c02. 2 Linear Dynamic Systems
What we experience of nature is in models, and all of nature’s models are so beautiful. 1
Buckminster Fuller ( 1895±1983 )
2.1 CHAPTER FOCUS
Models for Dynamic Systems. Since their introduction by Isaac Newton in the seventeenth century, differential equations have provided concise mathematical models for many dynamic systems of importance to humans. By this device, Newton was able to model the motions of the planets in our solars y stem with a small number of variables and parameters. Given a finite number of initial conditions (the initial positions and velocities of the sun and planets will do ) and these equations, one can uniquely determine the positions and velocities of the planets for all time. The finite-dimensional re presentation of a problem ( in this example, the problem of predicting the future course of the planets ) is the basis for the so-called state-space approach to the representation of differential equations and their solutions, which is the focus of this chapter. The de pendent variables of the differential equations become state variables of the dynamic system. They explicitly represent all the important characteristics of the dynamic system at any time.
The whole of dynamic system theory is a subject of considerably more scope than one needs for the present undertaking ( Kalman filtering). This chapter will stick to just those concepts that are essential for that purpose, which is the development of the state-space representation for dynamics y stems described by systems of linear differential equations. These are given a somewhat heuristic treatment, without the mathematical rigor often accorded the subject, omitting the development and use of the transform methods of functional analysis for solving differential equations when they serve no purpose in the derivation of the Kalman filter. The interested reader will find a more formal and thorough presentation in most upper-level and graduate-level textbooks on ordinary differential equations. The objective of the more engineering-oriented treatments of dynamic systems is usually to solve the controls problem, which is the problem of defining the in puts ( i.e., control settings ) that will bring the state of the dynamics y stem to a desirable condition. That is not the obj ective here, however.
2.1.1 Main Points to Be Covered
The objective in this chapter is to characterize the measurable outputs of dynamics y stems as functions of the internal states and inputs of the system. ( The italicized terms will be defined more precisel y further along.) The treatment here is deterministic, in order to define functional relationships between inputs and outputs. In the next chapter, the inputs are allowed to be nondeterministic ( i.e., random ) , and the objective of the following chapter will be to estimate the states of the dynamic system in this context.
Dynamic Systems and Differential Equations. In the context of Kalman ®lterin g , a d y namic s y stem has come to be s y non y mous with a s y stem of ordinar y differential e q uations describin g the evolution over time of the state of a p h y sical s y stem. This mathematical model is used to derive its solution, which s p eci®es the functional de p endence of the state variables on their initial values and the s y stem in p uts. This solution de®nes the functional de p endence of the measurable out p uts on the in p uts and the coef®cients of the model.
Mathematical Models for Continuous and Discrete Time. The p rinci p al d y namic s y stem models are summarized in Table 2.1. 2 For im p lementation in di g ital com p uters, the p roblem re p resentation is transformed from an analo g model ( functions of continuous time ) to a di g ital model ( functions de®ned at discrete times ) .
Observabilit y characterizes the f easibilit y of uni q uel y determinin g the state of a g iven d y namic s y stem if its out p uts are known. This characteristic of a d y namic s y stem is determinable from the p arameters of its mathematical model.
2.2 DYNAMIC SYSTEMS
2.2.1 Dynamic Systems Represented by Differential Equations
A s y stem is an assembla g e of interrelated entities that can be considered as a whole. If the attributes of interest of a s y stem are chan g in g with time, then it is called a d y namic s y stem. A p rocess is the evolution over time of a d y namic s y stem.
Our solar s y stem, consistin g of the sun and its p lanets, is a p h y sical exam p le of a d y namic s y stem. The motions of these bodies are g overned b y laws of motion that de p end onl y u p on their current relative p ositions and velocities. Sir Isaac Newton ( 1642±1727 ) discoveredthese laws and ex p ressed them as a s y stemof differential e q uationsÐanother of his discoveries. From the time of Newton, en g ineers and scientists have learned to de®ne d y namic s y stems in terms of the differential e q uations that g overn their behavior. The y have also learned how to solve man y of these differential e q uations to obtain formulas for p redictin g the future behavior of d y namic s y stems.
2
These include nonlinear models, which are discussed in Cha p ter 5. The p rimar y interest in this cha p ter will be in linear models.
EXAMPLE 2.1 ( below, left ) : Newton’s Model for a D y namic S y stem of n Massive Bodies For a p lanetar y s y stem with n bodies ( idealized as p oint masses ) , the acceleration of the ith bod y in an y inertial ( i.e., non-rotatin g and non-acceleratin g) Cartesian coordinate s y stem is g iven b y Newton’s third law as the second-order differential e q uation
d 2 r i P n m j r j À r3 i C g j ; 1 i n; dt 2 1 jr j À r i j j 6 i
where r j is the p osition coordinate vector of the j th bod y , m j is the mass of the j th bod y , and C g is the g ravitational constant. This set of n differential e q uations, p lus the associated initial conditions of the bodies ( i.e., their initial p ositions and velocities ) theoreticall y determines the future histor y of the p lanetar y s y stem.
m2
m1
m3
r2
r1
r3
m4
0
r4
Exam p le 2.1
EXAMPLE 2.2 ( above, ri g ht ) : The Harmonic Resonator with Linear Dam p in g Consider the accom p an y in g dia g ram of an idealized a pp aratus with a mass m attached throu g h a s p rin g to an immovable base and its frictional contact to its su pp ort base re p resented b y a dash p ot. Let d be the dis p lacement of the mass from its p osition at rest, dd=dt be the velocit y of the mass, and a t d 2 d=dt 2 its acceleration. The force F actin g on the mass can be re p resented b y Newton’s second law as
Exam p le 2.2
F t ma t 2 d d m 2 t dt Àk s d t À kd
dd
t; dt
where k s is the s p rin g constant and k d is the dra g coef®cient of the dash p ot. This relationshi p can be written as a differential e q uation
d d2 m Àk s d À kd dt2
dd
dt
in which time ( t ) is the differential variable and dis p lacement ( d ) is the de p endent variable. This e q uation constrains the d y namical behavior of the dam p ed harmonic resonator. The order of a differential e q uation is the order of the hi g hest derivative, which is 2 in this exam p le. This one is called a linear differential e q uation, because both sides of the e q uation are linear combinations of d and its derivatives. ( That of Exam p le 2.1 is a nonlinear differential e q uation. )
Not All Dynamic Systems Can Be Modeled by Differential Equations. There are other t yp es of d y namic s y stems, such as those modeled b y Petri nets or inference nets. However, the onl y t yp es of d y namic s y stems considered in this book will be modeled b y differential e q uations or b y discrete-time linear state d y namic e q uations derived from linear differential or difference e q uations.
2.2.2 State Variables and State Equations
The second-order differential e q uation of the p revious exam p le can be transformed to a s y stem of two ®rst-order differential e q uations in the two de p endent variables x 1 d and x 2 dd=dt. In this wa y , one can reduce the form of an y s y stem of hi g her order differential e q uations to an e q uivalent s y stem of ®rst-order differential e q uations. These s y stems are g enerall y classi®ed into the t yp es shown in Table 2.1, with the most g eneral t yp e bein g a time-var y in g differential e q uation for re p resentin g a d y namic s y stem with time-var y in g d y namic characteristics. This is re p resented in vector form as
_ x t f t; x t; u t;
2:1
where Newton’s ``dot’’ notation is used as a shorthand for the derivative with res p ect to time, and a vector-valued function f to re p resent a s y stem of n e q uations
x _ 1 f 1 t; x 1 ; x 2 ; x 3 ; … ; x n ; u 1 ; u 2 ; u 3 ; … ; u r ; t;
x _ 2 f 2 t; x 1 ; x 2 ; x 3 ; … ; x n ; u 1 ; u 2 ; u 3 ; … ; u r ; t;
x _ 3 f 3 t; x 1 ; x 2 ; x 3 ; … ; x n ; u 1 ; u 2 ; u 3 ; … ; u r ; t;
…
2:2
x _ n f n t; x 1 ; x 2 ; x 3 ; … ; x n ; u 1 ; u 2 ; u 3 ; … ; u r ; t
in the inde p endent variable t ( time ) , n de p endent variables fx i j1 i ng, and r known in p uts fu i j1 i rg. These are called the state e q uations of the d y namic s y stem.
