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An Infinite-Dimensional Discrete-Time Representation for Periodic Systems
AN INFINITE-DIMENSIONAL DISCRETE-TIME REPRESENTATION FOR PERlODIC SYSTEMS Joseph J. Yame, Raymond Hanus Service d'automatique,C.P.165 Faculte des sciences appliquees, Universite libre de Bruxelles, 50 av. F.D. Roosevelt, Brussels-l050, Belgium Fax: 32-2-650.26.11, e-mail: [email protected]
Abstract: This note deals with the representation of continuous-time periodic systems as discrete-time invariant systems in state-space form using the lifting technique. More precisely, we derive an infinite-dimensional state equation on a Hilbert space for continuous-time periodically time-varying systems and characterize the structure of the spectrum of the discrete semigroup generator. Copyright © 1998 IFAC Resume: Cet article traite de la modelisation des sytemes periodiques a temps continu sous forme de systemes a temps discret en representation d'etat grace a la technique du lifting. On montre que la dynamique des systemes periodiques est decrite par une equation aux differences du premier ordre sur un espace Hilbertien et on caracterise la structure du spectre du g€merateur de semigroupe discret. Keywords: Periodic systems, discrete-time systems, infinite-dimensional, modeling
1. INTRODUCTION
nique which is reminiscent of Floquet theory has the advantage to built the continuous-time behavior in the discrete model which is reflected in the infinite dimensionality of the input and output signals. In all previous works, the lifted models are specialized to sampled-data systems in the ''transfer function" approach and little attention has been paid to the "internal behavior". Except in (Yamamoto, 1994), these lifted models, although called infinite-dimensional systems, are actually finite-dimensional state-space systems. In fact the state is defined only at discrete instants and therefore the continuous-time internal behavior is lost and reduces to that of a purely discrete-time system. The purpose of this note is to focus our attention in the internal behavior of periodic systems by lifting the system to an infinite-dimensional state-space discrete-time system and to analyze the structure of the state-transition operator. To our present knowledge, the lifted model which is most related to ours is (Yamamoto, 1994). More precisely, in (Yamamoto, 1994), a continuous-
Systems described by periodic differential equations arise in a variety of unrelated fields and such systems have proved to be a subject of major interest in many scientific and technical areas promoting intensive research both in theory and practice. In control, they arise naturally when dealing with sampled-data feedback systems and, due to the increased use of digital computers for the purpose of implementing high performance control algorithms, this.has motivated a recent research in new modeling paradigms which take into account the continuous-time behavior of such systems . One of these paradigms, the so-called lifting technique developped independantly by Bamieh and Pearson (1992)and Yamamoto (1994)for the modeling and control of sampled-data systems is a general framework which applies to more general periodic systems. The lifting technique transforms the original continuous-time periodic system into a discrete-time shift-invariant system with infinitedimensional input and output signals. This tech-
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time time-invariant system is transformed into a discrete-time time-invariant system with the Banach space of continuous functions on (0, T] of uniform norm as the state space. However there are important differences between our lifted statespace model and that of Yamamoto (1994) . First, in our framework the classes of abstract spaces can be extended to larger spaces than the Banach space of continuous functions . Second, the state-transition operator in (Yamamoto, 1994)is actually a finite rank operator so that it is equivalent in a certain sense to an operator on a finite-dimensional space (which has the same dimension as that of the original continuous-time system) j this degeneracy is not present in our framework. The paper is structured as follows: section 1 contains some necessary mathematical background and notations. Section 2 develops the discrete-time infinite-dimensional statespace model of continuous-time periodic operators and section 3 derives a complete characterization of the spectrum of the state-transition operator. Finally, we conclude with some remarks for the applications of this framework to sampled-data control systems.
and we define the lifting operator (Bamieh and Pearson, 1992)
.c : L~ [0, 00) ~ l2 (X) x~x
= (xo, Xl, X2, ...) and xn (t) = x (t + nT) , < T , n EN.
