Linear time varying systems and sampled-data systems

345 Pages·2001·10.07 MB·English

by  Akira Ichikawa PhD

1. Introduction
The linear system theory offers basic concepts, design methods and opti-
mization problems. The underlying systems it deals with are usually time-
invariant continuous-time systems or discrete-time systems. There are many
excellent books ,1 19, 21, 46, 55, 66, 80, 39 for time-invariant systems which
cover stability theory, quadratic control, 2H control and ooH control. The
sampled-data system with sampler and zero-order hold is a linear system but
involves both continuous-time and discrete-time signals. The system theory
for it, which covers topics above, is usually developed via the system theory
for discrete-time systems after transforming the original problems to those
for discrete-time systems.
The purpose of this monograph is two fold. We introduce the linear sys-
tem theory for time-varying systems which covers H2 and ooH control. There
are some inherent features of time-varying systems, and not all arguments
used for time-invariant systems are easily extended to them. Thus we regard
the extension as important and hope that it gives some new insight into
the linear systems theory. Secondly we develop the H2 and H~ control the-
ory for sampled-data systems from the point of view of jump systems. The
jump system, which contains jumps in the state variable, is a natural state-
space representation of sampled-data systems and has an advantage that the
continuous-time nature and discrete-time signals of the original system are
maintained in the new system. Hence the system theory for jump systems can
be viewed as an extension of the theory of continuous-time or discrete-time
systems. In ooH control the initial conditions are usually taken to be zero, but
in this monograph initial uncertainty is incorporated and a general theory is
developed.
As the jump system may not be familiar to the reader, we shall introduce
below all the systems which appear in this monograph. We also introduce the
jump system which is obtained from a sampled-data system.
2 .1 Introduction
1.1 Continuous-time Systems and Discrete-time Systems
In Chapter 2 we consider continuous-time systems of the form
= A(t)x+Bl(t)w +B2(t)u,
z = Cl(t)x + D12(t)u, (1.1)
y = C2(t)x + D21(t)w
where x is the state of the system, w is a disturbance, u is a control input, z
is a controlled output and y is the output to be used for control.
In Chapter 3 we consider discrete-time systems of the form
x(k + 1) = A(k)x(k) +B,(k)w(k) +B2(k)u(k),
(1.2)
z(k) = Cl(k)x(k) + D12(k)u(k),
y(k) = C2(k)x(k) + D21(k)w(k).
We consider stability, quadratic control, disturbance attenuation problems,
differential games, H~ control and H2 control. When we introduce new prob-
lems, we sometimes start with the results for time-invariant systems. More-
over, all the results in the time-invariant case are given as corollaries. How-
ever, we give proofs only for time-varying systems. For proofs typical to time-
invariant systems we refer the reader to other books in the reference.
1.2 Jump Systems
A general form of jump systems is given by
:i: = Ax + BlW + B2u, kr < t < (k + 1)r,
x(kT +) = Adx(k~-) + Bldwd(k) + B2dud(k),
cZ = Clx + D12u, (1.3)
zd(k) = Cldx(k'r) + D12dUd(k),
Yc = C2x + D~lw,
ya(k) = C2ax(kr) + D21dWd(k )
where the continuous part satisfies (1.1) while the jump part satisfies (1.2).
We assume that all matrices in the system are constant. Then it is a T-
periodic system and a special case of time-varying systems. We can easily see
the following :
(a) If B1 ---- 0, B2 -- 0, Ad : I and Bid ---- 0, then it is a system with impulse
control.
(b) If Ad =- I, Bid -~ 0 and B2d = 0, it is a continuous-time system.
(c) If A = 0 B1 = 0 and B2 : 0, then it can be regarded as a discrete-time
system.
1.3. Sampled-data Systems 3
Hence the jump system is a natural extension of continuous-time and discrete-
time systems 37, 51, 65, 76 and we can expect some potential applications
in the areas of mechanical systems 83, chemical processes 56 and economic
systems 6 where impulsive inputs naturally appear. The system (1.3) is
often too general and for H2 or H~ control we shall restrict ourselves to the
system of the form
cJ = Ax n u Blw, kT < t 2< (k -t- 1)r,
x(kT +) =- Adx(kr) + B2u(k), (1.4)
cZ = 61 x,
zd(k) = D12u(k),
y(k) = C2x(kT) + D21wd(k).
This system still keeps the essential features of jump systems and covers
sampled-data systems. In Chapter 4 we consider stability and control prob-
lems for (1.4) as in Chapters 2 and 3.
1.3 Sampled-data Systems
In Chapter 5 we consider the sampled-data system ,8 61
= Ax(t) + Blw(t) + B2fi(t),
z(t) = Fix(t) 1 (1.5)
D12~(t) J '
y(k) = C2x(kr) + D21wd(k)
where T > 0 is a sampling period and fi is the control input realized through
the zero-order hold 18, 58
(t(t) = u(k), kr < t < (k + 1)T.