State Variables Represent the Degrees of Freedom of Dynamic Systems. The variables x 1 ; … ; x n are called the state variables of the d y namic s y stem de®ned b y E q uation 2.2. The y are collected into a sin g le n-vector
x t x 1 t
x 2 t
x 3 t
ÁÁÁ
x n tT
2:3
called the state vector of the d y namic s y stem. The n-dimensional domain of the state vector is called the state s p ace of the d y namic s y stem. Sub j ect to certain continuit y conditions on the functions f i and u i ; the values x i t 0 at some initial time t 0 will uni q uel y determine the values of the solutions x i t on some closed time interval t 2 t 0 ; t f with initial time t 0 and ®nal time t f [ 57 ] . In that sense, the initial value of each state variable re p resents an inde p endent de g ree of freedom of the d y namic s y stem. The n values x 1 t 0 ; x 2 t 0 ; x 3 t 0 ; … ; x n t 0 can be varied inde p endentl y , and the y uni q uel y determine the state of the d y namic s y stem over the time interval
t 2 t 0 ; t f .
EXAMPLE 2.3: State S p ace Model of the Harmonic Resonator For the second-order differential e q uation introduced in Exam p le 2.2, let the state variables _ x 1 d and x 2 d. The ®rst state variable re p resents the dis p lacement of the mass from static e q uilibrium, and the second state variable re p resents the instantaneous velocit y of the mass. The s y stem of ®rst-order differential e q uations for this d y namic s y stem can be ex p ressed in matrix form as d x 1 t x 1 2 t F c ; dt x 2 t x t ” # 0 1 k k F c ; À s À d m m
where F c is called the coe f® cient matrix of the s y stem of ®rst-order linear differential e q uations. This is an exam p le of what is called the com p anion f orm for hi g her order linear differential e q uations ex p ressed as a s y stem of ®rst-order differential e q uations.
2.2.3 Continuous Time and Discrete Time
The d y namic s y stem de®ned b y E q uation 2.2 is an exam p le of a continuous s y stem, so called because it is de®ned with res p ect to an inde p endent variable t that varies continuousl y over some real interval t 2 t 0 ; t f . For man y p ractical p roblems, however, one is onl y interested in knowin g the state of a s y stem at a discrete set of times t 2 ft 1 ; t 2 ; t 3 ; …g. These discrete times ma y , for exam p le, corres p ond to the times at which the out p uts of a s y stem are sam p led ( such as the times at which Piazzi recorded the direction to Ceres ) . For p roblems of this t yp e, it is convenient to order the times t k accordin g to their inte g er subscri p ts:
t 0 < t 1 < t 2 < Á Á Á t kÀ1 < t k < t k1 < Á Á Á :
That is, the time se q uence is ordered accordin g to the subscri p ts, and the subscri p ts take on all successive values in some ran g e of inte g ers. For p roblems of this t yp e, it suf®ces to de®ne the state of the d y namic s y stem as a recursive relation,
x t k1 f x t k ; t k ; t k1 ;
2:4
b y means of which the state is re p resented as a function of its p revious state. This is a de®nition of a discrete d y namic s y stem. For s y stems with uni f orm time intervals Dt
t k kDt:
Shorthand Notation for Discrete-Time Systems. It uses u p a lot of ink if one writes x t k when all one cares about is the se q uence of values of the state variable x. It is more ef®cient to shorten this to x k , so lon g as it is understood that it stands for x t k , and not the kth com p onent of x. If one must talk about a p articular com p onent at a p articular time, one can alwa y s resort to writin g x i t k to remove an y ambi g uit y . Otherwise, let us dro p t as a s y mbol whenever it is clear from the context that we are talkin g about discrete-time s y stems.
2.2.4 Time-Varying Systems and Time-Invariant Systems The term `` p h y sical p lant’’ or `` p lant’’ is sometimes used in p lace of ``d y namic s y stem,’’ es p eciall y for a pp lications in manufacturin g . In man y such a pp lications, the d y namic s y stem under consideration is literall y a p h y sical p lantÐa ®xed facilit y used in the manufacture of materials. Althou g h the in p ut u t ma y be a function of time, the functional de p endence of the state d y namics on u and x does not de p end u p on time. Such s y stems are called time invariant or autonomous. Their solutions are g enerall y easier to obtain than those of time-var y in g s y stems.
2.3 CONTINUOUS LINEAR SYSTEMS AND THEIR SOLUTIONS
2.3.1 Input±Output Models of Linear Dynamic Systems
The block dia g ram in Fi g ure 2.1 re p resents a linear continuous s y stem with three t yp es of variables:
In p uts, which are under our control, and therefore known to us, or at least measurable b y us. ( In the
next cha p ter, however, the y will be assumed to be known onl y statisticall y . That is, individual sam p les of u are random but with known statistical p ro p erties. )
State variables, which were described in the p revious section. In most a pp lications, these are ``hidden
variables,’’ in the sense that the y cannot g enerall y be measured directl y but must be somehow inferred from what can be measured.
Out p uts, which are those thin g s that can be known throu g h measurements.
These conce p ts are discussed in g reater detail in the followin g subsections.
2.3.2 Dynamic Coef®cient Matrices and Input Coupling Matrices
The d y namics of linear s y stems are re p resented b y a set of n ®rst-order linear differential e q uations ex p ressible in vector form as
d _ x t
x t dt F tx t C tu t;
2:5
where the elements and com p onents of the matrices and vectors can be functions of time:
2
6
6
6
6
6
6
6
4
2
6
6
6
6
6
6
6 4
3
F t
C t
f 11 t f 21 t f 31 t
. .
.
f n1 t c 11 t c 21 t c 31 t
…
c n1 t
f 12 t f 22 t f 32 t …
f n2 t c 12 t c 22 t c 32 t
. .
.
c n2 t
f 13 t f 23 t f 33 t …
f n3 t c 13 t c 23 t c 33 t
ÁÁÁ ÁÁÁ ÁÁÁ . .
.
ÁÁÁ
f 1n t f 2n t
f 3n t
…
f nn t
7
7
7
7
7
;
7
7
5
3
. .
.
c n3 t
ÁÁÁ ÁÁÁ ÁÁÁ . .
.
ÁÁÁ
c 1r t c 2r t c 3r t
…
c nr t
7
7
7
7
7
;
7
7 5
u t u 1 t
u 2 t
u 3 t
ÁÁÁ
u r t T :
The matrix F t is called the d y namic coe f® cient matrix, or sim p l y the d y namic matrix. Its elements are called the d y namic coe f® cients. The matrix C t is called the in p ut cou p lin g matrix, and its elements are called in p ut cou p lin g coe f® cients. The r-vector u is called the in p ut vector.
