolution of the system. Suppose that it is possible to define a norm on the space X. (If X ⊂ IRn the Euclidean norm can be defined.) Definition 2.4 [Lyapunov stability] Consider a system {X, φ, η} and an equilibrium point xe . The equilibrium is stable (in the sense of Lyapunov) if for every ǫ > 0 there exists a δ = δ(ǫ, t0 ) > 0 such that kx(t0 ) − xe k < δ implies kx(t) − xe k < ǫ, for all t ≥ t0 . In stability theory, the quantity x(t0 ) − xe is called initial perturbation, and x(t) = φ(t, t0 , x0 , 0) is called perturbed evolution. Therefore, an equilibrium xe is stable if for any neighborhood of xe (even very small) the perturbed evolution stays within this neighborhood for all initial perturbations belonging to a sufficiently small neighborhood of xe . The definition of stability can be interpreted as follows. An equilibrium point xe is stable if however we select a tolerable deviation ǫ, there exists a (sufficiently small) region with the equilibrium xe in its interior, such that all initial perturbations in this region give rise to trajectories which are within the tolerable deviation. Remark. The constant δ is in general a function of t0 . If it is possible to define a δ which does not depend upon t0 , we say that the equilibrium is uniformly stable. Note that if an equilibrium of a timeinvariant system is stable, it is also uniformly stable. ⋄ The property of stability dictates a condition on the free evolution of the system for all t ≥ t0 . Note, however, that in the definition of stability we have not requested that the perturbed evolution converges, for t → ∞, to xe . This convergence property is very important in applications, as it allows to characterize the situation in which not only the perturbed evolution remains close to the unperturbed evolution, but it also converges to the initial (unperturbed) evolution. To capture this property we introduce a new definition. Definition 2.5 [Asymptotic stability] Consider a system {X, φ, η} and an equilibrium point xe . The equilibrium is asymptotically stable if it is stable and if there exists a constant δa = δa (ǫ, t0 ) such that kx(t0 ) − xe k < δa
43
2.5. LYAPUNOV STABILITY implies lim kx(t) − xe k = 0.
t→∞
(2.23)
In summary, an equilibrium point is asymptotically stable if it is stable and whenever the initial perturbation is inside a certain neighborhood of xe the perturbed evolution converges, as t → ∞, to the equilibrium point, which is said to be attractive. From a physical point of view this means that all sufficiently small initial perturbations give rise to effects which can be apriori bounded (stability) and these vanish as t → ∞ (convergence).
It is important to realize that convergence does not imply stability: it is possible to have an equilibrium of a system which is not stable (i.e. it is unstable), yet for all initial perturbations the perturbed evolution converges to the equilibrium.
Remark. To define the notion of uniform asymptotic stability, it is important to understand the role of t0 in the convergence property. The existence of the limit in equation (2.23) implies that, for any κ > 0 there is a time ta such that kx(t) − xe k ≤ κ, for all t ≥ ta . The value ta is a function of κ (consistently with the definition of limit), but may be also function of t0 . Therefore, the convergence property depends upon t0 in two ways: firstly through the constant δa (ǫ, t0 ) and secondly on the fact that the speed of convergence of kx(t) − xe k, which can be measured by ta − t0 , depends upon t0 . If δa and ta − t0 are not function of t0 then the convergence is uniform. Finally, if the equilibrium is uniformly stable, and if the convergence property is uniform, the equilibrium is uniformly asymptotically stable. ⋄ Example 2.12 Consider the discretetime system xk+1 = −xk , with x ∈ IR. This system has the unique equilibrium xe = 0. Note that, for any initial condition x0 ∈ IR one has x2i−1 = −x0
x2i = x0 ,
for all i ≥ 1. This implies that the equilibrium is uniformly stable, but not attractive. Example 2.13 Consider the discretetime system ¨ +
x =
2x 0
if x < 1 if x ≥ 1,
with x ∈ IR. This system has the unique equilibrium xe = 0. This equilibrium is attractive, i.e. for any initial condition the trajectories converge to xe , but it is not stable.
44
CHAPTER 2. STABILITY
Example 2.14 Consider the continuoustime system x˙ 1 = ψ(t, x1 , x2 )x2
x˙ 2 = −ψ(t, x1 , x2 )x1 ,
with ψ(t, x1 , x2 ) > 0 for all (t, x1 , x2 ). The system has the unique equilibrium xe = 0. This equilibrium is stable, but not attractive. To see this note that, along the trajectories of the system, x1 x˙ 1 + x2 x˙ 2 = 0, and this implies that, along the trajectories of the system, x21 (t) + x22 (t) is constant for every t ∈ T , i.e. x21 (t) + x22 (t) = x21 (t0 ) + x22 (t0 ). Therefore the trajectory of the system with initial state (x1 (t0 ), x2 (t0 )) remains on a circle of radius x21 (t0 ) + x22 (t0 ) for all t, hence the condition for stability holds with δ(ǫ, t0 ) = ǫ. Finally, because δ does not depend upon t0 the equilibrium xe = 0 is uniformly stable. Definition 2.6 [Global asymptotic stability] Consider a system {X, φ, η} and an equilibrium point xe . The equilibrium is globally asymptotically stable if it is stable and if, for all x(t0 ) ∈ X, lim kx(t) − xe k = 0. t→∞
Remark. The property of global asymptotic stability is very strong, and it has important implications on the structure of the underlying state space realization. For example, in the case of finitedimensional, timeinvariant systems, it implies that X = IRn . This fact, which is the consequence of a very delicate theorem of J.W. Milnor, is sometimes informally explained with the sentence “it is not possible to comb the hair on a sphere without leaving a crown somewhere”. ⋄ Obviously, the property of global asymptotic stability is much stronger than the property of asymptotic stability (which is often referred to as local asymptotic stability), as it requires that the effect of all initial perturbations vanishes as t → ∞.
If an equilibrium xe is not globally asymptotically stable, it is possible to determine a region of X, containing xe , such that for all initial conditions in this region the free evolution converges to xe . This region is known as region of attraction of the equilibrium xe . Note that if xe is globally asymptotically stable then its region of attraction coincides with X. The property of asymptotic stability can be strengthened imposing conditions on the convergence speed of kx(t) − xe k.
45
2.5. LYAPUNOV STABILITY
Definition 2.7 [Exponential stability] Consider a system {X, φ, η} and an equilibrium point xe . The equilibrium is exponentially stable if there exists λ > 0 such that for all ǫ > 0 there exists a δ = δ(ǫ) > 0 such that kx(t0 ) − xe k < δ implies kx(t) − xe k < ǫe−λ(t−t0 ) ,
(2.24)
for all t ≥ t0 . Remark. The property of exponential stability implies the property of stability and the property of uniform asymptotic stability. ⋄ Definition 2.8 [Stability of motion] Consider a system {X, φ, η} and a motion M = {(t, x(t)) ∈ T × X : t ∈ F (t0 ), x(t) = φ(t, t0 , x0 , u)}. The motion is stable if for every ǫ > 0 there exists a δ = δ(ǫ, t0 ) > 0 such that kx(t0 ) − x0 k < δ implies kφ(t, t0 , x(t0 ), u) − φ(t, t0 , x0 , u)k < ǫ,
(2.25)
for all t ≥ t0 . The notion of stability of motion is substantially similar to the notion of stability of an equilibrium. The important issue is that the timeparameterization is important, i.e. a motion is stable if, for small initial perturbations, for any t ≥ t0 the perturbed evolution is close to the nonperturbed evolution. This does not mean that if the perturbed and unperturbed trajectories are close then the motion is stable: in fact the trajectories may be close, but may be followed with different timing, which means that for some t ≥ t0 condition (2.25) may be violated. Example 2.15 To illustrate, informally, the fact that an unperturbed and a perturbed trajectory can be close, yet the nominal unperturbed motion is not stable, consider two identical chemical reactions taking place one in the presence of a catalyst and the other without catalyst. The two reactions follow the same trajectory, but with a different time parameterization, i.e one is faster than the other. Then, the distance between the two motions may be larger than any preassigned bound.
46
2.5.2
CHAPTER 2. STABILITY
Stability of linear systems
The notion of stability relies on the knowledge of the trajectories of the system. As a result, even if this notion is very elegant, and useful in applications, it is in general very hard to assess stability of an equilibrium or of a motion. There are, however, classes of systems for which it is possible to give stability conditions without relying upon the knowledge of the trajectories. Linear systems belong to one such class. Therefore, in this section, we study the stability of linear systems ad we show that, because of the linear structure, it is possible to assess the properties of stability and attractivity in a simple way. To begin with, we recall some properties of linear representations. Proposition 2.4 Consider a system with a linear state space representation. Then (asymptotic) stability of one motion implies (asymptotic) stability of all motions. In particular, (asymptotic) stability of any motion implies and is implied by (asymptotic) stability of the equilibrium xe = 0. Proof. It is enough to prove the second claim. Consider a motion φ(t, t0 , x0 , u) and note that, by definition, the motion is stable if for every ǫ > 0 there exists a δ = δ(ǫ, t0 ) > 0 such that kx(t0 ) − x0 k < δ implies kφ(t, t0 , x(t0 ), u) − φ(t, t0 , x0 , u)k < ǫ, for all t ≥ t0 . However, by linearity of the state space representation φ(t, t0 , x(t0 ), u) − φ(t, t0 , x0 , u) = φ(t, t0 , x(t0 ) − x0 , 0). Hence, the motion is stable if and only if the equilibrium xe = 0 is stable. An analogous argument can be used to prove the asymptotic stability claim. ⊳ The above statement, together with the result in Proposition 2.1, implies the following important properties. Proposition 2.5 If the origin of the linear representation of a system is asymptotically stable, then necessarily, the origin is the only equilibrium of the system for u = 0. Moreover, asymptotic stability of the zero equilibrium is always global. Finally, uniform asymptotic stability implies exponential stability. The above discussion shows that the stability properties of a motion (e.g. an equilibrium) of a linear representation are inherited by all motions of the system. Moreover, for linear representations local properties are always global properties. This means that, with some
47
2.5. LYAPUNOV STABILITY
abuse of terminology, we can refer the stability properties to the linear representation, for example we say that a linear representation is stable to mean that all its motions are stable. Note that it does not make sense to say that a nonlinear representation or a nonlinear system is stable, despite the fact that this terminology is often used! The stability results discussed in this section apply to general linear representations, in particular it is not necessary to assume that X = IRn , for some n ≥ 1. However, in this case, i.e. in the case of a finitedimensional linear representation, it is possible to obtain very simple stability tests. To derive such tests, note that, by linearity of φ and by the finite dimensionality of X = IRn , we have that φ(t, t0 , x0 , 0) = Φ(t, t0 )x0 , where Φ(t, t0 ), defined for all t ≥ t0 , is a square matrix of dimension n × n, known as the state transition matrix. For finitedimensional linear representations this matrix plays a central role in the study of stability properties. Proposition 2.6 A linear, finitedimensional, representation is stable if and only if kΦ(t, t0 )k ≤ k,
(2.26)
for all t ≥ t0 and for some k > 0 possibly dependent on t0 . Proof. We prove only the sufficient part. For, suppose condition (2.26) holds. Then kx(t)k = kΦ(t, t0 )x0 k ≤ kkx0 k. Therefore, for any ǫ > 0 the selection δ=
ǫ k
is such that implies
kx0 k < δ kx(t)k < ǫ,
for all t ≥ t0 , which is the stability property for the equilibrium xe = 0.
⊳
A similar result, with conceptually similar proof, holds with respect to the property of asymptotic stability. Proposition 2.7 A linear, finitedimensional, representation is asymptotically stable if and only if kΦ(t, t0 )k ≤ k, for all t ≥ t0 , and for some k > 0 possibly dependent on t0 , and lim Φ(t, t0 ) = 0.
t→∞
48
CHAPTER 2. STABILITY
We conclude this section noting that the above statements can be further simplified, if we assume in addition that the linear representation is timeinvariant. To develop these tests, we need the following fact. Proposition 2.8 The equilibrium x0 of a linear, finitedimensional, timeinvariant representation is asymptotically stable if and only if it is attractive. This result implies that for linear, finitedimensional, timeinvariant representations attractivity of the zero equilibrium implies stability of the zero equilibrium. This implies that stability is a property of the matrix A (see equation (2.8)). Moreover, as the properties are uniform, attractivity implies uniform asymptotic stability, and hence exponential stability. Proposition 2.9 The equilibrium xe = 0 of a linear, finitedimensional, timeinvariant representation is stable if and only if the following conditions hold. • In the case of continuoustime systems, the eigenvalues of A with geometric multiplicity9 equal to one have nonpositive real part, and the eigenvalues of A with geometric multiplicity larger than one have negative real part. • In the case of discretetime systems, the eigenvalues of A with geometric multiplicity equal to one have modulo not larger than one, and the eigenvalues of A with geometric multiplicity larger than one have modulo smaller than one. Proof. Recall that, for the considered class of representations, stability implies and is implied by boundedness of the state transition matrix. For continuoustime systems the state transition matrix, with t0 = 0, is (see Footnote 6) eAt =
mi r X X i=1 k=1
Rik
tk−1 λi t e , (k − 1)!
where mi is the geometric multiplicity of the eigenvalue λi . This matrix is bounded if and only if the conditions in the statement hold. 9 To define the geometric multiplicity of an eigenvalue we need to recall a few facts. Consider a matrix A ∈ IRn×n and a polynomial p(λ). The polynomial p(λ) is a zeroing polynomial for A if p(A) = 0. Note that, by CayleyHamilton Theorem, the characteristic polynomial of A is a zeroing polynomial for A. Among all zeroing polynomials there is a unique monic polynomial pM (λ) with smallest degree. This polynomial is called the minimal polynomial of A. Note that the minimal polynomial of A is a divisor of the characteristic polynomial of A. If A has r distinct eigenvalues λ1 , ..., λr then
pM (λ) = (λ − λ1 )m1 (λ − λ2 )m2 · · · (λ − λr )mr , where the numbers mi denote by definition the geometric multiplicity of λi . This means that the geometric multiplicity of λi equals the multiplicity of λi as a root of pM (λ). Recall, finally, that the multiplicity of λi as a root of the characteristic polynomial is called algebraic multiplicity.
49
2.5. LYAPUNOV STABILITY
Similarly, for discretetime systems, the state transition matrix, for t0 = 0 and t ≥ 1, is t
A =
mi r X X
Rik
i=1 k=1
tk−1 t−k+1 λ , (k − 1)! i
and this is bounded if and only if the conditions in the statement hold.
⊳
Proposition 2.10 The equilibrium xe = 0 of a linear, finitedimensional, timeinvariant representation is asymptotically stable if and only if the following conditions hold. • In the case of continuoustime systems, the eigenvalues of A have all negative real part. • In the case of discretetime systems, the eigenvalues of A have all modulo smaller than one. Proof. The proof is similar to the one of the previous proposition, once it is noted that, for the considered class of representations, asymptotic stability implies and is implied by boundedness and convergence of the state transition matrix. ⊳ We conclude this discussion with an alternative characterization of asymptotic stability, the proof of which is outside the scope of these lecture notes. Proposition 2.11 The equilibrium xe = 0 of a linear, finitedimensional, timeinvariant representation is asymptotically stable if and only if the following conditions hold. • In the case of continuoustime systems, there exists a positive definite matrix10 P = P ′ such that A′ P + P A < 0. 10
A square and symmetric matrix P ∈ IRn×n is positive definite, denoted P > 0, if v ′ P v > 0,
for all nonzero vectors v ∈ IRn . Note that the symmetry condition is without loss of generality. In fact, a nonsymmetric matrix M is the sum of a symmetric matrix P and an antisymmetric matrix Q. Hence v ′ M v = v ′ (P + Q)v = v ′ P v. To test positivity of a symmetric matrix 2 6 6
P =6 6 4
p11 p12 p13 .. . p1n
p12 p22 p23 .. . p2n
p13 p23 p33 .. . . . . p3n
··· ··· ··· .. . ···
3
p1n p2n 7 7 p3n 7 7 5
pnn
50
CHAPTER 2. STABILITY • In the case of discretetime systems, there exists a positive definite matrix P = P ′ such that A′ P A − P < 0.
To complete our discussion we stress that stability properties are associated to the specific state space representation of the system that we consider. Thus, another state space representation of the same system may have different stability properties. Nevertheless, representations related by a change of coordinates with specific properties have the same stability properties. We discuss this issue with reference to linear representations and linear change of coordinates. To this end, consider a change of coordinates described by x(t) = L(t)ˆ x(t),
(2.27)
with L(t) invertible for all t, and note that the state transition matrix of the representation with state x ˆ is given by ˆ t0 ) = L−1 (t)Φ(t, t0 )L(t0 ). Φ(t, As a consequence, the following results hold. Proposition 2.12 Consider a state space representation of a linear, finitedimensional system, and assume it is (asymptotically) stable. Then any representation obtained by means of a change of variable of the form (2.27) is (asymptotically) stable if and only if kL−1 (t)k ≤ k2 ,
kL(t)k ≤ k1
(2.28)
for some constants k1 and k2 , and for all t. Corollary 2.2 Consider a state space representation of a linear, finitedimensional, timeinvariant system, and assume it is (asymptotically) stable. Then any representation obtained by means of a change of variable of the form (2.27) with L(t) constant and invertible is (asymptotically) stable. Proof. It is enough to note that if L(t) is constant and invertible then condition (2.28) holds. ⊳
Example 2.16 To show the importance of the bounds (2.28) consider the system described by x˙ = −x, we could use the Sylvester test, which states that P = P ′ > 0 if and only if p11 > 0
p11 p12
p12 >0 p22
p11 p12 p
13
p12 p22 p23
p13 p23 p33
>0
···
Note finally, that a matrix P is negative definite, denoted P < 0 is −P > 0.
detP > 0.
51
2.6. COORDINATES TRANSFORMATIONS with x ∈ IR, which is asymptotically stable, and the change of coordinates x ˆ(t) = eαt x(t).
Note that, for α 6= 0, this change of coordinates is such that conditions (2.28) do not hold. The state space representation with the state variable x ˆ is given by x ˆ˙ = (α − 1)ˆ x, and its stability depends upon the value of α!
2.6
Coordinates transformations
Because of the importance that it has in the forthcoming chapters, we elaborate on the operation of coordinates transformation. This operation is often used to simplify the state space representation of a given system or to highlight some specific property. Consider a continuoustime, finitedimensional, linear system described by the equations x˙ = A(t)x + B(t)u
y = C(t)x + D(t)u,
with x ∈ X = IRn , u(t) ∈ IRm , y(t) ∈ IRp , and the change of coordinates x(t) = L(t)ˆ x(t),
(2.29)
with L(t) invertible for all t. The state space representation in the new coordinates is given by x ˆ˙ = L−1 (t)(A(t)x + B(t)u) + L˙ −1 (t)x x=L(t)ˆx = (L(t)−1 A(t)L(t) + L˙ −1 (t)L(t))ˆ x + L−1 (t)B(t)u, and y = CL(t)ˆ x + D(t)u. Similarly, for a a discretetime, finitedimensional, linear system described by the equations x+ = A(t)x + B(t)u
y = C(t)x + D(t)u,
with x ∈ X = IRn , u(t) ∈ IRm , y(t) ∈ IRp , and the change of coordinates x(t) = L(t)ˆ x(t), with L(t) invertible for all t, the state space representation in the new coordinates is given by x ˆ+ = L−1 (t + 1)A(t)L(t)ˆ x + L−1 (t + 1)B(t)u, and y = CL(t)ˆ x + D(t)u.
52
CHAPTER 2. STABILITY
In the case of timeinvariant systems, we focus on timeinvariant coordinates transformations, i.e. x = Lˆ x, with L invertible. The transformed system is described by σˆ x = L−1 ALˆ x + L−1 Bu
y = CLˆ x + Du.
The matrices A and L−1 AL have the same eigenvalues, and the same characteristic and minimal polynomials, and they are said to be similar. Motivated by this discussion we introduce the following definition. Definition 2.9 (Algebraic equivalent systems) The state space representations σx = Ax + Bu
y = Cx + Du
with x ∈ IRn , u(t) ∈ IRm , y(t) ∈ IRp , and ˆx + B ˆu σˆ x = Aˆ ˆ
ˆu yˆ = Cˆ x ˆ + Dˆ
with x ˆ ∈ IRn , u ˆ(t) ∈ IRm , yˆ(t) ∈ IRp , are algebraically equivalent if there exists a nonsingular matrix L such that Aˆ = L−1 AL
ˆ = L−1 B B
Cˆ = CL
ˆ = D. D
Note that, for algebraically equivalent representations, one has ˆ CAi B = Cˆ Aˆi B and
ˆ ˆ ˆ At CeAt B = Ce B.
Example 2.17 (Lyapunov transformation) Consider a system, without input, described by x˙ = A(t)x, (2.30) with x ∈ IRn , and A(t) periodic of period T , i.e. A(t + T ) = A(t), for all t ∈ IR. A coordinates transformation x(t) = L(t)ˆ x(t), with L(t) periodic of period T , differentiable, bounded and nonsingular for every t, and with bounded inverse, is called a Lyapunov transformation if the matrix A˜ = L−1 (t)A(t)L(t) + L˙ −1 (t)L(t) has constant entries. Note that in general, for a given periodic matrix A(t) it is not possible to decide on the existence of a Lyapunov transformation. However, if such a transformation does exist, then it is possible to decide stability of the system (2.30) by ˜ computing the eigenvalues of A.