We introduce the following state space representation of the control ~(t):
~: = 0, ff:(kT +) = u(k), kT < t ~ (k -t- 1)T.
Then clearly ~i(t) = 2(t). Let xe(t) = x' 2'' (t) be the new state variable.
Then the system (1.5) is equivalent to the following jump system
~e(t) = A B2x~(t)+Bo1W(t), kT<t<(k+l)T,
l
z = zd(k) = v~D12u(k) '
y(k) = C20xe(kr)+D21wd(k)
4 .1 Introduction
where dZ -= v~D12u(k) is introduced when we consider
0/
i D12~(t)j2= i D12u(k) is= Ei v/-~D12u(k)12 .
k=0 0 k=0
The jump system is a natural state-space representation of sampled-data
systems and original signals and parameters are maintained in the new sys-
tem. The H2 and Hoo problems for jump systems can be treated in a unified
manner sm for time-invariant systems.
1.4 Infinite Dimensional Systems and Sampled-data
Fuzzy Systems
The infinite dimensional system in Chapter 6 is written as (1.4) but the
state, inputs and outputs lie in Hilbert spaces. We assume that A is the
infinitesimal generator of a strongly continuous semigroup in a Hilbert space
and other operators are bounded. Thus the system covers partial differential
equations and delay differential equations. It will be shown that the sampled-
data system with first-order hold can be expressed in this form and H2 and
Ha control problems are considered.
The sampled-data fuzzy system in Section 6.2 is given by
2(t) = ~ )h(z(t)){A~x(t) + B~(t)},
i=l
y(k) =
i=l
where ),~(z) > 0 , Ei=l r ~i(2:) = 1 and z is a premise variable in the IF-
THEN rules. It can represents certain nonlinear sampled-data systems. Using
local jump systems we give a design method of stabilizing output feedback
controllers.
1.5 Notation
Ix = xv/~x, x~R n.
IMI : norm of M E R ~(cid:141) induced by the Euclidean norm.
f;
< f,g> = f'(t)g(t)dt, f, g ~ L2(to, T; R ~)
where T can be finite or T -- .)xc
lS f 211 = vf~f,f:>.
1.5. Notation 5
N
<f,g> = ~ f'(k)g(k), f, g e/2(ko, N;R '~)
k=ko
where N can be finite or N = o0.
211/I = v/<f,f>.
x(s +)
= lim x(t).
tJ.s
= lim x(t).
tTs
2. Continuous-time Systems
In this chapter we are concerned with time-varying continuous-time systems
and consider stability, quadratic control, disturbance attenuation problems,
differential games, Ho~ control, ~oH filtering and H2 control. In ~oH control
and filtering we allow for initial uncertainty in the systems and develop the
general theory.
2.1 Stability
2.1.1 Lyapunov Equations
Consider
= A(t)x, x(to) = xo (2.1)
where x E R n and A E R ~(cid:141) is a piecewise continuous matrix with
I A(t) _<I a, vt _> to for some a > 0.
Let S(t, r) be the state transition matrix of the system (2.1). Then
d s(t, s) = A(t)S(t, s), S(s, s) = I.
If
A(t)A(s) = A(s)A(t) vt, s
then
s(t, s) = el: a(r)dr
If A is 0-periodic, i.e., A(t + )O = A(t), then
s(t + o, s + o) = s(t, s).
If A(t) = A, then S(t, s) = e A(t-s).
Definition 2.1 The system (2.1) (or simply A) is said to be exponentially
stable on to, co) if
I S(t,s) _<I Me -a(t-8) for any to < s < t < co
8 2. Continuous-time Systems
for some positive constants M and a independent of to and t. (The system
(2.1) is also called internally stable.)
If A(t) = A, then A is stable if and only if every eigenvalue of A has a negative
real part. The following result is also well-known.
Proposition 2.1 The following statements are equivalent.
(a) A is exponentially stable.
(b) There exists a positive definite matrix X satisfying
A'X + XA + I = 0. (2.2)
(c) There exists a positive definite matrix Y satisfying
AY + YA' + I = .O
The equation (2.2) is called the Lyapunov equation. We generalize this result
to the time-varying system. We need the following lemma.
Lemma 2.1 (a) I S(t,s) I< e a(t-s), to < s < t.
(b) For a given e ~ (0, 1), there exists a ~ > 0 such that
S'(t, s)S(t, s) > (1 - e)I for any 0 < t - s < 5.
proof. Since
s~ t
S(t, s) = I + A(r)S(r, s)dr
we have
s~ t
I S(t,s) ~<I 1 + a I S(r,s) I dr.
Hence by Gronwall's inequality, we obtain I S(t, s) I< e =(t-s) . We also have
// //
I A(r)S(r,s)dr I -< ~ IS(r,s) ldr
< aea(r-S)dr = e a(t-s) -- 1.