EXAMPLE 2.4: D y namic E q uation for a Heatin g /Coolin g S y stem Consider the tem p erature T in a heated enclosed room or buildin g as the state variable of a d y namic s y stem. A sim p li®ed p lant model for this d y namic s y stem is the linear e q uation
_ T t Àkc
T t À T o t k h u t;
where the constant ``coolin g coef®cient’’ k c de p ends on the q ualit y of thermal insulation from the outside, T o is the tem p erature outside, k h is the heatin g =coolin g rate coef®cient of the heater or cooler, and u is an in p ut function that is either u 0 ( off ) or u 1 ( on ) and can be de®ned as a function of an y measurable q uantities. The outside tem p erature T o , on the other hand, is an exam p le of an in p ut function which ma y be directl y measurable at an y time but is not p redictable in the future. It is effectivel y a random p rocess.
2.3.3 Companion Form for Higher Order Derivatives
In g eneral, the nth-order linear differential e q uation
d n y t
dt
n
t dnÀ1 f 1 t y dtnÀ1
d t Á Á Á f nÀ1 t y dt fn
t y t u t
2:6
can be rewritten as a s y stem of n ®rst-order differential e q uations. Althou g h the state variable re p resentation as a ®rst-order s y stem is not uni q ue [ 56 ] , there is a uni q ue wa y of re p resentin g it called the com p anion f orm.
Companion Form of the State Vector. For the nth-order linear d y namic s y stem shown above, the com p anion form of the state vector is
d 2
dt2
dnÀ1
dtnÀ1
T
d
x t
y t;
t; dt y
y
t;
…;
y
t
:
2:7
Companion Form of the Differential Equation. The nth-order linear differential e q uation can be rewritten in terms of the above state vector x t as the vector differential e q uation
2
6 d 6 6 6 dt 6 4
3
7
7
7
7
7
5
2
6
6
6
6
4
3
7
7
7
7
5
2
6
6
6
6
6
4
3
7
7
7
7
7
5
x 1 t x 2 t …
x nÀ1 t x n t
ÁÁÁ ÁÁÁ …
ÁÁÁ ÁÁÁ
x 1 t x 2 t x 3 t …
x n t
2
0 0 …
0 À f n t
1 0 …
0 À f nÀ1 t
0 1 …
0 À f nÀ2 t
0 0 …
1 À f 1 t
3 0 6 0 7 6 7 6 … 7 6 7 4 0 5
1
u t:
2:8
When E q uation 2.8 is com p ared with E q uation 2.5, the matrices F t and C t are easil y identi®ed.
The Companion Form is Ill-conditioned. Althou g h it sim p li®es the relationshi p between hi g her order linear differential e q uations and ®rst-order s y stems of differential e q uations, the com p anion matrix is not recommended for im p lementation. Studies b y Kenne y and Lie p nik [ 185 ] have shown that it is p oorl y conditioned for solvin g differential e q uations.
2.3.4 Outputs and Measurement Sensitivity Matrices
Measurable Outputs and Measurement Sensitivities. Onl y the in p uts and out p uts of the s y stem can be measured, and it is usual p ractice to consider the variables z i as the measured values. For linear p roblems, the y are related to the state variables and the in p uts b y a s y stem of linear e q uations that can be re p resented in vector form as
z t H tx t D tu t;
2:9
where
z t z 1 t
z 2 t
z 3 t
ÁÁÁ
z ` t T ;
2
6
6 6
6
6
6
6
4
3
h 11 t h 21 t
h 31 t …
h `1 t
d 11 t d 21 t
d 31 t …
d `1 t
H t
D t
h 12 t h 22 t h 32 t …
h `2 t
d 12 t d 22 t d 32 t …
d `2 t
h 13 t h 23 t h 33 t …
h `3 t
d 13 t d 23 t d 33 t …
d `3 t
ÁÁÁ ÁÁÁ ÁÁÁ . .
.
ÁÁÁ
h 1n t h 2n t h 3n t
…
h `n t
d 1r t d 2r t
d 3r t …
d `r t
7
7 7
7
7
;
7
7
5
2
6
6
6
6
6
6
6
4
3
ÁÁÁ ÁÁÁ ÁÁÁ . .
.
ÁÁÁ
7
7
7
7
7
7
7
5
:
The `-vector z t is called the measurement vector, or the out p ut vector of the s y stem. The coef®cient h i j t re p resents the sensitivit y ( measurement sensor scale factor ) of the ith measured out p ut to the j th internal state. The matrix H t of these values is called the measurement sensitivit y matrix, and D t is called the in p ut± out p ut cou p lin g matrix. The measurement sensitivities h i j t and in p ut=out p ut
cou p lin g coef®cients d i j t; 1 i `; 1 j r, are known functions of time. The state e q uation 2.5 and the out p ut e q uation 2.9 to g ether form the d y namic e q uations of the s y stem shown in Fi g ure 2.1.
2.3.5 Difference Equations and State Transition Matrices (STMs)
Di ff erence e q uations are the discrete-time versions of differential e q uations. The y are usuall y written in terms of f orward di ff erences x t k1 À x t k of the state variable ( the de p endent variable ) , ex p ressed as a function c of all inde p endent variables or of the forward value x t k1 as a function f of all inde p endent variables ( includin g the p revious value as an inde p endent variable ) :
x t k1 À x t k c t k ; x t k ; u t k ;
or
x t k1 f t k ; x t k ; u t k ;
f t k ; x t k ; u t k x t k c t k ; x t k ; u t k :
2:10
The second of these ( E q uation 2.10 ) has the same g eneral form of the recursive relation shown in E q uation 2.4, which is the one that is usuall y im p lemented for discrete-time s y stems.
For linear d y namic s y stems, the functional de p endence of x t k1 on x t k and u t k can be re p resented b y matrices:
x t k1 À x t k C t k x t k C t k u t k ; F k I C t k ; where the matrices C and F re p lace the functions c and f, res p ectivel y . The matrix F is called the state transition matrix ( STM ) . The matrix c is called the discrete-time in p ut cou p lin g matrix, or sim p l y the in p ut cou p lin g matrixÐif the discrete-time context is alread y established.
x k1 F k x k C k u k ;
2:11
2.3.6 Solving Differential Equations for STMs
A state transition matrix is a solution of what is called the ``homo g eneous’’ 3 matrix e q uation associated with a g iven linear d y namic s y stem. Let us de®ne ®rst what homo g eneous e q uations are, and then show how their solutions are related to the solutions of a g iven linear d y namic s y stem.
_ Homogeneous Systems. The e q uation x t F tx t is called the homo g e_ neous p art of the linear differential e q uation x t F tx t C tu t. The solution of the homo g eneous p art can be obtained more easil y than that of the full e q uation, and its solution is used to de®ne the solution to the g eneral ( nonhomo g eneous ) linear e q uation.
3
This terminolo gy comes from the notion that ever y term in the ex p ression so labeled contains the de p endent variable. That is, the ex p ression is homo g eneous with res p ect to the de p endent variable.
Fundamental Solutions of Homogeneous Equations. An n  n matrixvalued function F t is called a f undamental solution of the homo g eneous e q uation _ _ x t F tx t on the interval t 2 0; T if F t F tF t and F 0 I n , the n  n identit y matrix. Note that, for an y p ossible initial vector x 0, the vector x t F tx 0 satis®es the e q uation
d _ F tx 0 x t dt d • x 0 F t dt
F
tF tx 0 F tF tx 0 F tx t:
2:12 2:13 2:14 2:15 2:16
_ That is, x t F tx 0 is the solution of the homo g eneous e q uation x Fx with initial value x 0.
