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On Some Structural Properties of Sampled-Data Systems
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Copyright 10 IFAC System Structure and Control, Prague, Czech Republic, 2001
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ON SOME STRUCTURAL PROPERTIES OF SAMPLED-DATA SYSTEMS
Joseph-Julien Yame, Raymond Hanus Service d'Automatique et d'Analyse des Systemes, C.P. 165 Faculte des sciences appliquees, Universite libre de Broxelles, 50 av. F.D. Roosevelt, Brussels-1050, Belgium Fax:32-2-650.26.77, email: [email protected]
Abstract:
The study of some structural properties, namely controllability and spectrum distribution, in a hybrid system composed of a continuous-time system controlled by a digital controller is carried out in the lifting framework. The novelty of our approach lies in the infinite-dimensional nature of the state-space in the lifted domain due to the fact that the system evolves over a temporal continuum as opposed to the finite-dimensional state-space with infinite-dimensional input/output systems currently used in the literature. Different concepts of controllability related directly to the continuous-time behavior of sampled-data systems are introduced, namely exact, approximate and null controllability. It is shown that the two former notions never occur in sampled-data systems while the latter notion is a generic property for these systems. Some facts regarding the fundamental input/ output structure of sampled-data systems from the control viewpoint are drawn from these results. The spectrum distribution of sampled-data feedback systems is also characterized. Copyright ©2001 [FAC
Keywords: : sampled-data systems, controllability,
1. INTRODUCTION
lifting, spectrum distribution
viewpoint. In order to describe faithfully the internal behavior of sampled-data systems in the lifting framework, it is quite natural to consider the infinite-dimensional nature of the state-space due to the fact that the system evolves over a temporal continuum, the distributed parameter being the continuous-time variable. This is the approach taken in Yame and Hanus (1999) where a new representation of sampled-data systems is derived and it is shown that such systems can be considered as a class of infinite-dimensional state-space discrete-time systems with the following structure: (i) the state-space is a Hilbert space, (ii) the evolution of the state is governed by a discrete semigroup with bounded generator, (iii) the control space is finite-dimensional. In this paper, the notions of exact, approximate and function space null-controllability which are directly related to the continuous-time behavior of sampled-data systems are introduced via the above representation. Starting from the functional
Sampled-data systems are periodically time varying in continuous time and recently their analysis has been simplified by the use of the lifting technique (Bamieh and Pearson, 1992; Yamamoto, 1994; Chen and Francis, 1995). From an external point of view, the lifting provides a correspondence between continuous-time periodic systems and a kind of discrete-time shift-invariant systems, called the lifted systems, with infinitedimensional input and output spaces. In the litterature, except in Yamamoto (1994) , the state-space of the lifted models for sampled-data systems is chosen finite-dimensional at the outset, the state being the value of the continuous-time state at the sampling instants, that is, the state of the pure discrete-time models. Clearly, such a state rules out the internal intersample dynamics and this hides some important structural properties of sampled-data systems from the continuous-time
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for nT < t < nT + T , and tn = nT with n E N and where A E ff xN is the system matrix, B E Nx1 is the input matrix, h (t) is a T-periodic generalized hold real function and v [n] is the digital control applied at time instants tn . For clarity of exposition, the system is taken to be single input. It is shown in Yame and Hanus (1999), using a general state-space representation for periodic systems (Yame and Hanus, 1998), that the continuous-time dynamics of the hybrid system is faithfully described by an abstract difference equation on the Hilbert space X = L!j [0, T) given by
framework, it is shown that sampled-data systems are never exactly nor approximately controllable while null-controllability is the "generic" property for these systems if the original continuous-time systems are controllable in the classical sense. In spite of the research effort focused on the continuous-time behavior of sampled-data systems within the lifting framework, see e.g. Chen and Francis (1995) and references therein, to our knowledge there has been no study of the systemtheoretic concept of controllability from this point of view and this paper is intended to fill this gap by revisiting this important structural property. Note however that it has been conjectured in Kabamba and Hara (1993, remark 3.1) that intersampiing reachability, which is actually related to our notion of approximate controllability, can be achieved in a sampled-data system for almost all intersampling instants. Our study gives a definitive negative answer to such a conjecture. Even though the controllability results obtained seem to be trivial afterwards, they have rather profound implications on the characterization of the structure of sampled-data systems in the lifting framework with regard to control problems such as tracking and disturbance rejection. Other important implications are in problems related to spectrum assignment in continuous-time periodic systems within the paradigm of generalized sampled-data hold function control as evidenced by some misleading statements appeared in the litterature (Chen M-S and Y-Z. Chen, 1999). The closed-loop system operator of a sampled-data system is also characterized, it is shown that it is Fredholm and its spectrum is always a superset of the spectrum of the open-loop system operator.
e
(2)
where the state qn is a X -valued function and where the linear bounded operators :F and B are defined by :F : X -+ X, (:Fqn) (t) = eAT.qn (t)
B : C -+X, (Bv [n]) (t)
= loT eA(T+t-T) Bh (T) dT.V [n] (4)
for t E [0, T). Since the control space is onedimensional, the action of operator B reads simply as Bv [n] = bv [n], that is, (Bv In]) (t) = v [n] b Ct) on [0, T) where b E X and is given by
b(t)
= IoTeA(T+t-T)Bh(T)dT,
tE[O,T)
°
°
x[n+1]=~x[n]+,(T)v[n]
2. DISCRETE-TIME EVOLUTION OF SAMPLED-DATA SYSTEMS
{ x (t n ) =
+ Bh(t)v[n], lim x (tn + l)
Ax(t)
(6)
where ~ = eAT, ')' (T) = b(O) and x [n] = x (nT) . For any control signal v E £2, the "lifted state" sequence is determined from the initial state qo by iteration of the state evolution equation (2):
A finite N-dimensional continuous-time timeinvariant plant controlled by a digital computer at sampling rate l/T yields an overall hybrid system, called also a sampled-data system, with continuous-time dynamics described by the differential equation with boundary conditions
=
(5)
It is worth observing that the state at discretetime n is a trajectory segment qn (.) = {qn (t) , :s t < T}, the distributed parameter domain being the continuous-time interval [0, T), and therefore the intersample dynamics of the sampleddata system is built in the infinite-dimensional discrete-time equation (2) . The value of the state qn at t = 0, i.e. qn (0) , is the value of the original continuous-time state x at time nT and the sampled-behavior of the hybrid system (1) is described exactly by the abstract equation (2) at the boundary point t = of the distributed domain [0, T) , that is,
Notations. For two Hilbert spaces 7-£1,7-£2 and linear operator A : 7-£1 -+ 7-£2 , the domain and the kernel of A in 7-£1 and the range of A in 7-£2 are denoted respectively by 1) (A), ker (A) and n (A). The set of all linear bounded operators from 7-£1 into Ji2 is denoted by B (Ji 1, Ji2) . The inner product and norm in a Hilbert space Ji are denoted by (., .)1i and 1I.1I 1i . The space L~ [0, T) denotes the classical L 2 -Hilbert space of functions on [0, T) with values in eN.
x(t)
(3)
n-l
qn
= Pqo + L:FkBv[n -
k -1]
(7)
k=O
The original continuous-time state x on [0,00), Le. the unique solution to differential equation (1), is faithfully represented by the sequence {qn} n~O .