Now
z's'(t, s)S(t, s)z x'(I + A(r)S(r, s)dr)'(I + A(r)S(r, s)dr)x
> =I 12+2x' A(r)S(r,s)drz
> Ix 21 -2(e a(t-'~ - 1) Ix 51
> (1 - e) x I s
for any0 < t-s < where ~ = l l~ 1 + 3) so that 2(e a(t-s) - 1) < e,
O<t-s<& |
2.1. Stability 9
Proposition 2.2 The following statements are equivalent.
(a) The system (2.1) is exponentially stable.
(b) There exists a symmetric matrix X(t) such that
(i) cli _< X(t) _< c2I, vt > to for some ~c > ,O i = 1,2.
(ii) -J; = A'(t)X + XA(t) + I.
(c) f ~o S(t,s)z 2 dt < c x 2 , Vx, Vs > to and for some c > .O
If A is O-periodic, then X is also &periodic.
Proof. Suppose (a) holds. Then (c) also holds and
X(t) = S'(r, t)S(r, t)dr
is well-defined and bounded, i.e., X(t) <- c21. Let e e (0, 1) be given and
choose 5 > 0 as Lemma 2.1 such that
S'(t,s)S(t,s) _> (1 - e)I for 0 <_ t- s <_ 5.
Then
t+6
X(t) _> S'(r, t)S(r, t)dr _> (1 - e)5I.
at
Hence (i) of (b) has been shown. (ii) of (b) follows from differentiating X(t).
Now we assume (b). Then for x(t) = S(t, S)Xo
d_
dt (x'(t)X(t)x(t) ) = - x(t) 2< _lx,(t)X(t)x(t )
2C
Which implies
-~(t-~) ,
x'(t)x(t)x(t) < e ~ x (~)X(s)x(s).
Using the property (i) we have
cl Ix(t) 12-<,:=~ ^ -~(t-,) , I xo 2 (cid:12)9
Hence
I s(t, s) I < ,/-~Ae- ~ (~-')
V Cl
and (a) follows.
Finally let A(t) be &periodic. Then
X(t) = S'(r,t)S(r,t)dr
S
s
= S' (r + ,O t + O)S(r + 0, t + O)dr
= S'(s, t + O)S(s, t + O)ds
0
= x(t + o). t
01 .2 Continuous-time Systems
Definition 2.2 7he equation (ii) o (b) is called the Lyapunov equation or
the system (2.1).
If A is exponentially stable, we can show that any solution of the Lyapunov
equation coincides with X(t) given in the proof of Proposition 2.2. Hence the
Lyapunov equation has a unique solution. See also Theorem 2.4.
Example 2.1 Consider the periodic system with period :r72
0 1 xi ,23,
52 -1-0.5cost -1-cost x2
which is exponentially stable. In fact there exists a 27r-periodic nonnegative
solutionX(t)= X1 21X X12 X2 (t) of the condition (b) in Proposition 2.2 (Fig-
ures 1.2 and 2.2).
I " ' " I " ' ' I ' ' ' !
X2(~
X~(O
X2
0 I L 2 I ~ 4 I , , 6 I
time )ces(
Figure 2.1: The periodic solution X(t) of the Lyapunov equation
Consider the adjoint equation of (2.1)
(2.4)
-4 = A(t)~, ~(T) = .14
Let ~(t; T, )14 be the solution of (2.4).
Definition 2.3 The system (2.4) is said to eb exponentially stable if
I~(t;T,~l) l <_Me-~(T-t) 141 I for any to < t < T < oo
for some M > 0 and ~ > 0 independent of t, T and .14
2.1. Stability 11
(cid:12)9 , (cid:12)9 , ,
~2
m
0 I , 2 I , 4 I , 6 I
time (sec)
Figure 2.2: Eigenvalues of the periodic solution X(t)
Since ~(t; T, El) = St( T, t)~l, the system (2.4) is exponentially stable if and
only if the system (2.1) is exponentially stable.
We have a dual result to Proposition 2.2.
Proposition 2.3 The following statements are equivalent.
(a) The system (2.~) (and hence (2.1)) is exponentially stable.
(b) There exists a symmetric matrix Y(t) and a 5 > 0 such that
(i) 0 < Y(t), vt > to and OI < Y(t), vt > to + 5 for some cl > .O
(ii) Y(t) _< c2I, to _< vt < oo for some c2 > .O
(iii) Y = A(t)Y + YA'(t) + I, Y(to) = .O
(c) fT i S'(T,t)~ I 2 dt _< c l ~ 21 , Vs, T with to < s < T < o0 and for some
c>O.
Proof. Suppose (a) holds. Then (c) is true and
Y(t) = S(t, s)S'(t, s)ds
is well-defined, positive for t > to and bounded. Hence (ii) of (b) holds. To
show (i), let ~ E (0, 1) and choose 5 > 0 such that
S(t,s)S'(t,s) >> (1 - e)I for 0 < t - s <_ 5.
Now let t > 5, then
Y(t) _> s(t, s)S'(t, s)as _> (1 - ~)5i.
5