EXAMPLE 2.5
The unit u pp er trian g ular Toe p litz matrix
2
6
6
6
6
6
6
6
6
6 6
6
6
6
6
4
1
1
1
3
F t
ÁÁÁ t n À 1! 1 t 0 1 t 2 ÁÁÁ n À 2! 1 t 0 0 1 t ÁÁÁ n À 3! 1 t 0 0 0 1 ÁÁÁ n À 4! … … … . . .
2
3
1
nÀ1
t
t 2
t 1Á2Á 3 1 t 2
7 7 7 7 7 7 nÀ3 7
nÀ2
7 7 7 nÀ4 7
7
7
7
… …
5
0 0 0 0 ÁÁÁ 1 _ is the fundamental solution of x Fx for the strictl y u pp er trian g ular Toe p litz d y namic coef®cient matrix
2
0 6 0 6 6 … 6 4 0
F
1 0 …
0 1 …
3 ÁÁÁ 0 ÁÁÁ 0 7 7 … … 7
7 ; 0 0 ÁÁÁ 1 5 0 0 0 ÁÁÁ 0 _ which can be veri®ed b y showin g that F 0 I and F FF. This d y namic coef®cient matrix, in turn, is the com p anion matrix for the nth-order linear homo g eneous differential e q uation
d=dt n y
t 0.
Existence and Nonsingularity of Fundamental Solutions. If the elements of the matrix F t are continuous functions on some interval 0 t T, then the fundamental solution matrix F t is g uaranteed to exist and to be nonsin g ular on an interval 0 t 4 t for some t > 0. These conditions also g uarantee that F t will be nonsin g ular on some interval of nonzero len g th, as a conse q uence of the continuous de p endence of the solution F t of the matrix e q uation on its ( nonsin g ular ) initial conditions [ F 0 I ] [ 57 ] .
State Transition Matrices. Note that the fundamental solution matrix F t transforms an y initial state x 0 of the d y namic s y stem to the corres p ondin g state x t at time t. If F t is nonsin g ular, then the p roducts F À1 tx t x 0 and F tF À1 tx t x t. That is, the matrix p roduct
F t; t F tF À1 t
2:17
transforms a solution from time t to the corres p ondin g solution at time t, as dia g rammed in Fi g ure 2.2. Such a matrix is called the state transition matrix 4 for the associated linear homo g eneous differential e q uation. The state transition matrix F t; t re p resents the transition to the state at time t from the state at time t.
Properties of STMs and Fundamental Solution Matrices. The same s y mbol ( F ) has been used for f undamental solution matrices and for state transition matrices, the distinction bein g made b y the number of ar g uments. B y convention, then,
F t; 0 F t:
Other useful p ro p erties of F include the followin g :
1. F t; t F 0 I,
2. F À1 t; t F t; t,
3. F t; sF s; t F t; t,
4. @=@tF t; t F tF t; t,
Formall y , an o p erator F t; t 0 ; x t 0 such that x t F t; t 0 ; x t 0 is called an evolution o p erator for a d y namic s y stem with state x. A state transition matrix is a linear evolution o p erator.
and
5. @=@tF t; t ÀF t; tF t.
EXAMPLE 2.6: Fundamental Solution Matrix for the Underdam p ed Harmonic Resonator The g eneral solution o f the di ff erential e q uation. In Exam p les 2.2 and 2.3, the dis p lacement d of the dam p ed harmonic resonator was modeled b y the state e q uation
“
#
d
x
_ x Fx;
_ d
;
2
4
F
0
k À s
m
1
3 5 :
kd À m
The characteristic values of the d y namic coef®cient matrix F are the roots of its characteristic p ol y nomial
k k s l ; det lI À F l 2 d m m
which is a q uadratic p ol y nomial with roots
r !
1 l1 2
1 l2 2
k d
À m
k d
À
À m
k 2 d 4k À s ; m 2 m r !
k 2 d 4k À s : m 2 m
The g eneral solution for the dis p lacement d can then be written in the form
d t ae l 1 t be l 2 t ;
where a and b are (p ossibl y com p lex ) free variables.
The underdam p ed solution. The resonator is considered underdam p ed if the
discriminant
4k k 2 d À s < 0; m 2 m
in which case the roots are a con j u g ate p air of nonreal com p lex numbers and the g eneral solution can be rewritten in ``real form’’ as
d t ae Àt=t cos ot be Àt=t sin ot;
2m t ;
k d
o
r k s k 2 d À m 4m 2
;
where a and b are now real variables, t is the deca y time constant, and o is the resonator resonant fre q uenc y . This solution can be ex p ressed in state-s p ace form in terms of the real variables a and b:
2
4
3
5
cos ot
cos ot À À o sin ot t
sin ot
sin ot o cos ot À t
“
#
d t
_ d t
eÀt=t
a
:
b
Initial value constraints. The initial values
d 0 a;
_ a d 0 À
ob t
can be solved for a and b as
a
b
2 1 • 4 1
0
1
o
3 5 d 0 _ d 0
:
ot
This can then be combined with the solution for x t in terms of a and b to y ield the fundamental solution
x t
F
tx 0;
“
#
eÀt=t F t ot2
tot cos ot sin ot
À 1 o 2 t sin ot
t 2 sin ot
Àot cos ot sin ot
in terms of the dam p in g time constant and the resonant fre q uenc y .
2.3.7 Solution of Nonhomogeneous Equations
The solution of the nonhomo g eneous state e q uation 2.5 is g iven b y
t
x t F t; t 0 x t 0 F t; tC tu t dt
t 0 t F tF À1 t 0 x t 0 F t F À1 tC tu t dt;
2:18 2:19
t 0
where x t 0 • is the initial value and F t; t 0 is the state transition matrix of the d y namic s y stem de®ned b y F t. ( This can be veri®ed b y takin g derivatives and usin g the p ro p erties of STMs g iven above. )
2.3.8 Closed-Form Solutions of Time-Invariant Systems
In this case, the coef®cient matrix F is a constant function of time. The solution will still be a function of time, but the associated state transition matrices F t; t will onl y de p end on the differences t À t. In fact, one can show that
F t; t e F tÀt
1 t À ti P Fi ; i0 i!
2:20 2:21
where F 0 I, b y de®nition. The solution of the nonhomo g eneous e q uation in this case will be
t
x t e F tÀt x t
e F tÀs Cu s ds t
t e F tÀt x t eFt
e ÀFs Cu s ds:
2:22 2:23
t
The followin g methods have been used for com p utin g matrix ex p onentials:
1. The a pp roximation of e Ft b y a truncated p ower series ex p ansion is not a recommended g eneral- p ur p ose method, but it is useful if the characteristic values of Ft are well inside the unit circle in the com p lex p lane.
2. F t e Ft l À1 sI À F À1 ; t ! 0, where I is an n  n identit y matrix, lÀ1 is the inverse La p lacian o p erator, and s is the La p lace transform variable.
3. The ``scalin g and s q uarin g ‘’ method combined with a Pade  a pp roximation is the recommended g eneral- p ur p ose method. This method is discussed in g reater detail in Section 2.6.
Numerical inte g ration of the homo g eneous p art of the differential e q uation,
d
F t FF t; dt
2:24
with initial value F 0 I. ( This method also works for time-var y in g s y stems. )
There are man y other methods, 5 but these are the most im p ortant.