(1)
<-+0+
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3. REACHABLE SUBSET AND FUNCTION-SPACE COKTROLLABILITY Let us set
== qn - Fnqo , then the sequence
Zn
{Zn}n~O C X is given by
~oo : £dN -+ 0
n-l k Zn==Zn(.;v)=LF Bv[n-k-l]
(8)
[v]
k=O
Fk Bv [k]
\
Ilz(k) Ilx
(9)
Now, the reachability subset 0 with the topology induced by the norm 11.11 0 is viewed as the Hilbert space (0, (., ')0) which is a closed manifold in that topology. An important structure property of the input map (9) of sampled-data systems is the following: Proposition 2. The operator compact.
(10)
It is immediate from proposition 2 that sampleddata systems are never exactly controllable. Now, we show that the Hilbert space (0, (., ')0) cannot be densely embedded in X and for this, we need the following theorem (Conway, 1990):
Proof. Consider the closed linear manifold N
= ker~oo = {v E £2 : ~oov = O} C £2
Then the quotient space £2/N , which is the set of equivalence classes [v] = v + N, is closed (Conway, 1990) and
Theorem 4. If A E B (11. 1,11.2) is a compact operator and (A) is closed, then (A) is finite dimensional.
n
Il[v]II[2/N == /lw.,l IIJLIIl2 E
£2 and JL - v E
E B (£2, X) is
Definition 3. The sampled-data system (2) is exactly controllable if the reachability subset (10) is equal to X. It is approximately controllable if (0, (., ')0) is densely embedded in X. The sampled-data system (2) is function-space nullcontrollable in time n if every state can be transferred to the zero function of X in time n
Proposition 1. The reachability subset 0 c X is isometrically isomorphic to a Hilbert space and the injection i : 0 -+ X is continuous, that is, the identity map i from 0 into the Hilbert space X is a bounded operator.
: JL
~oo
We introduce the following notions of function space controllability:
This subset is clearly a linear submanifold of the Hilbert space X, and our aim is to provide it with a Hilbert space structure, Le. the structure of a closed linear manifold.
= inf {IIJLlI l2
Ilz(k) 110 :s
Ilz(k) IIx
n~O
N
[2/./If
:s
k=O
U On == n (~oo) c X
z( .;v)
and (0 , (., ')0) are isometrically isomorphic. Next, we prove that the natural injection i : 0 -+ X is continuous, i.e., lIi (z)lIx == IIzllx M IIzllo . Assume that this last inequality does not hold, then there is a sequence z(k) in 0 with norm = 1 in X such that 1/ k. Let [v(k)] == ~;;,lz(k) and JL(k) be the representative of [v(k)] with least minimum norm, then JL(k) -+ o in £2 and z(k) == ~ooJL(k). Clearly, z(k) = z (.; JL(k)) -+ 0 in X which contradicts the fact that == 1 •
This is a well-defined operator since convergence is not an issue on D (~oo) which is the subset of finitely non-zero sequences in £2. If ~oo is extendable as a bounded operator to the closure of Un>oD (~n) in £2 we say that the sampleddata system (2) is input stable. We assume in the sequel that the sampled-data system (F, B) is input stable or equivalently that the operator ~oo E B(£2'X), The set of all z E X reachable at discrete time n starting at initial time 0 is On ={zn(.;v) = ~nv:Vv E D(~n)} and the reachability subset of the sampled-data system (F, B) is defined as
o=
= n (~oo)
~oo an isometric map which implies that £2/N
00
=L
H
and it allows us to define a scalar product on 0 by (zr, Z2)0 = / ~;;,l Z t, ~;;,l Z2) . This makes
Introduce the maps whose domains of definition are D (~n) = {v E £2 : Vk == 0 for k ~ n} and defined by ~n : D (~n) -+ X with ~nv = Zn (.; v) as given by (8). Now, consider the operator ~oo defined as follows: its domain of definition is the set D (~oo) = Un>oD (~n) which is a dense subset of £2 and the action of ~oo : D (~oo) -+ X on an element v in this domain is given by
~oov
i 2-norm 11·11[2 ' Actually £2/N is a Hilbert space (Hilbert quotient space) and if JLI and IL2 are representatives of [VI] and [V2] , the inner product in £.}./N is defined as ([VI]' [V2])l2 /N = (JLI,IL2)[2 ' The following map is bijective
n
It is seen from the proof of proposition 1 that ~oo == i~ooQ where Q : £2 -+ ldN is the canonical surjection. Since i is the natural injection of (0, (., ')0) into X, the compactness of ~oo
N}
is a norm on £21N.Note that a representative of this coset is a control JL E £2 which has the least
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temporal nature and therefore the "preservation of controllability" from a continuous-time system to its pure discrete-time model, as evidenced by the equality of the state-space dimensions (N) of (1) and (6) , does not reflect that the original set of reachable continuous-time states has collapsed to a "very small" set of continuous-time signals which is R (Q3 oo ).
implies that either ~oo or Q or both operators are compact on the corresponding spaces. Assume first that the surjection Q is compact, then by the surjectivity of Q it follows that the Hilbert space f2/N is finite-dimensional and hence (0, (., ')rJ is finite-dimensional which in turn implies that sB oo is compact. Now, if we assume that ~oo is compact, since its range is the closed manifold (0, (., ')0)' theorem 4 implies that this range is finite-dimensional. Thus in either case, the reachability space (0, (., ')0) is finite-dimensional and this yields the desired negative result on approximate controllability which is stated as:
The following insightful facts for tracking and disturbance rejection problems in continuous-time with regard to the structure of sampled-data systems are drawn from proposition 5.
PropoSition 5. The sampled-data system (:F, B) is never approximately controllable.
Fact 1. Lifted exogenous signals that can be perfectly tracked by sampled-data systems are intrinsically finite-dimensional valued discrete-time signals.
