On Feedback Stabilizability of Linear Systems With State and Input ...
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On Feedback Stabilizability of Linear Systems With State and Input Delays in Banach Spaces Said Hadd and Qing-Chang Zhong, Senior Member, IEEE
Abstract—The feedback stabilizability of a general class of wellposed linear systems with state and input delays in Banach spaces is studied in this paper. Using the properties of infinite dimensional linear systems, a necessary condition for the feedback stabilizability of delay systems is presented, which extends the wellknown results for finite dimensional systems to infinite dimensional ones. This condition becomes sufficient as well if the semigroup of the delay-free system is immediately compact and the control space is finite dimensional. Moreover, under the condition that the Banach space is reflexive, a rank condition in terms of eigenvectors and control operators is proposed. When the delay-free state space and control space are all finite dimensional, a very compact rank condition is obtained. Finally, the abstract results are illustrated with examples. Index Terms—Banach spaces, feedback stabilizability, Hautus criterion, rank condition, regular systems, time-delay systems.
I. INTRODUCTION
very interesting and motivates many researchers to look for conditions guaranteeing stabilizability of such systems. By analyzing the existing theory in this area one can note that the results are obtained with some special techniques, which do not fit into a general theory as that developed for delay-free systems. Also, most of the works have been devoted to feedback stabilizability of state delay systems with a finite dimensional delay-free state space e.g., [4], [6], [20] or with an infinite dimensional space [25]. The feedback stabilizability of state-input delay systems is investigated in [19], [22], [26]–[29], where the delay-free state space is finite dimensional. It is shown by Olbrot [26] that the feedback stabilizability of the state-input delay system
is equivalent to the condition
A. Literature Review and Motivation
(1) is the dimension of the delay-free system and . We are interested in developing a more general approach to the feedback stabilizability of state-input delay systems in Banach spaces. We shall introduce a general class of feedback laws which stabilize the closed-loop system. This is based on the recent theory of regular linear systems [21], [31], [32], [37], and the recent work on state-input delay systems in Banach spaces [11], [24], where it has been proved that systems with general distributed state, input and output delays can be reformulated as regular linear systems.
where ELAY-DIFFERENTIAL systems arise in the study of many problems with theoretical and practical importance. The behavior of these systems was studied in the seventies, especially in the book by Hale [17], where the semigroup theory was applied to reformulate delay systems as distributed parameter systems (see also Bellman and Cooke [4] and Halanay [16] for an introduction to this approach). The recent well-developed theory for such systems is gathered in several books e.g., Bátkai and Piazzera [2], Bensoussan et al. [3], and Hale and Verduyn Lunel [18]. More recently it is shown in [10], [11], [14] that delay systems form a subclass of well-posed and regular infinite dimensional linear systems in the Salamon-Weiss sense [30]–[33], [37]. It has been recognized that the delay presence in the state and/or input could induce bad performance (even instability) and complicate controller design and system analysis [39]. This fact makes the investigation of the stability of delay systems
D
Manuscript received June 22, 2007; revised March 14, 2008 and August 07, 2008. Current version published March 11, 2009. This paper was presented in part at the 46th IEEE Conference on Decision and Control, New Orleans, LA, December 2007 and at the 18th International Symposium on Mathematical Theory of Networks and Systems (MTNS2008), July 2008. This work was supported by the EPSRC, UK under Grant EP/C005953/1. Recommended by Associate Editor G. Feng. The authors are with the Department of Electrical Engineering and Electronics, The University of Liverpool, Liverpool L69 3GJ, U.K. (e-mail: said. [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2009.2012969
Notation and Problem Statement Throughout this paper we use the following notation. For a Banach space , a real number , and a function , the history function of is for and . We dedefined as the space of all continuous functions from note by to . For we denote by the Lebesgue space of all -integrable functions [7], by the space of indefinite integrals of -inthe Banach space of all tegrable functions, and by linear and bounded operators from a Banach space to another Banach space with . In this paper we consider the state-input delay system
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.
(2)
HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS
439
Here is the generator of a -semigroup on a Banach space and the delay operators and are linear and bounded, where is a (control) Banach is the history function of the funcspace. The function and is the history function of tion . The initial conditions are the function and , which and are finite form the initial state of the system. If can take the dimensional spaces then the operators and explicit form of Riemann-Stieltjes integrals (3) for
and , where and are functions of bounded variations [3, p.249]. In this case the system (2) is well-posed in the sense that it can be rewritten as a well-posed open-loop system [11], [13]. is an arbitrary Banach space and the operIn this paper are not assumed to be of the form (3). In this ators and case, it is shown in [11] that the system (2) can be reformulated as a well-posed control system on the state space in the sense of [35] if and are observation operators of regular linear systems governed by the left shift semigroup (see [13] for more details on such systems). These conditions on and are always satisfied if and are finite dimensional, see [13]. B. Overview of the Major Contributions After discussing left shift semigroups, transforming the and system (2) into a well-posed open-loop system on based on [11] and characterizing the structure of feedback stabilizable operators, we introduce a necessary condition for the stabilizability of the system (2), using the generalization of the Hautus criterion for the stabilizability of distributed-parameter linear systems [38, Proposition 3.5]. This generalizes the condition (1) to the case of general Banach spaces with general delay operators (see Theorem 7). This necessary condition does not require any regularity on the semigroup or on the geometry of the state and control spaces. However, to show the sufficiency of this condition, we need to assume that the control space is finite dimensional and the semigroup is compact for , i.e., immediately compact. Under these conditions, the delay system (2) is feedback stabilizable if and only if
holds for all with , which belongs to the finite un, where stable spectrum of the operator is defined as with . Furthermore, a rank condition, derived from the above condition, is given in terms of eigenvectors and control operators in the case with a reflexive Banach space . This extends the result in [25] where a rank condition was given for systems with state delays only. When the delay-free state space and control space are finite dimensional, a compact
rank condition equivalent to that of Olbrot [26] is obtained. We illustrate our abstract results with examples involving a single delay, multiple delays, distributed delays and elliptic operators in bounded domains of the Euclidean space. C. Organization of the Paper The organization of the paper is as follows. In Section II we recall some preliminaries about well-posed and regular infinite dimensional systems. In Section III after we discuss left shift semigroups, state the main assumptions and reformulate the delay system (2) to an infinite dimensional open loop system, we present the definition of feedback stabilizability and characterize the structure of feedback stabilizable operators. Section IV is devoted to state and prove the main results on feedback stabilizability of the delay system (2) and in Section V we present some examples. Conclusions are made in Section VI, followed by two appendices, in which the frequently-cited known results are gathered. II. PRELIMINARIES: WELL-POSED AND REGULAR INFINITE DIMENSIONAL LINEAR SYSTEMS Here, we briefly recall the framework of infinite dimensional well-posed and regular linear systems in the Salamon-Weiss sense from [31], [32], [34]–[36], [33], [37]. A. Basic Concepts Let
be the generator of a -semigroup on a Banach space . We denote by the resolvent set of , i.e., the set of all such that is invertible. The spectrum of is by definition . We define the resolvent operator of as . The domain endowed , is a with the graph norm Banach space. We also define the norm for some . The completion of with respect to the is a Banach space denoted by , which is called norm the extrapolation space associated with and . Moreover, the continuous injection holds. The semigroup can be naturally extended to a strongly continuous semigroup on , of which the generator is the extension of from to ; see [8] for more details. The pair is called a control is a -semigroup on and system on , if , is a linear bounded operator satisand , fying, for (4) See [35], [36] for more details on the maps . By the representation theorem due to Weiss [35, Theorem 3.9], there ex, called an admissible ists a unique operator (or ), such that for any and control operator for , (5) . We say that is reprewhere the integral exists in sented by the control operator . Each control system
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with the representing control operator is completely determined by an abstract differential equation of the form (6) It is well-posed in the sense that it has a unique strong solution, , given by called the state trajectory with initial state (7) An operator vation operator for
The right hand side goes to 0 when , according to and Lemma 3.4 in [8, p.73]. Hence, for any . Note that the definition of does not require that . However, if this is the case, and for all and a.e. , (see [34, Theorem 4.5]). , denote its convolution with the semiFor as group
is called an admissible obser(or
) if (8)
and for some constants holds for any . For simplicity, denote
Then, we have the following result [9, Proposition 3.3]. and Proposition 1: Assume Define
.
and
Then
and
(9) For
, the linear system
for and a constant and approaches 0 as (10)
can is well-posed in the sense that the observation function . In fact, from (8), the be extended to a function in map
, which is independent of .