EXAMPLE 2.7: Solution of the Dam p ed Harmonic Resonator Problem with Constant Drivin g Function Consider a g ain the dam p ed resonator model of Exam p les 2.2, 2.3, and 2.6. The model can be written in the form of a secondorder differential e q uation
d
t 2zw n d t wn 2
_
d t u t;
where
d d2 d t dt2
r k s
_ dd d t ; dt
;
k d z p ; 2 mk s
on
:
m
The p arameter z is a unitless dam p in g coef®cient and w n the ``natural’’ ( i.e., undam p ed ) fre q uenc y of the resonator. This second-order linear differential e q uation can be rewritten in a state-s p ace _ _ form, with states x 1 d and x 2 d x 1 and p arameters z and o n ; as
x 1 t x 2 t
0
1
x 1 t x 2 t
d
dt
Àwn 2
À2zw n
x 1 2 t x t 0 0
0 u t 1
with initial conditions
:
As a numerical exam p le, let
u t 1;
w
n
1;
z 0:5;
so that the coef®cient matrix
F
0 À1
1 À1
:
5
See, for exam p le, Brockett [ 56 ] , DeRusso et al. [ 59 ] , or Kreindler and Sarachik [ 189 ] .
Therefore,
“
s 2 s 1
À1
s
#
sI À F
sI À F À1
;
1
s 1 ” 1
#
s 1
À1
1
s
F t e Ft
l À1
sI À FÀ1 2 s 1 6 s 2 s 1 l À1 6 4 À1
1
s 2 s 1 s
3
7
7
5
Àt=2
2
6
6
6
4
1
2
s 2 s 1 s 2 s 1 p p 1 p 1 1 3 cos 3t 3t sin 2 2 2
p
3
1 sin 2
3t
2e p 3
p
1
2
p
p
p
7 7 : 7 5
1 À sin 2
3t
1 3 cos 2
3t
1 1 À sin 2 2
3t
2.3.9 Time-Varying Systems If F t is not constant, the d y namic s y stem is called time-var y in g . If F t is a
p
iecewise smooth function of t, the n  n homo g eneous matrix differential e q uation
2.24 can be solved numericall y b y the fourth-order Run g e±Kutta method. 6
2.4 DISCRETE LINEAR SYSTEMS AND THEIR SOLUTIONS
2.4.1 Discretized Linear Systems
If one is onl y interested in the s y stem state at discrete times, then one can use the
formula
t
k
x t k F t k ; t kÀ1 x t kÀ1
F t k ; sC su s ds
2:25
t kÀ1
to p ro p a g ate the state vector between the times of interest.
6
Named after the German mathematicians Karl David Tolme Run g e ( 1856±1927 ) and Wilhelm Martin Kutta ( 1867±1944 ) .
Simpli®cation for Constant u. If u is constant over the interval t kÀ1 ; t k , then the above inte g ral can be sim p li®ed to the form
x t k F t k ; t kÀ1 x t kÀ1 G t kÀ1 u t kÀ1 t k G t kÀ1
F tk
;
sC s
ds:
2:26 2:27
t kÀ1
Shorthand Discrete-Time Notation. For discrete-time s y stems, the indices k in the time se q uence ft k g characterize the times of interest. One can save some ink b y usin g the shorthand notation:
def x k x t k ;
def z k z t k ;
def u k u t k ;
def H k H t k ;
def G k G t k for discrete-time s y stems, eliminatin g t entirel y . Usin g this notation, one can re p resent the discrete-time state e q uations in the more com p act form
def D k D t k ;
def F kÀ1 F t k ; t kÀ1 ;
x k F kÀ1 x kÀ1 G kÀ1 u kÀ1 ;
z k H k x k D k uk
2:28 2:29
2.4.2 Time-Invariant Systems
For continuous time-invariant s y stems that have been discretized usin g ®xed time intervals, the matrices F, G, H, and D are inde p endent of the discrete-time index as well. In that case, the solution can be written in closed form as
P x k F k x 0 k À1 F kÀiÀ1 Gu i ;
i0
2:30
where F k is the kth p ower of F. The matrix F k can also be com p uted as
F k z À1 zI À F À1 z; where z is the z-transform variable and z À1 is the inverse z-transform.
2.5 OBSERVABILITY OF LINEAR DYNAMIC SYSTEM MODELS
2:31
Observabilit y is the issue of whether the state of a d y namic s y stem is uni q uel y determinable from its in p uts and out p uts, g iven a model for the d y namic s y stem. It is essentiall y a p ro p ert y of the g iven s y stem model. A g iven linear d y namic s y stem
model with a g iven linear in p ut=out p ut model is considered observable if and onl y if its state is uni q uel y determinable from the model de®nition, its in p uts, and its out p uts. If the s y stem state is not uni q uel y determinable from the s y stem in p uts and out p uts, then the s y stem model is considered unobservable.
How to Determine Whether a Given Dynamic System Model Is Observable. If the measurement sensitivit y matrix is invertible at an y ( continuous or discrete ) time, then the s y stem state can be uni q uel y determined ( b y invertin g it ) as x H À1 z. In this case, the s y stem model is considered to be com p letel y observable at that time. However, the s y stem can still be observable over a time interval even if H is not invertible at an y time. In the latter case, the uni q ue solution for the s y stem state can be de®ned b y usin g the least-s q uares methods of Cha p ter 1, includin g those of Sections 1.2.2 and 1.2.3. These use the so-called Gramian matrix to characterize whether or not a vector variable is determinable from a g iven linear model. When a pp lied to the p roblem of the determinac y of the state of a linear d y namic s y stem, the Gramian matrix is called the observabilit y matrix of the g iven s y stem model.
The observabilit y matrix for d y namic s y stem models in continuous time has the form
t
f
o H; F; t 0 ; t f
F T tH T tH tF t dt
2:32
t 0
for a linear d y namic s y stem with fundamental solution matrix F t and measurement sensitivit y matrix H
t, de®ned over the continuous-time interval t 0 t t f . Note that this de p ends on the interval over which the in p uts and out p uts are observed but not on the in p uts and out p uts p er se. In fact, the observabilit y matrix of a d y namic s y stem model does not de p end on the in p uts u, the in p ut cou p lin g matrix C, or the in p ut±out p ut cou p lin g matrix DÐeven thou g h the out p uts and the state vector de p end on them. Because the fundamental solution matrix F de p ends onl y on the d y namic coef®cient matrix F, the observabilit y matrix de p ends onl y on H and F.
The observabilit y matrix of a linear d y namic s y stem model over a discrete-time interval t 0 t t k f has the g eneral form
( T ) P k f kÀ1 kÀ1 Q Q o H k ; F k ; 1 k k f F kÀi H k T H k F kÀi ;
2:33
k1
i0
i0
where H k is the observabilit y matrix at time t k and F k is the state transition matrix from time t k to time t k1 for 0 k k f . Therefore, the observabilit y of discrete-time s y stem models de p ends onl y on the values of H k and F k over this interval. As in the continuous-time case, observabilit y does not de p end on the s y stem in p uts.
The derivations of these formulas are left as exercises for the reader.
2.5.1 Observability of Time-Invariant Systems
The formulas de®nin g observabilit y are sim p ler when the d y namic coef®cient matrices or state transition matrices of the d y namic s y stem model are time invariant. In that case, observabilit y can be characterized b y the rank of the matrices
M HT
FT HT
F T 2 HT
F T 2 HT
ÁÁÁ
ÁÁÁ
F T nÀ1 H T
F T nÀ1 H T
2:34
for discrete-time s y stems and
M HT
F T H T
2:35
for continuous-time s y stems. The s y stems are observable if these have rank n, the dimension of the s y stem state vector. The ®rst of these matrices can be obtained b y re p resentin g the initial state of the linear d y namic s y stem as a function of the s y stem in p uts and out p uts. The initial state can then be shown to be uni q uel y determinable if and onl y if the rank condition is met. The derivation of the latter matrix is not as strai g htforward. O g ata [ 38 ] p resents a derivation obtained b y usin g p ro p erties of the characteristic p ol y nomial of F.