Afterwards, it is now a triviality to say that R ('Boo) given by (10) is closed in the topology of X. Moreover, proposition 2 is strengthened since 'l3 00 is revealed to be a finite-rank operator. As R('Boo) is a closed linear manifold of X, it follows from Kreyszig (1978, theorem 3.3-4) that X is decomposed as the direct sum
Moreover, from the lifted (infinite-dimensional) input/output perspective, this structure is exhibited by the following stronger fact: Fact 2. Sampled-data systems, from the control viewpoint, are essentially "finite-dimensional input" and "finite-dimensional output" discretetime systems.
(11) where Xc = R (Q3 oo ).L, which is the orthogonal complement of R (Q3 oo ) in X , is infinitedimensional. Note that approximate or exact controllability in the function space is indeed the usual controllability concept in continuoustime, that is, when the control signal is a L2 [0, T) -valued sequence then the set of reachable states might (densely) fill the function space X. Thus sampling destroys irremediably the usual controllability of continuous-time systems.
Proof. Note that fact 1 is a consequence of fact 2. Consider the following input-to-state/state-tooutput (infinite-dimensional) equations {
qn+l Yn
= :Fqn = Cqn
+Bv [n)
+ Bwwn
(12)
where w = {wn}:'o E £2 (x.:) is the lifting of a continuous-time exogenous signal w E Lz [0,00) and x.: =L 2 [0, T) . Assuming v = 0, the exogenous signal feeds the quantity q = '.Bw.oow = l:~o:Fk BwWk into the state space X. The map Q3w.oo is actually equivalent to the injective map 12 (x.:) / ker Q3w.oo -+ X since the same quantity q is obtained from any representative [w) E £2 (x.:) / ker Q3w.oo of w in £2 (x.:). Thanks to this equivalence, this injective map will be denoted by the same operator symbol Q3w,oo' The direct sum decomposition (11) of the state space implies that
Comments 6. The above fact has led the authors in something of a dilemna: on one hand it should be extremely well known, at least informally, and in the other hand the authors have not been able to find an adequate reference establishing proposition 5. Notwithstanding its apparent triviality, it has been conjectured in Kabamba and Hara (1993, remark 3.1) that intersampling reachability (which is actually the same notion as approximate controllability) is possible; this study proves that this conjecture is false even if the input to the continuous-time system is generated by a (optimal) generalized sampled-data hold function. Perhaps, this fact has not been emphasized in the litterature for the reason that controllability of a sampled-data system is usually understood in the sense of the controllability of its pure discrete-time model which is related solely to the controllability of the discrete states defined at the sampling instants (Hautus, 1969; Kalman et al, 1963). However, the state signals of systems (1) and (6) are actually not of the same
q = qc
+qc =
Q3w,oo
[w)
(13)
with qc E 0, (qc, qc) x = 0 and + is the direct sum operation. Note that a control signal v has no possible effect on the part qc resulting from the exogenous signal. The linearity of Q3w.oo and identity (13) imply that qc = Q3w.oo [wc] arld qc = Q3w.oo [Wc) for some [wc) ,[Wc) E £2 (x.:) / ker Q3w.oo such that [w) = [wc] + [Wc] (note that this is not a direct sum). The set of all [wc) is denoted by Wc, Le., Wc ={[w] E £2 (x.:) / ker Q3w.oo:'.B w•oo [w] E O} and it is clearly a closed linear manifold of
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£2 (x:.) I ker'Bw,oc' Let 0 1 ~ 0 denotes the set of all qe such that there is some [w] E Wc with qe = 'Bw,oc [wc]. Then there exists a map 'Bt,OQ : 0 1 -+ Wc such that 'Bt,oc 'Bw,oc [wc] = [wc], that is 'Bt,oc 'Bw,oo i:; the identity map on Wc' Since 'BLoc is compact (its domain 0 1 is finite-dimensional), the identity map is compact on Wc and hence Wc is finite-dimensional. Therefore each component of any sequence in "'e is a finite-dimensional vector and the set of all such vectors lies in a common finite-dimensional space. The assertion of fact 2 results from the fact that the "part" of the continuous-time output signal {Yn} which can be shaped by controls is a "static readout" map of the controllable (reachable) continuous-time state trajectory. •
Proposition 7. The sampled-data system (F, B) is function space null controllable if and only if rank ([4>N-1, (T) 14>''''-2, (T) 1...1, (T)]) = N
This proposition is the well-known necessary and sufficient conditions for "sampled-controllability" (Hautus, 1969), i.e. the controllability of the pure discrete-time model (6) so that function space null controllability is actually equivalent to "sampledcontrollability" and this equivalence is tight. Indeed, that "sampled-controllability" implies function space null controllability is long known since the early contribution of Kalman et al (1963) and this is pretty obvious. But that the converse holds true, i.e. that function space null controllability implies complete "sampled-controllability" (not only controllability to zero at sample-points), seems a little strong, however this cannot be weakened. Therefore, function space null controllability is seen to be the strongest controllability concept one could consider for sampled-data systems from a continuous-time viewpoint. Clearly, the property of null-controllability over the data set (A., E, h (t) ,T) is generic in the sense that any system in that class can be modified ever so slightly into one with that property.
:=0
From the F -invariance of the reachability subspace, the above proof leads to an obvious decomposition of the state equation (12) in which the controllable part of the continuous state trajectory "sees" a finite-dimensional exogenous input extracted from {w n }
:=0 .