2.7. STABILITY IN THE FIRST APPROXIMATION
2.7
53
Stability in the first approximation
In Section 2.4 we have seen that it is possible to approximatly describe a nonlinear system around a given operating condition by means of a linear system. This approximation makes sense only if the deviation variables δu and δx are small for all t, i.e. only if δu is selected sufficiently small and if a sort of stability property holds for δx . In this section we clarify this issue. In particular we show that, in several cases of practical interest, from properties of the linearized model it is possible to infer properties of the nominal motion of the nonlinear system. Proposition 2.13 Consider a nonlinear, timeinvariant, system σx = f (x), with x ∈ X. Let xe be an equilibrum of the system and let σδx = Aδx be the corresponding linearized system. • In the case of continuoustime systems, if A is nonsingular then the equilibrium xe is isolated. • In the case of discretetime systems, if A − I is nonsingular then the equilibrium xe is isolated. Remark. An equilibrum of a nonlinear system may be isolated even if the matrix A, or A − I, which describes the linearized system around the equilibrium is singular. For example, consider the discretetime system 2 x+ 1 = x1 + x2
3 x+ 2 = x2 + x1 ,
and the equilibrium (x1 , x2 ) = (0, 0). Note that this is the unique equilibrium of the system. Nevertheles, the matrix A of the linearized system around the equilibrium is equal to the identity, hence A − I is identically equal to zero, hence it is singular. ⋄ The result in Proposition 2.13 is important because only isolated equilibria may be asymptotically stable, hence it provides a necessary condition for asymptotic stability of the equilibrium of the nonlinear system. Sufficient conditions can be also developed, as given in the following statement. Proposition 2.14 Consider a nonlinear, timeinvariant, system σx = f (x),
54
CHAPTER 2. STABILITY
with x ∈ X. Let xe be an equilibrum of the system and let σδx = Aδx be the corresponding linearized system. • Asymptotic stability of the linearized system implies (local) asymptotic stability of the equilibrium x = xe of the nonlinear system. • In the case of continuoustime systems, if the linearized system is unstable because the matrix A has eigenvalues with positive real part then the equilibrium x = xe of the nonlinear system is unstable. • In the case of discretetime systems, if the linearized system is unstable because the matrix A has eigenvalues with modulo larger than one then the equilibrium x = xe of the nonlinear system is unstable. The second and third claims cannot be relaxed replacing the fact that one eigenvalue has positive real part, in the case of continuoustime systems, or modulo larger than one, in the case of discretetime systems, with the property that the linearized system is unstable, as illustrated in the following example. Example 2.18 Consider the system x˙ 2 = −x31 ,
x˙ 1 = x2
with x = (x1 , x2 ) ∈ IR2 . The origin is an equilibrium of the system, and the linearized system around such an equilibrium is
δ˙x =
0 1 0 0
δx ,
hence it is unstable. However, the zero equilibrium of the nonlinear system is stable. To prove this note that x31 x˙ 1 + x2 x˙ 2 = 0, hence
x41 (t) x22 (t) + 4 2 is constant along all trajectories of the system. As a results, using similar arguments as those in Example 2.14, we conclude stability of the equilibrium. Example 2.19 Consider the Newton iteration discussed in Example 2.10. Therein, it has been shown that the system linearized around any stationary point such that condition (2.19) holds is δx+ = 0,
55
2.7. STABILITY IN THE FIRST APPROXIMATION
hence the stationary point, which is an equilibrium of the nonlinear system, is locally asymptotically stable. This property justifies the fast convergence rate of Newton’s method whenever initialized close to a stationary point, provided the Hessian matrix of the function be nonsingular at the point. Example 2.20 Proposition 2.14 holds also for nontimeinvariant systems. For example, consider the system in Example 2.11. Suppose that, for all t ∈ IR, ∂g(x1 x2 ) ∂x1 are bounded and
e−t e x1 (0), x1 (0)
t
∂g(x1 x2 ) ∂x2
+1
e−t e x1 (0), x1 (0)
∂g(x1 x2 ) ∂x1
e−t e x1 (0), x1 (0)
t
t
+ 1 < −ǫ < 0,
for some constant ǫ > 0. Then, by direct integration of the linear differential equations, it is possible to conclude that the system (2.22) is asymptotically stable. This, in turn, implies that the motion associated to the trajectory (2.21) is locally asymptotically stable. Example 2.21 Newton’s iteration
−1
∂ 2 f (x) x =x− (x) ∂x2 +
∂f (x) (x). ∂x
is not welldefined, hence not applicable, for all x such that the matrix
∂ 2 f (x) (x) ∂x2
is singular. To avoid this difficulty it is sometimes modified into +
x = x − det
∂ 2 f (x) (x) ∂x2
−1
∂ 2 f (x) (x) ∂x2
∂f (x) (x). ∂x
(2.31)
This modification, known as discrete Branin method, is welldefined for all x. However, it has the disadvantage that there are equilibria of the system (2.31) which are not stationary points of the function f . (These points are called extraneous singularities). Moreover, the system linearized around a stationary point xe of f is given by
δx+
=
1 − det
∂ 2 f (x) (xe ) ∂x2
δx ,
hence it is not possible to claim anything on the local stability of the equilibrium xe .
56
CHAPTER 2. STABILITY
Chapter 3
The structural properties
58
3.1
CHAPTER 3. THE STRUCTURAL PROPERTIES
Introduction
In this chapter we focus on linear, timeinvariant, systems, namely systems described by equations of the form σx = Ax + Bu
y = Cx + Du,
(3.1)
with x ∈ X = IRn , u(t) ∈ IRm , y(t) ∈ IRp and A, B, C, and D matrices of appropriate dimensions and with constant entries. For such a class of systems we study the inputtostate and statetooutput interactions, which are characterized by means of the socalled structural properties: reachability and controllability, observability and reconstructability, respectively. These properties allow to quantify and describe in a precise way the effect of input signals on the state of the system and the ability to reconstruct the state of the system by means of measurements of the output variable. These properties are naturally defined in terms of properties of the trajectories of the system. However, because of linearity and timeinvariance, it is possible to characterize such properties in terms of properties of the matrices arising in the state space representation.
3.2
Reachability and controllability
When studying the inputtostate interaction we can take two different points of view. In the former, we assume that the initial state of the system, i.e. the state of the system at time t = 0, is fixed and we consider the problem of determining the states of the system that can be reached applying a certain input signal over a given period of time. In this case we study the socalled reachability property. In the latter, we assume that the final state of the system, at some time T , is fixed and we aim at determining all initial states that can be steered, by means of a certain input signal, to the selected final state. In this case we study the socalled controllability property. In the study of reachability and controllability, whenever the input signal that drives a certain initial state to a certain final state is not unique, we could impose constraints on such an input signal, e.g. we could consider the input signal with minimum energy, or with minimum amplitude, or the input signal which achieves the transfer in minimum time. If no constraint is imposed all input signals achieving the considered trasfer are equivalent. For linear systems the properties of reachability and controllability are referred to the state x = 0, hence we say that a state is reachable to mean that it is reachable from x = 0 and that a state is controllable to mean that it is controllable to x = 0. Note moreover that, because these properties are used to describe the inputtostate interaction they, trivially, depend only upon properties of the matrices A and B.
59
3.2. REACHABILITY AND CONTROLLABILITY
3.2.1
Reachability of discretetime systems
Consider a linear, timeinvariant, discretetime, system. Let x(0) = 0 and consider an input sequence u(0), u(1), u(2), · · · , u(k − 1). The state reached at t = k is given by 2
x(k) =
B AB · · ·
Ak−1 B
6 6 6 6 4
u(k − 1) u(k − 2) .. .
3
7 7 7. 7 5
u(0) This implies that the set of states that can be reached at t = k is a linear space, i.e. it is the subspace Rk spanned by all linear combinations of the columns of the matrix Rk =
B AB · · ·
Ak−1 B
,
i.e. Rk = ImRk . The set Rk is a vector space, denoted as the reachable subspace in k steps. If Rk = X, i.e. rankRk = n, then all states of the system are reachable in (at most) k steps and the system is said to be reachable in k steps. As k varies, we have a sequence of subspaces, namely R1 , R2 , · · · , Rk , · · · .
(3.2)
This sequence of subspaces is such that the following properties hold. Proposition 3.1 The sequence of subspaces (3.2) is such that R1 ⊆ R2 ⊆ · · · ⊆ Rk ⊆ · · · . ¯ Rk = R¯ . Finally, ¯ R¯ = R¯ , then, for all k ≥ k, Moreover, if for some k, k k+1 k R1 ⊆ R2 ⊆ · · · ⊆ Rn = Rn+1 . Proof. To prove the first claim note that if a state x ¯ is reached from zero in k steps, using the input sequence u(0), u(1), · · · , u(k − 1), then the same state is also reached from zero in k + 1 steps, using the input sequence 0, u(0), u(1), · · · , u(k − 1), hence, for all k ≥ 1, Rk ⊆ Rk+1 . To prove the second claim it is enough to show that
= Rk+2 ⇒ Rk+1 Rk¯ = Rk+1 ¯ , ¯ ¯
(3.3)
belongs also to Rk+1 ¯ ∈ Rk+2 or, equivalently that if Rk¯ = Rk+1 ¯ . For, let ¯ ¯ , then any x . This means that there is an input sequence which steers the x ¯ be an element of Rk+2 ¯
60
CHAPTER 3. THE STRUCTURAL PROPERTIES
state of the system from x(0) = 0 to x ¯ in k¯ + 2 steps. Consider now the state reached ¯ after k + 1 steps, using the same input sequence, which we denote with x ˜. By assumption, x ˜ ∈ Rk+1 = R , hence there is an input sequence which steers the state of the system ¯ ¯ k ¯ from x(0) = 0 to x ˜, in k steps. However, by definition of x ˜ it is possible to steer x ˜ to x ¯ in one step, hence there is an input sequence which steers x(0) = 0 to x ¯, in k¯ + 1 steps, which proves the claim. To prove the third claim note that if, for some k < n, Rk = Rk+1 then the claim follows from equation (3.3). Suppose now that, for all k, the dimension of Rk+1 is strictly larger than the dimension of Rk . This implies that the sequence dim Rk is strickly increasing at each step. However, this sequence is bounded (from above) by n, and this proves the claim. ⊳
Definition 3.1 Consider the discretetime system (3.1). The subspace R = Rn is the reachability subspace of the system. The matrix R = Rn is the reachability matrix of the system. The system is said to be reachable if R = X = IRn . Remark. By definition, R = ImR, hence the discretetime system (3.1) is reachable if and only if rankR = n.
(3.4)
Equation (3.4) is known as Kalman reachability rank condition, and was derived by R.E. Kalman in the 60’s. ⋄ Remark. From the above discussion it is obvious that, in a ndimensional, linear, discretetime system, if a state x ¯ is reachable, then it is reachable in at most n steps. This does not mean that n steps are necessarily required, i.e. the state x ¯ could be reached in less than n steps. In a reachable system, the smallest integer k⋆ such that rankRk⋆ = n is called the reachability index of the system. Note that, for singleinput reachable systems, necessarily, k⋆ = n. ⋄
61
3.2. REACHABILITY AND CONTROLLABILITY Example 3.1 Consider a discretetime system with x ∈ IR3 , 2 6
A=4
0 0 0 0 0 0 0 0 1
3
2
7 5
B=4
6
0 1 1
Then
3 7 5.
2
R1 = spanB
6
R2 = R3 = R = span 4
0 0 1 0 1 1
3 7 5.
Hence the system is not reachable, and its reachable subspace has dimension two. Example 3.2 Consider a discretetime system with x ∈ IR3 , 2 6
A=4 Then
0 1 0 0 0 1 0 0 0 2
R1 = spanB
6
R2 = span 4
3
2
7 5
B=4
α β β γ γ 0
6
α β γ
3 7 5.
3 7 5
2 6
R3 = span 4
α β γ β γ 0 γ 0 0
3 7 5.
As a result, the system is reachable if and only if γ 6= 0. Moreover, if γ = 0 and β 6= 0, the system is not reachable and the reachable subspace has dimension two. Finally, if γ = β = 0 and α 6= 0, the system is not reachable and the reachable subspace has dimension one. The reachability subspace R has the following important property, the proof of which is a simple consequence of the definition of the subspace. Proposition 3.2 The reachability subspace contains the subspace spanB, i.e. spanB ⊆ R, and it is Ainvariant, i.e. AR ⊆ R. We conclude this section noting that algebraically equivalent systems have the same reachability properties. In particular, consider two algebraically equivalent systems, with state x and x ˆ, respectively. Let L be the coordinates transformation matrix, as given in equaˆ k the reachability subspaces, and R and R ˆ the reachability matrices, tion (2.29), Rk and R respectively. Then ˆ k = L−1 Rk , R hence
ˆ = L−1 R, R
and one of the two systems is reachable if and only if the other is.
62
3.2.2
CHAPTER 3. THE STRUCTURAL PROPERTIES
Controllability of discretetime systems
The results established for the reachability property can be easily exploited to characterize the controllability property. In fact, for a linear, timeinvariant, discretetime system, a state x⋆ is controllable (to zero) in k steps if there exists an input sequence u(0), u(1), · · · , u(k − 1) that drives the state from x(0) = x⋆ to x(k) = 0, i.e. 2
k ⋆
0=A x +
B AB · · ·
Ak−1 B
6 6 6 6 4
u(k − 1) u(k − 2) .. .
3
7 7 7, 7 5
u(0) or equivalently
2 k ⋆
−A x =
B AB · · ·
k−1
A
B
6 6 6 6 4
u(k − 1) u(k − 2) .. .
3 7 7 7. 7 5
u(0) This last equation implies that x⋆ is controllable if the state −Ak x⋆ is reachable in k steps, hence if −Ak x⋆ ∈ Rk . (3.5) It is easy to see that the set of all x⋆ such that equation (3.5) holds is a vector space, denoted by Ck , and called the controllability subspace in k steps. A linear, discretetime, system is controllable in k steps if ImAk ⊆ Rk . Example 3.3 Consider the system in Example 3.1. The system is controllable in two steps. In fact A2 = A ∈ R2 = R. Example 3.4 Consider the system in Example 3.2. The system is controllable in three steps no matter the values of α, β, and γ. Note in fact that A3 = 0. The system is controllable in two steps if γ = 0 and α 6= 0 or β 6= 0. Finally, it is controllable in one step if γ = 0, and αβ 6= 0. Similarly to the reachability subspaces, as k varies we have a sequence of controllability subspaces, namely C1 , C2 , · · · , Ck , · · · . (3.6) This sequence of subspaces is such that the following properties hold.
3.2. REACHABILITY AND CONTROLLABILITY
63
Proposition 3.3 The sequence of subspaces (3.6) is such that C1 ⊆ C2 ⊆ · · · ⊆ Ck ⊆ · · · . ¯ Ck = C¯ . Finally, ¯ C¯ = C¯ , then, for all k ≥ k, Moreover, if for some k, k k+1 k C1 ⊆ C2 ⊆ · · · ⊆ Cn = Cn+1 . Proof. The proof of this statement is similar to the one of Proposition 3.1. We simply remark that, if a state is controllable in k steps, using the input sequence u(0), u(1), · · · , u(k − 1), then the same state is also controllable in k + 1 steps, using the input sequence u(0), u(1), · · · , u(k − 1), 0. ⊳ Definition 3.2 Consider the discretetime system (3.1). The subspace C = Cn is the controllability subspace of the system. The system is said to be controllable if C = X = IRn . The discretetime system (3.1) is controllable if and only if ImAn ⊆ R.
(3.7)
In particular, if A is nilpotent, i.e. Aq = 0, for some q ≤ n, then for any B (even B = 0) the system is controllable. Note that a reachable system is controllable, but the converse statement does not hold. In particular R ⊆ C ⊆ X = IRn .
3.2.3
Construction of input signals
The study of the properties of reachability and controllability leads to the following question. Is it possible to explicitly construct an input sequence which steers the state of the system from an initial condition x0 , i.e. x(0) = x0 , to a final condition xf in k steps, i.e. x(k) = xf ? To answer this question, consider the problem of determining an input sequence u(0), u(1), · · · , u(k − 1) such that xf − Ak x0 = Rk Uk−1 , (3.8) where
2
Uk−1 =
6 6 6 6 6 6 4
u(k − 1) u(k − 2) .. . u(1) u(0)
3 7 7 7 7 7 7 5
∈ IRk m .
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CHAPTER 3. THE STRUCTURAL PROPERTIES
To solve the problem we have to solve the linear system (3.8) in the unknown Uk−1 . This system has a solution if, and only if, xf − Ak x0 ∈ ImRk ,
(3.9)
which clearly shows the role of the matrices Rk in solving the considered problem. Note that the input sequence achieving the desired goal may not be unique. In particular, several solutions can be obtained as the linear combination of a particular solution of equation (3.9) and a solution of the homogeneous equation Rk Uk−1 = 0. In the special case of a reachable system, it is possible to obtain an explicit expression for one input sequence solving the considered problem in n steps. To this end, note that by reachability, rankRn = n, hence the condition expressed in equation (3.9), with k = n, holds. Consider now an input signal defined as Un−1 = Rn′ v, where v has to be determined. Using this definition, and setting k = n, equation (3.8) becomes xf − An x0 = Rn Rn′ v,
where the matrix Rn Rn′ is square and invertible. Hence, a control sequence solving the considered problem in n steps is given by Un−1 = Rn′ (Rn Rn′ )−1 (xf − An x0 ). It is possible to show that, among all input sequences steering the state of the system from x0 to xf in n steps the one constructed has minimal norm (energy).
3.2.4
Reachability and controllability of continuoustime systems
The properties of reachability and controllability for linear, timeinvariant, continuoustime systems can be assessed using the same ideas exploited in the case of discretetime systems. However, the tools are more involved as the inputstate relation is expressed by means of an integral (see equation (2.9)). Consider the reachability problem, i.e. the initial state of the system is x(0) = 0 and we want to characterize all states x ¯ that can be reached in some interval of time t, i.e. all states such that, for some input function u(t), Z t
x ¯=
0
eA(t−τ ) Bu(τ ) dτ.
Note now that, by CayleyHamilton Theorem eAt = α0 (t)I + α1 (t)A + · · · + αn−1 (t)An−1 ,
65
3.2. REACHABILITY AND CONTROLLABILITY for some scalar functions αi (t). Hence Z t
2
x ¯=
An−1 B
B AB · · ·
3
α0 (t − τ )u(τ ) dτ
6 0 6 6 Z t 6 α1 (t − τ )u(τ ) dτ 6 6 0 6 6 6 .. 6 . 6 4 Z t 0
7 7 7 7 7 7 7. 7 7 7 7 5
αn−1 (t − τ )u(τ ) dτ
This implies that a state x ¯ is reachable only if x ¯ ∈ Im
An−1 B
B AB · · ·
= ImR.
We now prove the converse fact, i.e. that if a state is in the image of R then it is reachable. To this end, define the controllability Gramian Z t
Wt =
0
′
eA(t−τ ) BB ′ eA (t−τ ) dτ,
with t > 0, and note that1 ImR = ImWt .
(3.10)
Selecting ′
u(τ ) = B ′ eA (t−τ ) β, where β is a constant vector, yields x ¯ = Wt β.
(3.11)
Hence, to assess reachability of the state x ¯ ∈ ImR it is sufficient to show that equation (3.11) has (at least) one solution β. However, this fact holds trivially by condition (3.10). Remark. Unlike the case of discretetime systems, where the set of reachable states depends upon the length of the input sequence, for continuoustime systems if a state is reachable, then it is reachable in any (possibly small) interval of time. ⋄ Definition 3.3 Consider the continuoustime system (3.1). The subspace R is the reachability subspace of the system. The matrix R is the reachability matrix of the system. The system is said to be reachable if R = X = IRn . We summarize the above discussion with a formal statement. 1
The proof of this property is not trivial.
66
CHAPTER 3. THE STRUCTURAL PROPERTIES
Proposition 3.4 Consider the continuoustime system (3.1). The following statements are equivalent. • The system is reachable. • rankR = n. • For all t > 0 the controllability Gramian Wt is positive definite. Remark. If a system is reachable then it is possible to explicitly determine one input signal which steers the state of the system from x(0) = 0 to any x ¯ in a give time t > 0. In fact, to determine one such input signal it is sufficient to solve the equation (3.11) which, if the system is reachable, has the unique solution β = Wt−1 x ¯, i.e. the input signal ′
¯ u(τ ) = B ′ eA (t−τ ) Wt−1 x
(3.12)
steers the state of the system from x(0) = 0 to x(t) = x ¯. Similarly to what discussed in Section 3.2.3, it is possible to prove that, among all input signals steering the state from 0 to x ¯ in time t, the input signal (3.12) is the one with minimum energy. Similar considerations can be done to determine an input signal steering a nonzero initial state to a given final state. ⋄ To discuss the property of controllability note that a state x ¯ is controllable (to zero) in time t > 0 if there exists an input signal such that At
Z t
0=e x ¯+
0
eA(t−τ Bu(τ ) dτ.