Practicality of the Formal De®nition of Observability. Sin g ularit y of the observabilit y matrix is a concise mathematical characterization of observabilit y . This can be too ®ne a distinction for p ractical a pp licationÐes p eciall y in ®nite- p recision arithmeticÐbecause arbitraril y small chan g es in the elements of a sin g ular matrix can render it nonsin g ular. The followin g p ractical considerations should be ke p t in mind when a pp l y in g the formal de®nition of observabilit y :
It is im p ortant to remember that the model is onl y an a pp roximation to a real s y stem, and we are p
rimaril y interested in the p ro p erties of the real s y stem, not the model. Differences between the real s y stem and the model are called model truncation errors. The art of s y stem modelin g de p ends on knowin g where to truncate, but there will almost surel y be some truncation error in an y model.
Com p utation of the observabilit y matrix is sub j ect to model truncation errors and roundo ff errors,
which could make the difference between sin g ularit y and nonsin g ularit y of the result. Even if the com p uted observabilit y matrix is close to bein g sin g ular, it is cause for concern. One should consider a s y stem as p oorl y observable if its observabilit y matrix is close to bein g sin g ular. For that p ur p ose, one can use the sin g ular-value decom p osition or the condition number of the observabilit y matrix to de®ne a more q uantitative measure of unobservabilit y . The reci p rocal of its condition number measures how close the s y stem is to bein g unobservable.
Real s y stems tend to have some amount of un p redictabilit y in their behavior, due to unknown or ne g
lected exo g enous in p uts. Althou g h such effects cannot be modeled deterministicall y , the y are not alwa y s ne g li g ible. Furthermore, the p rocess of measurin g the out p uts with p h y sical sensors introduces some
amount of sensor noise, which will cause errors in the estimated state. It would be better to have a q uantitative characterization of observabilit y that takes these t yp es of uncertainties into account. An a pp roach to these issues (p ursued in Cha p ter 4 ) uses a statistical characterization of observabilit y , based on a statistical model of the uncertainties in the measured s y stem out p uts and the s y stem d y namics. The de g ree of uncertaint y in the estimated values of the s y stem states can be characterized b y an in f ormation matrix, which is a statistical g eneralization of the observabilit y matrix.
EXAMPLE 2.8
Consider the followin g continuous s y stem:
“
#
“
The observabilit y matrix, usin g E q uation 2.35, is
#
0
_ x t 0 z t 1
1
0
x t 0 0 x t:
u t;
1
M
1
0
0 ; 1
rank of M 2:
Here, M has rank e q ual to the dimension of x t. Therefore, the s y stem is observable.
EXAMPLE 2.9
Consider the followin g continuous s y stem:
“
#
“
The observabilit y matrix, usin g E q uation 2.35, is
#
0
0
1
0
_ x t z t 0
x t 0 1x t:
u t;
1
M
0
1
0 ; 1
rank of M 1:
Here, M has rank less than the dimension of x t. Therefore, the s y stem is not observable.
EXAMPLE 2.10
Consider the followin g discrete s y stem:
6 7 6 7 x k 4 6 0 0 0 5 7 x kÀ1 4 6 1 5 7 u kÀ1 ;
2
3
2
3
0
1
0
1
0
0
1
0
z k 0
0
1x k :
The observabilit y matrix, usin g E q uation 2.34, is
2
4
M
0
0
1
1
1
0
0
0
0
3 5 ;
rank of M 2:
The rank is less than the dimension of x k . Therefore, the s y stem is not observable.
EXAMPLE 2.11
Consider the followin g discrete s y stem:
” # ” # 1 À1 2 x k x kÀ1 u kÀ1 ; 1 1 1
“
#
1
À1
0
z k
x k :
1
The observabilit y matrix, usin g E q uation 2.34, is
M
1
0
À1 ; 1
rank of M 2
The s y stem is observable.
2.5.2 Controllability of Time-Invariant Linear Systems
Controllability in Continuous Time. The conce p t of observabilit y in estimation theor y has al g ebraic relationshi p s to the conce p t of controllabilit y in control theor y . These conce p ts and their relationshi p s were discovered b y R. E. Kalman as what he called the dualit y and se p arabilit y of the estimation and control p roblems for linear d y namic s y stems. Kalman’s 7 dual conce p ts are p resented here and in the next subsection, althou g h the y are not issues for the estimation p roblem.
7
The dual relationshi p s between estimation and control g iven here are those ori g inall y de®ned b y Kalman. These conce p ts have been re®ned and extended b y later investi g ators to include conce p ts of reachabilit y and reconstructibilit y as well. The interested reader is referred to the more recent textbooks on ``modern’’ control theor y for further ex p osition of these other ``-ilities.’’
A d y namic s y stem de®ned on the ®nite interval t 0
t
t f b y the linear model
_ x t Fx t Cu t;
z t Hx t Du t
2:36
and with initial state vector x t 0 is said to be controllable at time t t 0 if, for an y desired ®nal state x t f , there exists a p iecewise continuous in p ut function u t that drives to state x t f . If ever y initial state of the s y stem is controllable in some ®nite time interval, then the s y stem is said to be controllable.
The s y stem g iven in E q uation 2.36 is controllable if and onl y if matrix S has n linearl y inde p endent columns,
S
C
FC
F 2
C
ÁÁÁ
F nÀ1
C:
2:37
Controllability in Discrete Time. Consider the time-invariant s y stem model g iven b y the e q uations
x k Fx kÀ1 Gu kÀ1 ;
z k Hx k Du k :
2:38 2:39
This s y stem model is considered controllable 8 if there exists a set of control si g nals u k de®ned over the discrete interval 0 k N that brin g the s y stem from an initial state x 0 to a g iven ®nal state x N in N sam p lin g instants, where N is a ®nite p ositive inte g er. This condition can be shown to be e q uivalent to the matrix
S G
FG
F2 G
ÁÁÁ
F NÀ1 G
2:40
havin g rank n:
EXAMPLE 2.12 Determine the controllabilit y of Exam p le 2.8. The controllabilit y matrix, usin g E q uation 2.37, is
1 0 S ; rank of S 2:
1 0
Here, S has rank e q ual to the dimension of x t. Therefore, the s y stem is controllable.
EXAMPLE 2.13 Determine the controllabilit y of Exam p le 2.10. The controllabilit y matrix, usin g E q uation 2.40, is 2 3 1 0 0 S • 4 1 0 0 5 ; rank of S 2: 0 2 0
The s y stem is not controllable.
8
This condition is also called reachabilit y , with controllabilit y restricted to x N 0.
2.6 PROCEDURES FOR COMPUTING MATRIX EXPONENTIALS
In a 1978 j ournal article titled ``Nineteen dubious wa y s to com p ute the ex p onential of a matrix’’ [ 205 ] , Moler and Van Loan re p orted their evaluations of methods for com p utin g matrix ex p onentials. Man y of the methods tested had serious shortcomin g s, and no method was considered universall y su p erior. The one p resented here was recommended as bein g more reliable than most. It combines several ideas due to Ward [ 233 ] , includin g settin g the al g orithm p arameters to meet a p res p eci®ed error bound. It combines Pade  a pp roximation with a techni q ue called ``scalin g and s q uarin g ‘’ to maintain a pp roximation errors within p res p eci®ed bounds.