B. Well-Posed and Regular Systems Here, we focus on a control system represented by and an observation system represented by . We say that the linear system
(11) (14) to can be extended to a linear bounded operator from , called the extended output map. Then we can set for any and almost every . If we define
with the initial state exists a family to
, is well-posed on , , if there of bounded linear operators from , satisfying
(12) (15) and , and 0 for , then for is a family of bounded linear operators from to . is an observation system We say that on , , and the observation equation in (10) satisfies for almost every (a.e.) and all . Note that is extended in the abstract way. In order to obtain a pointwise representation in terms of the observation operator , consider the Yosida extension[34] of with respect to defined as
(13) Let
be endowed with the graph norm. For
,
for , and . For more we refer to [36], [37]. In this case we details on operators also say that the quadruple is a well-posed system on , , . be the operator of truncation to , that is Let for and zero otherwise. The operators are compatible in the sense that for and zero otherwise. This property provides , a unique operator called the extended input-output map, which satisfies for . We recall from and a unique bounded [36, Theorem 3.6] that there exist and analytic function such that
if . Here and are the Laplace transform of and , respectively. Here, is called the transfer function of . Authorized licensed use limited to: University of Liverpool. Downloaded on March 13, 2009 at 12:45 from IEEE Xplore. Restrictions apply.
HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS
We say that the well-posed system feedthrough) if the limit
441
is regular (with zero
for the control function is a control system and , which is represented by the (strictly) on unbounded admissible control operator [13] (19)
exists in for the constant input . The following definitions will be used throughout this paper. Definition 1: Let and be the admissible control and oband , respectively. servation operators issued from We say that the triple generates a regular system if there exists a bounded operator such that is regular on , , . Definition 2: Let be a regular system with input-output . An operator is called an admissible operators has uniformly bounded inverse. feedback for if Note that in the case of Hilbert spaces and one can use transfer functions instead of input-output operators for the definition of admissible feedback operators [36]. We now state a very general perturbation theorem due to Weiss in Hilbert spaces [37] and due to Staffans in general Banach spaces [32, Chap.7]. generates a regular system Theorem 3: Assume that with admissible feedback operator . Then the operator
with the sum defined in . Moreover, for a.e. ,
generates a ,
-semigroup and, for
on
(16)
III. FEEDBACK STABILIZABILITY OF SYSTEM (2) A. Left Shift Semigroups and Main Assumptions Here, we follow Section II-A to present two left shift semigroups and then state the main assumptions on delay operators and of the system (2). Define
where is the generator of the extrapolation semigroup associated with and is the linear bounded operator
In fact, is the adjoint of the delta operator at zero. Similarly, and . we can have operators of For the control function with for a.e. , the state trajectory of is the history function of given by
Similarly, for the control system we have
represented by
,
with for a.e. . According to (11), if is an admissible observation operator of , we define
and then according to (12). Similarly, we define and for operator . The input-output operator associated with and will be defined according to (15), replacing with . The input-output operator is similarly defined for operator . From now on we assume that the operators and in system (2) satisfy: and the triple gen(A1) , on erates a regular system . and the triple gen(A2) erates a regular system , on . Remark 4: Here, we give an example of operators and that satisfy the above assumptions. Let and be functions of bounded variations such that the total variations of and approach 0 when the interval goes to 0. If the operators and take the form of
(17) . It is well known that and, similarly, ates the left shift semigroup
gener-
(18) for have
and . The pair
. Similarly, we with
for and are not necessarily finite dimensional, then satisfies (A2); see [13].
, where and satisfies (A1) and
B. Reformulation of System (2) We recall from [11], [15] how the delay system (2) can be reformulated as a linear distributed parameter system on some appropriate Banach spaces.
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Consider the Banach space
en-
dowed with the norm
with
given by
. It is well-known
(20) It is not difficult to show that
(27)
is closely
(see e.g., [2]) that the delay system (2) with related to the following linear operator
if and only if
for and , where is the extended output map associated with and . Furthermore, we have the following result from [11, Proposition 3.5] and [15, Proposition 3.2]. Proposition 2: Assume that satisfies (23) and satisfies the condition (A2). Then the operator defined in (25) is an defined in (24). admissible control operator for According to (5), the control maps associated with the control operator are given by
(21) for
and some constants
and
[9],
is defined as
where
for and 3.2], this is equivalent to
. According to [11, Proposition
(22)
otherwise.
(28)
According to the Hölder inequality, (21) implies that (23) for
and
with
for and , where is the extended . Now, the open loop input-output map of the regular system system
.
generates a strongly continuous semiHence, the operator group on , according to [1]. The following remark is important for the proof of the main results in this paper. Remark 5: From the discussion above, we have seen that for the estimate (23) holds with as . If in addition the operator generates an immediately compact (i.e., compact for ), then generates semigroup on an eventually compact semigroup on (i.e., compact for , with in this paper) [23]. In order to consider the well-posedness of the state-input delay systems (2), define the operators
(29) is well-posed according to Proposition 2. Combining (26) and (28), one can see that the state trajectory of the system (29), for , is given by the initial state
according to (7). This is the same as the state trajectory of the system (2), see [11, Proposition 3.3]. This gives a natural connection between the systems (2) and (29). Let and be sufficiently large. According to [13], the Laplace transin is equal to and that of in form of is exactly . Hence, by taking the Laplace transform on both sides of (28), we have
(24) and
(30) C. The Structure of Feedback Stabilizable Operators
(25) It has been shown in [11, Theorem 3.1] that when condition (23) and , the operator the following -semigroup on
satisfies the generates ,
(26)
According to the reformulation of the delay system above, Proposition 2 and [38, Definition 2] (see also Appendix B), we introduce the following definition. Definition 6: Let the conditions (A1) and (A2) be satisfied. We say that the state-input delay system (2) is feedback stabilizable if there exists an operator such that: generates a regular system on i) The triple , , . is an admissible feedii) The identity operator back operator for .
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HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS
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iii) The semigroup generated by the operator endowed with the domain
is exponentially stable. In this case we say that the operator stabilizes the delay system (2). The following result shows an explicit structure of such operators . Proposition 3: Let the conditions (A1) and (A2) be satisfied. stabilizes the delay system (2) then it is If of the form
with , and . be the regular system Now prove the second part. Let . According to [11, Thegenerated by the triple orem 5.1], at least should be an observation operator of the generated by the regular system on and such that triple is an admissible feedback operator. Hence, . This ends the proof. IV. MAIN RESULTS A. A Necessary Condition For each
with , , and with the triple generating a regular linear system with as an admissible feedback operator. stabilizes the Proof: Assume that . Decomsystem (2), then at least we have pose into with and . Then, for any and , we have
, we redefine (33)
for the system (2). From [2, p.56], we have (34) defined in (20). In view of this equivalence, is with called the characteristic operator. The result below gives a necessary condition for the feedback stabilizability of the system (2). Theorem 7: Assume the conditions (A1) and (A2) are satisfied. The feedback stabilizability of the delay system (2) implies such that the existence of (35)
for . This means . In order to characterize the operators into with pose
From [1], the operator
is the generator of the
, decom-
-semigroup
with for any Proof: Denote
.
According to the generalized Hautus criterion [38, Prop. 3.5], which is reproduced in Appendix B for the reader’s convenience, the feedback stabilizability of system (29), and hence such that of (2), implies that there exists
(31) for , where the operators are defined in (22). on implies that . Thus, the fact that According to the invariance of admissibility of observation operators [12, Theorem 3.1], which is reproduced in Appendix A . This for the reader’s convenience, there is . Using (31), allows us to concentrate on operators if and only if one can easily show that an operator it is of the form
for any
with . This indicates, for any and hence for any , there exist and such that, for , (36)
with Let According to (30), we have
be fixed1.
and let
(32) with and . This shows that the have operators stabilizing the open loop system at least the following form 1This
A
is because does not always belong to (
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).