4. CHARACTERIZATION OF FUNCTION SPACE NULL CONTROLLABILITY Null-controllability at time n in the functional space framework means that the continuous-time output is identically zero for almost all t E [kT, kT + T) with k 2: n. From equation (7), it is seen that the sampled-data system (F, B) is null controllable at time n , i.e. qk = 0 for k 2: n, if and only if there exists a control signal 1/ = {I/[k]}~;:~ E £2 such that n-1 Pqo = p- k - 1 BI/ [k] (14)
5. SPECTRUM DISTRIBUTION Consider a static state feedback bounded operator on X, K : X -+ C such that the input to system (2) is given by 1/ [n] = Kqn + {[n] and resulting in the closed-loop system
L
k=O
As equation (14) holds pointwise for almost all t E [0, T) , it yields the next identity in
([f
n-1
4>nqo (t)
=L
4>n-k-1 eAt, (T).v [k]
(15)
k=O
where we have used the defining equations of operators F and B and v [k] = -1/ [k] ,or explicitly
4>nqo (t)
= eAt [4>n-1, (T) l4>n-2, (T) I... h (T)] .:!L (16)
noticing that eAt commutes with all4>k (k=0,1,2, ... ) and :!L is the vector whose transpose is :!LT = (v [0], v [1], ...v [n - 1]) . From the non-singularity of the N x N matrix eAt and using standard arguments, i.e. the Caley-Hamilton theorem related to the N x N matrix 4>, the existence of a control signal {v [k]} :=~1 E £2 which realizes the identity (16) depends on the invertibility of the bloc matrix in the right hand side for n = N . We obtain the following proposition:
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qn+1 = (F + Bx:.) qn + B{ [n] (17) Usually, it is desirable to choose K such that stability in some sense is obtained in (17) . Let say that the sampled-data system (2) is strongly stable if its discrete seInigroup converges to the zero operator in the strong operator topology, i.e., IIFnqllx -+ 0 as n -+ 00, for all q E X. In Yame and Hanus (1999, theorem 4) it is shown that, under some mild conditions which turn out to be null-controllability, the sampled-data system (2) can be stabilized in the sense of strong stability. The resulting feedback system as given by (5) has a discrete semigroup generator :FBK. = F + BK where operator BK is a compact operator on X. We first notice that the closed-loop discrete seInigroup :FBK. belongs to a very special class of the Banach algebra of bounded operators on X. Proposition 8. The closed-loop discrete semigroup :Fm::: is Fredholm
Proof. It is known from Yame and Hanus (1998) that the open-loop discrete semigroup generator F is always nonsingular and boundedly invertible , thus we have F-1 FBK. = 1 + K1 and
FBlCF- l = I + K2 where Kl = F- l BK and K2 = BKF- l are obviously bounded compact operators on X. Clearly operator FBlc is invertible modulo compact operators, and therefore it is Fredholm (Conway, 1990) . • We would like to characterize the structure of the spectrum U (FBK.) of the closed-loop discrete semigroup generator and for this, we quote a proposition from Yame and Hanus (1998) which describes the spectrum of the open-loop semigroup generator F on X . Proposition 9. Each point in the spectrum is an eigenvalue of infinite multipliCity
U
(F)
Now, from the compactness of the perturbation BK, one has immediately a partial information on the structure of the spectrum of FBK.. Proposition 10. The spectrum U (FBK.) of the closed-loop semigroup generator is a superset of u(F)
Proof. It is a direct application of Weyl theorem (see Kato (1980, Theorem 5.35)) • We need the next lemma (Kato, 1980, p. 243) to characterize the whole structure of the spectrum of FBK. Lemma 11. If A is a linear bounded operator on X and the complement in C of its essential spectrum U ess (A) is connected, then U (A) \u w (A) consists of isolated eigenvalues with finite algebraic multiplicity The essential spectrum Uess (FBK.) of FBK. is the set of all A E C such that )J - F BK. is not Fredholm and thus UW (FBK.) = Uess (FBK. +C) holds true for all bounded compact operators C on X. From FBK. = F + BK and proposition 9, we have Uess(FBK.) = Uess(FBK.+C) = uess(F+C 1 ) = U (F) where Cl = BK + C is compact. Therefore the essential spectrum of FBK. is exactly U (F), and since U (F) consists solely of isolated eigenvalues (Yame and Hanus, 1998), it is obvious that C\u eS8 (FBK.) is a connected set and consequently U (FBd \u m (FBK.) is a set of finite type eigenvalues. We are now able to give the structure of the spectrum of F BK. in the proposition below: Proposition 12. The spectrum of the closed-loop semigroup generator consists in the union of an essential spectrum and a discrete spectrum (Le., a set of isolated eigenvalues of finite multiplicity)
Remark 13. The significance of proposition 12 is that sampled feedback controllers, although they have a stabilizing property, do not have arbitrary
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spectrum relocation capabilities since the closedloop semigroup generator has its spectrum which always contains the spectrum of the open-loop semigroup generator. The best one can hope is to introduce only some discrete spectral values and nothing else in the original spectrum. It is worth noticing that proposition 12 is closely related to the lack of approximate controllability. As the development of this paper extends mutatis mutandis to periodic systems, and from the fact that the set of characteristic exponents of such systems are mirrored in the discrete (lifted) domain by the spectrum U (F), one is led to the following conclusion: it is impossible to assign the characteristic exponents in continuous-time periodic systems by sampled output feedback controllers no matter what the generalized hold function might be, contrary to the result recently stated by Chen M-S and Y-Z. Chen (1999, theorem 1) .
6. REFERENCES Bamieh, B. and J.B. Pearson (1992). A general framework for linear periodic systems with applications to 1i. oo sampled-data control. IEEE 7rans. Automat. Contr., Vol. 37, pp. 418-435 Chen M-S. and Y-Z. Chen (1999). Static Output Feedback Control for Periodically Time-Varying Systems. IEEE 7rans. Automat. Contr., Vol. 44, pp. 218-222 Chen, T. and B.A. Francis (1995). Optimal Sampled-Data Control Systems Springer-Verlag, Berlin Conway, J.B. (1990). A Course in Functional Analysis Springer-Verlag, New York Hautus, M. L. J . (1969). Controllability and Observability Conditions of Linear Autonomous Systems Proc. Koninkl. Nederl. Akademie Van Wetenschappen series A, vol. 72, pp. 443-448 Kabamba, P.T. and S. Hara (1993). Worst-Case Analysis and Design of Sampled-Data Control Systems. IEEE 7rans. Automat. Contr.Vol. 38, pp. 1337-1357 Kalman, R.E., Ho, Y.C. and K.S. Narendra (1963). Controllability of Linear Dynamical Systems. Contributions To Differential Equations, Vol. 1, pp. 189-213 Kato, T . (1980). Perturbation Thwry for Linear Operators Springer-Verlag,Berlin Kreyszig, E. (1978) . Introductory Functional Analysis with Applications John Wiley & Sons, New York Yamamoto, Y. (1994). A Function Space Approach to Sampled-Data. Control Systems and Tracking Problems. IEEE 7rans. Automat. Contr.Vol. 39, pp. 703713
Yame, J.J. and R. Hanus (1998) . An infinite-dimensional discrete-time representation for periodic systems. Proc. of the IFAC Conference on System Structure and Control Nantes, France, vel. 2, pp. 501-506 Yame, J.J. and R . Hanus (1999). Representation and Stabilizability of ContinuoU5-time Systems under Digital Controls. Proc. IEEE ConJ. on Decision and Contro~ Phoenix, Arizona, pp. 3434-3439
Copyright 10 IFAC System Structure and Control, Prague, Czech Republic, 2001
c:
0
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Publications www.elsevier.comllocate/ifac
ON SOME STRUCTURAL PROPERTIES OF SAMPLED-DATA SYSTEMS
Joseph-Julien Yame, Raymond Hanus Service d'Automatique et d'Analyse des Systemes, C.P. 165 Faculte des sciences appliquees, Universite libre de Broxelles, 50 av. F.D. Roosevelt, Brussels-1050, Belgium Fax:32-2-650.26.77, email: [email protected]