with x
°: ;
t
It is easy to see that .c is a unitary operator and hence preserves all the structures of Hilbert spaces. From this it follows that if G E B (L2 [0,00» , then under the lifting transformation it goes over into G = .cG.c- l and G E B (l2 (H» where we have set H =L2 [0, T) . For a fixed T > O,we denote the (right) T-translation operator on L2 [0, 00) by Vr , and the (right) shift operator on l2 (H) by Q. An operator on L2 [0, 00) is said to be T-periodic if its commutes with Vr , and an operator on l2 (H) is said to be shiftinvariant if it commutes with Q. We state the following key lemma (Bamieh and Pearson, 1992)
Lemma 1. If G E B (L2 [0,00» and Vr G = GV r then QG = GQ on l2 (H) and moreover
IIGIIB(Lz[O ,oo)) = IIGIIB(12(H)) . 2. MATHEMATICAL PRELIMINARIES
This lemma establishes an input/output correspondence between continuous-time periodic operators on L2 [0, 00) and shift-invariant operators on l2 (H) .
The natural integers and the complex numbers are denoted by Nand e. For the sake of definiteness, we work in the Hilbert space setting, however our framework extend without difficulties to Banach spaces of Lp type (1 ::; p ::; 00). Let H be a Hilbert space, the set of all linear bounded operators on H is denoted by B (H) . The norm on H will be denoted by II·II H or 11·11 when there is no confusion. The range of the operator G is denoted by R (G) , its closure by R (G) , I is the identity operator on H and when written with a subscript like IN it denotes the identity matrix on eN. We consider the set of finite energy signals x : [a, b] -+ eN as the Hilbert space L~ [a, b] defined by
f Ix
3 . PERIODIC OPERATORS AS ABSTRACT DISCRETE-TIME STATE-SPACE SYSTEMS Let G be a T-periodic and linear continuoustime systems described by the following standard equations
x=
{x:
IIxllif[a,b] =
y = C 2 (t) x 2
(t)1 dt <
oo}
a
where 1.1 denotes the usual Euclidian norm on eN and the integral is in the Lebesgue sense. For notation convenience, we set X = L~ [0, T) for some positive real T . Next we consider the set of finite energy sequences with range in X, x : N --+ X, i.e., x = (xn):'=o as the l2-space over the Hilbert space X
l2 (X)
= {x = (xn):'=o jXn EX : 00
Ilxll~2(X)
=
+ Bl (t) W + B2 (t) u
z=Cl (t)x+D ll (t)w+D 12 (t)U
b
Lf [a , b] =
A (t) x
L IIXnlli < oo} n=O
480
(1)
+ D2l (t) W + D22 (t) U
where x denotes the N -dimensional state of G and the matrices in the realization have dimension compatible with the appropriate signals w, u, z, y, and x. For simplicity and for clarity of exposition we restrict our discussion to one-dimensional signals w, u, z and y. We assume that all the matrices are T-periodic and depend continuously on the parameter t and we view G as a linear operator on L2 [0,00). From the key lemma, G is input/output equivalent to a shift invariant system Gon l2 (H) . The question we intend to solve in this section is how to describe the full internal behavior of G as reflected in G. For an initial state x(O), the continuous-time state evolution is given by the variation of constants formula
J~(t,T)B1(T)W(T)dT t
x(t)=~(t,O)x(O)+
defined by
y= rx ~ y(t) = (rx) (t) =
o
(7)
t
+
J~(t,T)B2(T)u(T)dT
a.e on 0 ~ t < T . Clearly F is a linear operator on X and moreover it belongs to B (X) . For the inhomogeneous periodic differential equation of system (1), one obtains the following autonomous first order inhomogeneous difference equation (see the appendix)
(2)
o where the transition matrix ~ (t, T) is a fundamental matrix of the state equation of system (1). Since A(t) is T-periodic,
such that (Brauer and Nohel, 1989) x (t)
T
qn+1 = Rn
zn = &qn
~n
(T, 0) =
(9) (10)
where 81 , 82,Cl, C2 , 13 11 , 1312 , 1321 , 1322 are bounded linear operators defined by &:X--+ H qn --+ &qn
(5)
(&lin) (t) =
Consider the zero-input state response, at time t + T, the state is given by