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As we are only interested in any with , into (41), then
, substitute
and
Now, by (36), we obtain Hence, the condition (35) holds. This completes the proof. B. A Necessary and Sufficient Condition (37) It is clear that
Theorem 8: Assume the conditions (A1) and (A2) are satisfied and the control space is finite dimensional. Moreover, asis compact for . The delay system (2) is sume that feedback stabilizable if and only if (42)
which implies that (38)
with . holds for all Proof: The necessity has been proved in Theorem 7. Only the sufficiency will be shown here. At first, we will transform the system (2) into an equivalent system different from (29). and For the system (2), take , respectively, and introduce the Banach space
and, as a result
0 (A =
=
z '
0A (
z '
)
0A
)
z '
+ R(;
A
+e
A
)
+e
u
+ R(;
+ R(;
0
A
)
)
Be u 0
and the operator
u Be u 0
Be u
+(
0
0
0 )e
B(
u
+e
u)
0
:
which generates the
-semigroup
(39)
Here, the identity
on . Since is compact for
was used. Note that
is compact for . Define
and
is compact,
(40) By combining (37), (39) and (40), we have where is assumed to be smooth. Then the system (2) is converted into By using the expression of
in (20), we obtain
(43) (41) Here the new control operator . Denote
and
The last identity implies
, or
and define
Similarly, we have
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is defined by
HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS
445
with
Furthermore, take any define
Then, the operator
then
and
such that
and
(46)
is the generator of the following semigroup on
:
Note that on . In particular, the assumptions . Hence generates a (A1) and (A2) show that strongly continuous semigroup on , according to [2, p.67], [9]. Define
which is bounded, then the delay system (43), and hence (2), is equivalent to the open loop system (44) . where the state is defined as We have now converted the feedback stabilizability of the system (43) or system (2) to that of the above system, where is linear bounded. . According to [2, Now let us characterize the spectrum of p.56], we have
Substitute (46), after rearrangement, into (45), then
Putting the last two identities together, we have
where
. This is equivalent to
which means that the system (44), or system (2), is feedback stabilizable, according to [5, Theorem 1]. This ends the proof. Remark 9: Denote by the adjoint space of . Then the adjoint of the characteristic operator defined in (33) is given by (47) where is the complex conjugate of . Using Theorem 7, [25, Prop.5.1] and the proof of Theorem 8 one can see that
(48)
with where
Hence, by (34) we obtain
A simple computation shows that
By assumption that is compact for generates an eventually compact semigroup on , according to Remark 5. As a result, the following unstable set is finite:
and (49) Since
Take any and let . The condition (42) implies the existence of and such that , which yields
(50) we have
(45)
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(51)
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Combining (49), (51) and (48), the following holds under the conditions of Theorem 8: (56) (52)
with
C. A Rank Condition for a Special Case in Banach Spaces In addition to the assumptions in Theorem 8 we assume that is a reflexive Banach space. As the control space is finite dimensional, denote as
.. .
.. .
.. . (57)
with
is linear and bounded for all . , we define
For
with As
for , we have
and
, given in (54). Proof: We only need to prove that the condition (52) is equivalent to (56) under the extra conditions. Since for
. we have (53)
for and . We say that satgenerates a regular linear isfies (A2) if the triple system on , and . Naturally, satisfies (A2) satisfies (A2) for all . if every is compact for , then generates We recall that if an eventually compact semigroup on . In this case, the set of unstable eigenvalues can be denoted by (54)
.. . and, as a result, (58) Now, assume that the condition (56) does not hold. This means that there exists such that , i.e.,
which is finite with finite algebraic multiplicity. Because of (34), defined in (47) with , we for as can set the dimension of (55) by . and the basis of Throughout the following we denote the duality pairing beby and the orthogonal space of a set in tween and by
When is reflexive and , the adjoint space of is the product space with satisfying . with Theorem 10: Assume that (i) and satisfy the conditions (A1) and (A2), respectively, (ii) the Banach space is reflexive and the control space is finite dimenis compact for . The system (2) is sional and (iii) feedback stabilizable if and only if
.. .
.. .
Now since
is a basis of
(59)
, we have
On the other hand, by using (58) and (59) one can obtain . This is a contradiction, which shows that the equivalence (52) implies the rank conditions (56). The converse can be obtained in a direct way using (58) . This ends the proof. and the basis
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D. A Rank Condition for Delay Systems With Finite-Dimensional State (When Delay-Free) and Control Spaces
On the other hand, without loss of generality (otherwise carrying out some elementary operations), assume that
Consider the delay system (60) where
and
are are
Here,
real matrices, and matrices, and is the delay. and . Let the control function for . We have
which gives the following
:
Now with the partition of , we have (62) as follows: patible with
Since (61) holds,
with
having rank
the above with
has full column rank
com-
. Multiply
from the left, then
.. . We note that the operator and satisfy the conditions (A1) and (A2), see [13, Remark 2]. Since is a finite dimensional is compact for . In this case, matrix,
Since has full column rank , the leftmost big matrix in the above identity has full column rank as well. This means has full column rank. In other that words, the condition (63) holds. V. EXAMPLES
with
Denote the
. The dimension of and the basis of is matrix formed by the basis as
is
for .
According to Theorem 10, we have the following: Corollary 1: The system (60) is feedback stabilizable if and only if
We start from giving two examples which illustrate the result of Corollary 1. In the third example we treat the case of systems with multiple delays. We end this section by an example of distributed delay systems governed by an uniformly elliptic differential operator of second order on a bounded domain of the Euclidean space. Example 1: Consider the system (60) with
(61) Then,
and
. Hence
As this condition is obtained from (42), which is an extension of (1), it should be equivalent to the condition (1) proposed in [26]. This is indeed true. In fact, we have (62) Assume that for each
which implies that duced to
. Now (61) is re-
, (63)
which implies that and
has full column rank Hence, the delay system is feedback stabilizable if and only if . In other words, the system is not stabilizable if
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Example 2: Consider the system (60) with
This was studied in [26] (with a typo in
Example 4: Let be a bounded open set of with boundary and outer unit normal . We consider the elliptic operator
). Here where
with for some and all , with for , and is essentially bounded in , i.e., . We now consider the controlled partial differential equation with state and input delays and
Now we have
Thus, is stabilizable but is not stabilizable. Example 3: Consider the state-input delay system
(64) Here the operator generates an immediately compact semigroup on a reflexive Banach space , the family , and for each ,
(65) Here we assume that is a bounded variation function on and it satisfies for all
where . As a result,
is linear and continuous for any . We assume that and the control takes values in
.
Here,
for and , which satisfy the conditions (A1) and (A2), respectively, by [13, Theorem 3, Remark 2]. In this example,
is not in. The dimension of is for and the basis of is . According to Theorem 10, the system (64) is feedback stabiliz, the rank of the following able if and only if, for matrix is :
for some constant . Similarly, we assume that bounded variations function of for each satisfies
is the , and
for any and a constant . On the other hand, and the history we assume that the initial state functions and for , the control function . We now introduce the following realization operator
with vertible and
B (e B (e
;' ;'
h
)
i
h
h
)
i
h
B (e
h
.. .
;'
)
i
B (e B (e
B (e
h
;' ;'
B (e B (e
;' ;'
)
i
111
h
)
i
)
i
111
h
)
i
.. .
;'
)
i
111
h
B (e
.. .
;'
)
It is known that generates an immediately compact semigroup on . Furthermore the spectrum , where is isolated with finite multiplicity for . We now define the linear operators any
i
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HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS
for orem 3] the operators (A2). In this example,
449
.. .
APPENDIX A ADMISSIBILITY OF OBSERVATION OPERATORS FOR PERTURBED SEMIGROUPS
and . By [13, Theand satisfy the conditions (A1) and
Here, we recall some results on admissibility of observation operators for perturbed semigroups. Let , be two Banach be a -semigroup on with generator spaces, , and be a real number. is called a Miyadera-Voigt An operator perturbation for if (66)
with vertible and
is not in. The dimension of is for and the basis of is . According to Theorem 10, the system (65) is feedback stabiliz, see the equation able if and only if, for each shown at the bottom of the page.
VI. CONCLUSION In this paper, we have presented an analytic approach to address the feedback stabilizability of state-input delay systems in Banach spaces. Based on the transformation of the delay system into a delay-free open loop system, we have introduced a large class of feedback operators that stabilize the delay system. Making use of the recent results on infinite dimensional wellposed and regular linear systems, we have extended the well known necessary condition for the feedback stabilizability of state-input delay systems when the delay-free state space is finite dimensional to the infinite dimensional case. To make it sufficient we have introduced extra conditions that the initial delayfree system is governed by an immediately compact semigroup and that the control space is finite dimensional. This shows that the state-delay equation is governed by an immediately compact semigroup on a product space, which allows us to use the well-known results on feedback stabilizability of distributed-parameter linear systems. Moreover, when the state space is reflexive, we have presented a rank condition for the feedback stabilizability of the state-input delay system in terms of control operators and eigenvectors. Some examples are given to show the applications.
.. . .. .
for all known that if
and some constants . It is satisfies the estimate (66) then the operator
generates a -semigroup on , see [8, p.196]. Now if we use the notation (9) then by Hölder inequality it is clear that every satisfies (66), and then is a generator on . The following theorem, taken from [12], shows the invariance of admissibility of observation operators. then generates a Theorem 11: Let -semigroup on and (67)
APPENDIX B GENERALIZED HAUTUS CRITERION Here, we recall from [38] the generalized Hautus criterion. be the generator of a strongly continuous semiLet , be another Banach group on a Banach space . The open-loop system despace, and fined by and is denoted by . Let be the . The operator Yosida extension of is endowed with its natural domain defined by . The following definition is taken from [38, Def.2.1]. is feedback stabilizable if there exists Definition 12: such that generates a regular system on , , ; i) ii) the identity operator is an admissible feedback operator for ;
.. . .. .
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.. . .. .