Abstract:
The study of some structural properties, namely controllability and spectrum distribution, in a hybrid system composed of a continuous-time system controlled by a digital controller is carried out in the lifting framework. The novelty of our approach lies in the infinite-dimensional nature of the state-space in the lifted domain due to the fact that the system evolves over a temporal continuum as opposed to the finite-dimensional state-space with infinite-dimensional input/output systems currently used in the literature. Different concepts of controllability related directly to the continuous-time behavior of sampled-data systems are introduced, namely exact, approximate and null controllability. It is shown that the two former notions never occur in sampled-data systems while the latter notion is a generic property for these systems. Some facts regarding the fundamental input/ output structure of sampled-data systems from the control viewpoint are drawn from these results. The spectrum distribution of sampled-data feedback systems is also characterized. Copyright ©2001 [FAC
Keywords: : sampled-data systems, controllability,
1. INTRODUCTION
lifting, spectrum distribution
viewpoint. In order to describe faithfully the internal behavior of sampled-data systems in the lifting framework, it is quite natural to consider the infinite-dimensional nature of the state-space due to the fact that the system evolves over a temporal continuum, the distributed parameter being the continuous-time variable. This is the approach taken in Yame and Hanus (1999) where a new representation of sampled-data systems is derived and it is shown that such systems can be considered as a class of infinite-dimensional state-space discrete-time systems with the following structure: (i) the state-space is a Hilbert space, (ii) the evolution of the state is governed by a discrete semigroup with bounded generator, (iii) the control space is finite-dimensional. In this paper, the notions of exact, approximate and function space null-controllability which are directly related to the continuous-time behavior of sampled-data systems are introduced via the above representation. Starting from the functional
Sampled-data systems are periodically time varying in continuous time and recently their analysis has been simplified by the use of the lifting technique (Bamieh and Pearson, 1992; Yamamoto, 1994; Chen and Francis, 1995). From an external point of view, the lifting provides a correspondence between continuous-time periodic systems and a kind of discrete-time shift-invariant systems, called the lifted systems, with infinitedimensional input and output spaces. In the litterature, except in Yamamoto (1994) , the state-space of the lifted models for sampled-data systems is chosen finite-dimensional at the outset, the state being the value of the continuous-time state at the sampling instants, that is, the state of the pure discrete-time models. Clearly, such a state rules out the internal intersample dynamics and this hides some important structural properties of sampled-data systems from the continuous-time
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for nT < t < nT + T , and tn = nT with n E N and where A E ff xN is the system matrix, B E Nx1 is the input matrix, h (t) is a T-periodic generalized hold real function and v [n] is the digital control applied at time instants tn . For clarity of exposition, the system is taken to be single input. It is shown in Yame and Hanus (1999), using a general state-space representation for periodic systems (Yame and Hanus, 1998), that the continuous-time dynamics of the hybrid system is faithfully described by an abstract difference equation on the Hilbert space X = L!j [0, T) given by
framework, it is shown that sampled-data systems are never exactly nor approximately controllable while null-controllability is the "generic" property for these systems if the original continuous-time systems are controllable in the classical sense. In spite of the research effort focused on the continuous-time behavior of sampled-data systems within the lifting framework, see e.g. Chen and Francis (1995) and references therein, to our knowledge there has been no study of the systemtheoretic concept of controllability from this point of view and this paper is intended to fill this gap by revisiting this important structural property. Note however that it has been conjectured in Kabamba and Hara (1993, remark 3.1) that intersampiing reachability, which is actually related to our notion of approximate controllability, can be achieved in a sampled-data system for almost all intersampling instants. Our study gives a definitive negative answer to such a conjecture. Even though the controllability results obtained seem to be trivial afterwards, they have rather profound implications on the characterization of the structure of sampled-data systems in the lifting framework with regard to control problems such as tracking and disturbance rejection. Other important implications are in problems related to spectrum assignment in continuous-time periodic systems within the paradigm of generalized sampled-data hold function control as evidenced by some misleading statements appeared in the litterature (Chen M-S and Y-Z. Chen, 1999). The closed-loop system operator of a sampled-data system is also characterized, it is shown that it is Fredholm and its spectrum is always a superset of the spectrum of the open-loop system operator.
e
(2)
where the state qn is a X -valued function and where the linear bounded operators :F and B are defined by :F : X -+ X, (:Fqn) (t) = eAT.qn (t)
B : C -+X, (Bv [n]) (t)
= loT eA(T+t-T) Bh (T) dT.V [n] (4)
for t E [0, T). Since the control space is onedimensional, the action of operator B reads simply as Bv [n] = bv [n], that is, (Bv In]) (t) = v [n] b Ct) on [0, T) where b E X and is given by
b(t)
= IoTeA(T+t-T)Bh(T)dT,
tE[O,T)
°
°
x[n+1]=~x[n]+,(T)v[n]
2. DISCRETE-TIME EVOLUTION OF SAMPLED-DATA SYSTEMS
{ x (t n ) =
+ Bh(t)v[n], lim x (tn + l)
Ax(t)
(6)
where ~ = eAT, ')' (T) = b(O) and x [n] = x (nT) . For any control signal v E £2, the "lifted state" sequence is determined from the initial state qo by iteration of the state evolution equation (2):
A finite N-dimensional continuous-time timeinvariant plant controlled by a digital computer at sampling rate l/T yields an overall hybrid system, called also a sampled-data system, with continuous-time dynamics described by the differential equation with boundary conditions
=
(5)
It is worth observing that the state at discretetime n is a trajectory segment qn (.) = {qn (t) , :s t < T}, the distributed parameter domain being the continuous-time interval [0, T), and therefore the intersample dynamics of the sampleddata system is built in the infinite-dimensional discrete-time equation (2) . The value of the state qn at t = 0, i.e. qn (0) , is the value of the original continuous-time state x at time nT and the sampled-behavior of the hybrid system (1) is described exactly by the abstract equation (2) at the boundary point t = of the distributed domain [0, T) , that is,
Notations. For two Hilbert spaces 7-£1,7-£2 and linear operator A : 7-£1 -+ 7-£2 , the domain and the kernel of A in 7-£1 and the range of A in 7-£2 are denoted respectively by 1) (A), ker (A) and n (A). The set of all linear bounded operators from 7-£1 into Ji2 is denoted by B (Ji 1, Ji2) . The inner product and norm in a Hilbert space Ji are denoted by (., .)1i and 1I.1I 1i . The space L~ [0, T) denotes the classical L 2 -Hilbert space of functions on [0, T) with values in eN.
x(t)
(3)
n-l
qn
= Pqo + L:FkBv[n -
k -1]
(7)
k=O
The original continuous-time state x on [0,00), Le. the unique solution to differential equation (1), is faithfully represented by the sequence {qn} n~O .