This, however, implies that hence
eAt x ¯∈R x ¯ ∈ e−At R.
This implies that the set of controllable states in time t > 0 is the set Ct = e−At R, which has the same dimension as R, by invertibility of e−At for all t, and it is contained in R, by the fact that R is Ainvariant, hence it is trivially e−At invariant. As a consequence, for all t > 0, Ct = R,
which shows that the set C of controllable states does not depend upon t > 0 and that a continuoustime system is controllable if and only if it is reachable (unlike what happens for discretetime systems, for which reachability implies, but it is not implied by, controllability).
67
3.2. REACHABILITY AND CONTROLLABILITY
Example 3.5 Consider the linear electric network in Figure 3.1. Assume R1 > 0, R2 > 0, L > 0 and C > 0. The input u is the driving voltage and the output y is the current supplied. y=i L +
R2
u
R1

C
Figure 3.1: A linear electric network. Let x1 be the current through L and x2 the voltage across C. By Kirchhoff ’s laws we have x˙ 1 = −
1 1 R1 x1 − x2 + u L L L
and y = x1 + Therefore
"
A=
− RL1 1 C
− L1
#
" 1 #
B=
" 1
R=
L
0
1 x1 C
1 u. R2
0
The reachability matrix is
x˙ 2 =
1 −R L2
L
0
.
#
1 LC
hence the system is reachable (and controllable) for any L and C. Example 3.6 Consider the linear electric network in Figure 3.2. Assume R1 > 0, R2 > 0, L > 0 and C > 0. The input u is the driving voltage and the output y is the current supplied. Let x1 be the voltage across C and x2 the current through L. By Kirchhoff ’s laws we have x˙ 1 = −
1 1 x1 + u R1 C R1 C
and y=−
x˙ 2 = −
1 1 x1 + x2 + u. R1 R1
R2 1 x2 + u L L
68
CHAPTER 3. THE STRUCTURAL PROPERTIES y=i
R1
R2
C
L
u
Figure 3.2: A linear electric network.
The reachability matrix is
"
R= and
1 detR = R1 CL
#
− R21C 2
1 R1 C 1 L
1
2 −R L2
R2 1 , − R1 C L
hence the system is reachable (and controllable) provided R1 R2 C 6= L.
3.2.5
A canonical form for reachable systems
In this section we focus on singleinput systems and we show that the property of reachability allows to write the system in a special form, known as reachability canonical form. Consider the system (3.1), with m = 1, and suppose the system is reachable, i.e. the rank of the reachability matrix is equal to n. By reachability, there is a (row) vector l such that lB = 0
lAB = 0
lAn−2 B = 0
lAn−1 B = 1.
(3.13)
In fact, conditions (3.13) can be rewritten as lR = hence l=
0 0 ···
0 0 ···
0 1
0 1
is well defined. The vector l has the following important property.
R−1
(3.14)
69
3.2. REACHABILITY AND CONTROLLABILITY Lemma 3.1 Let l be as in equation (3.14). Then the square matrix 2
T =
6 6 6 6 6 6 4
l lA .. .
3
7 7 7 7 7 7 lAn−2 5
lAn−1 is invertible. Proof. Consider the matrix 2
TR =
6 6 6 6 6 6 4
3
l lA .. .
7 7 7 7 7 7 n−2 5 lA
An−2 B An−1 B
B AB · · ·
lAn−1 2
=
6 6 6 6 6 6 6 6 4
lB
lAB
lAB .. .
lA2 B .. .
···
··· . ..
lAn−2 B lAn−1 B lAn−1 B .. .
lAn B .. .
··· ···
··· ···
lAn−2 B lAn−1 B · · · lAn−1 B ··· ···
3 7 7 7 7 7 7 7 7 5
and note that, by conditions (3.13), 2
TR =
6 6 6 6 6 6 4
··· ··· . .. 0 1 ··· 1 lAn B · · · 0 0 .. .
0 0 .. .
0 1 .. .
1 lAn B .. .
··· ···
··· ···
3 7 7 7 7 7 7 5
which shows that det(T R) = 1, hence T is invertible.
⊳
The matrix T can be used to define a new set of coordinates x ˆ such that x ˆ = T x. To derive the state space representation of the system in the x ˆ coordinates we could use the general discussion in Section 2.6. However, it is easier to proceeed in an alternative way. For, consider the auxiliary signal x ˆ1 = lx
70
CHAPTER 3. THE STRUCTURAL PROPERTIES
and note that σˆ x1 = lAx = x ˆ2 , σˆ xi = lAi x = x ˆi+1 , for i = 1, · · · , n − 1 and
σˆ xn = lAn x + u = lAn T −1 x ˆ + u.
As a result, in the new coordinates x ˆ one has σˆ x = Ar x ˆ + Br x ˆ, where
2
Ar =
and
6 6 6 6 6 6 4
0 0 .. .
1 0 .. .
··· ··· .. .
0 1 .. .
0 0 .. .
0 0 0 ··· −α0 −α1 −α2 · · ·
1 −αn−1
−α0 −α1 −α2 · · ·
(3.15)
3
2
7 7 7 7 7 7 5
6 6 6 6 6 6 4
Br =
−αn−1
0 0 .. . 0 1
3 7 7 7 7, 7 7 5
= lAn T −1 .
Note that for any αi , the system (3.15) is reachable, hence a system described by equation (3.15) is said to be in reachability canonical form. The matrix Ar is in companion form. It is worth noting that its characteristic polynomial is p(s) = sn + αn−1 sn−1 + αn−2 sn−2 + · · · + α1 s + α0 , i.e. it depends only upon the elements of the last row.
Example 3.7 Consider the system in Example 3.5. This system is reachable (and controllable) for any L and C. To write the system in reachability canonical form we have to find a (row) vector l such that conditions (3.13) hold, namely " 1 #
l
L
"
=0
0
l
yielding l= Finally
"
T =
l lA
"
=
= 1,
1 LC
0 LC #
#
1 −R L2
.
0 LC L 0
#
and the system in the transformed coordinates is described by "
x ˆ˙ =
0
1
1 − LC
− RL1
i.e. it is in reachability canonical form.
#
"
x ˆ+
0 1
#
u,
71
3.2. REACHABILITY AND CONTROLLABILITY
3.2.6
Description of nonreachable systems
In this section we study systems which are not reachable, i.e. systems described by the equation (3.1) and such that rankR = ρ < n. Under this assumption, consider a set of coordinates x ˆ such that x = Lˆ x, and the matrix L is constructed as follows. The first ρ columns of L are ρ linearly independent columns of the matrix R, and the last n − ρ columns are selected in such a way that the matrix L is invertible2 . The system in the x ˆ coordinates, which is algebraically equivalent to the system in the x coordinates, is described by the equations ˆx + Bu ˆ = L−1 ALˆ σˆ x = Aˆ x + L−1 Bu. We now show that, because of the way in which L has been constructed, the matrices Aˆ ˆ have a special structure. To this end note that and B LAˆ = AL
ˆ=B LB
ˆ B and B ˆ as and partition the matrices L, A, A,
L=
L11 L12 L12 L22
,
A=
B=
B1 B2
A11 A12 A12 A22
Aˆ =
,
,
ˆ= B
ˆ1 B ˆ2 B
Aˆ11 Aˆ12 Aˆ21 Aˆ22
,
,
where L11 , A11 and Aˆ11 have dimensions ρ×ρ; L12 , A12 and Aˆ12 have dimensions ρ×n−ρ; L21 , A21 and Aˆ21 have dimensions n−ρ×ρ; L22 , A22 and Aˆ22 have dimensions n−ρ×n−ρ; ˆ1 have dimensions ρ × m and B2 and B ˆ2 have dimensions n − ρ × m. B1 and B Recall that, by construction, R is spanned by the first ρ columns of the matrix L, and ˆ2 have that R is Ainvariant and contains B. This implies that the submatrices Aˆ21 and B to be identically zero, i.e. in the x ˆ coordinates the system is described by
σˆ x=
σˆ x1 σˆ x2
=
Aˆ11 Aˆ12 0 Aˆ22
x ˆ+
ˆ1 B 0
u,
(3.16)
where x ˆ1 ∈ IRρ and x ˆ2 ∈ IRn−ρ . The reachability matrix of the system in the x ˆ coordinates is ˆ1 Aˆ11 B ˆ1 · · · Aˆn B ˆ1 B 11 ˆ R= , 0 0 0 0 2 It is always possible to determine such n − ρ vectors. Moreover it is possible to select them among the vectors ei of the canonical basis.
72
CHAPTER 3. THE STRUCTURAL PROPERTIES
ˆ it has rank ρ. This implies that the subsystem and, because R = LR, ˆ1 u σˆ x1 = Aˆ11 x ˆ1 + B
(3.17)
is reachable. The subsystem (3.17) is called the reachable subsystem of the system (3.1), whereas the subsystem σˆ x2 = Aˆ22 x ˆ2 , (3.18) which is clearly not affected by the input, is called the unreachable subsystem of the system (3.1). The eigenvalues of the matrix Aˆ22 , which are a subset of the eigenvalues of the matrix A, are called unreachable modes. Example 3.8 Consider the system in Example 3.6, and assume R1 R2 C = L, which implies that the system is not reachable (and not controllable), as the rank of the reachability matrix is equal to one. To obtain a decomposition into reachable and unreachable subsystems, let " 1 # R1 C 1 L= 1 0 L yielding
"
Aˆ = L
3.2.7
−1
AL =
− RL2 0
0
#
"
ˆ=L B
− R11 C
−1
B=
1 0
#
.
PBH reachability test
The decomposition of a system in reachable and unreachable parts allows to derive an alternative test for reachability. Proposition 3.5 (PopovBelevichHautus (PBH) reachability test) Consider the system (3.1). The system is reachable if and only if rank
sI − A B
=n
for all s ∈ C I. Remark. The matrix
sI − A B
is called reachability pencil. Note that the rank condition in Proposition 3.5 holds trivially for all s which are not eigenvalues of A, hence the condition has to be checked only for the n (complex) numbers which are eigenvalues of A. ⋄ Proof. (Necessity) We prove the necessity by contradiction. Suppose the system is reachable and that, for some s⋆ ∈ C I, rank
s⋆ I − A B
< n.
73
3.2. REACHABILITY AND CONTROLLABILITY Then there is a vector w such that w′
s⋆ I − A B
= 0,
hence w′ B = 0
w ′ A = s⋆ w ′ .
As a result w′ AB = 0
w′ A2 B = 0
w′ An−1 B = 0
···
or, equivalently, w′ R = 0,
which implies that the system is not reachable, hence the contradiction. (Sufficiency) Again, we prove the statement by contradiction. Suppose rank
sI − A B
=n
for all s ∈ C I and the system is not reachable. Consider the change of coordinates which transforms the system into the form (3.16) and note that, for the transformed system one has3 ˆ1 B sI − Aˆ11 −Aˆ12 = n. rank 0 sI − Aˆ22 0 However, this is not true as this matrix loses rank for all s which are eigenvalues of Aˆ22 , hence the contradiction. ⊳
Remark. The PBH test allows to compute the unreachable modes of a system without performing the decomposition into reachable and unreachable parts. In fact, the unreachable modes are all the complex numbers for which the reachability pencil loses rank. ⋄ Example 3.9 Consider again the system in Example 3.6. The reachability pencil is "
s+
1 R1 C
0
0 s+
R2 L
# 1 R1 C . 1 L
This matrix has rank two for all s 6= − R11C and s 6= − RL2 . For s = − R11C or s = − RL2 the reachability pencil loses rank, i.e. the system is not reachable, if −
R2 1 + = 0, R1 C L
and this condition is the same as the one derived in Example 3.6. 3
Note that the rank of the reachability pencils of algebraically equivalent systems is the same for all s∈C I. This is a consequence of the identity
sI − Aˆ
ˆ B
=
sI − L−1 AL
L−1 B
= L−1
sI − A
B
L 0
0 I
.
74
CHAPTER 3. THE STRUCTURAL PROPERTIES
Example 3.10 Consider a continuoustime system described by the equation x˙ = Ax + Bu
(3.19)
and its Euler approximate discretetime model x+ = x + T Ax + T Bu,
(3.20)
where T denotes the samplingtime. Suppose the continuoustime system is controllable, i.e. rank sI − A B = n, for all s ∈ C I. Consider now the reachability pencil of the discretetime model (3.20), i.e.
sI − (I + T A) T B
and note that, by reachability of the continuoustime system, rank
sI − (I + T A) T B
= rank
s−1 I −A B T
= rank
zI − A B
= n,
where z = s−1 T . Hence, reachability of system (3.19) implies, and it is implied by, reachability of system (3.20). Suppose now that the continuoustime system is not reachable, and that all nonreachable modes are in the open left part of the complex plane, i.e. they have negative real part hence they are asymtotically stable. This implies that the discretetime Euler approximate model is not reachable, however the nonreachable modes belong to the set described by §
z∈C I  Real(z) < −
1 T
ª
,
hence it is not possible to decide apriori if they are stable or otherwise.
3.3
Observability and reconstructability
In this section we study the statetooutput relation, and we focus on the problem of determining the state of a system, at a given time instant, from measurements of the input and output signals. This problem, of great importance in applications, can be addressed from two different perspectives. In the former, we assume that the state at time t has to be determined on the basis of current and future measurements. In this case we deal with a socalled observability problem. In the latter, we assume that the state at time t has to be determined from past and current measurements. In this case we have a socalled reconstructability problem. Observability problems typically arise in realtime problems, where one has to determine the state of a system from the actual measurements, whereas a classical reconstructability problem arises in weather forecast, where one wishes to determine the current weather from past measurements.
3.3. OBSERVABILITY AND RECONSTRUCTABILITY
3.3.1
75
Observability of discretetime systems
Consider a linear, timeinvariant, discretetime system. Two states xa and xb are indistinguishable in the future in k steps if for any input sequence u(0), u(1), . . . , u(k − 1) the corresponding output sequences ya and yb , coincide for the first k steps, i.e. ya (t) = yb (t)
(3.21)
for all t ∈ [0, k]. By equation (2.12), condition (3.21) is equivalent to CAt xa = CAt xb
(3.22)
for all t ∈ [0, k]. This implies that the property of indistinguishability in the future does not depend upon the input sequence, i.e. it is a property of the free response of the output of the system, hence of the matrices A and C. Note, moreover, that condition (3.22) can be rewritten as xa − xb ∈ kerOk
where
2
Ok =
6 6 6 6 6 6 4
C CA CA2 .. .
3 7 7 7 7. 7 7 5
CAk We say that two states are indistinguishable in the future if they are indistinguishable in the future in k steps for all k ≥ 0. Note that, by CayleyHamilton Theorem, two states xa and xb are indistinguishable in the future if xa − xb ∈ kerOn−1 Definition 3.4 A state x is not observable in k steps if it is not distinguishable in the future in k steps from the zero state. It is not observable if it is not observable in k steps for all k. As a consequence of the above discussion, we conclude that a state x is not observable in k steps if x ∈ ker Ok and it is not observable if
x ∈ ker On−1 .
Note that the set of nonobservable (in k steps) states is a subspace. Definition 3.5 Consider the discretetime system (3.1). The subspace kerOn−1 is the unobservable subspace of the system. The matrix O = On−1 is the observability matrix of the system. The system is said to be observable if kerO = {0}.
76
CHAPTER 3. THE STRUCTURAL PROPERTIES
Remark. The discretetime system (3.1) is observable if and only if rankO = n.
(3.23)
Equation (3.23) is known as Kalman observability rank condition, and was derived by R.E. Kalman in the 60’s. ⋄ Example 3.11 Consider a discretetime system with x ∈ IR3 , y ∈ IR, 2 6
A=4
0 0 0 0 0 0 0 0 1
3 7 5
C=
Then
O0 = C
O1 =
α β γ 0 0 γ
α β γ
2
6
.
O2 = 4
.
α β γ 0 0 γ 0 0 γ
3 7 5.
Hence the system is not observable. The unobservable subspace has dimension one if αγ 6= 0 or βγ 6= 0, it has dimension two either if γ = 0 and α 6= 0 or β 6= 0, or if γ 6= 0 and α = 0 and β = 0.
Remark. The existence of a nonempty unobservable subspace, implies that from current and future output and input measurements it is not possible to determine the current state. In fact this can be determined modulo an element in kerO. ⋄ Example 3.12 Consider the system in Example 3.11 and assume α 6= 0, β = 0 and γ 6= 0. Thus the unobservable subspace has dimension one and 2 6
ker O = span 4
0 1 0
3
7 5.
Therefore it is not possible to distinguish, using current and future measurements, between states which have the same x1 and x3 components, but different x2 component. This means that, for example, from current and past measurements, it is possible to conclude that the initial state belongs to the straight line in Figure 3.3, but it is not possible to determine at which point on the line the initial state is. The unobservable subspace ker O has the following important property, the proof of which is a simple consequence of the definition of the subspace.
77
3.3. OBSERVABILITY AND RECONSTRUCTABILITY x2
x1 x3
Figure 3.3: Admissible initial states for an unobservable system.
Proposition 3.6 The unobservable subspace contains the subspace ker C, i.e. ker C ⊆ ker O, and it is Ainvariant, i.e. A ker O ⊆ ker O. We conclude this section noting that algebraically equivalent systems have the same observability properties. In particular, consider two algebraically equivalent systems, then one of the two systems is observable if and only if the other is.
3.3.2
Reconstructability of discretetime systems
In this section we study the property of reconstructability. Consider a linear, timeinvariant, discretetime system, assume that the input sequence u(0), u(1), . . . , u(k − 1) and the output sequence y(0), y(1), . . . , y(k) are known and consider the problem of determining the state of the system at time k, i.e. x(k). Of course, if the system is observable in k steps, then the considered problem is solvable. In fact, the input and output sequences determine a unique state x(0), from which it is possible to compute x(k). If the system is not observable in k steps, then the initial state cannot be uniquely determined, but it can determined modulo an element in the unobservable subspace. This means that, if x(0) is an initial state which is consistent with the input and output sequences, then all states described by x ˜(0) = x(0) + kerOk
78
CHAPTER 3. THE STRUCTURAL PROPERTIES
are also consistent with the same input and output sequences. Consider now the initial state x ˜(0) and the resulting state at time k, namely x ˜(k) = Ak x(0) +
k−1 X
Ak−t−1 Bu(t) + Ak kerOk = x(k) + Ak kerOk .
t=0
As a result, for given input and output sequences, the state of the system at time k is uniquely defined if Ak kerOk = {0}, or, equivalently, if kerOk ⊆ ker Ak
(3.24)
whereas it is not uniquely defined if the above condition does not hold. A system is said reconstructable in k steps if condition (3.24) holds. It is said reconstructable if it is reconstructable in k steps for all k. Recalling that ker Ak = ker An for all k ≥ n, a system is reconstructable if and only if kerO ⊆ kerAn .
(3.25)
Note that, the observability rank condition (3.23) implies, but it is not implied, by condition (3.25). Example 3.13 Consider a discretetime system with x ∈ IR3 , y ∈ IR, 2 6
A=4
1 1 0 1 0 0 0 1 0
3 7 5
C=
1 0 0
,
i.e. x+ 1 = x1 + x2
x+ 2 = x1
x+ 3 = x2 ,
and y = x1 . The observability matrix is 2 6
O=4
1 0 0 1 1 0 2 1 0
3 7 5,
hence the system is not observable. The unobservable subspace has dimension one and is given by 2 3 0 6 7 kerO = span 4 0 5 . 1
3.3. OBSERVABILITY AND RECONSTRUCTABILITY Note now that
2 6
A3 = 4
3 2 0 2 1 0 1 1 0
79
3 7 5,
hence kerA3 = kerO, which implies that the system is reconstructable. This fact can be shown directly. Suppose we want to determine x(2) from y(0), y(1) and y(2). To this end, note that x1 (2) = y(2), x2 (2) = x1 (1) = y(1) and x3 (2) = x2 (1) = x1 (0) = y(0), which proves the claim. Example 3.14 Consider again the system in Example 3.13. Note that kerA2 = kerO, hence the system is reconstructable in two steps. This property can be shown directly. In fact, suppose we want to determine x(1) from y(0) and y(1). This can be achieved noting that x1 (1) = y(1), x2 (1) = x1 (0) = y(0) and x3 (1) = x2 (0) = x1 (1) − x1 (0) = y(1) − y(0).
3.3.3
Computation of the state
The property of observability highlights the ability to determine the state of a system from present and future measurements. From a practical point of view it is however important not only to characterize the observability property, but also to have a procedure that allows to effectively compute the unknown state. Consider a linear, timeinvariant, discretetime system, and note that, by equation (2.12), the output response of the system is a linear combination of the free response and of the forced response. The latter is known, once the input signal is known, whereas the former depends upon the initial state, which has to be determined. This means that the problem of determining the initial state of a system from current and future (input and output) measurements is equivalent to the problem of determining the initial state of the system
80
CHAPTER 3. THE STRUCTURAL PROPERTIES
from the knowledge of its free output response, i.e. it is possible to assume, without loss of generality, that the input sequence is identically equal to zero. To solve this problem note that 2
Yk =
6 6 6 6 4
y(0) y(1) .. . y(k)
3
2
7 7 7 7 5
6 6 6 6 4
=
C CA .. .