8j : ~
(11)
Ci (t) qn (t)
(12)
~~
(13)
--+
rn --+ Bjr n
x (t + T) =
=
(8j rn) (t)
and moreover for n ;::: 1
=
J
(t , () Bj (() rn (() d( (14)
o
x (nT + t) =
13ij : ~
--+ ~ H~ rn --+ 'Dijrn
The significance of equation (6) is that if a solution is known over one full period, then the solution is known for all time. This is in fact a restatement of Floquet theorem for periodic differential equations in matrix form. A symbolic way of writing equation (6) is to view x (t) as a trajectory composed of successive "segments" over one full line period. Precisely, let denote the lifting of x , i.e.,
i,j = 1,2 and 0 ::; t
Xn (t) = x (t + nT), 0 ~ t < T, nE N then from equation (6), the components of the lifted state verify the abstract first-order homogeneous difference equation
Xn+1
= rxn
X
-+
X
1-+
Xn+l
Remark 1. A different infinite-dimensional model specialized to sampled-data systems is proposed in (Yamamoto, 1994)where the state-transition operator is an operator on C (0, T] (the Banach space of continuous functions with the uniform norm). The state-transition operator as described
with the state-transition operator, which we call the discrete semigroup generator of the system
xn
(15)
and rn stands for wn or Un. The state variable qn in equation (8) is the lifting of a new variable (see appendix - equation (27) ) and in fact , it embeds the original state xn of the homogeneous system since for zero-inputs wn = Un = 0, qn reduces to xn . It is also interesting to note that the values of qn and xn match exactly at t = O. Since all the signals and the state in equations (8), (9) and (10) are distributed discrete-time signals, the distributed parameter being the continuoustime variable t (0 ::; t < T), it is clear that these equations establish a ''faithfull'' correspondence with the full behavior of the original continuoustime periodic system.
x
F:
+ 13 n wn + 1312 un
(4)
(8)
the output equations are given by
(3)
= 0, (3) yields
+ 81wn + ihfin
481
°: :;
x
for (t) i= 0 almost everywhere on t < T, if and only if A = Aj , j = 1, 2, ... ,p; so the set of eigenvalues of the matrix cP is the point spectrum of the discrete semigroup generator J . Let A i= Aj, for j = 1,2, ... , p , then (16) is solvable and
in (Yamamoto, 1994)contains a delta like distribution (in fact, an N-tuple functionals on C (0, TJ) and acts as a "point operator" in the point t = T; actually this operator is finite-dimensional and it would be unbounded if it is to be defined on Lp spaces. Thus compared to Yamamoto (1994), our model is less restrictive and it is well suited to modem control problems when considering energy cost on the state.
x= (AI -
J) -1 Y
{=}
x(t) = (>..IN - cp)-l y(t) (18)
J)
-1 defines for 0 :::; t < T.Clearly (>..I _ a bounded linear operator on all of X, i.e.,
4. STRUCTURE OF THE SPECTRUM
J)
(>..I -1 E B (X) .Thus p and the spectrum is
We study the distribution of the complex values A for which the operator-valued function >..I - J has an inverse. First, we give the following definitions (Naylor and Sell, 1982) , (Yoshino, 1993)
Observe that J is not a compact operator on X since cP being always a non-singular matrix and the point spectrum being a finite set, A = 0 is neither an eigenvalue nor a point of accumulation of a sequence of eigenvalues.The main result of this section is the characterization of the spectrum of the discrete semigroup generator in the following theorem.
Definition 1. Given A E B (X) , a point A E C is called a regular point of A if >..I - A is invertible, i.e., (AI - A)-l exists as a bounded operator with domain X . The set p (A) of regular points is called the resolvent set of A. The spectrum i:T (A) of A is the complement of p (A) . The spectrum can be sorted into more precise classes as follows
Theorem 2. Each point in the spectrum an eigenvalue of infinite multiplicity.