(1)
<-+0+
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3. REACHABLE SUBSET AND FUNCTION-SPACE COKTROLLABILITY Let us set
== qn - Fnqo , then the sequence
Zn
{Zn}n~O C X is given by
~oo : £dN -+ 0
n-l k Zn==Zn(.;v)=LF Bv[n-k-l]
(8)
[v]
k=O
Fk Bv [k]
\
Ilz(k) Ilx
(9)
Now, the reachability subset 0 with the topology induced by the norm 11.11 0 is viewed as the Hilbert space (0, (., ')0) which is a closed manifold in that topology. An important structure property of the input map (9) of sampled-data systems is the following: Proposition 2. The operator compact.
(10)
It is immediate from proposition 2 that sampleddata systems are never exactly controllable. Now, we show that the Hilbert space (0, (., ')0) cannot be densely embedded in X and for this, we need the following theorem (Conway, 1990):
Proof. Consider the closed linear manifold N
= ker~oo = {v E £2 : ~oov = O} C £2
Then the quotient space £2/N , which is the set of equivalence classes [v] = v + N, is closed (Conway, 1990) and
Theorem 4. If A E B (11. 1,11.2) is a compact operator and (A) is closed, then (A) is finite dimensional.
n
Il[v]II[2/N == /lw.,l IIJLIIl2 E
£2 and JL - v E
E B (£2, X) is
Definition 3. The sampled-data system (2) is exactly controllable if the reachability subset (10) is equal to X. It is approximately controllable if (0, (., ')0) is densely embedded in X. The sampled-data system (2) is function-space nullcontrollable in time n if every state can be transferred to the zero function of X in time n
Proposition 1. The reachability subset 0 c X is isometrically isomorphic to a Hilbert space and the injection i : 0 -+ X is continuous, that is, the identity map i from 0 into the Hilbert space X is a bounded operator.
: JL
~oo
We introduce the following notions of function space controllability:
This subset is clearly a linear submanifold of the Hilbert space X, and our aim is to provide it with a Hilbert space structure, Le. the structure of a closed linear manifold.
= inf {IIJLlI l2
Ilz(k) 110 :s
Ilz(k) IIx
n~O
N
[2/./If
:s
k=O
U On == n (~oo) c X
z( .;v)
and (0 , (., ')0) are isometrically isomorphic. Next, we prove that the natural injection i : 0 -+ X is continuous, i.e., lIi (z)lIx == IIzllx M IIzllo . Assume that this last inequality does not hold, then there is a sequence z(k) in 0 with norm = 1 in X such that 1/ k. Let [v(k)] == ~;;,lz(k) and JL(k) be the representative of [v(k)] with least minimum norm, then JL(k) -+ o in £2 and z(k) == ~ooJL(k). Clearly, z(k) = z (.; JL(k)) -+ 0 in X which contradicts the fact that == 1 •
This is a well-defined operator since convergence is not an issue on D (~oo) which is the subset of finitely non-zero sequences in £2. If ~oo is extendable as a bounded operator to the closure of Un>oD (~n) in £2 we say that the sampleddata system (2) is input stable. We assume in the sequel that the sampled-data system (F, B) is input stable or equivalently that the operator ~oo E B(£2'X), The set of all z E X reachable at discrete time n starting at initial time 0 is On ={zn(.;v) = ~nv:Vv E D(~n)} and the reachability subset of the sampled-data system (F, B) is defined as
o=
= n (~oo)
~oo an isometric map which implies that £2/N
00
=L
H
and it allows us to define a scalar product on 0 by (zr, Z2)0 = / ~;;,l Z t, ~;;,l Z2) . This makes
Introduce the maps whose domains of definition are D (~n) = {v E £2 : Vk == 0 for k ~ n} and defined by ~n : D (~n) -+ X with ~nv = Zn (.; v) as given by (8). Now, consider the operator ~oo defined as follows: its domain of definition is the set D (~oo) = Un>oD (~n) which is a dense subset of £2 and the action of ~oo : D (~oo) -+ X on an element v in this domain is given by
~oov
i 2-norm 11·11[2 ' Actually £2/N is a Hilbert space (Hilbert quotient space) and if JLI and IL2 are representatives of [VI] and [V2] , the inner product in £.}./N is defined as ([VI]' [V2])l2 /N = (JLI,IL2)[2 ' The following map is bijective
n
It is seen from the proof of proposition 1 that ~oo == i~ooQ where Q : £2 -+ ldN is the canonical surjection. Since i is the natural injection of (0, (., ')0) into X, the compactness of ~oo
N}
is a norm on £21N.Note that a representative of this coset is a control JL E £2 which has the least
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temporal nature and therefore the "preservation of controllability" from a continuous-time system to its pure discrete-time model, as evidenced by the equality of the state-space dimensions (N) of (1) and (6) , does not reflect that the original set of reachable continuous-time states has collapsed to a "very small" set of continuous-time signals which is R (Q3 oo ).
implies that either ~oo or Q or both operators are compact on the corresponding spaces. Assume first that the surjection Q is compact, then by the surjectivity of Q it follows that the Hilbert space f2/N is finite-dimensional and hence (0, (., ')rJ is finite-dimensional which in turn implies that sB oo is compact. Now, if we assume that ~oo is compact, since its range is the closed manifold (0, (., ')0)' theorem 4 implies that this range is finite-dimensional. Thus in either case, the reachability space (0, (., ')0) is finite-dimensional and this yields the desired negative result on approximate controllability which is stated as:
The following insightful facts for tracking and disturbance rejection problems in continuous-time with regard to the structure of sampled-data systems are drawn from proposition 5.
PropoSition 5. The sampled-data system (:F, B) is never approximately controllable.
Fact 1. Lifted exogenous signals that can be perfectly tracked by sampled-data systems are intrinsically finite-dimensional valued discrete-time signals.