3 7 7 7 x(0) 7 5
= Ok x(0),
CAk
with Yk ∈ IRp (k+1) . If the system is observable then On−1 = O is full rank, hence the equation Yn−1 = Ox(0) (3.26) has the (unique) solution x(0) = (O′ O)−1 O′ Yn−1 ,
(3.27)
which provides a simple way to compute the state x(0) of the system. Remark. Equation (3.27) shows how to compute the initial state of a system from current and future measurements. This information, in turn, can be used to determine the state at time k using equation (2.11). ⋄ Remark. Measurements are naturally affected by noise, i.e. in practice equation (3.26) has to be replaced by Yn−1 + ν = Ox(0), where ν represents a vector of additive noise affecting the output measurements. (For simplicity we assume that the input sequence is identically zero.) It is possible to prove that the initial state given by equation (3.27) yields a free output response which minimizes the norm (energy) of the error between the actual free output response and the calculated one. ⋄
3.3.4
Observability and reconstructability for continuoustime systems
The properties of observability and reconstructability for continuoustime systems can be assessed using the same ideas exploited in the case of discretetime systems. Consider a linear, timeinvariant, continuoustime system. Two states xa and xb are indistinguishable in the future over the interval [0, t] if for any input signal u the corresponding output responses coincide in the interval [0, t]. This property, recalling equation (2.10), and noting that the forced responses do not depend upon the initial states, is equivalent to the condition CeAτ xa = CeAτ xb ,
81
3.3. OBSERVABILITY AND RECONSTRUCTABILITY for all τ ∈ [0, t], or to the condition CeAτ (xa − xb ) = 0,
(3.28)
for all τ ∈ [0, t].
The function CeAτ is analytic, hence condition (3.28) is equivalent to CeAτ (xa −xb )τ =0 = 0
d CeAτ (xa −xb )τ =0 = 0 dτ
···
di CeAτ (xa −xb )τ =0 = 0 dτ i
··· ,
yielding, by a property of the matrix exponential, C(xa − xb ) = 0
CA(xa − xb ) = 0
···
CAi (xa − xb ) = 0
··· .
(3.29)
Note finally that, by CayleyHamilton Theorem, (3.29) is equivalent to (recall the definition of the observability matrix O) O(xa − xb ) = 0. (3.30) In summary, if two states xa and xb are indistinguishable in the future over the interval [0, t] then they have to be such that condition (3.30) holds. Remark. Unlike discretetime systems, in the continuoustime case if two states are indistinguishabe in the future over an interval [0, t] then they are indistinguishabe in the future over any interval [0, t¯], with t¯ > 0. ⋄ From the above discussion, we conclude that, for a continuoustime system, all states which are indistinguishable in the future from the zero state, i.e. the unobservable states, are those, and only those, belonging to ker O, as expressed in the following statement. Proposition 3.7 Consider the continuoustime system (3.1). The following statements are equivalent. • The system is observable. • rankO = n. • For all t > 0 the observability Gramian Z t
Vt =
0
′
eA τ C ′ CeAτ dτ
is positive definite. Proof. We have only to prove the claim on the observability Gramian. Note first that Vt = Vt′ ≥ 0, hence we only need to show that rankVt = n, for all t > 0, if and only if the system is observable.
82
CHAPTER 3. THE STRUCTURAL PROPERTIES
Suppose the system is observable and rankVt < n, for some t > 0. This implies that there exists a vector w such that w′ Vt w = 0, which implies
Z t 0
hence
′
w′ eA τ C ′ CeAτ wdτ = 0 Z t 0
kCeAτ wkdτ = 0.
This last equality implies that the function CeAτ w is identically equal to zero on the interval [0, t]. As a result, by analyticity of the function and a property of the matrix exponential, Cw = 0
···
CAw = 0
CAn−1 w = 0,
or equivalently Ow = 0, which contradicts the observability assumption. Suppose now that Vt > 0 for all t > 0. To show that the system is observable consider the function Z t
δ(t) =
0
′
eA τ C ′ y(τ )dτ
and note that δ(t) = Vt x(0). Hence, by positivity of Vt we can uniquely determine x(0) processing future measurements, i.e. the system is observable. ⊳ We conclude this section studying the property of reconstructability for linear, continuoustime systems. For, note that two states xa and xb are indistinguishable in the past over the interval [−t, 0] if for any input signal u the corresponding output responses coincide in the interval [−t, 0]. Using arguments similar to the ones used in the case of discretetime systems, we conclude that all states are distinguishable in the past, over the interval [−t, 0], from the zero state if eAt ker O = {0}. However, because the matrix eAt is invertible for all t, the above condition is equivalent to ker O = {0}, i.e. a continuoustime system is reconstructable if and only if it is observable.
3.3. OBSERVABILITY AND RECONSTRUCTABILITY
83
Example 3.15 Consider the system in Example 3.5. The observability matrix is "
O=
1
0
− RL1
− L1
#
hence the system is observable for any zero L (recall that L > 0). Example 3.16 Consider the system in Example 3.6. The observability matrix is 2
O=4
− R11 1
R21 C
and det O =
1 − RL2
3 5
1 R2 − 2 , R1 L R1 C
hence the system is observable if, and only if, R1 R2 C 6= L.
3.3.5
Duality
In the study of the structural properties there is a strong relation between the results on reachability and observability and the results on controllability and reconstructability. This relation can be defined formally by introducing the notion of dual system. Definition 3.6 Consider the system, with state x ∈ X = IRn , input u ∈ IRm , and output y ∈ IRp , described by the equations σx = Ax + Bu
y = Cx + Du.
(3.31)
The system, with state ξ ∈ Ξ = IRn , input v ∈ IRp , and output η ∈ IRm , described by the equations σξ = A′ ξ + C ′ v η = B ′ξ + D′ v (3.32) is called the dual system of system (3.31), which is called the primal system of system (3.32). To understand the importance and the usefulness of this notion of duality, let R and O be the reachability and observability matrix, respectively, of the system (3.31) and let R⋆ and O⋆ be the reachability and observability matrix, respectively, of the dual system (3.32). Then, trivially, R⋆ = O ′
O ⋆ = R′ .
Moreover, for discretetime systems, the following implications hold ImAn ⊆ R ⇔ (ImAn )⊥ ⊇ R⊥ ⇔ ker An ⊇ ker R′ = ker O⋆ . As a result, the following statement holds.
84
CHAPTER 3. THE STRUCTURAL PROPERTIES
Proposition 3.8 Consider the system (3.31) and its dual (3.32). • System (3.31) is reachable if and only if system (3.32) is observable. • System (3.31) is observable if and only if system (3.32) is reachable. • System (3.31) is controllable if and only if system (3.32) is reconstructable. • System (3.31) is reconstructable if and only if system (3.32) is controllable. The notion of duality and the above statement allow to derive results similar to those presented in Sections 3.2.5, 3.2.6 and 3.2.7, but with respect to the observability property. Proposition 3.9 Consider the system (3.1) with p = 1. Suppose the system is observable. Then the system is algebraically equivalent to a system of the form σˆ x = Ao x ˆ + Bo u with
2
A0 =
6 6 6 6 6 6 6 6 4
0 0 ··· 1 0 ··· 0 1 ··· .. .. . . . . . 0 0 ··· 0 0 ···
y = Co x ˆ + Do u
0 0 −α0 0 0 −α1 0 0 −α2 .. .. .. . . . 1 0 −αn−2 0 1 −αn−1
(3.33)
3 7 7 7 7 7 7 7 7 5
C0 =
0 ···
0 1
.
Note that for any αi , the system (3.33) is observable, hence a system described by the equations (3.33) is said to be in observability canonical form. Proposition 3.10 Consider the system (3.1). Suppose the system is not observable. Then the system is algebraically equivalent to a system of the form
σˆ x=
σˆ x1 σˆ x2
Aˆ11 0 Aˆ21 Aˆ22
= y=
Cˆ1 0
x ˆ+
ˆ1 B ˆ2 B
u,
ˆ x ˆ + Du.
Moreover, the system ˆ1 u σˆ x1 = Aˆ11 x ˆ1 + B is observable.
ˆ y = Cˆ1 x ˆ1 + Du
(3.34)
85
3.3. OBSERVABILITY AND RECONSTRUCTABILITY
The subsystem (3.34) is called the observable subsystem of the system (3.1), whereas the subsystem ˆ2 u σˆ x2 = Aˆ22 x ˆ2 + B y = 0, (3.35) which clearly does not contribute to the output, is called the unobservable subsystem of the system (3.1). The eigenvalues of the matrix Aˆ22 , which are a subset of the eigenvalues of the matrix A, are called unobservable modes. Proposition 3.11 (PBH observability test) Consider the system (3.1). The system is observable if and only if sI − A =n rank C for all s ∈ C I. Example 3.17 (Kalman decomposition) Consider the system (3.1), the associated reachable subspace R and unobservable subspace ker O. Let X1 = R ∩ ker O, with dim X1 = n1 , and note that, by Ainvariance of R and ker O, also X1 is Ainvariant. Let X2 , X3 and X4 be subspaces, of dimension n2 , n3 , and n4 , respectively, such that4 X1 ⊕ X2 = R
X1 ⊕ X2 ⊕ X3 ⊕ X4 = X = IRn .
X1 ⊕ X3 = ker O
Note that, by construction, n1 + n2 + n3 + n4 = n. Consider now a change of coordinates described by x = Lˆ x where the matrix L is constructed as follows. The first n1 columns of L are a basis for X1 . The subsequent n2 columns are a basis for X2 , the subsequent n3 columns are a basis for X3 , and finally, the last n4 columns are a basis for X4 . In the coordinates x ˆ the system is described by 2
σˆ x =
y =
6 6 6 4
Aˆ11 Aˆ12 Aˆ13 0 Aˆ22 0 ˆ 0 0 A33 0 0 0 0 Cˆ2 0 Cˆ4
Aˆ14 Aˆ24 Aˆ34 Aˆ44
3
2
7 6 7 6 ˆ+6 7x 5 4
ˆ1 B ˆ2 B 0 0
3 7 7 7u 5
(3.36)
ˆ x ˆ + Du.
The state space representation (3.36) is known as Kalman’s canonical form. From this representation it is possible to single out the following subsystems. 4
The symbol ⊕ denotes the direct sum between subspaces. The direct sum of two subspaces is defined as follows. Let X and Y be two subspaces of a vector space W. The sum of these subspaces, denoted X + Y, is the set of all vectors x + y, with x ∈ X and y ∈ Y. If X ∩ Y = {0} the sum is a direct sum and it is denoted X ⊕ Y.
86
CHAPTER 3. THE STRUCTURAL PROPERTIES • The reachable and unobservable subsystem ˆ1 u σˆ x1 = Aˆ11 x ˆ1 + B
y = 0.
• The reachable and observable subsystem ˆ2 u σˆ x2 = Aˆ22 x ˆ2 + B
ˆ y = Cˆ2 x ˆ2 + Du.
• The unreachable and unobservable subsystem σˆ x3 = Aˆ33 x ˆ3
y = 0.
• The unreachable and observable subsystem σˆ x4 = Aˆ44 x ˆ4
y = Cˆ4 x ˆ4 .
These subsystems, and their role in the overall system, are illustrated in Figure 3.4. Reachable Observable
u
Unreachable Observable
Reachable Unobservable
+
y
Unreachable Unobservable
Figure 3.4: Illustrative representation of a system in Kalman’s canonical form.
We conclude this example noting that a state space representation which is reachable and observable is called minimal. From a nonminimal state space representation it is always possible to obtain a (reduced) minimal representation. This is constructed computing Kalman’s canonical form and then disregarding the unreachable and/or unobservable parts. Note that only the reachable and observable subsystem contribute to the inputoutput behaviour of the system. This can be proved, for example, noting that the transfer function of the system (3.36) is ˆ2 + D. ˆ G(s) = Cˆ2 (sI − Aˆ22 )−1 B
3.3. OBSERVABILITY AND RECONSTRUCTABILITY
87
Example 3.18 Consider the system x˙ 1 = x2 + u x˙ 2 = αx1 + u y = x1 − x2 with x1 ∈ IR, x2 ∈ IR, u ∈ IR, y ∈ IR and α a constant parameter. The reachability matrix is
R=
1 1 1 α
,
hence the system is reachable (and controllable) if α 6= 1. If α = 1 the reachability pencil is s −1 1 sI − A B = −1 s 1 and this has rank one for s = −1 and rank two for any other s. Therefore, for α = 1, the unreachable mode is s = −1. The observability matrix is
O=
1 −1 −α 1
,
hence the system is observable if α 6= 1. If α = 1 the observability pencil is
sI − A C
2 6
=4
s −1 −1 s 1 −1
3 7 5
and this has rank one for s = 1 and rank two for any other s. Therefore, for α = 1, the unobservable mode is s = 1. Note that, if α = 1 the system does not have a subsystem which is reachable and observable, hence the transfer function is identically equal to zero, i.e. there is no direct inputoutput connection. To see this, note that CB = 0
CAB = 1 − α
and, by CayleyHamiton Theorem, CAi B, with i > 1, can be written in terms of CB and CAB.
88
CHAPTER 3. THE STRUCTURAL PROPERTIES
Chapter 4
Design tools
90
4.1
CHAPTER 4. DESIGN TOOLS
Introduction
In Chapter 3 we have considered the problem of determining an input signal driving the state of the system to a given final condition and the problem of determining the state of the system from measurements of the input and output signals. The proposed solutions rely upon information on the system over a finite and predetermined time interval and generate the required input signal or the required state estimate offline. Typically, the input signal is computed a priori and the state estimate a posteriori. This approach is unsatisfactory for various reasons: the effect of disturbances, model errors and uncertainties is not taken into consideration and the actual evolution of the system is not considered in the solution of the problems, hence these solutions are openloop. This implies that it makes sense to seek methods, to control or estimate the state of the system, which are based on current information. Such methods provide closedloop solutions to the considered problems. Closedloop solutions can be constructed in several ways. In the case of linear, finitedimensional, timeinvariant systems it is natural to solve the above problems considering the interconnection, of the system to be controlled or of the system the state of which has to be estimated, with another linear, finitedimensional, timeinvariant system. This system, which has to be designed, processes the current information and generates the current input to be applied to the system to achieve a specific goal, or the current estimate of the state of the system.
4.2
The notion of feedback
Consider a linear, finitedimensional, timeinvariant system and the problem of determining an input signal such that a certain objective is achieved. Typically, the input signal has to be such that the state of the system has to be driven to zero with a given speed of convergence (state regulation) or the output of the system has to follow a preassigned reference value (output tracking). We are interested in determining the input signal in closedloop form, i.e. the input signal at time t has to be a function of the available information (state or output) at the same time instant. This implies that the input signal is generated by means of a feedback mechanism. This mechanism can be instantaneous, i.e. the input signal is generated instantaneously by processing the available information. In this case we have a static feedback. Alternatively, the input signal can be generated processing the available information through a dynamical device. In this case we have a dynamic feedback. Assume that the system to be controlled is described by equations of the form σx = Ax + Bu
y = Cx + Du,
(4.1)
with x ∈ X = IRn , u(t) ∈ IRm , y(t) ∈ IRp and A, B, C, and D matrices of appropriate
91
4.2. THE NOTION OF FEEDBACK
dimensions and with constant entries. Assume, in addition, that the system which generates the input signal is linear, finitedimensional and timeinvariant. Then we may have the following four configurations. • Static output feedback. The input signal is generated via the relation u = Ky + v,
(4.2)
with K a constant matrix of appropriate dimensions and v a new external signal. The resulting closedloop system is described by the equations σx = (A + B(I + KD)−1 KC)x + Bv = (I + D(I + KD)−1 K)Cx + Dv,
y
(4.3)
which are welldefined provided the matrix I + KD is invertible. Note that this is always the case if D = 0. • Static state feedback. The input signal is generated via the relation u = Kx + v,
(4.4)
with K a constant matrix of appropriate dimensions and v a new external signal. The resulting closedloop system is described by the equations σx = (A + BK)x + Bv y
= (C + DK)x + Dv.
(4.5)
• Dynamic output feedback. The input signal is generated by the system1 σξ = F ξ + Gy
u = Kξ + v
(4.6)
with F , G and K constant matrices of appropriate dimensions and v a new external signal. The resulting closedloop system is described by the equations σx = Ax + BKξ + Bv σξ
= (F + GDK)ξ + GCx + GDv
y
= Cx + DKξ + Dv.
(4.7)
• Dynamic state feedback. The input signal is generated by the system2 σξ = F ξ + Gx 1
u = Kξ + v
(4.8)
We consider the simplest version of dynamic output feedback. A more general form is given by σξ = F ξ + Gy + Hv, u = Kξ + Jy + Lv. 2 We consider the simplest version of dynamic state feedback. A more general form is given by σξ = F ξ + Gx + Hv, u = Kξ + Jx + Lv.
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CHAPTER 4. DESIGN TOOLS with F , G and K constant matrices of appropriate dimensions and v a new external signal. The resulting closedloop system is described by the equations σx = Ax + BKξ + Bv σξ
= F ξ + Gx
(4.9)
y
= Cx + DKξ + Dv.
In what follows we study in detail the static state feedback (Section 4.3) and the dynamic output feedback (Section 4.6) configurations. This is mainly due to the fact that these two configurations allow to solve most control problems for linear systems. Moreover, while the use of static output feedback is very appealing in practice because it results in a simple to implement control strategy, the study of the properties of system (4.3) as a function of K is very difficult3 . Finally, dynamic state feedback is useful only in very specific problems, such as the socalled noninteracting control problem with stability4 , which are not the subject of these notes.
4.3
State feedback
Consider a system described by the equations (4.1), the state feedback control law (4.4) and the resulting closedloop system (4.5). The use of state feedback modifies the inputtostate interaction, i.e. the system σx = Ax + Bu is replaced by the system σx = (A + BK)x + Bv. As a result, it makes sense to study which properties of the system are left unchanged by the application of state feedback (i.e. are feedback invariant) and which properties can be modified as a function of K. Proposition 4.1 The system (4.1) is reachable if and only if the system (4.5) is reachable. Proof. Note that rank
sI − A B
= rank
sI − A B
Hence, by Hautus test, the claim holds.
I 0 −K I
= rank
sI − (A + BK) B
.
⊳
3 In the case m = p = 1 and D = 0 the root locus method can be used to study the eigenvalues of the matrix A + BKC. 4 The noninteracting control problem with stability, in the case m = p > 1, can be informally described as the problem of designing a system described by equations of the form (4.8) such that the closedloop system (4.9) is asymptotically stable and composed of m decoupled systems, e.g. the Markov parameters are diagonal matrices.
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4.3. STATE FEEDBACK
Proposition 4.2 Consider the system (4.1). Suppose the system is not reachable and it is described by equations of the form (3.16). Then the system (4.5) is described by equations of the same form. Moreover, the systems (4.1) and (4.5) have the same unreachable modes. Proof. By assumption, system (4.1) is described by equations of the form
σx =
A11 A12 0 A22
B1 0
x+
u.
As a result, system (4.5) is described by
A11 + B1 K1 A12 + B1 K2 0 A22
σx = where
K=
K1 K2
x+
B1 0
v,
,
which proves the claim.
⊳
This result implies that state feedback does not modify unreachable modes, hence to evaluate the effect of the feedback on the dynamics of the system it is sufficient to consider only the reachable subsystem. We focus initially on singleinput reachable systems. Proposition 4.3 Consider system (4.1). Assume m = 1 (i.e. the system has only one input) and suppose the system is reachable. Let p(s) be a monic polynomial of degree n. Then there is a (unique) K such that the characteristic polynomial of A + BK is equal to p(s). Proof. By reachability of the system it is possible to write the system in reachability canonical form (see Section 3.2.5). Let T be the transformation matrix defined in Section 3.2.5, i.e. let Ar = T AT −1 Br = T B. Let p(s) = sn + α ˜ n−1 sn−1 + α ˜ n−2 sn−2 + · · · + α ˜1 s + α ˜0 and define Kr =
α0 − α ˜ 0 α1 − α ˜1 · · ·
Note that
2
Ar + Br Kr =
6 6 6 6 6 6 4
0 0 .. .
1 0 .. .
0 1 .. .
αn−1 − α ˜ n−1 ··· ··· .. .
0 0 0 ··· −α ˜ 0 −α ˜ 1 −α ˜2 · · ·
0 0 .. . 1 −α ˜ n−1
.