Definition 2. i:Tc
(A)
t::. = {>.. E C; R(>..I -
A)
(A) ~ {>.. E C; R (>..I - A)
i= X
and (>..I - A) is injective} i:Tp
(A) ~ {A E C; (AI - A) is not injective }
are said to be the continuous spectrum, the residual spectrum and the point spectrum of A respectively. These sets are mutually disjoint and i:T
(A) =
i:Tc
i:T
(J) is
Proof. Indeed pick one eigenvalue of J , say AI , and let us assume that it has finite multiplicity. Thus the corresponding eigenspace EA! is a finite-dimensional subspace of X. Let tP1' tP2' ... , tPr be any orthonormal basis in it. By the Riesz's lemma (Naylor and Sell, 1982)and the Hilbert space structure of X , using the Gram-Schmidt process it is possible to construct an infinite orthonormal sequence {tPn}~ such that the first r vectors are tP1' tP2' ... , tPr . Since by hypothesis Al is of finite multiplicity, there exist a compact operator on X such that Al E P + Then
= X,
(>..I - A) -1 exists and is unbounded}
i:T r
(J) = C\ {AI, ... , Ap}
fC (J fC) . All -J -fC is invertible and in particular we have
(Yoshino, 1993) , for some 8 > 0
(A) U i:T r (A) U i:Tp (A)
811xl~ :::; 11 ( All - J -
and A E i:TP (A) if and only if there exists a nonzero vector x E X such that Ax = ,Xx.
fC) xii,
for all x E X (19)
Now consider the sequence {tPn}~ c X , we obtain with (19)
Let AI, A2, ... , Ap ( p :::; N) be an enumeration of the eigenvalues of the matrix cP (with possible multiple eigenvalues) . Consider the following equation on X
811tPn - tPmll :::; 11 (All - J) (tP n - tPm)11 + IIfC (tPn - tPm)1I (20)
(16)
and for n, m > r , the sequence vectors tPn ~ and therefore there exist "Y > such that
°
Now (( >..I -
J) x) (t) = (>..IN -
CP) x(t)
= 0 (17) 482
EAl
We can choose 'Y such that 'Y < 6, and the following inequality follows from (20) and (21)
6. REFERENCES Barnieh, B. and Pearson J.B. (1992), A general framework for linear periodic systems with applications to ?too sampled-data control, IEEE Trans. Automat. Contr. , vol. 37 , pp. 418-435 Brauer, F. and Nohel J.A. (1989) The Qualitative The
But this contradicts the compactness of operator
JC and thus the finite multiplicity of the eigenvalue ).1 ' Since ).1 has been chosen arbitrarily, the same reasoning applied to all others eigenvalues in the spectrum. In the course of the proof of the above theorem, we have thus shown the following
Theorem 3. For every ). E
(J'
(1),
there is a
sequence {
(i) lI
()'I - 1)
A sequence {
(1)
are verified is called the essential spectrum of 1 and it is denoted by (J' ess Thus a complete characterization of the spectrum can be stated as
7. APPENDIX-DERIVATION OF THE DISCRETE-TIME MODEL
(1).
To avoid cumbersome notations, define an input r = (w ,u)T and an input matrix B = (Bl B2) . The zero-state response of system (1) at time (t + nT) is given by (see formula (2))
Corollary 4. The spectrum of the discrete semigroup generator 1 consists entirely in an essential spectrum.
J
t+nT
x(t+nT)=
o
5. CONCLUDING REMARKS
(23)
In this note, we have presented a state-space representation for continuous-time periodic systems on an abstract space and the emphasis has been made on the internal behavior as opposed to the external approach. A complete characterization of the spectrum of the system has been obtained. Our motivation for introducing such a framework comes from a desire to understand system theoretic properties in the study of sampled-data control systems from a continuous-time viewpoint. Indeed, the assumption of finite-dimensionality of the state in the lifting framework hides important questions regarding systems theoretic concepts such as stabilizability and controllability and it is not at all clear what are the limitations of the performance of sampled-data controls inherent to their structure. These questions remain important subjects under current investigation and we believe that the state-space representation provided in this note is an appropriate one for the study of such questions.