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iii)
generates an exponentially stable semigroup on . . In this case, we say that stabilizes We denote by the subspace of which consists of all vectors of the form , and . where The following proposition, taken from [38, Prop.3.5] (combined with [38, Remark 1]), is an extension of the Hautus criterion for stabilizability of infinite-dimensional systems. is feedback stabilizable, then there Proposition 4: If such that exists a
for all
with
.
ACKNOWLEDGMENT The authors greatly appreciate the reviewers’ comments, which have helped improve the quality of the paper.
REFERENCES [1] A. Bátkai and S. Piazzera, “Semigroup and linear partial differential equations with delay,” J. Math. Anal. Appl., vol. 264, no. 1, pp. 1–20, 2001. [2] A. Bátkai and S. Piazzera, Semigroups for Delay Equations. Wellesley, MA: A K Peters, Ltd., 2005, vol. 10. [3] A. Bensoussan, G. Da Prato, M. C. Delfour, and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems. Boston, MA: Birkhäuser, 2007. [4] R. Bellman and K. L. Cooke, Differential-Difference Equations. New York: Academic Press, 1963. [5] K. P. M. Bhat and W. M. Wonham, “Stabilizability and detectability for evolution systems on Banach spaces,” in Proc. IEEE Conf. Decision Control 15th Symp. Adaptive Processes, 1976, pp. 1240–1243. [6] G. Da Prato and A. Lunardi, “Stabilizability of integrodifferential parabolic equations,” J. Integral Equations Appl., no. 2, pp. 281–304, 1990. [7] J. Diestel and J. J. Uhl, Vector Measures. Providence, RI: Amer. Math. Soc., 1977, vol. 15. [8] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. New York: Springer-Verlag, 2000. [9] S. Hadd, “Unbounded perturbations of C –semigroups in Banach spaces and applications,” Semigroup Forum, vol. 70, no. 3, pp. 451–465, 2005. [10] S. Hadd, “An evolution equation approach to non-autonomous linear systems with state, input and output delays,” SIAM J. Control Optim., vol. 45, no. 1, pp. 246–272, 2006. [11] S. Hadd and A. Idrissi, “Regular linear systems governed by systems with state, input and output delays,” IMA J. Math. Control Inform., vol. 22, no. 4, pp. 423–439, 2005. [12] S. Hadd and A. Idrissi, “On the admissibility of observation for perturbed C –semigroups on Banach spaces,” Syst. Control Lett., vol. 55, no. 1, pp. 1–7, 2006. [13] S. Hadd, A. Idrissi, and A. Rhandi, “The regular linear systems associated to the shift semigroups and application to control delay systems,” Math. Control Signals Syst., vol. 18, no. 3, pp. 272–291, 2006. [14] S. Hadd, A. Rhandi, and R. Schnaubelt, “Feedbacks for non-autonomous delay control linear systems,” IMA J. Math. Control Inform., vol. 25, no. 1, pp. 85–110, 2008. [15] S. Hadd and A. Rhandi, “Feedback theory for neutral equations in infinite dimensional spaces,” Note Matematica, vol. 28, no. 1, pp. 43–68, 2008. [16] A. Halanay, Differential Equations, Stability, Oscillations, Time Lag. New York: Academic, 1966. [17] J. K. Hale, Functional Differential Equations. Berlin, Germany: Springer-Verlag, 1971.
[18] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993, vol. 99. [19] A. Ichikawa, “Quadratic control of evolution equations with delays in control,” SIAM J. Control Optim., vol. 20, pp. 645–668, 1982. [20] J.-M. Jeong, “Retarded functional differential equations with L –valued controller,” Funkcial Ekvac., vol. 36, pp. 71–93, 1993. [21] A. Manitius and R. Triggiani, “Sufficient conditions for function space controllability and feedback stabilizability of linear retarded systems,” IEEE Trans. Automat. Control, vol. AC-23, no. 4, pp. 659–665, Aug. 1978. [22] A. Manitius and R. W. Olbrot, “Finite spectrum assignment problem for systems with delays,” IEEE Trans. Automat. Control, vol. AC–24, no. 4, pp. 541–552, Aug. 1979. [23] T. Mátrai, “On Perturbations of eventually compact semigroups preserving eventual compactness,” Semigroup Forum, vol. 69, no. 3, pp. 317–340, 2004. [24] S. I. Nakagiri, “Structural properties of functional equations in Banach spaces,” Osaka J. Math., vol. 25, no. 2, pp. 353–398, 1988. [25] S. I. Nakagiri and M. Yamamoto, “Feedback stabilization of linear retarded systems in Banach spaces,” J. Math. Anal. Appl., vol. 262, no. 1, pp. 160–178, 2001. [26] A. W. Olbrot, “Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays,” IEEE Trans. Automat. Control, vol. AC-23, no. 5, pp. 887–890, Oct. 1978. [27] L. Pandolfi, “Feedback stabilization of functional differential equations,” Bollettino Unione Matematica Italiana, vol. 11, no. 3, pp. 626–635, 1975. [28] L. Pandolfi, “Stabilization of neutral functional equations,” J. Optim. Theory Appl., vol. 20, no. 2, pp. 191–204, 1996. [29] L. Pandolfi, “Dynamic stabilization of systems with input delays,” Automatica, vol. 27, no. 6, pp. 1047–1050, 1991. [30] D. A. O’Connor and T. J. Tran, “On stabilization by state feedback for neutral differential-difference equations,” IEEE Trans. Automat. Control, vol. AC-28, no. 5, pp. 615–618, May 1983. [31] D. Salamon, “Infinite-dimensional linear system with unbounded control and observation: A functional analytic approach,” Trans. Amer. Math. Soc., vol. 300, no. 2, pp. 383–431, 1987. [32] O. J. Staffans, Well-Posed Linear Systems. Cambridge, U.K.: Cambridge Univ. Press, 2005. [33] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups. Boston, MA: Birkhäuser, 2008. [34] G. Weiss, “Admissible observation operators for linear semigroups,” Israel J. Math., vol. 65, no. 1, pp. 17–43, 1989. [35] G. Weiss, “Admissibility of unbounded control operators,” SIAM J. Control Optim., vol. 27, no. 3, pp. 527–545, 1989. [36] G. Weiss, “Transfer functions of regular linear systems. Part I: Characterization of regularity,” Trans. Amer. Math. Soc., vol. 342, no. 2, pp. 827–854, 1994. [37] G. Weiss, “Regular linear systems with feedback,” Math. Control Signals Syst., vol. 7, no. 1, pp. 23–57, 1994. [38] G. Weiss and R. Rebarber, “Optimizability and estimatability for infinite-dimensional linear systems,” SIAM J. Control Optim., vol. 39, no. 4, pp. 1204–1232, 2000. [39] Q.-C. Zhong, Robust Control of Time-Delay Systems. London, U.K.: Springer-Verlag, 2006.
Said Hadd received the M.Sc degree in pure mathematics and the Ph.D. degree in mathematics from University Cadi Ayyad, Morocco, in 1999 and 2005, respectively. He held a postdoctoral position at INRIA, Grenoble, France, and also had several research stays at the Functional Analysis Group (AGFA), Department of Mathematics, The University of Tübingen, Germany. From July 2006 to August 2008 he was a Research Associate with the Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool, U.K. His current research interests include functional analysis of operator semigroups, control theory of general infinite-dimensional systems, and time-delay systems.
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HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS
Qing-Chang Zhong (M’04–SM’04) received the M.Sc. degree in electrical engineering from Hunan University, Hunan, China, in 1997, the Ph.D. degree in control theory and engineering from Shanghai Jiao Tong University, China, in 1999, and the Ph.D. degree in control and power engineering from Imperial College London, London, U.K., in 2004. He started working in the area of control engineering after graduating from the Hunan Institute of Engineering, Hunan, China, in 1990. He was a Postdoctoral Research Fellow at the Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, Israel, from 2000 to 2001, and then a Research Associate at Imperial College London from 2001 to 2003. He took up a Senior Lectureship at the School of Electronics, University of Glamorgan, U.K., in January 2004 and was subsequently promoted to Reader in May 2005. He joined the Department of Electrical Engineering and Electronics, The University of Liverpool, Liverpool, U.K., in August 2005 as a Senior Lecturer. He is currently leading an EPSRC-funded Network for New Academics in Control Engineering (New-ACE, www.newace.org.uk), which has attracted more than 60 members from academia and industry. He is the author of the monograph Robust Control of Time-delay Systems (New York: Springer-Verlag, Ltd., 2006). His current research interests cover control theory (including H-infinity control, time-delay systems and infinite-dimensional systems) and control engineering (including process control, power electronics, renewable energy, embedded control, rapid control prototyping and hardware-in-the-loop, engine control, hybrid vehicles and control using delay elements such as input-shaping technique and repetitive control). Dr. Zhong received the Best Doctoral Thesis Prize from Imperial College London.