3 7 7 7 7, 7 7 5
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hence p(s) is the characteristic polynomial of Ar + Br Kr . Finally, let K = Kr T and note that Ar + Br Kr = T (A + BK)T −1 , which shows that p(s) is the characteristic polynomial of A + BK. To prove unicity, let ˆ 6= K and define K ˆ r = KT ˆ −1 , K yielding ˆ r 6= Ar + Br Kr , Ar + Br K
ˆ r is not p(s). hence the characteristic polynomial of Ar + Br K
⊳
The main disadvantage of the above result is that, to compute the feedback gain K which assigns the eigenvalues of the closedloop system, it is necessary to transform the system in reachability canonical form. This transformation is however not needed, as shown in the following statement. Proposition 4.4 (Ackermann’s formula) Consider system (4.1). Assume m = 1 (i.e. the system has only one input) and suppose the system is reachable. Let p(s) be a monic polynomial of degree n. Then K=−
0 ···
0 1
R−1 p(A)
is such that the characteristic polynomial of A + BK is equal to p(s). Remark. Propositions 4.3 and 4.4 provide a constructive way to assign the characteristic polynomial, hence the eigenvalues, of system (4.5). Note that, for low order systems, i.e. if n = 2 or n = 3, it may be convenient to compute directly the characteristic polynomial of A + BK and then compute K using the principle of identity of polynomials, i.e. K should be such that the coefficients of the polynomials det(sI − (A + BK)) and p(s) coincide. ⋄ The result summarized in Proposition 4.3 can be extended to multiinput systems. Proposition 4.5 Consider system (4.1) and suppose the system is reachable. Let p(s) be a monic polynomial of degree n. Then there is a K such that the characteristic polynomial of A + BK is equal to p(s). Note that in the case m > 1 the feedback gain K assigning the characterisitc polynomial of the matrix A+ BK is not unique. Finally, to compute such feedback gain we may either use the direct approach discussed in the above Remark or exploit the following fact.
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4.3. STATE FEEDBACK
Lemma 4.1 (Heymann) Consider system (4.1) and suppose the system is reachable. Let bi be a nonzero column of the matrix B. Then there is a matrix G such that the singleinput system σx = (A + BG)x + bi v (4.10) is reachable. Exploiting Lemma 4.1 it is possible to design a matrix K such that the characteristic polynomial of A + BK equals some monic polynomial p(s) of degree n in two steps. First we compute a matrix G such that the system (4.10) is reachable, and then we use Ackermann’s formula to compute a matrix k such that the characteristic polynomial of A + BG + bi k is p(s). We conclude this section noting that the property of reachability implies the existence of a state feedback gain assigning the eigenvalues of system (4.5). Conversely, it is possible to conclude reachability of system (4.1) by a property of system (4.5). Proposition 4.6 System (4.1) is reachable if and only if it is possible to arbitrarily assign the eigenvalues of A + BK.
4.3.1
Stabilizability
The main goal of a state feedback control law is to render the closedloop system asymptotically stable. This goal may be achieved, as discussed in the previous section, if the system is reachable. However, reachability is not necessary to achieve this goal. In fact, as highlighted in Proposition 4.2, the unreachable modes are not modified by the application of state feedback. This implies that there exists a matrix K such that system (4.5) is asymptotically stable if and only if the unreachable modes of system (4.1) have negative real part, in the case of continuoustime systems, or have modulo smaller than one, in the case of discretetime systems. To capture this situation we introduce a new definition. Definition 4.1 (Stabilizability) System (4.1) is stabilizable if its unreachable modes have negative real part, in the case of continuoustime systems, or have modulo smaller than one, in the case of discretetime systems. Example 4.1 (Deadbeat control) Consider a discretetime system described by equations of the form x+ = Ax + Bu,
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and the problem of designing a static, state feedback control law, described by the equation (4.4), yielding a closedloop system (4.5) such that, for any initial condition x(0) and for v = 0, one has x(k) = 0, for all k ≥ N , and for some N > 0. A control law achieving this goal is called a deadbeat control law. To achieve this goal it is necessary to select K such that (A + BK)N = 0 or, equivalently, such that the matrix A + BK has all eigenvalues equal to 0. Therefore, there exists a deadbeat control law for the considered system if and only if the system is controllable. Note that N ≤ n.
4.4
The notion of filtering
Consider a linear, finitedimensional, timeinvariant system and the problem of estimating its state from measurements of the input and output signals. We are interested in determining an online estimate, i.e. the estimate at time t has to be a function of the available information (input and output) at the same time instant. This implies that the estimate is generated by means of a device (known as filter) processing the current input and output of the system and generating a state estimate. The filter may be instantaneous, i.e. the estimate is generated instantaneously by processing the available information. In this case we have a static filter. Alternatively, the state estimate can be generated processing the available information through a dynamical device. In this case we have a dynamic filter. Assume that the system to be controlled is described by equations of the form (4.1) and assume5 that D = 0. Assume, in addition, that the filter which generates the online estimate is linear, finitedimensional and timeinvariant. Then we may have the following two configurations. • Static filter. The state estimate is generated via the relation xe = M y + N u,
(4.11)
with M and N constant matrices of appropriate dimensions. The resulting interconnected system is described by the equations σx = Ax + Bu xe
= M Cx + N u.
(4.12)
5 This assumption is without loss of generality. In fact, if y = Cx + Du and u are measurable then also y˜ = Cx is measurable.
97
4.5. STATE OBSERVER • Dynamic filter. The state estimate is generated by the system σξ = F ξ + Ly + Hu
xe = M ξ + N y + P u
(4.13)
with F , L, H, M , N and P constant matrices of appropriate dimensions. The resulting interconnected system is described by the equations σx = Ax + Bu σξ
= F ξ + LCx + Hu
xe
= M ξ + N Cx + P u.
(4.14)
In what follows we study in detail the dynamic filter configuration. This is mainly due to the fact that this configuration allows to solve most estimation problems for linear systems. Moreover, while the use of a static filter is very appealing, it provides a useful alternative only in very specific situations.
4.5
State observer
A state observer is a filter that allows to estimate, asymptotically or in finite time, the state of a system from measurements of the input and output signals. The simplest possible observer can be constructed considering a copy of the system, the state of which has to be estimated. This means that a candidate observer for system (4.1) is given by σξ = Aξ + Bu xe = ξ. (4.15) To assess the properties of this candidate state observer let e = x − xe be the estimation error and note that σe = Ae. As a result, if e(0) = 0 then e(t) = 0 for all t and for any input signal u. However, if e(0) 6= 0 then, for any input signal u, e(t) will be bounded only if the system (4.1) is stable, and will converge to zero only if the system (4.1) is asymptotically stable. If these conditions do not hold, the estimation error will not be bounded and system (4.15) does not qualify as a state observer for system (4.1). The intrinsic limitation of the observer (4.15) is that it does not use all the available information, i.e. it does not use the knowledge of the output signal y. This observer is therefore an openloop observer. To exploit the knowledge of y we modify the observer (4.15) adding a term which depends upon the available information on the estimation error, which is given by ye = Cxe − y.
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This modification yields a candidate state observer described by σξ = Aξ + Bu + Lye
xe = ξ.
(4.16)
To assess the properties of this candidate state observer note that e = x − xe is such that σe = (A + LC)e.
(4.17)
The matrix L (known as output injection gain) can be used to shape the dynamics of the estimation error. In particular, we may select L to assign the characteristic polynomical p(s) of A + LC. To this end, note that p(s) = det(sI − (A + LC)) = det(sI − (A′ + C ′ L′ )). Hence, there is a matrix L which arbitrarily assigns the characteristic polynomial of A+LC if and only if the system σξ = A′ ξ + C ′ v is reachable, or equivalently, if and only if the system (4.1) is observable. We summarize the above discussion with two formal statements. Proposition 4.7 Consider system (4.1) and suppose the system is observable. Let p(s) be a monic polynomial of degree n. Then there is6 a matrix L such that the characteristic polynomial of A + LC is equal to p(s). Proposition 4.8 System (4.1) is observable if and only if it is possible to arbitrarily assign the eigenvalues of A + LC.
4.5.1
Detectability
The main goal of a state observer is to provide an online estimate of the state of a system. This goal may be achieved, as discussed in the previous section, if the system is observable. However, observability is not necessary to achieve this goal. In fact, similarly to what discussed in Proposition 4.2, the unobservable modes are not modified by the output injection gain. This implies that there exists a matrix L such that system (4.17) is asymptotically stable if and only if the unobservable modes of system (4.1) have negative real part, in the case of continuoustime systems, or have modulo smaller than one, in the case of discretetime systems. To capture this situation we introduce a new definition. Definition 4.2 (Detectability) System (4.1) is detectable if its unobservable modes have negative real part, in the case of continuoustime systems, or have modulo smaller than one, in the case of discretetime systems. 6
For singleoutput systems the matrix L assigning the characteristic polynomial of A + LC is unique.
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4.5. STATE OBSERVER
Example 4.2 (Deadbeat observer) Consider a discretetime system described by equations of the form x+ = Ax + Bu y = Cx, and the problem of designing a state observer, described by the equation (4.16), such that, for any initial condition x(0) and for any u, e(k) = 0, for all k ≥ N , and for some N > 0. A state observer achieving this goal is called a deadbeat state observer. To achieve this goal it is necessary to select L such that (A + LC)N = 0 or, equivalently, such that the matrix A + LC has all eigenvalues equal to 0. Therefore, there exists a deadbeat state observer for the considered system if and only if the system is reconstructable. Note that N ≤ n.
4.5.2
Reduced order observer
We have shown that, under the hypotheses of observability or detectability, it is possible to design an asymptotic observer of order n for the system (4.1). However, this observer is somewhat oversized, i.e. it gives an estimate for the n components of the state vector, without making use of the fact that some of these components can be directly determined from the output function, e.g. if y = x1 there is no need to reconstruct x1 . Therefore, it makes sense to design a reduced order observer, i.e. a device that estimates only the part of the state vector which is not directly attainable from the output. To this end, consider the system (4.1) with D = 0 and assume that the matrix C has p independent columns7 . Then there exists a matrix Q such that QC = [I C2 ] . Let v = Qy = QCx = x1 + C2 x2 , in which x1 ∈ IRp and x2 ∈ IRn−p denote the first p and the last n − p components of the state x. Observe that the vector v is measurable. From the definition of v we conclude that if v and x2 are known then x1 can be easily computed, i.e. there is no need to construct a dynamic observer for x1 . Define now the new coordinates
7
x ˆ1 x ˆ2
= Tx =
I C2 0 I
x1 x2
This is the case if rankC = p, whereas if rankC < p it is always possible to eliminate redundant lines.
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CHAPTER 4. DESIGN TOOLS
and note that, by construction, v = Qy = x ˆ1 . In the new coordinates the system, with output v, is described by equations of the form ˜1 u σˆ x1 = A˜11 x ˆ1 + A˜12 x ˆ2 + B ˜2 u σˆ x2 = A˜21 x ˆ1 + A˜22 x ˆ2 + B v
= x ˆ1 .
In order to construct an observer for x ˆ2 consider the system σξ = F ξ + Hv + Gu, with state ξ, driven by u and v, and with output w = ξ + Lv. The idea is to select the matrices F , H, G and L in such a way that w be an estimate for x ˆ2 . Let w − x ˆ2 be the observation error. Then
˜1 u − A˜21 x ˜2 u ˆ1 + A˜12 x ˆ2 + B ˆ1 + A˜22 x ˆ2 + B σw − σˆ x2 = F ξ + Hv + Gu + L A˜11 x
˜1 − B ˜2 u. = F ξ + H + LA˜11 − A˜12 x ˆ1 + LA˜12 − A˜22 x ˆ 2 + G + LB (4.18) To have convergence of the estimation error to zero, regardless of the initial conditions and of the input signal, we must have σ(w − x ˆ2 ) = F (w − x ˆ2 )
(4.19)
and F must have all eigenvalues with negative real part, in the case of continuoustime systems, or with modulo smaller than one, in the case of discretetime systems. Comparing equations (4.18) and (4.19), we obtain that the matrices F , H, G, and L must be such that LA˜12 − A˜22 = −F H + LA˜11 − A˜21 = F L ˜1 − B ˜2 = 0. G + LB
We now show how the previous equations can be solved and how the stability condition of F can be enforced. Detectability of the system implies that the (reduced system) σ ξ˜ = A˜22 ξ˜
y˜ = A˜12 ξ
is detectable. As a result, there exists a matrix L such that the matrix F = A˜22 − LA˜12
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4.6. THE SEPARATION PRINCIPLE
u
y
System
v
Q
T Reduced order observer
1
xe
w
Figure 4.1: A reduced order observer.
has all all eigenvalues with negative real part, in the case of continuoustime systems, or with modulo smaller than one, in the case of discretetime systems. Then the remaining equations are solved by H = F L − LA˜11 + A˜21
˜1 + B ˜2 . G = −LB
Finally, from x ˆ1 = v and the estimate w of x ˆ2 we build an estimate xe of the state x inverting the transformation T , i.e.
x1e x2e
=
I −C2 0 I
v w
.
The (conceptual) structure of the reduced order observer is shown in Figure 4.1.
4.6
The separation principle
In Section 4.3 it has been shown that system (4.1) can be stabilized by means of a state feedback control law, provided the system is stabilizable. Moreover, in Section 4.5 it has been shown that the state of system (4.1) can be (asymptotically) estimated provided the system is detectable. It is therefore natural to discuss the properties resulting from the use of a state feedback control law in which the state is replaced by an estimate generated by a state observer. To this end, consider system (4.1) with D = 0, the state feedback control law u = Kx + v, the state observer σξ = (A + LC)ξ + Bu − Ly
xe = ξ,
and the control law obtained replacing the state x with its estimate xe , namely u = Kxe + v.
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The overall system is described by equations of the form
σx σξ
=
y
A BK −LC A + LC + BK
+
B B
v (4.20)
= Cx.
To study this system consider the coordinate e = x − xe and note that the system can be rewritten in the form
σx σe
=
y
A + BK −BK 0 A + LC
+
B 0
v (4.21)
= Cx.
From this representation it is possible to draw the following conclusions. The characteristic polynomial of the matrix
A + BK −BK 0 A + LC
is given by the product of the characeristic polynomials of the matrices A + BK and A + LC. This result, known as the separation principle, implies that the designs of the state feedback and of the state observer can be carried out independently. Therefore, the problem of asymptotic stabilization of the system (4.1) by means of a dynamic output feedback control law can be solved provided system (4.1) is stabilizable and detectable. The control law stabilizing the system is described by the equations σξ = (A + BK + LC)ξ − Ly
u = Kξ + v,
i.e. it is a dynamic output feedback control law. System (4.21), hence system (4.20), is not reachable. In fact, by Hautus test, we note that the unreachable modes are all the eigenvalues of A + LC. This implies that the state observer does not contribute to the inputoutput behaviour of the closedloop system, i.e.
C 0
and, similarly,
A + BK −BK 0 A + LC
C 0
e
t
A + BK −BK 0 A + LC
B 0
= C(A + BK)t B,
t
B 0
= Ce(A+BK)t B.
Therefore the inputoutput behaviour of the closedloop system resulting from the use of an output feedback controller designed on the basis of the separation principle coincides with the inputoutput behaviour of the closedloop system resulting from the use of the underlying state feedback controller.
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4.7. TRACKING AND REGULATION
4.7
Tracking and regulation
In the previous sections we have considered the simplest possible design problems, namely the stabilization and observation problems. Practical problems, however, present themselves in a more complex form. In particular, the system to be controlled may be affected by disturbances, and the output of the system does not have to be regulated to zero, but should asymptotically track a certain, prespecified, reference signal. In this section we discuss this control problem and present possible solutions. To begin with, consider a system to be controlled described by equations of the form σx = Ax + Bu + P d
e = Cx + Qd,
(4.22)
with x ∈ X = IRn , u(t) ∈ IRm , e(t) ∈ IRp , d(t) ∈ IRr , and A, B, P , C and Q matrices of appropriate dimensions and with constant entries. The signal d(t), denoted exogeneous signal, is in general composed of two components: the former models a set of disturbances acting on the system to be controlled, the latter a set of reference signals. In what follows we assume that the exogeneous signal is generated by a linear system, denoted exosystem, described by the equation σd = Sd,
(4.23)
with S a matrix with constant entries. Note that, under this assumption, it is possible to generate, for example, constant or polynomial references/disturbances and sinusoidal references/disturbances with any given frequency. The variable e(t), denoted tracking error, is a measure of the error between the ideal behaviour of the system and the actual behaviour. Ideally, the variable e(t) should be regulated to zero, i.e. should converge asymptotically to zero, despite the presence of the disturbances. If this happens we say that the tracking error is regulated to zero, i.e. converges asymptotically to zero, hence the disturbances are not affecting the asymptotic behaviour of the system and the output Cx(t) is asymptotically tracking the reference signal −Qd(t).
In general, the tracking error does not naturally converge to zero, hence it is necessary to determine an input signal u(t) which drives it to zero. The simplest possible way to construct such an input signal is to assume that it is generated via static feedback of the state x(t) of the system to be controlled and of the state d(t) of the exosystem, i.e. u = Kx + Ld.
(4.24)
In practice, it is unrealistic to assume that both x(t) and d(t) are measurable, hence it may be more natural to assume that the input signal u(t) is generated via dynamic feedback of the error signal only, i.e. it is generated by the system σχ = F χ + Ge
u = Hχ,
with χ(t) ∈ IRν , for some ν > 0, and F , G and H matrices with constant entries.
In summary, it is possible to formally pose the regulator problem as follows.
(4.25)
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Definition 4.3 (Full information regulator problem) Consider the system (4.22), driven by the exosystem (4.23) and interconnected with the controller (4.24). The full information regulator problem is the problem of determining the matrices K and L of the controller such that8 (S) the system σx = (A + BK)x is asymptotically stable; (R) all trajectories of the system σd = Sd
σx = (A + BK)x + (BL + P )d
e = Cx + Qd
(4.26)
are such that lim e(t) = 0.
t→∞
Definition 4.4 (Error feedback regulator problem) Consider the system (4.22), driven by the exosystem (4.23) and interconnected with the controller (4.25). The error feedback regulator problem is the problem of determining the matrices F , G and H of the controller such that (S) the system σx = Ax + BHχ
σχ = F χ + GCx
is asymptotically stable; (R) all trajectories of the system σd = Sd
σx = Ax + BHχ + P d
σχ = F χ + G(Cx + Qd)
e = Cx + Qd (4.27)
are such that lim e(t) = 0.
t→∞
4.7.1
The full information regulator problem
Consider the full information regulator problem and assume the following. Assumption 4.1 The matrix S of the exosystem has all eigenvalues with nonnegative real part, in the case of continuoustime systems, or with modulo not smaller than one, in the case of discretetime systems. Assumption 4.2 The system (4.22) with d = 0 is reachable. 8
(S) stands for stability and (R) for regulation.
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4.7. TRACKING AND REGULATION
Assumption 4.1 implies that there are no initial conditions d(0) such that the signal d(t) converges (asymptotically) to zero. This assumption is not restrictive. In fact, disturbances converging to zero do not have any effect on the asymptotic behaviour of the system, and references which converge to zero can be tracked simply by driving the state of the system to zero, i.e. by stabilizing the system. Assumption 4.2 implies that it is possible to arbitrarily assign the eigenvalues of the matrix A+BK by a proper selection of K. Note that, in practice, this assumption can be replaced by the weaker assumption that the system (4.22) with d = 0 is stabilizable. We now present a preliminary result which is instrumental to derive a solution to the full information regulator problem. Lemma 4.2 Consider the full information regulator problem. Suppose Assumption 4.1 holds. Suppose, in addition, that there exists matrices K and L such that condition (S) holds. Then condition (R) holds if and only if there exists a matrix Π ∈ IRn×r such that the equations ΠS = (A + BK)Π + (P + BL) 0 = CΠ + Q (4.28) hold. Proof. Consider the system (4.26) and the coordinates transformation dˆ = d where Π is the solution of the equation9
x ˆ = x − Πd,
ΠS = (A + BK)Π + (P + BL). Note that, by condition (S) and Assumption 4.1, there is a unique matrix Π which solves this equation. In the new coordinates x ˆ and dˆ the system is described by the equations σ dˆ = S dˆ
σx ˆ = (A + BK)ˆ x
ˆ e = Cx ˆ + (CΠ + Q)d.
Note now that, by condition (S) limt→∞ x ˆ = 0, hence condition (R) holds, by Assumption 4.1, if and only if CΠ + Q = 0. In summary, under the state assumptions, condition (R) holds if and only if there exists a matrix Π such that equations (4.28) hold. ⊳ We are now ready to state and prove the result which provides conditions for the solution of the full information regulator problem. 9 This equation is a socalled Sylvester equation. The Sylvester equation is a (matrix) equation of the form A1 X = XA2 + A3 , in the unknown X. This equation has a unique solution, for any A3 , if and only if the matrices A1 and A2 do not have common eigenvalues.