2.6.1 Pade  Approximation of the Matrix Exponential
Pade  approximations. These a pp roximations of functions b y rational functions 9 ( ratios of p ol y nomials ) date from a 1892 p ublication [ 206 ] b y H. Pade  . The y have been used in derivin g solutions of differential e q uations, includin g Riccati e q uations 10 [ 69 ] . The y can also be a pp lied to functions of matrices, includin g the matrix ex p onential. In the matrix case, the p ower series is a pp roximated as a ``matrix fraction’’ of the form d À1 n, with the numerator matrix ( n ) and denominator matrix ( d ) re p resented as p ol y nomials with matrix ar g uments. The ``order’’ of the Pade  a pp roximation is two dimensional. It de p ends on the orders of the p ol y nomials in the numerator and denominator of the rational function. The Ta y lor series is the s p ecial case in which the order of the denominator p ol y nomial of the Pade  a pp roximation is zero. Like the Ta y lor series a pp roximation, the Pade  a pp roximation tends to work best for small values of its ar g ument. For matrix ar g uments, it will be some matrix norm of the ar g ument that will be re q uired to be small.
Pade  approximation of exponential function. The ex p onential function with ar g ument z has the p ower series ex p ansion
1 P e z k0
1
zk k!
:
The p ol y nomials n z and d q z such that p
p P n z a k z k ; p
k0
q P d q z b k z k ;
k0
1 P
e z d q z À n z p
c k z k
k p q 1
9
Pronounced p ah-DAY..
10 The order of the numerator and denominator of the matrix fraction are reversed here from the order used
in linearizin g the Riccati e q uation in Cha p ter 4.
are the numerator and denominator p ol y nomials, res p ectivel y , of the Pade  a pp roximation of e z . The ke y feature of the last e q uation is that there are no terms of order p q on the ri g ht-hand side. This constraint is suf®cient to determine the coef®cients a k and b k of the p ol y nomial a pp roximants, exce p t for a common constant factor. The solution ( within a common constant factor ) will be [ 69 ]
- !
À k! a k p p q ; k!
p À k!
- À1k !
p q À k! b k q k! q À k!
:
Application to Matrix Exponential. The above formulas ma y be a pp lied to p ol y nomials with scalar coef®cients and s q uare matrix ar g uments. For an y n  n matrix X,
q P q ! i0
p q À i!
À1
fpq
X
- ÀXi i!
q À i!
p P p ! i0
p q À i!
- X i!
p À i!
i
% e X
- is the Pade  a pp roximation of e X of order
p ; q .
Bounding Relative Approximation Error. The bound g iven here is from Moler and Van Loan [ 205 ] . It uses the 1-norm of a matrix, which can be com p uted 11 as
!
kXk 1 max
1 i n
n P j 1
jx i j j
for an y n  n matrix X with elements x i j . The relative a pp roximation error is de®ned as the ratio of the matrix 1-norm of the a pp roximation error to the matrix 1-norm of the ri g ht answer. The relative Pade  a pp roximation error is derived as an anal y tical function of X in Moler and Van Loan [ 205 ] . It is shown in Golub and Van Loan [ 89 ] that it satis®es the ine q ualit y bound
k f pq X À e X k1 e
p ; q ; Xe e p ; q ;X ; ke X k1
p
! q !2 3À p À q
- e
p ; q ; X
- kXk1
p q ! p q 1!
:
Note that this bound de p ends onl y on the sum p q . In that case, the com p utational com p lexit y of the Pade  a pp roximation for a g iven error tolerance is minimized when p q , that is, if the numerator and denominator p ol y nomials have the same order.
11 This formula is not the de®nition of the 1-norm of a matrix, which is de®ned in A pp endix B. However, it is a conse q uence of the de®nition, and it can be used for com p utin g it.
Bounding the Argument. The p roblem with the Pade  a pp roximation is that the error bound g rows ex p onentiall y with the norm kXk 1 . Ward [ 233 ] combined scalin g ( to reduce kXk 1 and the Pade  a pp roximation error ) with s q uarin g ( to rescale the answer ) to obtain an a pp roximation with a p redetermined error bound. In essence, one chooses the p ol y nomial order to achieve the g iven bound.
2.6.2 Scaling and Squaring
Note that, for an y nonne g ative inte g er N,
e X e 2 ÀN X 2 N
f
Á Á Á e 2 ÀN X Á Á Á 2 2 g 2 : |{z}
N s q uarin g s
Conse q uentl y , X can be ``downscaled’’ b y 2 ÀN to obtain a g ood Pade  a pp roximation of e 2 ÀN X , then ``u p scaled’’ a g ain ( b y N s q uarin g s ) to obtain a g ood a pp roximation to e X .
2.6.3 MATLAB Implementations
The built-in MATLAB function ex p m ( M ) is essentiall y the one recommended b y Moler and Van Loan [ 205 ] , as im p lemented b y Golub and Van Loan [ 89, Al g orithm 11.3.1, p a g e 558 ] . It combines scalin g and s q uarin g with a Pade  a pp roximation for the ex p onential of the scaled matrix, and it is desi g ned to achieve a s p eci®ed tolerance of the a pp roximation error. The MATLAB m-®le ex p m1.m ( Section A.4 ) is a scri p t im p lementation of ex p m.
MATLAB also includes the functions ex p m2 ( Ta y lor series a pp roximation ) and ex p m3 ( alternative im p lementation usin g ei g envalue±ei g envector decom p ositions ) , which can be used to test the relative accuracies and s p eeds relative to ex p m of these alternative im p lementations of the matrix ex p onential function.
2.7 SUMMARY
Systems and Processes. A s y stem is a collection of interrelated ob j ects treated as a whole for the p ur p ose of modelin g its behavior. It is called d y namic if attributes of interest are chan g in g with time. A p rocess is the evolution over time of a s y stem.
Continuous and Discrete Time. Althou g h it is sometimes convenient to model time as a continuum, it is often more p ractical to consider it as takin g on discrete values. ( Most clocks, for exam p le, advance in discrete time ste p s. )
State Variables and Vectors. The state of a d y namic s y stem at a g iven instant of time is characterized b y the instantaneous values of its attributes of interest. For
the p roblems of interest in this book, the attributes of interest can be characterized b y real numbers, such as the electric p otentials, tem p eratures, or p ositions of its com p onent p artsÐin a pp ro p riate units. A state variable of a s y stem is the associated real number. The state vector of a s y stem has state variables as its com p onent elements. The s y stem is considered closed if the future state of the s y stem for all time is uni q uel y determined b y its current state. For exam p le, ne g lectin g the g ravit y ®elds from other massive bodies in the universe, the solar s y stem could be considered as a closed s y stem. If a d y namic s y stem is not closed, then the exo g enous causes are called ``in p uts’’ to the s y stem. This state vector of a s y stem must be com p lete in the sense that the future state of the s y stem is uni q uel y determined b y its current state and its f uture in p uts. 12 In order to obtain a com p lete state vector for a s y stem, one can extend the state variable com p onents to include derivatives of other state variables. This allows one to use velocit y ( the derivative of p osition ) or acceleration ( the derivative of velocit y) as state variables, for exam p le.
State-Space Models for Dynamic Systems. In order that the future state of a s y stem ma y be determinable from its current state and future in p uts, the d y namical behavior of each state variable of the s y stem must be a known function of the instantaneous values of other state variables and the s y stem in p uts. In the canonical exam p le of our solar s y stem, for instance, the acceleration of each bod y is a known function of the relative p ositions of the other bodies. The state-s p ace model for a d y namic s y stem re p resents these functional de p endencies in terms of ®rst-order di ff erential e q uations ( in continuous time ) or di ff erence e q uations ( in discrete time ) . The differential or difference e q uations re p resentin g the behavior of a d y namic s y stem are called its state e q uations. If these can be re p resented b y linear functions, then it is called a linear d y namic s y stem.