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On Feedback Stabilizability of Linear Systems With State and Input Delays in Banach Spaces Said Hadd and Qing-Chang Zhong, Senior Member, IEEE
Abstract—The feedback stabilizability of a general class of wellposed linear systems with state and input delays in Banach spaces is studied in this paper. Using the properties of infinite dimensional linear systems, a necessary condition for the feedback stabilizability of delay systems is presented, which extends the wellknown results for finite dimensional systems to infinite dimensional ones. This condition becomes sufficient as well if the semigroup of the delay-free system is immediately compact and the control space is finite dimensional. Moreover, under the condition that the Banach space is reflexive, a rank condition in terms of eigenvectors and control operators is proposed. When the delay-free state space and control space are all finite dimensional, a very compact rank condition is obtained. Finally, the abstract results are illustrated with examples. Index Terms—Banach spaces, feedback stabilizability, Hautus criterion, rank condition, regular systems, time-delay systems.
I. INTRODUCTION
very interesting and motivates many researchers to look for conditions guaranteeing stabilizability of such systems. By analyzing the existing theory in this area one can note that the results are obtained with some special techniques, which do not fit into a general theory as that developed for delay-free systems. Also, most of the works have been devoted to feedback stabilizability of state delay systems with a finite dimensional delay-free state space e.g., [4], [6], [20] or with an infinite dimensional space [25]. The feedback stabilizability of state-input delay systems is investigated in [19], [22], [26]–[29], where the delay-free state space is finite dimensional. It is shown by Olbrot [26] that the feedback stabilizability of the state-input delay system
is equivalent to the condition
A. Literature Review and Motivation
(1) is the dimension of the delay-free system and . We are interested in developing a more general approach to the feedback stabilizability of state-input delay systems in Banach spaces. We shall introduce a general class of feedback laws which stabilize the closed-loop system. This is based on the recent theory of regular linear systems [21], [31], [32], [37], and the recent work on state-input delay systems in Banach spaces [11], [24], where it has been proved that systems with general distributed state, input and output delays can be reformulated as regular linear systems.
where ELAY-DIFFERENTIAL systems arise in the study of many problems with theoretical and practical importance. The behavior of these systems was studied in the seventies, especially in the book by Hale [17], where the semigroup theory was applied to reformulate delay systems as distributed parameter systems (see also Bellman and Cooke [4] and Halanay [16] for an introduction to this approach). The recent well-developed theory for such systems is gathered in several books e.g., Bátkai and Piazzera [2], Bensoussan et al. [3], and Hale and Verduyn Lunel [18]. More recently it is shown in [10], [11], [14] that delay systems form a subclass of well-posed and regular infinite dimensional linear systems in the Salamon-Weiss sense [30]–[33], [37]. It has been recognized that the delay presence in the state and/or input could induce bad performance (even instability) and complicate controller design and system analysis [39]. This fact makes the investigation of the stability of delay systems
D
Manuscript received June 22, 2007; revised March 14, 2008 and August 07, 2008. Current version published March 11, 2009. This paper was presented in part at the 46th IEEE Conference on Decision and Control, New Orleans, LA, December 2007 and at the 18th International Symposium on Mathematical Theory of Networks and Systems (MTNS2008), July 2008. This work was supported by the EPSRC, UK under Grant EP/C005953/1. Recommended by Associate Editor G. Feng. The authors are with the Department of Electrical Engineering and Electronics, The University of Liverpool, Liverpool L69 3GJ, U.K. (e-mail: said. [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2009.2012969
Notation and Problem Statement Throughout this paper we use the following notation. For a Banach space , a real number , and a function , the history function of is for and . We dedefined as the space of all continuous functions from note by to . For we denote by the Lebesgue space of all -integrable functions [7], by the space of indefinite integrals of -inthe Banach space of all tegrable functions, and by linear and bounded operators from a Banach space to another Banach space with . In this paper we consider the state-input delay system
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.
(2)
HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS
439
Here is the generator of a -semigroup on a Banach space and the delay operators and are linear and bounded, where is a (control) Banach is the history function of the funcspace. The function and is the history function of tion . The initial conditions are the function and , which and are finite form the initial state of the system. If can take the dimensional spaces then the operators and explicit form of Riemann-Stieltjes integrals (3) for
and , where and are functions of bounded variations [3, p.249]. In this case the system (2) is well-posed in the sense that it can be rewritten as a well-posed open-loop system [11], [13]. is an arbitrary Banach space and the operIn this paper are not assumed to be of the form (3). In this ators and case, it is shown in [11] that the system (2) can be reformulated as a well-posed control system on the state space in the sense of [35] if and are observation operators of regular linear systems governed by the left shift semigroup (see [13] for more details on such systems). These conditions on and are always satisfied if and are finite dimensional, see [13]. B. Overview of the Major Contributions After discussing left shift semigroups, transforming the and system (2) into a well-posed open-loop system on based on [11] and characterizing the structure of feedback stabilizable operators, we introduce a necessary condition for the stabilizability of the system (2), using the generalization of the Hautus criterion for the stabilizability of distributed-parameter linear systems [38, Proposition 3.5]. This generalizes the condition (1) to the case of general Banach spaces with general delay operators (see Theorem 7). This necessary condition does not require any regularity on the semigroup or on the geometry of the state and control spaces. However, to show the sufficiency of this condition, we need to assume that the control space is finite dimensional and the semigroup is compact for , i.e., immediately compact. Under these conditions, the delay system (2) is feedback stabilizable if and only if
holds for all with , which belongs to the finite un, where stable spectrum of the operator is defined as with . Furthermore, a rank condition, derived from the above condition, is given in terms of eigenvectors and control operators in the case with a reflexive Banach space . This extends the result in [25] where a rank condition was given for systems with state delays only. When the delay-free state space and control space are finite dimensional, a compact
rank condition equivalent to that of Olbrot [26] is obtained. We illustrate our abstract results with examples involving a single delay, multiple delays, distributed delays and elliptic operators in bounded domains of the Euclidean space. C. Organization of the Paper The organization of the paper is as follows. In Section II we recall some preliminaries about well-posed and regular infinite dimensional systems. In Section III after we discuss left shift semigroups, state the main assumptions and reformulate the delay system (2) to an infinite dimensional open loop system, we present the definition of feedback stabilizability and characterize the structure of feedback stabilizable operators. Section IV is devoted to state and prove the main results on feedback stabilizability of the delay system (2) and in Section V we present some examples. Conclusions are made in Section VI, followed by two appendices, in which the frequently-cited known results are gathered. II. PRELIMINARIES: WELL-POSED AND REGULAR INFINITE DIMENSIONAL LINEAR SYSTEMS Here, we briefly recall the framework of infinite dimensional well-posed and regular linear systems in the Salamon-Weiss sense from [31], [32], [34]–[36], [33], [37]. A. Basic Concepts Let
be the generator of a -semigroup on a Banach space . We denote by the resolvent set of , i.e., the set of all such that is invertible. The spectrum of is by definition . We define the resolvent operator of as . The domain endowed , is a with the graph norm Banach space. We also define the norm for some . The completion of with respect to the is a Banach space denoted by , which is called norm the extrapolation space associated with and . Moreover, the continuous injection holds. The semigroup can be naturally extended to a strongly continuous semigroup on , of which the generator is the extension of from to ; see [8] for more details. The pair is called a control is a -semigroup on and system on , if , is a linear bounded operator satisand , fying, for (4) See [35], [36] for more details on the maps . By the representation theorem due to Weiss [35, Theorem 3.9], there ex, called an admissible ists a unique operator (or ), such that for any and control operator for , (5) . We say that is reprewhere the integral exists in sented by the control operator . Each control system
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with the representing control operator is completely determined by an abstract differential equation of the form (6) It is well-posed in the sense that it has a unique strong solution, , given by called the state trajectory with initial state (7) An operator vation operator for
The right hand side goes to 0 when , according to and Lemma 3.4 in [8, p.73]. Hence, for any . Note that the definition of does not require that . However, if this is the case, and for all and a.e. , (see [34, Theorem 4.5]). , denote its convolution with the semiFor as group
is called an admissible obser(or
) if (8)
and for some constants holds for any . For simplicity, denote
Then, we have the following result [9, Proposition 3.3]. and Proposition 1: Assume Define
.
and
Then
and
(9) For
, the linear system
for and a constant and approaches 0 as (10)
can is well-posed in the sense that the observation function . In fact, from (8), the be extended to a function in map
, which is independent of .