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CHAPTER 4. DESIGN TOOLS
Theorem 4.1 Consider the full information regulator problem. Suppose Assumptions 4.1 and 4.2 hold. There exists a full information control law described by the equation (4.24) which solves the full information regulator problem if and only if there exist two matrices Π and Γ such that the equations ΠS = AΠ + BΓ + P 0 = CΠ + Q (4.29) hold. Proof. (Necessity) Suppose there exist two matrices K and L such that conditions (S) and (R) of the full information regulator problem hold. Then, by Lemma 4.2, there exists a matrix Π such that equations (4.28) hold. As a result, the matrices Π and Γ = KΠ + L are such that equations (4.29) hold. (Sufficiency) The proof of the sufficiency is constructive. Suppose there are two matrices Π and Γ such that equations (4.29) hold. The full information regulator problem is solved selecting K and L as follows. The matrix K is any matrix such that the system σx = (A + BK)x is asymptotically stable. By Assumption 4.2 such a matrix K does exist. The matrix L is selected as L = Γ − KΠ. This selection is such that condition (S) of the full information regulator problem holds, hence to complete the proof we have only to show that, with K and L as selected above, the equations (4.28) hold. This is trivially the case. In fact, replacing L in (4.28) yields the equations (4.29), which hold by assumption. As a result, also condition (R) of the full information regulator problem holds, and this completes the proof. ⊳ The proof of Theorem 4.1 implies that a controller (it is not the only one) which solves the full information regulator problem is described by the equation u = Kx + (Γ − KΠ)d, with K such that a stability condition holds, and Π and Γ such that equations (4.29) hold. By Assumption 4.2 the stability condition can be always satisfied. As a result, the solution of the full information regulator problem relies upon the existence of a solution of equations (4.29).
4.7.2
The FBI equations
Equations (4.29), known as the FrancisByrnesIsidori (FBI) equations, are linear equations in the unknown Π and Γ, for which the following statement holds.
107
4.7. TRACKING AND REGULATION
Lemma 4.3 (Hautus) The equations (4.29), in the unknown Π and Γ, are solvable for any P and Q if and only if
rank
sI − A B C 0
= n + p,
(4.30)
for all s which are eigenvalues of the matrix S. Remark. The equations (4.29) can be rewritten in compact form as
A B C 0
Π Γ
−
I 0 0 0
Π Γ
S=
−P −Q
, ⋄
which is a socalled generalized Sylvester equation.
For singleinput, singleoutput systems (i.e. m = p = 1) the condition expressed by Lemma 4.3 has a very simple interpretation. In fact, the complex number s such that
rank
sI − A B C 0
are the zeros of the system σx = Ax + Bu
y = Cx,
which coincides with the roots of the numerator polynomial of the transfer function W (s) = C(sI − A)−1 B, i.e. the zeros of W (s). This implies that, for singleinput, singleoutput systems the full information regulator problem is solvable if and only if the eigenvalues of the exosystem are not zeros of the transfer function of the system (4.22), with input u, output e and d = 0.
4.7.3
The error feedback regulator problem
To provide a solution to the error feedback regulator problem we need to introduce a new assumption. Assumption 4.3 The system
is observable.
σx σd
=
A P 0 S
x d
e=
C Q
x d
(4.31)
108
CHAPTER 4. DESIGN TOOLS
Note that Assumption 4.3 implies observability of the system σx = Ax
y = Cx.
(4.32)
To prove this property note that observability of the system (4.31) implies that 2 6 6 rank 6 6 4
This, in turn, implies
C CA .. .
Q ⋆ .. .
CAn+r−1
⋆
2 6 6 rank 6 6 4
3 7 7 7 7 5
= n + r.
3
C CA .. .
7 7 7 7 5
=n
CAn+r−1 and, by CayleyHamilton Theorem, 2 6 6 rank 6 6 4
C CA .. .
3 7 7 7 7 5
= n,
CAn−1 which implies observability of system (4.32). Similarly to what discussed in the case of Assumption 4.2, Assumption 4.3 can be replaced by the weaker assumption that the system (4.31) is detectable. We are now ready to state and prove the result which provides conditions for the solution of the error feedback regulator problem. Theorem 4.2 Consider the error feedback regulator problem. Suppose Assumptions 4.1, 4.2 and 4.3 hold. There exists an error feedback control law described by the equation (4.25) which solves the full information regulator problem if and only if there exist two matrices Π and Γ such that the equations ΠS = AΠ + BΓ + P 0 = CΠ + Q (4.33) hold. Remark. Theorem 4.2 can be alternatively stated as follows. Consider the error feedback regulator problem. Suppose Assumptions 4.1, 4.2 and 4.3 hold. Then the error feedback regulator problem is solvable if and only if the full information regulator problem is solvable. ⋄
109
4.7. TRACKING AND REGULATION
Proof. (Necessity) The proof of the necessity is similar to the proof of the necessity of Theorem 4.1, hence omitted. (Sufficiency) The proof of the sufficiency is constructive. Suppose there are two matrices Π and Γ such that equations (4.33) hold. Then, by Theorem 4.1 the full information control law u = Kx + (Γ − KΠ)d, with K such that the system σx = (A + BK)x is asymptoticaly stable, solves the full information regulator problem. This control law is not implementable, because we only measure e. However, by Assumption 4.3, it is possible to build asymptotic estimates ξ and δ of x and d, hence implement the control law u = Kξ + (Γ − KΠ)δ.
(4.34)
To this end, consider an observer described by the equation
σξ σδ
=
A P 0 S
ξ G1 + G2 δ
C Q
ξ δ
B −e + 0
K Γ − KΠ
ξ δ
.
Note that the estimation errors ex = x − ξ and ed = d − δ are such that
σex σed
=
A P 0 S
+
G1 G2
C Q
ex ed
,
(4.35)
hence, by Assumption 4.3, there exist G1 and G2 that assign the eigenvalues of this error system. Note now that the control law (4.34) can be rewritten as u = Kx + (Γ − KΠ)d − (Kex + (Γ − KΠ)ed ) , hence the control law is composed of the full information control law, which solves the considered regulator problem, and of an additive disturbance which decays exponentially to zero. Such a disturbance does not affect the regulation requirement, provided the closedloop system is asymptotically stable. Therefore, to complete the proof we need to show that condition (S) holds. For, note that, in the coordinates x, ex and ed the closedloop system, with d = 0, is described by the equations 2 6 4
σx σex σed
3
2
7 5
=4
6
A + BK −BK −B(Γ − KΠ) 0 A + G1 C P + G1 Q 0 G2 C S + G2 Q
32 76 54
x ex ed
3 7 5.
(4.36)
Recall that the matrices G1 and G2 have been selected to render system (4.35) asymptotically stable, and that K is such that the system σx = (A + BK)x is asymptotically stable. As a result, system (4.36) is asymptotically stable. ⊳
110
4.7.4
CHAPTER 4. DESIGN TOOLS
The internal model principle
The proof of Theorem 4.2 implies that a controller (it is not the only one) which solves the error feedback regulator problem is described by equations of the form (4.25) with ′ χ= ξ δ ,
F =
A + G1 C + BK P + G1 Q + B(Γ − KΠ) G2 C S + G2 Q
G=
G1 G2
,
H=
K Γ − KΠ
, (4.37)
,
K, G1 and G2 such that a stability condition holds, and Π and Γ such that equations (4.33) hold. This controller, and in particular the matrix F , possesses a very interesting property. Proposition 4.9 (Internal model property) The matrix F in equation (4.37) is such that F Σ = ΣS, for some matrix Σ of rank r. In particular, any eigenvalue of S is also an eigevalue of F . Proof. Let
Σ=
Π I
and note that rankΣ = r, by construction, and that
FΣ =
=
AΠ + G1 CΠ + BKΠ + P + G1 Q + B(Γ − KΠ) −G2 CΠ + S − G2 Q (AΠ + BΓ + P ) + G1 (CΠ + Q) S − G2 (CΠ + Q)
=
ΠS S
= ΣS,
hence the first claim. To prove the second claim, let λ be an eigenvalue of S and v the corresponding eigenvector. Then Sv = λv, hence F Σv = ΣSv = λΣv, which shows that λ is an eigenvalue of F with eigenvector Σv, and this proves the second claim. ⊳ It is possible to prove that the property highlighted in Proposition 4.9 is shared by all error feedback control laws which solve the considered regulation problem, and not only the proposed controller. This property, which is often referred to as the internal model
4.7. TRACKING AND REGULATION
111
principle, can be interpreted as follows. The control law solving the regulator problem has to contain a copy of the exosystem, i.e. it has to be able to generate, when e = 0, a copy of the exogeneous signal.
112
CHAPTER 4. DESIGN TOOLS
Exercises Exercise 1 Consider the matrix 2 6
A=4
7 −6 2 8.8 −7.6 2.8 9.6 −7.2 2.6
3 7 5.
1. Compute the characteristic polynomial of A and hence show that the eigenvalues of A are 1, −1 and 2. 2. Compute three linearly independent eigenvectors. 3. Find a similarity transformation L such that Aˆ = L−1 AL is a diagonal matrix. 4. Determine exp(At) as a function of t. 5. Determine sin(At) as a function of t. Solution 1 1. The characteristic polynomal is det(sI − A) = s3 − 2s2 − s + 2 = (s2 − 1)(s − 2), hence the claim. 2. The matrix A has three distinct eigenvalues, hence has three linearly independent eigenvectors. These are computed solving the equation Av = λv with λ = 1, −1, 2. The eigenvector associated to λ = 1 is v1 = [1/2, 1, 3/2]′ ; the eigenvector associated to λ = −1 is v2 = [1, 4/3, 0]′ ; the eigenvector associated to λ = 2 is v3 = [1, 1/2, 2]′ . Note that, as suggested by the theory, det([v1 , v2 , v3 ]) 6= 0, hence the three vectors are linearly independent. I
II
EXERCISES 3. Let V = [v1 , v2 , v3 ] and
2 6 Aˆ = 4
1 0 0 0 −1 0 0 0 2
3 7 5.
Note that ˆ AV = V A, hence, L = V is such that L−1 AL is a diagonal matrix. 4. By a property of the matrix exponential ˆ
eAt = e(V AV where
2 6
ˆ
eAt = 4
−1 )t
ˆ
= V eAt V −1 ,
et 0 0 −t 0 e 0 0 0 e2t
3 7 5.
5. Similarly to the definition of the matrix exponential we have sin(At) = At − hence
(At)3 (At)5 + − ··· , 3! 5!
−1 ˆ sin(At) = V sin(At)V ,
with
2 6
ˆ =4 sin(At)
sin t 0 0 0 − sin t 0 0 0 sin 2t
3 7 5.
Exercise 2 Let A be a square matrix of dimension n and let λ1 , · · · λn denote its eigenvalues. Assume for simplicity that A has n distinct eigenvalues. 1. Show that the determinant of A is equal to the product of the eigenvalues of A: det(A) = λ1 λ2 · · · λn . 2. Show that the trace of A is equal to the sum of the eigenvalues of A: tr(A) = λ1 + λ2 + · · · + λn . (tr(A) = a11 + a22 + · · · + ann .) 3. Show that if tr(A) > 0, the system x˙ = Ax is unstable. Is the converse true? 4. Show that if the system x˙ = Ax is asymptotically stable then tr(A) < 0.
III
EXERCISES
5. Assume n = 3, tr(A) = T and det(A) = ω 2 T . Assume moreover that A has two purely imaginary eigenvalues. Discuss the stability of the system x˙ = Ax as a function of T and ω and explain how to compute exp(At) as a function of t if the eigenvectors of A are known. Solution 2 If A has n distinct eigenvalues then there exists a nonsingular matrix L such that 2 3 λ1 0 · · · 0 6 λ2 · · · 0 7 6 0 7 −1 6 = A˜ L AL = 6 . .. . . .. 7 . . . 7 4 . 5 . 0
0
···
λn
for some λi ∈ C. I 1. Note that ˜ = λ1 λ2 · · · λn . det(A) = det(L) det(L−1 ) det(A) 2. The characteristic polynomial of A is given by det(sI − A) = (s − a11 )(s − a22 ) · · · (s − an ) + · · · = sn + αn−1 sn−1 + · · · with αn−1 = −a11 − a22 − · · · − ann = −tr(A). Moreover det(sI − A) = (s − λ1 )(s − λ2 ) · · · (s − λn ) = sn − (λ1 + λ2 + · · · + λn )sn−1 + · · · . Hence tr(A) = λ1 + λ2 + · · · + λn . 3. If tr(A) > 0 then there exists a λi with positive real part. Hence the system x˙ = Ax is unstable. Instability of the system x˙ = Ax does not imply tr(A) > 0, as the sum of the real parts of the eigenvalues may be negative even if some of the real parts are positive. 4. If the system x˙ = Ax is asymptotically stable then the real parts of all eigenvalues are negative, hence tr(A) < 0. 5. Let n = 3 and, as stated, λ1 = −iω
λ2 = iω
λ3 = α,
then tr(A) = λ1 + λ2 + λ3 = T
det(A) = λ1 λ2 λ3 = ω 2 T.
IV
EXERCISES As a result, α = T . Hence the system x˙ = Ax is stable, not asymptotically if T ≤ 0 and it is unstable if T > 0. Finally let v1 , v2 and v3 be the eigenvectors of A corresponding to the eigenvalues λ1 , λ2 and λ3 . Hence 2
AV = A
v1 v2 v3
=
6 v3 4
v1 v2
−iω 0 0 0 iω 0 0 0 α
3 7 5
= V Λ.
As a result A = V ΛV −1 , hence eAt = V eΛt V −1 with
2 6
eΛt = 4
e−iωt 0 0 0 eiωt 0 0 0 eαt
3 7 5.
Exercise 3 Consider the discretetime system xk+1 = Axk . 1. Let
A= Consider the initial state
−1 1 0 −1
x(0) =
0 1
.
and plot x(k) on the state space for k = 1, 2, 3, 4. Exploiting the obtained result discuss the stability of the equilibrium x = 0. 2. Let
A=
−1 1 0 1
Consider the initial state
x(0) =
0 1
.
and plot x(k) on the state space for k = 1, 2, 3, 4. Exploiting the obtained result discuss the stability of the equilibrium x = 0. Solution 3 1. Note that x(1) =
1 −1
x(2) =
−2 1
x(3) =
3 −1
x(4) =
−4 1
,
and these are indicated in the figure with ⋆ signs. This implies that the equilibrium x = 0 is unstable. (Note that to decide instability of an equilibrium it is enough that one trajectory does not satisfy the “ǫ − δ” argument in the definition of stability.)
V
EXERCISES 2. Note that
x(1) =
1 1
x(2) =
0 1
x(3) =
1 1
x(4) =
0 1
,
and these are indicated in the figure with X signs. This trajectory is such that the “ǫ − δ” argument holds, however we cannot conclude stability of the equilibrium x = 0 only from properties of one trajectory.
1
x2
0.9
0.8
0.7
0.6
*
x
*
*
1
x
x1
1 0.5
0.4
*
*
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Exercise 4 Consider the discretetime system
xk+1 = Axk =
1 1 a −1
xk
with a ∈ IR. 1. Show that the system is asymptotically stable for all a ∈ (−2, 0) and it is unstable for a < −2 and a > 0.
VI
EXERCISES 2. Let a = 0. Discuss the stability properties of the system. 3. Let a = −2. Discuss the stability properties of the system.
Solution 4 The characteristic polynomial of the matrix A is det(sI − A) = (s − 1)(s + 1) − a = s2 − 1 − a. Hence the eigenvalues of A are √ λ1 = + 1 + a
√ λ2 = − 1 + a.
1. The system is asymptotically stable if (and only if) λ1  < 1
λ2  < 1.
Observe that λ1 and λ2 are real if a ≥ −1 and are imaginary if a < −1. Moreover, λ1  = λ2  and this is smaller than one if (and only if) a ∈ (−2, 0). 2. If a = 0 the eigenvalues are real and have modulo equal to one. Moreover they are distinct, hence they have geometric and algebraic multiplicity equal to one. Therefore the system is stable, not asymptotically. 3. If a = −2 the eigenvalues are imaginary and have modulo equal to one. Moreover they are distinct, hence they have geometric and algebraic multiplicity equal to one. Therefore the system is stable, not asymptotically. Exercise 5 Consider two systems σxi = Ai xi + Bi u, yi = Ci xi , with i = 1, 2 and
A1 =
A2 =
1 1 0 2 1 0 1 2
B1 =
B2 =
1 0 1 1
C1 =
C2 =
1 1
1 0
.
Note that the systems have the same input. 1. Show that if x1 (0) = x2 (0) = 0 then y1 (t) = y2 (t) for all t. (This is equivalent to C1 Ak1 B1 = C2 Ak2 B2 for all k ≥ 0, in discretetime and to C1 eA1 t B1 = C2 eA2 t B2 for all t ≥ 0, in continuoustime.) 2. The two systems are not algebraically equivalent.
VII
EXERCISES Solution 5 1. If x1 (0) = x2 (0) = 0 then (suppose the systems are discretetime) y1 (k) =
k−1 X
C1 A1k−1−i B1 u(i)
y2 (k) =
i=0
k−1 X
C2 A2k−1−i B2 u(i)
i=0
Note that A1 = A′2 , B1 = C2′ and C1 = B2′ . Hence C1 Ak1 B1 = (C1 Ak1 B1 )′ = B1′ (A′1 )k C1′ = C2 Ak2 B2 which proves that the output responses of the two systems coincide. 2. The systems are algebraically equivalent if and only if there exists a nonsingular L such that A1 = L−1 A2 L
B1 = L−1 B2
(or LA1 = A2 L)
Setting
L= yields
LB1 = C1 =
1 1
L1 L3
L1 L2 L3 L4
(or LB1 = B2 )
= B2 =
= C2 L =
C1 = C2 L.
1 1
L1 L2
,
hence L1 = L2 = L3 = 1. Finally, for any L4 ,
LA1 =
1 3 1 1 + 2L4
6= A2 L =
1 1 1 3 + 2L4
,
which shows that the two systems are not algebraically equivalent. It is worth noting that the system with state x1 is not reachable but observable, whereas the system with state x2 is reachable but not observable (it is the dual of the system with state x1 ). Finally, if two systems are reachable and observable and the forced responses of the output coincide then the systems are algebraically equivalent. Exercise 6 The function y = x − log x has a minimum for x = 1. This minimum can be computed using Newton’s algorithm which yields the discretetime system xk+1 = 2xk − x2k . 1. Compute the equilibrium points of this system. 2. Study the stability properties of the equilibrium points.
VIII
EXERCISES
3. Show that (a) if x0 = 2 or x0 = 0 then xk = 0, for all k ≥ 1; (b) if x0 > 2 then xk+1 < xk < 0, for all k ≥ 1;
(c) if x0 ∈ (0, 2) then xk ∈ (0, 1), for all k ≥ 1, and limk→∞ xk = 1. Solution 6 1. At an equilibrium point of a discretetime system we have xk+1 = xk . Hence, the equilibrium points of the system are the solutions of the equation x = 2x − x2 . We have therefore two equilibria x=0
x = 1.
2. The linearization of the system around an equilibrium point xeq is described by (δx )k+1 = (2 − 2xeq )(δx )k . Hence, for xeq = 0 we have (δx )k+1 = 2(δx )k , which is an unstable system, while for xeq = 1 we have (δx )k+1 = 0, which is a stable system. As a result, by the principle of stability in the first approximation, the equilibrium point x = 0 is unstable and the equilibrium point x = 1 is locally asymptotically stable. 3. If x0 = 2, then x1 = 0, and by definition of equilibrium xk = 0 for all k > 1. If x0 = 0 then, again by definition of equilibrium, xk = 0, for all k > 0. Note now that the relation xk+1 = 2xk − x2k , implies xk+1 < xk if and only if xk < 0 or xk > 1. If x0 > 2 then x1 < 0, hence x2 < x1 < 0 and, in general, xk+1 < xk < · · · < x2 < x1 < 0. This shows that, for any x0 > 2 the sequence {xk } diverges, i.e. limk→∞ xk = −∞. If x0 ∈ (0, 2) then x1 ∈ (0, 1). Moreover, x1 ∈ (0, 1) implies that x2 ∈ (0, 1), and so on. Therefore, if x0 ∈ (0, 2) then xk ∈ (0, 1) for all k > 0. Note now that if xk ∈ (0, 1) then xk+1 > xk . As a result, for any x0 ∈ (0, 2) we have 0 < x1 < x2 < · · · < xk < xk+1 < 1, which states that the sequence {xk } is monotonically increasing and bounded from above, hence should have a limit. However, any limit of the sequence has to be an equilibrium point of the system, which implies that, for any x0 ∈ (0, 2) we have that limk→∞ xk = 1.
IX
EXERCISES Exercise 7 Consider the discretetime system 1 x1,k + α(x2,k )x2,k 2 = x2,k − α(x2,k )x1,k − r(x2,k )x2,k
x1,k+1 = x2,k+1 with
r(x2,k ) =
2 (α(x2,k ))2 , 3
and α(·) a differentiable function. 1. Show that if
√
3 2 the origin is a (locally) asymptotically stable equilibrium. 0 < α(x2,k ) <
Solution 7 To study the stability of the origin we use the principle of stability in the first approximation. 1. The linearization of the system around the zero equilibrium is described by
δx (k + 1) = Aδx (k) =
1 2
α −α 1 − 23 α2
δx (k).