Afterwards, it is now a triviality to say that R ('Boo) given by (10) is closed in the topology of X. Moreover, proposition 2 is strengthened since 'l3 00 is revealed to be a finite-rank operator. As R('Boo) is a closed linear manifold of X, it follows from Kreyszig (1978, theorem 3.3-4) that X is decomposed as the direct sum
Moreover, from the lifted (infinite-dimensional) input/output perspective, this structure is exhibited by the following stronger fact: Fact 2. Sampled-data systems, from the control viewpoint, are essentially "finite-dimensional input" and "finite-dimensional output" discretetime systems.
(11) where Xc = R (Q3 oo ).L, which is the orthogonal complement of R (Q3 oo ) in X , is infinitedimensional. Note that approximate or exact controllability in the function space is indeed the usual controllability concept in continuoustime, that is, when the control signal is a L2 [0, T) -valued sequence then the set of reachable states might (densely) fill the function space X. Thus sampling destroys irremediably the usual controllability of continuous-time systems.
Proof. Note that fact 1 is a consequence of fact 2. Consider the following input-to-state/state-tooutput (infinite-dimensional) equations {
qn+l Yn
= :Fqn = Cqn
+Bv [n)
+ Bwwn
(12)
where w = {wn}:'o E £2 (x.:) is the lifting of a continuous-time exogenous signal w E Lz [0,00) and x.: =L 2 [0, T) . Assuming v = 0, the exogenous signal feeds the quantity q = '.Bw.oow = l:~o:Fk BwWk into the state space X. The map Q3w.oo is actually equivalent to the injective map 12 (x.:) / ker Q3w.oo -+ X since the same quantity q is obtained from any representative [w) E £2 (x.:) / ker Q3w.oo of w in £2 (x.:). Thanks to this equivalence, this injective map will be denoted by the same operator symbol Q3w,oo' The direct sum decomposition (11) of the state space implies that
Comments 6. The above fact has led the authors in something of a dilemna: on one hand it should be extremely well known, at least informally, and in the other hand the authors have not been able to find an adequate reference establishing proposition 5. Notwithstanding its apparent triviality, it has been conjectured in Kabamba and Hara (1993, remark 3.1) that intersampling reachability (which is actually the same notion as approximate controllability) is possible; this study proves that this conjecture is false even if the input to the continuous-time system is generated by a (optimal) generalized sampled-data hold function. Perhaps, this fact has not been emphasized in the litterature for the reason that controllability of a sampled-data system is usually understood in the sense of the controllability of its pure discrete-time model which is related solely to the controllability of the discrete states defined at the sampling instants (Hautus, 1969; Kalman et al, 1963). However, the state signals of systems (1) and (6) are actually not of the same
q = qc
+qc =
Q3w,oo
[w)
(13)
with qc E 0, (qc, qc) x = 0 and + is the direct sum operation. Note that a control signal v has no possible effect on the part qc resulting from the exogenous signal. The linearity of Q3w.oo and identity (13) imply that qc = Q3w.oo [wc] arld qc = Q3w.oo [Wc) for some [wc) ,[Wc) E £2 (x.:) / ker Q3w.oo such that [w) = [wc] + [Wc] (note that this is not a direct sum). The set of all [wc) is denoted by Wc, Le., Wc ={[w] E £2 (x.:) / ker Q3w.oo:'.B w•oo [w] E O} and it is clearly a closed linear manifold of
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£2 (x:.) I ker'Bw,oc' Let 0 1 ~ 0 denotes the set of all qe such that there is some [w] E Wc with qe = 'Bw,oc [wc]. Then there exists a map 'Bt,OQ : 0 1 -+ Wc such that 'Bt,oc 'Bw,oc [wc] = [wc], that is 'Bt,oc 'Bw,oo i:; the identity map on Wc' Since 'BLoc is compact (its domain 0 1 is finite-dimensional), the identity map is compact on Wc and hence Wc is finite-dimensional. Therefore each component of any sequence in "'e is a finite-dimensional vector and the set of all such vectors lies in a common finite-dimensional space. The assertion of fact 2 results from the fact that the "part" of the continuous-time output signal {Yn} which can be shaped by controls is a "static readout" map of the controllable (reachable) continuous-time state trajectory. •
Proposition 7. The sampled-data system (F, B) is function space null controllable if and only if rank ([4>N-1, (T) 14>''''-2, (T) 1...1, (T)]) = N
This proposition is the well-known necessary and sufficient conditions for "sampled-controllability" (Hautus, 1969), i.e. the controllability of the pure discrete-time model (6) so that function space null controllability is actually equivalent to "sampledcontrollability" and this equivalence is tight. Indeed, that "sampled-controllability" implies function space null controllability is long known since the early contribution of Kalman et al (1963) and this is pretty obvious. But that the converse holds true, i.e. that function space null controllability implies complete "sampled-controllability" (not only controllability to zero at sample-points), seems a little strong, however this cannot be weakened. Therefore, function space null controllability is seen to be the strongest controllability concept one could consider for sampled-data systems from a continuous-time viewpoint. Clearly, the property of null-controllability over the data set (A., E, h (t) ,T) is generic in the sense that any system in that class can be modified ever so slightly into one with that property.
:=0
From the F -invariance of the reachability subspace, the above proof leads to an obvious decomposition of the state equation (12) in which the controllable part of the continuous state trajectory "sees" a finite-dimensional exogenous input extracted from {w n }
:=0 .