B. Well-Posed and Regular Systems Here, we focus on a control system represented by and an observation system represented by . We say that the linear system
(11) (14) to can be extended to a linear bounded operator from , called the extended output map. Then we can set for any and almost every . If we define
with the initial state exists a family to
, is well-posed on , , if there of bounded linear operators from , satisfying
(12) (15) and , and 0 for , then for is a family of bounded linear operators from to . is an observation system We say that on , , and the observation equation in (10) satisfies for almost every (a.e.) and all . Note that is extended in the abstract way. In order to obtain a pointwise representation in terms of the observation operator , consider the Yosida extension[34] of with respect to defined as
(13) Let
be endowed with the graph norm. For
,
for , and . For more we refer to [36], [37]. In this case we details on operators also say that the quadruple is a well-posed system on , , . be the operator of truncation to , that is Let for and zero otherwise. The operators are compatible in the sense that for and zero otherwise. This property provides , a unique operator called the extended input-output map, which satisfies for . We recall from and a unique bounded [36, Theorem 3.6] that there exist and analytic function such that
if . Here and are the Laplace transform of and , respectively. Here, is called the transfer function of . Authorized licensed use limited to: University of Liverpool. Downloaded on March 13, 2009 at 12:45 from IEEE Xplore. Restrictions apply.
HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS
We say that the well-posed system feedthrough) if the limit
441
is regular (with zero
for the control function is a control system and , which is represented by the (strictly) on unbounded admissible control operator [13] (19)
exists in for the constant input . The following definitions will be used throughout this paper. Definition 1: Let and be the admissible control and oband , respectively. servation operators issued from We say that the triple generates a regular system if there exists a bounded operator such that is regular on , , . Definition 2: Let be a regular system with input-output . An operator is called an admissible operators has uniformly bounded inverse. feedback for if Note that in the case of Hilbert spaces and one can use transfer functions instead of input-output operators for the definition of admissible feedback operators [36]. We now state a very general perturbation theorem due to Weiss in Hilbert spaces [37] and due to Staffans in general Banach spaces [32, Chap.7]. generates a regular system Theorem 3: Assume that with admissible feedback operator . Then the operator
with the sum defined in . Moreover, for a.e. ,
generates a ,
-semigroup and, for
on
(16)
III. FEEDBACK STABILIZABILITY OF SYSTEM (2) A. Left Shift Semigroups and Main Assumptions Here, we follow Section II-A to present two left shift semigroups and then state the main assumptions on delay operators and of the system (2). Define
where is the generator of the extrapolation semigroup associated with and is the linear bounded operator
In fact, is the adjoint of the delta operator at zero. Similarly, and . we can have operators of For the control function with for a.e. , the state trajectory of is the history function of given by
Similarly, for the control system we have
represented by
,
with for a.e. . According to (11), if is an admissible observation operator of , we define
and then according to (12). Similarly, we define and for operator . The input-output operator associated with and will be defined according to (15), replacing with . The input-output operator is similarly defined for operator . From now on we assume that the operators and in system (2) satisfy: and the triple gen(A1) , on erates a regular system . and the triple gen(A2) erates a regular system , on . Remark 4: Here, we give an example of operators and that satisfy the above assumptions. Let and be functions of bounded variations such that the total variations of and approach 0 when the interval goes to 0. If the operators and take the form of
(17) . It is well known that and, similarly, ates the left shift semigroup
gener-
(18) for have
and . The pair
. Similarly, we with
for and are not necessarily finite dimensional, then satisfies (A2); see [13].
, where and satisfies (A1) and
B. Reformulation of System (2) We recall from [11], [15] how the delay system (2) can be reformulated as a linear distributed parameter system on some appropriate Banach spaces.
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Consider the Banach space
en-
dowed with the norm
with
given by
. It is well-known
(20) It is not difficult to show that
(27)
is closely
(see e.g., [2]) that the delay system (2) with related to the following linear operator
if and only if
for and , where is the extended output map associated with and . Furthermore, we have the following result from [11, Proposition 3.5] and [15, Proposition 3.2]. Proposition 2: Assume that satisfies (23) and satisfies the condition (A2). Then the operator defined in (25) is an defined in (24). admissible control operator for According to (5), the control maps associated with the control operator are given by
(21) for
and some constants
and
[9],
is defined as
where
for and 3.2], this is equivalent to
. According to [11, Proposition
(22)
otherwise.
(28)
According to the Hölder inequality, (21) implies that (23) for
and
with
for and , where is the extended . Now, the open loop input-output map of the regular system system
.
generates a strongly continuous semiHence, the operator group on , according to [1]. The following remark is important for the proof of the main results in this paper. Remark 5: From the discussion above, we have seen that for the estimate (23) holds with as . If in addition the operator generates an immediately compact (i.e., compact for ), then generates semigroup on an eventually compact semigroup on (i.e., compact for , with in this paper) [23]. In order to consider the well-posedness of the state-input delay systems (2), define the operators
(29) is well-posed according to Proposition 2. Combining (26) and (28), one can see that the state trajectory of the system (29), for , is given by the initial state
according to (7). This is the same as the state trajectory of the system (2), see [11, Proposition 3.3]. This gives a natural connection between the systems (2) and (29). Let and be sufficiently large. According to [13], the Laplace transin is equal to and that of in form of is exactly . Hence, by taking the Laplace transform on both sides of (28), we have
(24) and
(30) C. The Structure of Feedback Stabilizable Operators
(25) It has been shown in [11, Theorem 3.1] that when condition (23) and , the operator the following -semigroup on
satisfies the generates ,
(26)
According to the reformulation of the delay system above, Proposition 2 and [38, Definition 2] (see also Appendix B), we introduce the following definition. Definition 6: Let the conditions (A1) and (A2) be satisfied. We say that the state-input delay system (2) is feedback stabilizable if there exists an operator such that: generates a regular system on i) The triple , , . is an admissible feedii) The identity operator back operator for .
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iii) The semigroup generated by the operator endowed with the domain
is exponentially stable. In this case we say that the operator stabilizes the delay system (2). The following result shows an explicit structure of such operators . Proposition 3: Let the conditions (A1) and (A2) be satisfied. stabilizes the delay system (2) then it is If of the form
with , and . be the regular system Now prove the second part. Let . According to [11, Thegenerated by the triple orem 5.1], at least should be an observation operator of the generated by the regular system on and such that triple is an admissible feedback operator. Hence, . This ends the proof. IV. MAIN RESULTS A. A Necessary Condition For each
with , , and with the triple generating a regular linear system with as an admissible feedback operator. stabilizes the Proof: Assume that . Decomsystem (2), then at least we have pose into with and . Then, for any and , we have
, we redefine (33)
for the system (2). From [2, p.56], we have (34) defined in (20). In view of this equivalence, is with called the characteristic operator. The result below gives a necessary condition for the feedback stabilizability of the system (2). Theorem 7: Assume the conditions (A1) and (A2) are satisfied. The feedback stabilizability of the delay system (2) implies such that the existence of (35)
for . This means . In order to characterize the operators into with pose
From [1], the operator
is the generator of the
, decom-
-semigroup
with for any Proof: Denote
.
According to the generalized Hautus criterion [38, Prop. 3.5], which is reproduced in Appendix B for the reader’s convenience, the feedback stabilizability of system (29), and hence such that of (2), implies that there exists
(31) for , where the operators are defined in (22). on implies that . Thus, the fact that According to the invariance of admissibility of observation operators [12, Theorem 3.1], which is reproduced in Appendix A . This for the reader’s convenience, there is . Using (31), allows us to concentrate on operators if and only if one can easily show that an operator it is of the form
for any
with . This indicates, for any and hence for any , there exist and such that, for , (36)
with Let According to (30), we have
be fixed1.
and let
(32) with and . This shows that the have operators stabilizing the open loop system at least the following form 1This
A
is because does not always belong to (
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).