The characteristic polynomial of the matrix A is
2 2 2 1 3 + = s2 + sη + (η + 2) det(sI − A) = s + s α2 − α + 3 2 3 2 2
with η = 23 α2 − 23 . Note that we are interested in values of α2 in the set (0, 3/4) and hence of η in the set (−3/2, −1). We need to show that, for η ∈ (−3/2, −1), the roots of the characteristic polynomial above have modulo not larger than one. The simplest way to perform this test, without computing the roots, is to consider the change of variable 1+ξ , s→ 1−ξ and then apply Routh test to the resulting polynomial. Note in fact that is s < 1 then the real part of ξ is negative. The polynomial to study is thus p(ξ) = (1 + ξ)2 + (1 + ξ)(1 − ξ)η + (1 − ξ)2 (η + 2) = 3ξ 2 + 2(−1 − η)ξ + (3 + 2η). By Routh test, this polynomial has all roots with negative real part if and only if η ∈ (−3/2, −1). Hence, the matrix A has all roots with modulo smaller than one if and only if α2 ∈ (0, 3/4). By the principle of stability in the first approximation, for all α2 ∈ (0, 3/4) the zero equilibrium of the nonlinear system is locally asymptotically stable. Note that, if α2 = 0 or α2 = 3/4, the linearized system is stable, not asymptotically, but we cannot conclude anything on the stability properties of the zero equilibrium of the nonlinear system.
X
EXERCISES
Exercise 8 Consider the nonlinear model of a bioreactor described by the equations X˙ = µ(S)X − Xu S˙
= −µ(S)X + (Sin − S)u
y
= S + kS 3 ,
in which S ≥ 0, X ≥ 0, Sin > 1 is a constant, k is a constant, u is an external signal and µ(S) = 2
S . 1 + 2S + S 2
1. Suppose u is constant and determine all equilibrium points of the system in the following cases (a) u ∈ (0, 1/2); (b) u = 1/2; (c) u > 1/2. Sketch the position of the equilibrium points on the (X, S)plane as a function of u. 2. Write the linearized model of the system around the equilibrium point with both nonzero components determined in part 1.(b). Study the stability of such linearized system. Solution 8 1. The equilibrium points are the solutions of the equations X˙ = 0 and S˙ = 0. From the first equation we obtain X = 0 or u = µ(S). The first alternative yields, exploiting the second equation, S = Sin . The second alternative yields, exploiting again the second equation, X = Sin − S, where S is the solution of u = µ(S). Note now that the function µ(S) is always nonnegative, it is zero for S = 0, it tends to zero as S → ∞, and it has a maximum for S = 1 equal to 1/2. As a result we obtain the following equilibria. (a) u ∈ (0, 1/2). There are three equilibria: (0, Sin ), (Sin −S1 , S1 ) and (Sin −S2 , S2 ), where S1 and S2 are the two solutions of u = µ(S) with u ∈ (0, 1/2). (b) u = 1/2. There are two equilibria: (0, Sin ) and (Sin − 1, 1). (c) u > 1/2. There is one equilibrium: (0, Sin ). 2. The system linearized around the equilibrium point (Sin − 1, 1) is described by (note dµ (1) = 0) that dS
δ˙x = Aδx + Bu =
0 0 −1/2 −1/2
δy =
0 1 + 3k
δx +
1 − Sin Sin − 1
δu
δx .
The linearized system is stable, not asymptotically, but nothing can be concluded on the stability of the equilibrium of the nonlinear system.
XI
EXERCISES
Exercise 9 The (simplified and normalized) model of a patient in the presence of an infectious disease is described by the equations x˙ = 1 − x − T xy
y˙ = T xy − y − ISy,
in which x represents the number of noninfected cells, y represents the number of infected cells, T represents the effect of the therapy and IS ∈ (0, 1) represents the action of the immune system. 1. Determine the two equilibrium points of the system. Show that one equilibrium corresponds to a healthy patient, i.e. the number of infected cells is zero, and one equilibrium corresponds to an ill patient, i.e. the number of infected cells is nonzero. 2. Write the linearized models of the system around the two equilibrium points. 3. Without therapy T > 1 + IS. Show that the equilibrium point associated to a healthy patient is unstable and the one associated to an ill patient is locally asymptotically stable. 4. With therapy T < 1 + IS. Show that the equilibrium point associated to a healthy patient is locally asymptotically stable and the one associated to an ill patient is unstable. Solution 9 1. The equilibrium points of the system are the solutions of the equations x˙ = 0 and y˙ = 0. From the second equation we have y = 0 or x = 1+IS T . From the equation T −1−IS x˙ = 0, the former yields x = 1, and the latter yields y = T (1+IS) . Hence, the equiT −1−IS librium (1, 0) corresponds to a healty patient, and the equilibrium ( 1+IS T , T (1+IS) ) corresponds to an ill patient.
2. The linearized model of the system around the first equilibrium point is
δ˙x δ˙y
=
−1 −T 0 T − 1 − IS
δx δy
.
The linearized model of the system around the second equilibrium point is
δ˙x δ˙y
2
=
4
T − 1+IS
T −1−IS 1+IS
−1 − IS 0
3 5
δx δy
.
3. Suppose T > 1 + IS. The system linearized around the equilibrium associated to a healthy patient has an eigenvalue with positive real part, hence the equilibrium is unstable. On the contrary, the system linearized around the equilibrium associated to a ill patient has both eigenvalues with negative real part, hence the equilibrium is (locally) asymptotically stable.
XII
EXERCISES
4. Suppose T < 1 + IS. The system linearized around the equilibrium associated to a healthy patient has both eigenvalues with negative real part, hence the equilibrium is (locally) asymptotically stable. On the contrary, the system linearized around the equilibrium associated to a ill patient has one eigenvalue with positive real part, hence the equilibrium is unstable. Exercise 10 Consider the model of a simple robot arm given by φ¨ + m sin φ = T, in which φ is the angle of the arm with respect to a vertical line directed upward, T is the applied torque, and m is a positive parameter. 1. Let T = 0. Determine the equilibrium points of the system. 2. Let T = 0. Consider the energy of the system E = 21 φ˙ 2 − m cos φ. Compute E˙ and show that E remains constant for all t. ˙ = (0, 0) is locally 3. Let T = −kφ˙ with k > 0. Show that the equilibrium (φ, φ) asymptotically stable. Solution 10 1. The equilibrium points of the system are such that φ(t) is constant for all t, hence are such that φ˙ = φ¨ = 0. As a result, the equilibrium points are the solutions of the equation sin φ = 0. This equation has infinitely many solutions, however, from a physical point of view, we have only two solutions φ = 0 (the arm is directed upward) and φ = π (the arm is directed downward). 2. Note that
E˙ = φ˙ φ¨ + m sin φφ˙
and replacing the equation describing the motion of the arm yields ˙ E˙ = T φ. Hence, if T = 0 we have that E˙ = 0, hence E(t) is constant for all t. 3. Setting T = −kφ˙ yields
φ¨ + kφ˙ + m sin φ = 0.
To study the stability of the equilibrium (0, 0) we linearize this equation around this point and we obtain δ¨φ + kδ˙φ + mδφ = 0. The characteristic polynomial associated with this linear system is (Laplace transform the equation, and factor L(φ(s))) s2 + ks + m,
which has all roots with negative real part, by Routh test. As a result, the equilibrium (0, 0) of the nonlinear system is locally asymptotically stable.
XIII
EXERCISES
Exercise 11 Consider the nonlinear system modelling the dynamics of the angular velocities of a rigid body in space (for example a satellite), described by the equations (known as Euler’s equations) I2 − I3 x2 x3 x˙ 1 = I1 I3 − I1 x˙ 2 = x3 x1 I2 I1 − I2 x˙ 3 = x1 x2 , I3 with I1 6= 0, I2 6= 0 and I3 6= 0. 1. Determine the equilibrium points of the system. 2. Consider the equilibrium x ˜ = (α, 0, 0) with α 6= 0. Compute the linearized model around the equilibrium x ˜, and study its stability as a function of I1 , I2 and I3 . Discuss the stability properties of the equilibrium x ˜ of the nonlinear system. Solution 11 1. The equilibrium points of the system are the solutions of the equations x˙ 1 = x˙ 2 = x˙ 3 = 0. From the equation x˙ 1 = 0 we have x2 = 0 or x3 = 0. Replacing x2 = 0 in the second and third equations yields the constraint x3 x1 = 0. As a result, we have the two families of equilibrium points described by (0, 0, ⋆)
(⋆, 0, 0),
where ⋆ is any real number. Finally, from x3 = 0 we have another family of equilibrium points described by (0, ⋆, 0). All the above equilibria describe steady rotations around an axis of symmetry (these are sometimes called, in mechanics, relative equilibria). 2. The system linearized around the given equilibrium is described by 2 6 δ˙x = 4
0 0 0
0 0 I1 −I2 I3 α
0
3
7 I3 −I1 I2 α 5 δx .
0
As a result, if
I3 − I1 I1 − I2 <0 I2 I3 the linearized system is stable, but no conclusions can be drawn on the stability properties of the given equilibrium of the nonlinear system. If I3 − I1 I1 − I2 >0 I2 I3 the linearized system is unstable, hence the given equilibrium of the nonlinear systems is unstable.
XIV
EXERCISES
Exercise 12 A continuoustime system described by the equations x˙ = Ax + Bu
y = Cx
is passive if there exists a matrix P = P ′ > 0 such that A′ P + P A ≤ 0
P B = C ′.
Consider the continuoustime system
x˙ =
0 1 −2 0
x+
0 1
u
y=
0 1
x.
1. Show that the system is passive. 2. Let u = −Ky. Write the equations of the closedloop system and show that the system is asymptotically stable for all K > 0. 3. The zeros of a system with x ∈ IRn , u ∈ IR and y ∈ IR, are the complex numbers s¯ such that s¯I − A B < n + 1. rank C 0 Show that the considered system has a zero at s¯ = 0. Finally, show that a passive system does not have zeros s¯ with positive real part. Solution 12 1. Let
P = and note that
PB =
2 0 0 1 0 1
= C′
and A′ P + P A = 0, hence the system is passive. 2. The closedloop system is described by the equations
x˙ = (A − KBC)x =
0 1 −2 −K
x = Acl x.
The characteristic polynomial of the matrix Acl is given by det(sI − Acl ) = s2 + Ks + 2, hence, by Routh test, its roots have negative real part if and only if K > 0, hence the closedloop system is asymptotically stable for all K > 0.
XV
EXERCISES 3. Note that
s¯I − A B C 0
2 6
=4
s¯ −1 0 −2 s¯ 1 0 1 0
3 7 5.
Setting s¯ = 0 yields the matrix 2 6 4
0 −1 0 −2 0 1 0 1 0
3 7 5.
The determinant of this matrix is equal to zero, hence the matrix has rank smaller than three, which proves that the system has a zero for s¯ = 0. (This definition of zeros for singleinput, singleoutput systems is consistent with the definition based on the transfer function of the system. For example, for the considered system, the transfer function is W (s) = C(sI − A)−1 B =
s2
s , +2
hence the system has a zero at zero.) To prove that a passive, singleinput, singleoutput system does not have zeros with positive real part we proceed as follows. Suppose that the system has a zero at some s¯, hence s¯I − A B rank < n + 1. C 0 This implies that there exist a nonzero (complex) vector v and a scalar w such that (¯ sI − A)v + Bw = 0
Cv = 0 (or v ⋆ C ′ = 0),
where v ⋆ is the adjoint of v. Then, left multiplying by P , 0 = (¯ sP − P A)v + P Bw = (¯ sP − P A)v + C ′ w, hence, left multiplying by v ⋆ ,
sP − P A)v + C ′ w = v ⋆ (¯ sP − P A)v. 0 = v ⋆ (¯ As a result s¯v ⋆ P v = v ⋆ P Av and s¯⋆ v ⋆ P v = v ⋆ A′ P v. Adding these two last equations yields (¯ s + s¯⋆ )v ⋆ P v = v ⋆ (A′ P + P A)v. Finally, note that v ⋆ P v > 0, v ⋆ (A′ P + P A)v ≤ 0 and (¯ s + s¯⋆ ) is equal to twice the real part of s¯, which has to be nonpositive, as claimed.
XVI
EXERCISES
Exercise 13 Consider the discretetime system x(k + 1) = Ax(k) + Bu(k). Let 2 6
A=4
0 1 0 −1 0 0 0 2 0
3
2
7 5
B=4
6
1 −1 2
3 7 5.
1. Compute the reachability matrix R. 2. Determine if the system is reachable and compute the set of reachable states. 3. Determine all states xI such that x(0) = xI and x(1) = 0. Solution 13 1. The reachability matrix is 2
R=
B AB A2 B
6
=4
1 −1 −1 −1 −1 1 2 −2 −2
3 7 5.
2. The first two columns of the reachability matrix are linearly independent, and det(R) = 0, hence the system is not reachable. The set of reachable states is two dimensional and it is described by the linear combination of the first two columns of the reachable matrix. 3. We have to determine all states which are controllable in one step. Instead of using the definition of controllable states in one step, we perform a direct calculation. Let 2 6
x(0) = xI = 4 and note that
xI,1 xI,2 xI,3
2 6
x(1) = Ax(0) + Bu(0) = 4
3 7 5
xI,2 + u(0) −xI,1 + u(0) 2(xI,2 + u(0))
3 7 5.
The condition x(1) = 0 implies xI,1 = u(0), xI,2 = −u(0), hence all states that can be controlled to zero in one step are given by 2 6
xI = 4
u(0) −u(0) xI,3
3 7 5,
and this is a two dimensional set. Note that this implies that the considered system has an eigenvalue at zero.
XVII
EXERCISES Exercise 14 Consider the discretetime system x(k + 1) = Ax(k) + Bu(k). Let 2 6
A=4
0 1 0 −1 0 0 0 0 0
3
2
7 5
B=4
1 −1 0
6
3 7 5.
1. Compute the reachable subspaces in one step, two steps and three steps. 2. Using Hautus test determine the unreachable modes. 3. Show that the system is controllable in two steps. Solution 14 The reachability matrix is 2
R=
B AB A2 B
6
=4
1 −1 −1 −1 −1 1 0 0 0
3 7 5,
and it has rank equal to two. 1. The set of reachable states in one step is 2 6
R1 = spanB = span 4
1 −1 0
3 7 5.
The set of reachable states in two steps is 2 6
R2 = span[B, AB] = span 4
1 −1 −1 −1 0 0
3
2
7 5
= span 4
6
1 0 0 1 0 0
3 7 5.
The set of reachable states in three steps is R3 = span[B, AB, A2 B] = R2 . 2. The reachability pencil is 2
sI − A B
6
=4
s −1 0 1 1 s 0 −1 0 0 0 s
3 7 5.
This matrix has rank three for all s 6= 0, hence the unreachable mode is s = 0.
XVIII
EXERCISES
3. The system is controllable, since the unreachable modes are at s = 0. To show that it is controllable in two steps note that 2 6
A2 = 4
−1 0 0 0 −1 0 0 0 0
3 7 5,
hence ImA2 ⊆ R2 , which proves the claim. Exercise 15 Consider the continuoustime system x˙ = Ax + Bu. Let
A=
1 0 0 1
B=
1 0
.
Compute the set of states that can be reached from the state
x0 =
0 1
.
Solution 15 Note that Z t
At
x(t) = e x0 +
0
A(t−τ )
e
0 et
Bu(τ )dτ =
+
R t t−τ e u(τ )dτ 0
0
.
Note that, by a proper selection of u(τ ) in the interval [0, t) it is possible to assign Therefore, the states that can be reached at time t from x0 are described 0 by 0 + λB, x(t) = et R t t−τ e u(τ )dτ.
with λ ∈ IR. Exercise 16 Consider the discretetime system x(k + 1) = Ax(k), y(k) = Cx(k). Let 2 6
A=4
0 −4 0 1 4 0 0 −4 2
3 7 5,
C=
0 1 0
.
Determine if the system is observable and compute the unobservable subspace. Solution 16 The observability matrix is 2 6
O=4
0 1 0 1 4 0 4 12 0
3 7 5.
XIX
EXERCISES
This matrix has rank two, hence the system is not observable. The unobservable subspace ker O is spanned by the vector 2 3 0 6 7 4 0 5, 1 this means that it is not possible to obtain information on the third component of the state from measurements of the output. Exercise 17 Consider the continuoustime system x˙ = Ax + Bu
y = Cx
and its Euler discretetime approximate model xk+1 = xk + T (Axk + Buk )
yk = Cxk
where T > 0 is the sampling time. Show that the continuoustime system is observable if and only if the Euler discretetime approximate model is observable. Solution 17 By observability of the continuoustime system we have that
rank
sI − A C
= n,
for all s ∈ C I. Consider now the observability pencil of the Euler model, namely
sI − (I + T A) C
Note now that, for all s ∈ C I,
n = rank
sI − A C
2
=
6 rank 4
.
s−1 I −A T C
3 7 5
= rank
zI − A C
,
where z = s−1 T . Hence, observability of the continuoustime system implies, and it is implied by, observability of the Euler discretetime approximate model. Exercise 18 Consider the system σx = Ax, y = Cx, with
A=
A11 A12 A21 A22
C=
I 0
,
and Aij matrices of appropriate dimensions. Show that the system is observable if and only if the (simpler) system σξ = A22 ξ with output η = A12 ξ is observable.
XX
EXERCISES
Solution 18 Observability of the system σx = Ax, y = Cx implies, and is implied by, 2 6
rank 4
sI − A11 −A12 A21 sI − A22 I 0
for all s ∈ C. I Suppose that y(t) ∈ IRp , then 2
6 rank 4
sI − A11 −A12 A21 sI − A22 I 0
Note now that the matrix
3 7 5
=n
3 7 5
= p + rank
−A12 sI − A22
−A12 sI − A22
.
has full rank for all s ∈ C I if and only if the system σξ = A22 ξ with output η = A12 ξ is observable, and this proves the claim. Exercise 19 Consider the continuoustime system
x˙ = y =
3 −1 + ǫ 1 2−ǫ
−1 1
x +
0 1
u
x.
1. Show that the system is reachable for all ǫ 6= 1. 2. Let ǫ = 1. Write the system in the canonical form for nonreachable systems, i.e. determine coordinates x ˆ, defined by x = Lˆ x for some matrix L, such that, in the x ˆ coordinates, the system is described by equations of the form ˜1 A˜11 A˜12 B ˙x ˆ= x ˆ+ u, 0 0 A˜22 ˜1 such that x ˜1 u is reachable. (Compute explicitly L, with A˜11 and B ˆ˙ 1 = A˜11 x ˆ1 + B ˜1 .) A˜11 , A˜12 , A˜22 and B 3. Show that the system is observable for all ǫ 6= 1/2. 4. Let ǫ = 1/2. Determine, using Hautus test, the unobservable modes.
XXI
EXERCISES Solution 19 1. The reachability matrix is
0 ǫ−1 1 2−ǫ
R=
.
If ǫ 6= 1 then det(R) 6= 0, hence the system is reachable. 2. If ǫ = 1 the system is not reachable. Let L be constructed setting the first column equal to the first column of the reachable matrix (which is nonzero) and selecting the second column to render L nonsingular, for example
L= (Any matrix L of the form
L=
0 1 1 0
.
0 L12 1 L21
,
with L12 6= 0, can be used.) Note now that (recall that ǫ = 1 and note that L−1 = L)
x ˆ˙ = L−1 y =
3 0 1 1
−1 1
Lˆ x + L−1
Lˆ x=
1 −1
0 1
u=
1 1 0 3
x ˆ+
1 0
u
x ˆ.
˜1 = 1 and the system x ˜1 u = x Hence A˜11 = 1, A˜12 = 1, A˜22 = 3, B ˆ˙ 1 = A˜11 x ˆ1 + B ˆ1 + u is reachable. (Note that the system is not stabilizable, and the unreachable mode is s = 3.) 3. The observability matrix is
O=
−1 1 −2 3 − 2ǫ
.
Note that det(O) = 2ǫ − 1. Therefore the system is observable if ǫ 6= 1/2. 4. The observability pencil, for ǫ = 1/2, is 2 6 4
s−3 1/2 −1 s − 3/2 1 −1
3 7 5.
As the system is not observable we know that the observability pencil loses rank, i.e. it has rank equal to one, for some s. To compute such an s consider the submatrix
−1 s − 3/2 1 −1
.
This has rank equal to one for s = 5/2, which is therefore the unobservable mode.
XXII
EXERCISES
y=i
R1
R2
C
L
u
Figure A: The electrical network for Exercise 20.
Exercise 20 Consider the linear electrical network in Figure A. Let u be the driving voltage. 1. Using Kirchhoff ’s laws, or otherwise, express the dynamics of the circuit in the standard statespace form x˙ = Ax + Bu
y = Cx + Du
Take x1 to be the voltage across the capacitor, x2 to be the current through the inductor and the output to be the current supplied by the generator. 2. Derive a condition on the parameters R1 , R2 , C and L under which the pair (A, B) is controllable. 3. Derive a condition on the parameters R1 , R2 , C and L under which the pair (A, C) is observable. 4. Assume R1 R2 C = L and R1 6= R2 . Derive the Kalman canonical form for the system. 5. Assume R1 R2 C = L and R1 6= R2 . Define the controllable subspace and the unobservable subspace. Illustrate these subspaces as lines in R2 . Solution 20 Let x1 denote the voltage across C and x2 the current through L. 1. Kirchhoff’s laws yield u = x1 + R1 C1 x˙ 1
u = R2 x2 + lx˙ 2
XXIII
EXERCISES and
u − x1 . R1
y = i = x2 + As a result,
x˙ =
x˙ 1 x˙ 2
= Ax + Bu =
and y = Cx + Du =
− R11C 0
− R11
0 − RL2
1
"
x+
x+
1 u. R1
2. The reachability matrix is "
R= and
1 R1 C 1 L
1 det(R) = R1 CL
#
− R21C 2 1
2 −R L2
R2 1 − . R1 C L
Hence the system is reachable and controllable if, and only if, R1 R2 C 6= L. 3. The observability matrix is 2
O=4 and
− R11
5
− RL2
1 R21 C
1 det(O) = R1
3
1
1 R2 − L R1 C
Hence the system is observable if, and only if, R1 R2 C 6= L. 4. If R1 R2 C = L then the reachable subspace is
R = span
R2 1
and the unobservable subspace is
kerO = span
R1 1
.