4. CHARACTERIZATION OF FUNCTION SPACE NULL CONTROLLABILITY Null-controllability at time n in the functional space framework means that the continuous-time output is identically zero for almost all t E [kT, kT + T) with k 2: n. From equation (7), it is seen that the sampled-data system (F, B) is null controllable at time n , i.e. qk = 0 for k 2: n, if and only if there exists a control signal 1/ = {I/[k]}~;:~ E £2 such that n-1 Pqo = p- k - 1 BI/ [k] (14)
5. SPECTRUM DISTRIBUTION Consider a static state feedback bounded operator on X, K : X -+ C such that the input to system (2) is given by 1/ [n] = Kqn + {[n] and resulting in the closed-loop system
L
k=O
As equation (14) holds pointwise for almost all t E [0, T) , it yields the next identity in
([f
n-1
4>nqo (t)
=L
4>n-k-1 eAt, (T).v [k]
(15)
k=O
where we have used the defining equations of operators F and B and v [k] = -1/ [k] ,or explicitly
4>nqo (t)
= eAt [4>n-1, (T) l4>n-2, (T) I... h (T)] .:!L (16)
noticing that eAt commutes with all4>k (k=0,1,2, ... ) and :!L is the vector whose transpose is :!LT = (v [0], v [1], ...v [n - 1]) . From the non-singularity of the N x N matrix eAt and using standard arguments, i.e. the Caley-Hamilton theorem related to the N x N matrix 4>, the existence of a control signal {v [k]} :=~1 E £2 which realizes the identity (16) depends on the invertibility of the bloc matrix in the right hand side for n = N . We obtain the following proposition:
583
qn+1 = (F + Bx:.) qn + B{ [n] (17) Usually, it is desirable to choose K such that stability in some sense is obtained in (17) . Let say that the sampled-data system (2) is strongly stable if its discrete seInigroup converges to the zero operator in the strong operator topology, i.e., IIFnqllx -+ 0 as n -+ 00, for all q E X. In Yame and Hanus (1999, theorem 4) it is shown that, under some mild conditions which turn out to be null-controllability, the sampled-data system (2) can be stabilized in the sense of strong stability. The resulting feedback system as given by (5) has a discrete semigroup generator :FBK. = F + BK where operator BK is a compact operator on X. We first notice that the closed-loop discrete seInigroup :FBK. belongs to a very special class of the Banach algebra of bounded operators on X. Proposition 8. The closed-loop discrete semigroup :Fm::: is Fredholm
Proof. It is known from Yame and Hanus (1998) that the open-loop discrete semigroup generator F is always nonsingular and boundedly invertible , thus we have F-1 FBK. = 1 + K1 and
FBlCF- l = I + K2 where Kl = F- l BK and K2 = BKF- l are obviously bounded compact operators on X. Clearly operator FBlc is invertible modulo compact operators, and therefore it is Fredholm (Conway, 1990) . • We would like to characterize the structure of the spectrum U (FBK.) of the closed-loop discrete semigroup generator and for this, we quote a proposition from Yame and Hanus (1998) which describes the spectrum of the open-loop semigroup generator F on X . Proposition 9. Each point in the spectrum is an eigenvalue of infinite multipliCity
U
(F)
Now, from the compactness of the perturbation BK, one has immediately a partial information on the structure of the spectrum of FBK.. Proposition 10. The spectrum U (FBK.) of the closed-loop semigroup generator is a superset of u(F)
Proof. It is a direct application of Weyl theorem (see Kato (1980, Theorem 5.35)) • We need the next lemma (Kato, 1980, p. 243) to characterize the whole structure of the spectrum of FBK. Lemma 11. If A is a linear bounded operator on X and the complement in C of its essential spectrum U ess (A) is connected, then U (A) \u w (A) consists of isolated eigenvalues with finite algebraic multiplicity The essential spectrum Uess (FBK.) of FBK. is the set of all A E C such that )J - F BK. is not Fredholm and thus UW (FBK.) = Uess (FBK. +C) holds true for all bounded compact operators C on X. From FBK. = F + BK and proposition 9, we have Uess(FBK.) = Uess(FBK.+C) = uess(F+C 1 ) = U (F) where Cl = BK + C is compact. Therefore the essential spectrum of FBK. is exactly U (F), and since U (F) consists solely of isolated eigenvalues (Yame and Hanus, 1998), it is obvious that C\u eS8 (FBK.) is a connected set and consequently U (FBd \u m (FBK.) is a set of finite type eigenvalues. We are now able to give the structure of the spectrum of F BK. in the proposition below: Proposition 12. The spectrum of the closed-loop semigroup generator consists in the union of an essential spectrum and a discrete spectrum (Le., a set of isolated eigenvalues of finite multiplicity)
Remark 13. The significance of proposition 12 is that sampled feedback controllers, although they have a stabilizing property, do not have arbitrary
584
spectrum relocation capabilities since the closedloop semigroup generator has its spectrum which always contains the spectrum of the open-loop semigroup generator. The best one can hope is to introduce only some discrete spectral values and nothing else in the original spectrum. It is worth noticing that proposition 12 is closely related to the lack of approximate controllability. As the development of this paper extends mutatis mutandis to periodic systems, and from the fact that the set of characteristic exponents of such systems are mirrored in the discrete (lifted) domain by the spectrum U (F), one is led to the following conclusion: it is impossible to assign the characteristic exponents in continuous-time periodic systems by sampled output feedback controllers no matter what the generalized hold function might be, contrary to the result recently stated by Chen M-S and Y-Z. Chen (1999, theorem 1) .
6. REFERENCES Bamieh, B. and J.B. Pearson (1992). A general framework for linear periodic systems with applications to 1i. oo sampled-data control. IEEE 7rans. Automat. Contr., Vol. 37, pp. 418-435 Chen M-S. and Y-Z. Chen (1999). Static Output Feedback Control for Periodically Time-Varying Systems. IEEE 7rans. Automat. Contr., Vol. 44, pp. 218-222 Chen, T. and B.A. Francis (1995). Optimal Sampled-Data Control Systems Springer-Verlag, Berlin Conway, J.B. (1990). A Course in Functional Analysis Springer-Verlag, New York Hautus, M. L. J . (1969). Controllability and Observability Conditions of Linear Autonomous Systems Proc. Koninkl. Nederl. Akademie Van Wetenschappen series A, vol. 72, pp. 443-448 Kabamba, P.T. and S. Hara (1993). Worst-Case Analysis and Design of Sampled-Data Control Systems. IEEE 7rans. Automat. Contr.Vol. 38, pp. 1337-1357 Kalman, R.E., Ho, Y.C. and K.S. Narendra (1963). Controllability of Linear Dynamical Systems. Contributions To Differential Equations, Vol. 1, pp. 189-213 Kato, T . (1980). Perturbation Thwry for Linear Operators Springer-Verlag,Berlin Kreyszig, E. (1978) . Introductory Functional Analysis with Applications John Wiley & Sons, New York Yamamoto, Y. (1994). A Function Space Approach to Sampled-Data. Control Systems and Tracking Problems. IEEE 7rans. Automat. Contr.Vol. 39, pp. 703713
Yame, J.J. and R. Hanus (1998) . An infinite-dimensional discrete-time representation for periodic systems. Proc. of the IFAC Conference on System Structure and Control Nantes, France, vel. 2, pp. 501-506 Yame, J.J. and R . Hanus (1999). Representation and Stabilizability of ContinuoU5-time Systems under Digital Controls. Proc. IEEE ConJ. on Decision and Contro~ Phoenix, Arizona, pp. 3434-3439