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As we are only interested in any with , into (41), then
, substitute
and
Now, by (36), we obtain Hence, the condition (35) holds. This completes the proof. B. A Necessary and Sufficient Condition (37) It is clear that
Theorem 8: Assume the conditions (A1) and (A2) are satisfied and the control space is finite dimensional. Moreover, asis compact for . The delay system (2) is sume that feedback stabilizable if and only if (42)
which implies that (38)
with . holds for all Proof: The necessity has been proved in Theorem 7. Only the sufficiency will be shown here. At first, we will transform the system (2) into an equivalent system different from (29). and For the system (2), take , respectively, and introduce the Banach space
and, as a result
0 (A =
=
z '
0A (
z '
)
0A
)
z '
+ R(;
A
+e
A
)
+e
u
+ R(;
+ R(;
0
A
)
)
Be u 0
and the operator
u Be u 0
Be u
+(
0
0
0 )e
B(
u
+e
u)
0
:
which generates the
-semigroup
(39)
Here, the identity
on . Since is compact for
was used. Note that
is compact for . Define
and
is compact,
(40) By combining (37), (39) and (40), we have where is assumed to be smooth. Then the system (2) is converted into By using the expression of
in (20), we obtain
(43) (41) Here the new control operator . Denote
and
The last identity implies
, or
and define
Similarly, we have
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is defined by
HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS
445
with
Furthermore, take any define
Then, the operator
then
and
such that
and
(46)
is the generator of the following semigroup on
:
Note that on . In particular, the assumptions . Hence generates a (A1) and (A2) show that strongly continuous semigroup on , according to [2, p.67], [9]. Define
which is bounded, then the delay system (43), and hence (2), is equivalent to the open loop system (44) . where the state is defined as We have now converted the feedback stabilizability of the system (43) or system (2) to that of the above system, where is linear bounded. . According to [2, Now let us characterize the spectrum of p.56], we have
Substitute (46), after rearrangement, into (45), then
Putting the last two identities together, we have
where
. This is equivalent to
which means that the system (44), or system (2), is feedback stabilizable, according to [5, Theorem 1]. This ends the proof. Remark 9: Denote by the adjoint space of . Then the adjoint of the characteristic operator defined in (33) is given by (47) where is the complex conjugate of . Using Theorem 7, [25, Prop.5.1] and the proof of Theorem 8 one can see that
(48)
with where
Hence, by (34) we obtain
A simple computation shows that
By assumption that is compact for generates an eventually compact semigroup on , according to Remark 5. As a result, the following unstable set is finite:
and (49) Since
Take any and let . The condition (42) implies the existence of and such that , which yields
(50) we have
(45)
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(51)
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Combining (49), (51) and (48), the following holds under the conditions of Theorem 8: (56) (52)
with
C. A Rank Condition for a Special Case in Banach Spaces In addition to the assumptions in Theorem 8 we assume that is a reflexive Banach space. As the control space is finite dimensional, denote as
.. .
.. .
.. . (57)
with
is linear and bounded for all . , we define
For
with As
for , we have
and
, given in (54). Proof: We only need to prove that the condition (52) is equivalent to (56) under the extra conditions. Since for
. we have (53)
for and . We say that satgenerates a regular linear isfies (A2) if the triple system on , and . Naturally, satisfies (A2) satisfies (A2) for all . if every is compact for , then generates We recall that if an eventually compact semigroup on . In this case, the set of unstable eigenvalues can be denoted by (54)
.. . and, as a result, (58) Now, assume that the condition (56) does not hold. This means that there exists such that , i.e.,
which is finite with finite algebraic multiplicity. Because of (34), defined in (47) with , we for as can set the dimension of (55) by . and the basis of Throughout the following we denote the duality pairing beby and the orthogonal space of a set in tween and by
When is reflexive and , the adjoint space of is the product space with satisfying . with Theorem 10: Assume that (i) and satisfy the conditions (A1) and (A2), respectively, (ii) the Banach space is reflexive and the control space is finite dimenis compact for . The system (2) is sional and (iii) feedback stabilizable if and only if
.. .
.. .
Now since
is a basis of
(59)
, we have
On the other hand, by using (58) and (59) one can obtain . This is a contradiction, which shows that the equivalence (52) implies the rank conditions (56). The converse can be obtained in a direct way using (58) . This ends the proof. and the basis
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HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS
447
D. A Rank Condition for Delay Systems With Finite-Dimensional State (When Delay-Free) and Control Spaces
On the other hand, without loss of generality (otherwise carrying out some elementary operations), assume that
Consider the delay system (60) where
and
are are
Here,
real matrices, and matrices, and is the delay. and . Let the control function for . We have
which gives the following
:
Now with the partition of , we have (62) as follows: patible with
Since (61) holds,
with
having rank
the above with
has full column rank
com-
. Multiply
from the left, then
.. . We note that the operator and satisfy the conditions (A1) and (A2), see [13, Remark 2]. Since is a finite dimensional is compact for . In this case, matrix,
Since has full column rank , the leftmost big matrix in the above identity has full column rank as well. This means has full column rank. In other that words, the condition (63) holds. V. EXAMPLES
with
Denote the
. The dimension of and the basis of is matrix formed by the basis as
is
for .
According to Theorem 10, we have the following: Corollary 1: The system (60) is feedback stabilizable if and only if
We start from giving two examples which illustrate the result of Corollary 1. In the third example we treat the case of systems with multiple delays. We end this section by an example of distributed delay systems governed by an uniformly elliptic differential operator of second order on a bounded domain of the Euclidean space. Example 1: Consider the system (60) with
(61) Then,
and
. Hence
As this condition is obtained from (42), which is an extension of (1), it should be equivalent to the condition (1) proposed in [26]. This is indeed true. In fact, we have (62) Assume that for each
which implies that duced to
. Now (61) is re-
, (63)
which implies that and
has full column rank Hence, the delay system is feedback stabilizable if and only if . In other words, the system is not stabilizable if
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Example 2: Consider the system (60) with
This was studied in [26] (with a typo in
Example 4: Let be a bounded open set of with boundary and outer unit normal . We consider the elliptic operator
). Here where
with for some and all , with for , and is essentially bounded in , i.e., . We now consider the controlled partial differential equation with state and input delays and
Now we have
Thus, is stabilizable but is not stabilizable. Example 3: Consider the state-input delay system
(64) Here the operator generates an immediately compact semigroup on a reflexive Banach space , the family , and for each ,
(65) Here we assume that is a bounded variation function on and it satisfies for all
where . As a result,
is linear and continuous for any . We assume that and the control takes values in
.
Here,
for and , which satisfy the conditions (A1) and (A2), respectively, by [13, Theorem 3, Remark 2]. In this example,
is not in. The dimension of is for and the basis of is . According to Theorem 10, the system (64) is feedback stabiliz, the rank of the following able if and only if, for matrix is :
for some constant . Similarly, we assume that bounded variations function of for each satisfies
is the , and
for any and a constant . On the other hand, and the history we assume that the initial state functions and for , the control function . We now introduce the following realization operator
with vertible and
B (e B (e
;' ;'
h
)
i
h
h
)
i
h
B (e
h
.. .
;'
)
i
B (e B (e
B (e
h
;' ;'
B (e B (e
;' ;'
)
i
111
h
)
i
)
i
111
h
)
i
.. .
;'
)
i
111
h
B (e
.. .
;'
)
It is known that generates an immediately compact semigroup on . Furthermore the spectrum , where is isolated with finite multiplicity for . We now define the linear operators any
i
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HADD AND ZHONG: FEEDBACK STABILIZABILITY OF LINEAR SYSTEMS
for orem 3] the operators (A2). In this example,
449
.. .
APPENDIX A ADMISSIBILITY OF OBSERVATION OPERATORS FOR PERTURBED SEMIGROUPS
and . By [13, Theand satisfy the conditions (A1) and
Here, we recall some results on admissibility of observation operators for perturbed semigroups. Let , be two Banach be a -semigroup on with generator spaces, , and be a real number. is called a Miyadera-Voigt An operator perturbation for if (66)
with vertible and
is not in. The dimension of is for and the basis of is . According to Theorem 10, the system (65) is feedback stabiliz, see the equation able if and only if, for each shown at the bottom of the page.
VI. CONCLUSION In this paper, we have presented an analytic approach to address the feedback stabilizability of state-input delay systems in Banach spaces. Based on the transformation of the delay system into a delay-free open loop system, we have introduced a large class of feedback operators that stabilize the delay system. Making use of the recent results on infinite dimensional wellposed and regular linear systems, we have extended the well known necessary condition for the feedback stabilizability of state-input delay systems when the delay-free state space is finite dimensional to the infinite dimensional case. To make it sufficient we have introduced extra conditions that the initial delayfree system is governed by an immediately compact semigroup and that the control space is finite dimensional. This shows that the state-delay equation is governed by an immediately compact semigroup on a product space, which allows us to use the well-known results on feedback stabilizability of distributed-parameter linear systems. Moreover, when the state space is reflexive, we have presented a rank condition for the feedback stabilizability of the state-input delay system in terms of control operators and eigenvectors. Some examples are given to show the applications.
.. . .. .
for all known that if
and some constants . It is satisfies the estimate (66) then the operator
generates a -semigroup on , see [8, p.196]. Now if we use the notation (9) then by Hölder inequality it is clear that every satisfies (66), and then is a generator on . The following theorem, taken from [12], shows the invariance of admissibility of observation operators. then generates a Theorem 11: Let -semigroup on and (67)
APPENDIX B GENERALIZED HAUTUS CRITERION Here, we recall from [38] the generalized Hautus criterion. be the generator of a strongly continuous semiLet , be another Banach group on a Banach space . The open-loop system despace, and fined by and is denoted by . Let be the . The operator Yosida extension of is endowed with its natural domain defined by . The following definition is taken from [38, Def.2.1]. is feedback stabilizable if there exists Definition 12: such that generates a regular system on , , ; i) ii) the identity operator is an admissible feedback operator for ;
.. . .. .
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.. . .. .