.
# 1 R1 C u 1 L
XXIV
EXERCISES
Note that, as R1 6= R2 these two subspaces are independent. Let
R2 R1 1 1
L=
and note that the matrices of the system in Kalman canonical form are
A˜ = L−1 AL = and
− RL2 0
0 − RL2
C˜ = CL =
˜ = L−1 B = B
1−
R2 R1
0
1 L
0
.
5. The subspaces are indicated in the figure. 1
x2
0.9
kerO
0.8
R
1
0.7 0.6
R2
R1
0.5
x1
0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Exercise 21 Consider the electrical network depicted in Figure B. 1. Using Kirchoff ’s laws, or otherwise, write a state space description of the system. 2. Let R1 = R2 = R and C1 = C2 = C. Compute the controllability and observability matrices and their ranks. 3. Let R1 = R2 = R and C1 = C2 = C. Compute the Kalman canonical form of the system. Solution 21 Let xj be the voltage across the capacitor Cj , considered positive “from left to right”, and ij the current through the capacitor Cj , considered positive “from left to right”. The input current i is positive in the upward direction, and the output voltage is the voltage, positive “upward” between the two open terminals.
XXV
EXERCISES
C1
R1 C3 u=i
y=V R2
C2
Figure B: The electrical network for Exercise 21.
1. Using the above conventions we have i1 = C1 v˙ 1
i2 = C2 v˙ 2
i3 = C3 v˙ 3
v3 − v2 = R2 (i − i1 − i3 )
i1 + i2 + i3 = 0
v1 − v3 = R1 (i − i1 ).
As a result 2 6 4
1 1 1 −1 0 −1 −1 0 0
32 76 54
3
C1 v˙ 1 C2 v˙ 2 C3 v˙ 3
2
7 6 5+4
0 1 1
3 7 5i
2 6
=4
0 0 0 1 0 − R12 R2 1 1 0 − R1 R1
32 76 54
v1 v2 v3
3 7 5
yielding 2 6 4
v˙ 1 v˙ 2 v˙ 3
3
2
7 5
=6 4
6
− C11R1
0 − C21R2
0
1 C3 R2
1 C3 R2
32
1 C1 R1 1 C2 R2
− C31R1 −
Finally
76 74 5 1 C3 R2
2
y=
1 −1 0
6 4
v1 v2 v3
v1 v2 v3
2
3
6 7 5+6 4
− C11 1 C2
3 7 7 i. 5
0
3 7 5.
2. Setting R1 = R2 = R, C1 = C2 = C, RC = 1/α, RC3 = 1/β, and u = i/C yields 2 6 4
v˙ 1 v˙ 2 v˙ 3
3
2
7 5
=4
6
−α 0 α 0 −α α β β −2β
32 76 54
v1 v2 v3
3
2
7 6 5+4
−1 1 0
3 7 5 u.
XXVI
EXERCISES
The reachability matrix is 2 6
R=4
3
−1 α −α2 1 −α α2 0 0 0
7 5.
R has rank one, hence the system is not reachable and not controllable. The observability matrix is 2 3 1 −1 0 6 7 α 0 5. O = 4 −α α2 −α2 0 O has rank one, hence the system is not observable.
3. The reachability subspace is 2 6
R = spanB = span 4
−1 1 0
3 7 5.
The unobservable subspace is 2 6
kerO = span 4
1 0 1 0 0 1
3 7 5.
Note that X1 = R ∩ kerO = ∅
X2 = R
hence x = Lˆ x, with
2 6
L=4
X3 = kerO
−1 1 0 1 1 0 0 0 1
X4 = ∅,
3 7 5.
Finally, Kalman canonical form is given by the equations 2 6 x ˆ˙ = 4
y=
−α 0 0 0 −α α 0 2β −2β −2 0 0
3
2
7 6 ˆ+4 5x
1 0 0
3 7 5u
x ˆ.
Exercise 22 The linearized model of an orbiting satellite about a circular orbit of radius r0 > 0 and angular velocity ω0 6= 0 is described by the equations 2 6 6 4
x˙ = Ax + Bu = 6
0 1 0 3ω02 0 0 0 −2ω0 /r0
0 0 0 2r0 ω02 0 1 0 0
3
2
7 6 7 6 7x + 6 5 4
0 0 1 0 0 0 0 1/r0
3
7 7 7u 5
XXVII
EXERCISES
y = Cx =
1 0 0 0 0 0 1 0
x.
The output components are variations in radius and angle of the orbit and the input components are radial and tangential forces. 1. Show that the system is controllable. 2. Design a state feedback control law
u = Kx + Gv =
k11 k12 k13 k14 k21 k22 k23 k24
x+
g11 0 0 g22
v
such that • the matrix A + BK has all eigenvalues equal to −1 and it is block diagonal, i.e.
A + BK =
F1 0 0 F2
with Fi ∈ IR2×2 ;
• the closedloop system has unity DC gain, i.e.
−1
−C(A + BK)
BG =
1 0 0 1
.
Solution 22 1. Consider the following submatrix of the reachability matrix 2 6
˜ = [B AB] = 6 R 6 4
0 0 1 0 1 0 0 2ω02 0 0 0 1/r0 0 1/r0 −2ω0 /r0 0
3 7 7 7 5
and note that its determinant is −1/r02 6= 0. Hence the system is reachable and controllable. 2. Consider the closedloop system x˙ = (A + BK)x + BGv and note that 2 6 6 4
A + BK = 6
0 1 0 0 k12 k13 2r0 ω02 + k14 3ω02 + k11 0 0 0 1 k21 /r0 −2ω0 /r0 + k22 /r0 k23 /r0 k24 /r0
3 7 7 7. 5
XXVIII
EXERCISES
Hence, selecting k11 = −3ω02 − 1 k21 = 0
k14 = −2r0 ω02
k12 = −2 k13 = 0 k23 = −r0
k22 = 2ω0
yields
2 6 6 4
A + BK = 6
k24 = −2r0
0 1 0 0 −1 −2 0 0 0 0 0 1 0 0 −1 −2
3 7 7 7 5
which shows that the first condition has been achieved. To achieve the second condition note that −1
−C(A + BK)
BG =
g11 0 0 g22 /r0
.
Hence, it suffices to select g11 = 1
g22 = r0 .
Exercise 23 Consider the continuoustime system x˙ = Ax + Bu. Let
A=
1 2 3 4
B=
1 1
.
Find a matrix K such that σ(A + BK) = {−1, −2}. Solve the problem in two ways: 1. using the general theory discussed in the lectures; 2. using a direct computation, i.e. without computing the reachability matrix of the system. Solution 23 1. The general theory states that the state feedback is given by K=−
0 1
R−1 p(A),
where R is the reachability matrix and p(s) is the desired closedloop characteristic polynomial, in this case p(s) = (s + 1)(s + 2) = s2 + 3s + 2. As a result, K=−
0 1
1 4
7 −3 −1 1
yielding
A + BK =
12 16 24 36
−2 −3 0 −1
,
which has eigenvalues equal to −1 and −2, as requested.
=−
3 5
,
XXIX
EXERCISES 2. Let K= and note that
K1 K2
1 + K1 2 + K2 3 + K1 4 + K2
A + BK =
.
The characteristic polynomial of A + BK is det(sI − (A + BK)) = s2 + s(−5 − K1 − K2 ) + (−2K2 + 2K1 − 2), and this should be equal to p(s) = s2 + 3s + 2. As a result, K1 and K2 should be such that −5 − K1 − K2 = 3 − 2K2 + 2K1 − 2 = 2, which yields K1 = −3 and K2 = −5. Note that, because the system has only one input and it is reachable, the state feedback assigning the eigenvalues is unique! Exercise 24 Consider the continuoustime system
x˙ = y =
1 −2 3 −1 3 −1
x+
1 −1
u
x.
1. Show that the system is controllable and observable. 2. Assume zero initial state. Compute the response of the system when u is a unity step applied at t = 0. 3. Design a state feedback control law u = Kx + Gr such that the closedloop system has two eigenvalues at −3. Solution 24 1. The reachability matrix is
R=
1 3 −1 4
,
which is full rank, hence the system is reachable and controllable. The observability matrix is 3 −1 O= , 0 −5 which is full rank, hence the system is observable.
XXX
EXERCISES
2. We have to the forced response of the output of the system. Note that √ compute √ σ(A) = {i 5, −i 5}, hence the forced response of the output of the system has the form √ y(t) = y0 + y1 sin( 5t + φ). Note now that (recall that u = 1, for t ≥ 0, and that x(0) = 0) y(0) =
y(0) ˙ = y¨(0) =
3 −1
0 −5
−15 5
x(0) = 0
x(0) + 4u = 4
x(0) + 5u + 4u˙ = 5.
Therefore, we have to determine y0 , y1 and φ from the equations √ 5y1 cos φ = 4 − 5y1 sin φ = 5. y0 + y1 sin φ = 0 This yields r
y1 =
21 5
√
5 φ = arctan − 4
3. Let K= and note that
A + BK =
K1 K2
r
y0 =
21 . 5
1 + K1 −2 + K2 3 + K1 −1 − K2
.
The characteristic polynomial of A + BK is det(sI − (A + BK)) = s2 + s(−K1 + K2 ) + (−3K1 + 5 − 4K2 ), and this should be equal to p(s) = (s + 3)2 = s2 + 6s + 9. As a result, K1 and K2 should be such that −K1 + K2 = 6
− 3K1 + 5 − 4K2 = 9,
which yields K1 = −4 and K2 = 2. Exercise 25 Consider the continuoustime system
x˙ = y =
3 −1 + ǫ 1 2−ǫ −1 1
x+
0 1
u
x.
1. Show that the system is controllable for any ǫ 6= 1. Study the stabilisability of the system for ǫ = 1.
XXXI
EXERCISES
2. Show that the system is observable for any ǫ 6= 1/2. Study the detectability of the system for ǫ = 1/2. 3. Assume ǫ = 0. Design a state feedback control law u = Kx + Gr such that the closedloop system has two eigenvalues equal to −2. 4. Show that the state feedback control law designed above stabilizes the system for any ǫ ∈ (−4, 1/7). Solution 25 1. The reachability matrix is
0 ǫ−1 1 2−ǫ
R=
,
and det(R) = 1 − ǫ. As a result the system is reachable and controllable for ǫ 6= 1. Let ǫ = 1 and consider the reachability pencil
sI − A B
=
s−3 0 0 −1 s − 1 1
,
which has rank equal to one for s = 3. The system is therefore not stabilizable. Note that it is possible to obtain this conclusion without computing the reachability pencil. In fact, for ǫ = 1 the eigenvalues of A are {3, 1}, hence if there is an unreachable mode this is associated to a value of s with positive real part. 2. The observability matrix is
O=
−1 1 −2 −2ǫ + 3
,
and det(O) = 2ǫ − 1. As a result the system is observable for ǫ 6= 1/2. Let ǫ = 1/2 and consider the observability pencil
sI − A C
2 6
=4
s−3 1/2 −1 s − 3/2 −1 1
3 7 5.
Because the system is not observable, this matrix has to have rank equal to one for some s. To find such an s consider the submatrix
−1 s − 3/2 −1 1
.
Its determinant is s − 5/2, hence the unobservable mode is s = 5/2 and the system is not detectable. Note that it is possible to obtain this conclusion without computing the observability pencil. In fact, for ǫ = 1/2 the eigenvalues of A are {5/2, 2}, hence if there is an unobservable mode this is associated to a value of s with positive real part.
XXXII
EXERCISES
3. If ǫ = 0 we have
3 −1 1 2
A=
.
Setting K = [K1 , K2 ] yields det(sI − (A + BK)) = s2 + s(−5 − K2 ) + (K1 + 3K2 + 7), which should be equal to (s + 2)2 . This is achieved setting K2 = −9.
K1 = 24 4. Consider now the matrix
A + BK =
3 ǫ−1 25 −7 − ǫ
.
Its characteristic polynomial is det(sI − (A + BK)) = s2 + s(4 + ǫ) + (4 − 28ǫ), which has both roots with negative real part (by Routh test) if and only if ǫ ∈ (−4, 1/7). Exercise 26 Consider the continuoustime system x˙ = Ax + Bu, y = Cx, with 2 6
A=4
0 1 0 0 0 1 0 −5 −6
3
2
7 5
B=4
6
0 0 2
3 7 5
C=
1 0 0
.
Design an asymptotic observer having three eigenvalues at −10. Solution 26 An asymptotic observer is described by ξ˙ = Aξ + L(Cξ − y) + Bu = (A + LC)ξ − Ly + Bu for some L = [L1 L2 L3 ]′ , where ξ is the asymptotic estimate of x provided the matrix A + LC has all eigenvalues with negative real part. Note that 2 6
A + LC = 4
L1 1 0 L2 0 1 L3 −5 −6
3 7 5,
and its characteristic polynomial is s3 + s2 (6 − L1 ) + s(5 − 6L1 − L2 ) + (−L3 − 5L1 − 6L2 ). This should be equal to (s + 10)3 = s3 + 30s2 + 300s + 1000. As a result, L1 = −24
L2 = −151
L3 = 26.
XXXIII
EXERCISES Exercise 27 Consider the continuoustime system x˙ = Ax, y = Cx. Let
A=
0 1 0 −α
,
C=
1 0
.
1. Show, using Hautus test, that the system is observable for all α. 2. Design an asymptotic observer for the system. Select the output injection gain L such that the matrix A − LC has two eigenvalues equal to 3. 3. Suppose that one can measure y(t) and a delayed copy of y(t) given by y(t − τ ), with τ > 0. Assume (for simplicity) that α 6= 0. For t ≥ τ , express the vector
Y (t) =
y(t) y(t − τ )
from x(0). Show that the relation determined above can be used, for any τ > 0, to compute x(0) as a function of Y (t), where t ≥ τ . Argue that the above result can be used to determine x(t) from Y (t), for t ≥ τ , exactly. Solution 27 1. The observability pencil is
2 6 4
s −1 0 s+α 1 0
3 7 5,
which has rank two for any s and any α. Hence the system is observable. 2. An asymptotic observer is described by ξ˙ = Aξ + L(Cξ − y) = (A + LC)ξ − Ly for some L = [L1 L2 ]′ , where ξ is the asymptotic estimate of x provided the matrix A + LC has all eigenvalues with negative real part. Note that
A + LC =
L1 1 L2 −α
and its characteristic polynomial is s2 + s(α − L1 ) − αL1 − L2 . This should be equal to (s + 3)2 = s2 + 6s + 9, yielding L1 = α − 6
L2 = −9 + (6 − α)α.
XXXIV
EXERCISES
3. Note that y(t) = CeAt x(0) and replacing t with t − τ one has y(t − τ ) = CeA(t−τ ) x(0). Then, for t ≥ τ ,
Y (t) =
y(t) y(t − τ )
=
C Ce−Aτ
eAt x(0).
For the given A and C (using α 6= 0) we have
C Ce−Aτ
=
1 0 ατ 1 − e α−1
which is invertible for all α 6= 0 and all τ > 0. Hence
1 0 eατ −1 1 − α
At
x(t) = e x(0) =
−1
Y (t).
The above relation implies that, for all t ≥ τ , it is possible to obtain exactly x(t). Exercise 28 Consider the discretetime system
xk+1 = yk
=
1 −2 3 −1 3 −1
xk +
1 −1
uk ,
xk .
1. Show that the system is observable. 2. Design an asymptotic observer, with state x ˆk , such that ek = xk − x ˆk = 0 for all k ≥ N . Determine the smallest value of N for which the above condition can be satisfied. 3. Let uk = K x ˆk + vk with K = [3/4, 3/4]. Write the equations of the closedloop system, with state [xk , x ˆk ], input vk and output yk , and determine the eigenvalues of this system. Solution 28 1. The observability matrix is
O=
3 −1 0 −5
,
which has rank equal to two. The system is therefore observable.
XXXV
EXERCISES 2. An asymptotic observer is described by ξk+1 = Aξk + L(Cξk − yk ) + Buk = (A + LC)ξk − Lyk + Buk
for some L = [L1 L2 ]′ , where ξk is the asymptotic estimate of x provided the matrix A + LC has all eigenvalues with negative real part. To obtain a deadbeat observer L should be such that both eigenvalues of A + LC are zero. Note that
A + LC =
1 + 3L1 −2 − L1 3 + 3L2 −1 − L2
,
and det(sI − (A + LC)) = s2 + s(L2 − 3L1 ) + 5(1 + L2 ). Hence, L1 = −1/3
L2 = −1.
With this selection of L we have (A + LC)2 = 0, hence N = 2. To prove that the smallest N for which the considered condition holds is N = 2 it is enough to observe that there is no selection of L such that (A + LC)1 = 0. 3. By the separation principle the eigenvalues of the closedloop system are the eigenvalues of the observer and of the matrix
A + BK =
7/4 −5/4 9/4 −7/4
.
Hence, the eigenvalues of the closedloop system are {1/2, −1/2, 0, 0}. Exercise 29 Consider the simplified model of a ship described by the equation M θ¨ + dθ˙ + cα = w α˙ + α = u where θ denotes the heading angle error (the angle between the ship’s heading and the desired heading), α denotes the rudder angle, w denotes a disturbance due to wind, and u is the control input. M and c are positive parameters, and d is a nonnegative parameter. ˙ α), input (w, u) and output θ in 1. Write the equation of the system, with state (θ, θ, standard state space form. 2. Let w = 0. Show that the system is controllable. 3. Show that the system is observable. 4. Let w = 0, M = 1, c = 1 and d = 0. Design an output feedback controller applying the separation principle. In particular, select the state feedback gain K such that the matrix A − BK has three eigenvalues equal to −1 and the output injection gain L such that the matrix A − LC has three eigenvalues equal to −3.
XXXVI
EXERCISES
Solution 29 ˙ α)′ ) 1. The description of the system in standard state space form is (set x = (θ, θ, 2
x˙ =
6 4
0 1 0 −d/M 0 0
y=
0 −c/M −1
1 0 0
3
2
7 6 5x + 4
0 1/M 0
3
0 0 1
7 5
w u
x.
2. The reachability matrix is 2 6
R=4
0 0 0 −c/M 1 −1
−c/M c/M (d/M + 1) 1
3 7 5
and this has full rank for all positive c and M . The system is reachable and controllable. 3. The observability matrix is 2 6
O=4
1 0 0 1 0 −d/M
0 0 −c/M
3 7 5
and this has full rank for all positive c and M . The system is observable. 4. Let K = [K1 K2 K3 ] and note that 2 6
A + BK = 4
0 1 0 0 0 −1 K1 K2 −1 + K3
3 7 5,
and that the characteristic polynomial of this matrix is s3 +(1−K3 )s2 +(K2 )s+(K1 ). Hence the selection K1 = 1 K2 = 3 K3 = −2
is such that all eigenvalues of A + BK are equal to −1. Let L = [L1 L2 L3 ]′ and note that 2 3 L1 1 0 6 7 A + LC = 4 L2 0 −1 5 , L3 0 −1
and that the characteristic polynomial of this matrix is s3 + (1 − L1 )s2 + (−L1 − L2 )s + (−L2 + L3 ). Hence the selection L1 = −8
L2 = −19
L3 = 8
is such that all eigenvalues of A + LC are equal to −3. Finally, the controller is ξ˙ = (A + BK + LC)ξ − Ly, u = Kξ.
XXXVII
EXERCISES Exercise 30 Consider the continuoustime system
x˙ = y =
1 −2 3 −1 3 −1
x+
1 −1
u,
x
1. Design an asymptotic observer with a double pole at −6. 2. Suppose x0 is the observer state evaluated in part 1. Let u = Kx0 + Gr with K = [−4, 2]. Compute the eigenvalues of the closedloop system. Solution 30 1. An asymptotic observer is described by ξ˙ = Aξ + L(Cξ − y) + Bu = (A + LC)ξ − Ly + Bu for some L = [L1 L2 ]′ , where ξ is the asymptotic estimate of x provided the matrix A + LC has all eigenvalues with negative real part. Note that
A + LC =
1 + 3L1 −2 − L1 3 + 3L2 −1 − L2
,
and its characteristic polynomial is s2 + s(L2 − 3L1 ) + 5(1 + L2 ). This should be equal to (s + 6)2 = s2 + 12s + 36. As a result, L1 = −29/15
L2 = 31/5.
2. By the separation principle the eigenvalues of the closedloop system are the eigenvalues of the observer and of the matrix
A + BK =
−3 0 7 −3
.
Hence, the eigenvalues of the closedloop system are {−3, −3, −6, −6}.
XXXVIII
EXERCISES