Linear Dynamic System Models. The model for a linear d y namic s y stem in continuous time can be ex p ressed in g eneral form as a ®rst-order vector differential e q uation
d
x t F tx t C tu t; dt
where x t is the n-dimensional s y stem state vector at time t, F t is its n  n d y namic coe f® cient matrix, u t is the r-dimensional s y stem in p ut vector, and C t is
the n  r in p ut cou p lin g matrix. The corres p ondin g model for a linear d y namic s y stem in discrete time can be ex p ressed in the g eneral form
x k F kÀ1 x kÀ1 G kÀ1 u kÀ1 ;
12
This conce p t in the state-s p ace a pp roach will be g eneralized in the next cha p ter to the ``state of knowled g e’’ about a s y stem, characterized b y the p robabilit y distribution of its state variables. That is, the f uture p robabilit y distribution of the s y stem state variables will be uni q uel y determined b y their p resent p robabilit y distribution and the p robabilit y distributions of f uture in p uts.
where x kÀ1 is the n-dimensional s y stem state vector at time t kÀ1 , x k is its value a time t k > t kÀ1 , F kÀ1 is the n  n state transition matrix for the s y stem at time t k , u k is the in p ut vector to the s y stem a time t k , and G k is the corres p ondin g in p ut cou p lin g matrix.
Time-Varying and Time-Invariant Dynamic Systems. If F and C ( or F and C ) do not de p end u p on t ( or k ) , then the continuous ( or discrete ) model is called time invariant. Otherwise, the model is time-var y in g .
Homogeneous Systems and Fundamental Solution Matrices. The e q uation
d
x t F tx t dt
is called the homo g eneous p art of the model e q uation
d
x t F tx t C tu t: dt
A solution F t to the corres p ondin g n  n matrix e q uation
d
F t F tF t dt
on an interval startin g at time t t 0 and with initial condition
F t 0 I
(
the identit y matrix )
is called a f undamental solution matrix to the homo g eneous e q uation on that interval. It has the p ro p ert y that, if the elements of F t are bounded, then F t cannot become sin g ular on a ®nite interval. Furthermore, for an y initial value x t 0 ;
x t F tx t 0
is the solution to the corres p ondin g homo g eneous e q uation.
Fundamental Solution Matrices and State Transition Matrices. For a homo g enous s y stem, the state transition matrix F kÀ1 from time t kÀ1 to time t k can be ex p ressed in terms of the fundamental solution F t as
F kÀ1 F t k F À1 t kÀ1
for times t k > t kÀ1 > t 0 .
Transforming Continuous-Time Models to Discrete Time. The model for a d y namic s y stem in continuous time can be transformed into a model in discrete time usin g the above formula for the state transition matrix and the followin g formula for the e q uivalent discrete-time in p uts:
t
k
u kÀ1 f t k
F À1 tC tu t dt:
t kÀ1
Linear System Output Models and Observability. An out p ut of a d y namic s y stem is somethin g we can measure directl y , such as directions of the lines of si g ht to the p lanets ( viewin g conditions p ermittin g) or the tem p erature at thermocou p le. A d y namic s y stem model is said to be observable from a g iven set of out p uts if it is feasible to determine the state of the s y stem from those out p uts. If the de p endence of an out p ut z on the s y stem state x is linear, it can be ex p ressed in the form
z Hx; where H is called the measurement sensitivit y matrix. It can be a function of continuous time [ H t ] or discrete time ( H k ) . Observabilit y can be characterized b y the rank of an observabilit y matrix associated with a g iven s y stem model. The observabilit y matrix is de®ned as
8 t > > FT tH T tH tF t dt > > < t 0 ” # o > > P m i Q i Q T > > À1 F T H i T H i À1 F T : k k i0 k0 k0
for continuous-time models,
for discrete-time models.
The s y stem is observable if and onl y if its observabilit y matrix has full rank ( n ) for some inte g er m ! 0 or time t > t 0 . ( The test for observabilit y can be sim p li®ed for time-invariant s y stems. ) Note that the determination of observabilit y de p ends on the ( continuous or discrete ) interval over which the observabilit y matrix is determined.
Reliable Numerical Approximation of Matrix Exponential. The closedform solution of a s y stem of ®rst-order differential e q uations with constant coef®cients can be ex p ressed s y mbolicall y in terms of the ex p onential function of a matrix, but the p roblem of numerical a pp roximation of the ex p onential function of a matrix is notoriousl y ill-conditioned.
PROBLEMS
d t 2.1 What is a state vector model for the linear d y namic s y stem y u t, dt ex p ressed in terms of y ? ( Assume the com p anion form of the d y namic coef®cient matrix. )
2.2 What is the com p anion matrix for the nth-order differential e q uation d=dt n y t 0? What are its dimensions?
2.3 What is the com p anion matrix of the above p roblem when n 1? For n 2?
2.4 What is the fundamental solution matrix of Exercise 2.2 when n 1? When n 2?
2.5 What is the state transition matrix of the above p roblem when n 1? For n 2?
2.6 Find the fundamental solution matrix F t for the s y stem 1 d x 1 t 0 0 x 1 t dt x 2 t À1 À2 x 2 t 1
and also the solution x t for the initial conditions
x1
0 1 and x 2 0 2:
2.7 Find the total solution and state transition matrix for the s y stem À1 5 d x 1 t 0 x 1 t • dt x 2 t 0 À1 x 2 t 1
with initial conditions x 1 0 1 and x 2 0 2. 2.8 The reverse p roblem: f rom a discrete-time model to a continuous-time model. For the discrete-time d y namic s y stem model 0 1 0 x k x kÀ1 ; À1 2 1
®nd the state transition matrix for continuous time and the solution for the continuous-time s y stem with initial conditions 1 x 0 • : 2 2.9 Find conditions on c 1 ; c 2 ; h 1 ; h 2 such that the followin g s y stem is com p letel y
observable and controllable:
1 c1 2 d x 1
t 1 x t • u t; dt x 2 t 0 1 x t c 1 2 1 2 t
2.10 Determine the controllabilit y and observabilit y of the d y namic s y stem model g iven below: 1 1 d x 1 t 0 x 1 2 t 0 u 1 • ; dt x 2 t 1 0 x t 0 À1 u 2 t x1 2 z t 0 1 : x t
2.11 Derive the state transition matrix of the time-var y in g s y stem
_ x t
t 0
0 t
x t:
2.12 Find the state transition matrix for
F
0
1
1 : 0
2.13 For the s y stem of three ®rst-order differential e q uations
_ x 1 x 2 ;
_ x 2 x 3 ;
_ x 3 0
( a ) What is the com p anion matrix F? ( b ) What is the fundamental solution matrix F t such that d=dtF t FF t and F 0 I?
2.14 Show that the matrix ex p onential of an antis y mmetric matrix is an ortho g onal matrix.
2.15 Derive the formula of E q uation 2.32 for the observabilit y matrix of a linear d y namic s y stem model in continuous time. ( Hint: Use the a pp roach of Exam p le 1.2 for estimatin g the initial state of a s y stem and E q uation 2.19 for the state of a s y stem as a linear function of its initial state and its in p uts. )
2.16 Derive the formula of E q uation 2.33 for the observabilit y matrix of a d y namic s y stem in discrete time. ( Hint: Use the method of least s q uares of Exam p le
1.1 for estimatin g the initial state of a s y stem, and com p are the resultin g Gramian matrix to the observabilit y matrix of E q uation 2.33. )