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iii)
generates an exponentially stable semigroup on . . In this case, we say that stabilizes We denote by the subspace of which consists of all vectors of the form , and . where The following proposition, taken from [38, Prop.3.5] (combined with [38, Remark 1]), is an extension of the Hautus criterion for stabilizability of infinite-dimensional systems. is feedback stabilizable, then there Proposition 4: If such that exists a
for all
with
.
ACKNOWLEDGMENT The authors greatly appreciate the reviewers’ comments, which have helped improve the quality of the paper.
REFERENCES [1] A. Bátkai and S. Piazzera, “Semigroup and linear partial differential equations with delay,” J. Math. Anal. Appl., vol. 264, no. 1, pp. 1–20, 2001. [2] A. Bátkai and S. Piazzera, Semigroups for Delay Equations. Wellesley, MA: A K Peters, Ltd., 2005, vol. 10. [3] A. Bensoussan, G. Da Prato, M. C. Delfour, and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems. Boston, MA: Birkhäuser, 2007. [4] R. Bellman and K. L. Cooke, Differential-Difference Equations. New York: Academic Press, 1963. [5] K. P. M. Bhat and W. M. Wonham, “Stabilizability and detectability for evolution systems on Banach spaces,” in Proc. IEEE Conf. Decision Control 15th Symp. Adaptive Processes, 1976, pp. 1240–1243. [6] G. Da Prato and A. Lunardi, “Stabilizability of integrodifferential parabolic equations,” J. Integral Equations Appl., no. 2, pp. 281–304, 1990. [7] J. Diestel and J. J. Uhl, Vector Measures. Providence, RI: Amer. Math. Soc., 1977, vol. 15. [8] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. New York: Springer-Verlag, 2000. [9] S. Hadd, “Unbounded perturbations of C –semigroups in Banach spaces and applications,” Semigroup Forum, vol. 70, no. 3, pp. 451–465, 2005. [10] S. Hadd, “An evolution equation approach to non-autonomous linear systems with state, input and output delays,” SIAM J. Control Optim., vol. 45, no. 1, pp. 246–272, 2006. [11] S. Hadd and A. Idrissi, “Regular linear systems governed by systems with state, input and output delays,” IMA J. Math. Control Inform., vol. 22, no. 4, pp. 423–439, 2005. [12] S. Hadd and A. Idrissi, “On the admissibility of observation for perturbed C –semigroups on Banach spaces,” Syst. Control Lett., vol. 55, no. 1, pp. 1–7, 2006. [13] S. Hadd, A. Idrissi, and A. Rhandi, “The regular linear systems associated to the shift semigroups and application to control delay systems,” Math. Control Signals Syst., vol. 18, no. 3, pp. 272–291, 2006. [14] S. Hadd, A. Rhandi, and R. Schnaubelt, “Feedbacks for non-autonomous delay control linear systems,” IMA J. Math. Control Inform., vol. 25, no. 1, pp. 85–110, 2008. [15] S. Hadd and A. Rhandi, “Feedback theory for neutral equations in infinite dimensional spaces,” Note Matematica, vol. 28, no. 1, pp. 43–68, 2008. [16] A. Halanay, Differential Equations, Stability, Oscillations, Time Lag. New York: Academic, 1966. [17] J. K. Hale, Functional Differential Equations. Berlin, Germany: Springer-Verlag, 1971.
[18] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993, vol. 99. [19] A. Ichikawa, “Quadratic control of evolution equations with delays in control,” SIAM J. Control Optim., vol. 20, pp. 645–668, 1982. [20] J.-M. Jeong, “Retarded functional differential equations with L –valued controller,” Funkcial Ekvac., vol. 36, pp. 71–93, 1993. [21] A. Manitius and R. Triggiani, “Sufficient conditions for function space controllability and feedback stabilizability of linear retarded systems,” IEEE Trans. Automat. Control, vol. AC-23, no. 4, pp. 659–665, Aug. 1978. [22] A. Manitius and R. W. Olbrot, “Finite spectrum assignment problem for systems with delays,” IEEE Trans. Automat. Control, vol. AC–24, no. 4, pp. 541–552, Aug. 1979. [23] T. Mátrai, “On Perturbations of eventually compact semigroups preserving eventual compactness,” Semigroup Forum, vol. 69, no. 3, pp. 317–340, 2004. [24] S. I. Nakagiri, “Structural properties of functional equations in Banach spaces,” Osaka J. Math., vol. 25, no. 2, pp. 353–398, 1988. [25] S. I. Nakagiri and M. Yamamoto, “Feedback stabilization of linear retarded systems in Banach spaces,” J. Math. Anal. Appl., vol. 262, no. 1, pp. 160–178, 2001. [26] A. W. Olbrot, “Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays,” IEEE Trans. Automat. Control, vol. AC-23, no. 5, pp. 887–890, Oct. 1978. [27] L. Pandolfi, “Feedback stabilization of functional differential equations,” Bollettino Unione Matematica Italiana, vol. 11, no. 3, pp. 626–635, 1975. [28] L. Pandolfi, “Stabilization of neutral functional equations,” J. Optim. Theory Appl., vol. 20, no. 2, pp. 191–204, 1996. [29] L. Pandolfi, “Dynamic stabilization of systems with input delays,” Automatica, vol. 27, no. 6, pp. 1047–1050, 1991. [30] D. A. O’Connor and T. J. Tran, “On stabilization by state feedback for neutral differential-difference equations,” IEEE Trans. Automat. Control, vol. AC-28, no. 5, pp. 615–618, May 1983. [31] D. Salamon, “Infinite-dimensional linear system with unbounded control and observation: A functional analytic approach,” Trans. Amer. Math. Soc., vol. 300, no. 2, pp. 383–431, 1987. [32] O. J. Staffans, Well-Posed Linear Systems. Cambridge, U.K.: Cambridge Univ. Press, 2005. [33] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups. Boston, MA: Birkhäuser, 2008. [34] G. Weiss, “Admissible observation operators for linear semigroups,” Israel J. Math., vol. 65, no. 1, pp. 17–43, 1989. [35] G. Weiss, “Admissibility of unbounded control operators,” SIAM J. Control Optim., vol. 27, no. 3, pp. 527–545, 1989. [36] G. Weiss, “Transfer functions of regular linear systems. Part I: Characterization of regularity,” Trans. Amer. Math. Soc., vol. 342, no. 2, pp. 827–854, 1994. [37] G. Weiss, “Regular linear systems with feedback,” Math. Control Signals Syst., vol. 7, no. 1, pp. 23–57, 1994. [38] G. Weiss and R. Rebarber, “Optimizability and estimatability for infinite-dimensional linear systems,” SIAM J. Control Optim., vol. 39, no. 4, pp. 1204–1232, 2000. [39] Q.-C. Zhong, Robust Control of Time-Delay Systems. London, U.K.: Springer-Verlag, 2006.
Said Hadd received the M.Sc degree in pure mathematics and the Ph.D. degree in mathematics from University Cadi Ayyad, Morocco, in 1999 and 2005, respectively. He held a postdoctoral position at INRIA, Grenoble, France, and also had several research stays at the Functional Analysis Group (AGFA), Department of Mathematics, The University of Tübingen, Germany. From July 2006 to August 2008 he was a Research Associate with the Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool, U.K. His current research interests include functional analysis of operator semigroups, control theory of general infinite-dimensional systems, and time-delay systems.
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Qing-Chang Zhong (M’04–SM’04) received the M.Sc. degree in electrical engineering from Hunan University, Hunan, China, in 1997, the Ph.D. degree in control theory and engineering from Shanghai Jiao Tong University, China, in 1999, and the Ph.D. degree in control and power engineering from Imperial College London, London, U.K., in 2004. He started working in the area of control engineering after graduating from the Hunan Institute of Engineering, Hunan, China, in 1990. He was a Postdoctoral Research Fellow at the Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, Israel, from 2000 to 2001, and then a Research Associate at Imperial College London from 2001 to 2003. He took up a Senior Lectureship at the School of Electronics, University of Glamorgan, U.K., in January 2004 and was subsequently promoted to Reader in May 2005. He joined the Department of Electrical Engineering and Electronics, The University of Liverpool, Liverpool, U.K., in August 2005 as a Senior Lecturer. He is currently leading an EPSRC-funded Network for New Academics in Control Engineering (New-ACE, www.newace.org.uk), which has attracted more than 60 members from academia and industry. He is the author of the monograph Robust Control of Time-delay Systems (New York: Springer-Verlag, Ltd., 2006). His current research interests cover control theory (including H-infinity control, time-delay systems and infinite-dimensional systems) and control engineering (including process control, power electronics, renewable energy, embedded control, rapid control prototyping and hardware-in-the-loop, engine control, hybrid vehicles and control using delay elements such as input-shaping technique and repetitive control). Dr. Zhong received the Best Doctoral Thesis Prize from Imperial College London.
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