Split the integration in (23) from 0 to t+(n - 1) T and t+ (n -1) T to t+nT and use relation (3) to obtain
x (t + nT) =
+ (n -
1) T)
t+nT
J
+
t+(n-l)T (24)
for all t 2: 0, and n E N. Now assume that 0 $ t < T , and split the integration in (24) as the sum of integrals over the intervals [(n -1) T + t , nT] and [nT, nT + tj to yield nT+t
x(t+nT) -
J
nT
=
483
nT
+
J
T
q,(t+nT,T)B(T)r(T)dT
J
1Pq, (t + (n - 1) T, ( + (n - 1) T) .
(25)
o
(n-l)T+t
B (( + (n - 1) T) r (( + (n - 1) T) d(
Note that the left hand side of equation (25) is related to information up to time nT + t whereas the right hand side is related to information up to time nT. The integral in the right hand side of (25) can be written as (the integrand has been omitted for simplicity) nT
nT
Thanks to the T-periodicity of matrices B (t) and A (t), equation (28) reduces to q (t
+ nT) = IP . q (t + (n - 1) T) T
+
(n-l)T+t
J (-) J (-) - J (.) (n - l)T
(29)
(n-l)T
Equation (29) describes the same dynamics as (23) with a new state variable defined by (27) . For the output equations of system (1) , introduce the output vector s= (z, and matrices C =
and using relation (3) in the second integral of the right hand side of the above identity, equation (25) can be written as
( g~ ) and D
nT+t
J
{x(t+nT)-
1Pq,(t, ()B(()r((+(n-1)T)d(
o
=
(n-l)TH
J
=
yf (g~~ ~~~) , then at time
t + nT, one obtains the output equation in terms
q,(t+nT, T)B(T)r(T)dT}
of the new state q
nT
= IP . {x (t
+ (n -
1) T) -
s (t
(n-l)TH
J J
+ nT) =
+ nT)
nT+t
q,(t+(n-1)T,T)B(T)r(T ) dT}
+
(n-l)T
J
C(t)q,(t+nT,T)B(T)r(T)dT
nT
nT
+
C (t) q (t
+D (t) r (t + nT) q,(t+nT, T)B(T)r(T)dT
(26) which is written, after the change of variable T = nT + ( and assuming that 0 ::; t < T,as
(n-l)T
A key observation here is that equation (26) can be viewed as an evolution equation for the quantities in the parentheses. Define a new variable q as
s (t + nT) = C (t) q (t + nT) + t
J
{C (t) q, (t , () B (()
+ D (t) c5 (t -
()) r (( + nT) d(
o q (t
+ nT) =
+ nT)
x (t
- j+tq, (t
(30) where c5 (t) is the Dirac distribution. Let denote the lifting of q, rand s , i.e.,
+ nT, T) B (T) r (T) dT
nT
, qn (t) = q (t + nT) , Tn (t) = r (t + nT) , sn (t) = S (t + nT)
(27) and write simply equation (26) as q (t
+ nT) =
IPq (t
+ (n -
J (n-l)T
q,r, S
with 0 ::; t < T, n E N. Now, in terms of the lifted variables, equations (29) and (30) are written as
1) T)
nT
+
qn+l = Fqn + Brn sn = Cqn + 15rn
q,(t+nT,T)B(T)r(T)dT
(28)
(31)
where F, 8, Cand 15 are bounded linear operators defined on appropriate Hilbert spaces (see equations (11)-(15) for the details) .
Now, it is straightforward to show that the kernel of the integral operator in the right hand side of equation (28) is independant of n for all n E N. Indeed, use relation (3) and make the change of variable T = (n - 1) T + ( , dT = d( to express the integral in equation (28) as
484