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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 4, APRIL 2011
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[9] D. Henrion, O. Bachelier, and M. Sebek, “ -stability of polynomial matrices,” Int. J. Contr., vol. 74, no. 8, pp. 845–856, 2001. [10] D. Henrion, D. Arzelier, D. Peaucelle, and M. Sebek, “An LMI condition for robust stability of polynomial matrix polytopes,” Automatica, vol. 37, no. 3, pp. 461–468, 2001. [11] D. Henrion, D. Arzelier, and D. Peaucelle, “Positive polynomial matrices and improved LMI robustness conditions,” Automatica, vol. 39, no. 8, pp. 1479–1485, 2003. [12] Y. Ebigara, K. Maeda, and T. Hagiwara, “Robust -stability analysis of uncertain polynomial matrices via polynomial-type multipliers,” in Proc. 16th IFAC World Congr., Prague, Czechoslovakia, 2005. [13] D. C. W. Ramos and P. L. D. Peres, “An LMI condition for the robust stability of uncertain continuous-time linear systems,” IEEE Trans. Autom. Control, vol. 47, no. 4, pp. 675–678, Apr. 2002. [14] R. C. L. F. Oliveira and P. L. D. Peres, “Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations,” IEEE Trans. Autom. Control, vol. 52, no. 7, pp. 1334–1340, Jul. 2007. [15] P. A. Parrilo, “Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization,” Ph.D. dissertation, California Institute of Technology, Pasadena, CA, 2000. [16] P.-A. Bliman, “A convex approach to robust stability for linear systems with uncertain scalar parameters,” SIAM J. Contr. Optimiz., vol. 42, no. 6, pp. 2016–2042, 2004. [17] G. Chesi, A. Garulli, A. Tesi, and A. Vicino, “Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: An LMI approach,” IEEE Trans. Autom. Control, vol. 50, no. 3, pp. 65–370, Mar. 2005. [18] Y. Y. Cao and Z. Lin, “A discriptor approach to robust stability analysis and controller synthesis,” IEEE Trans. Autom. Control, vol. 49, no. 11, pp. 2081–2084, Nov. 2004. [19] R. C. L. F. Oliveira, M. C. de Oliveira, and P. L. D. Peres, “Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions,” Syst. Control Lett., vol. 57, no. 8, pp. 680–689, 2008. [20] T. M. Guerra, A. Kruszewski, and M. Bernal, “Control law proposition for the stabilization of discrete Takagi-Sugeno models,” IEEE Trans. Fuzzy Syst., vol. 17, no. 3, pp. 724–731, Jun. 2009. [21] R. E. Skelton, T. Iwasaki, and K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design. New York: Taylor & Francis, 1998. [22] D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, “Vector spaces of lizarizations for matrix polynomials,” SIAM J. Matrix Anal. Appl., vol. 28, no. 4, pp. 971–1004. [23] I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials. New York: Academic, 1982. [24] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985. [25] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox. Natick, MA: MathWorks, 1995. [26] Polyx, Ltd., The Polynomial Toolbox for Matlab Release 2.5.0. [Online]. Available: www.polyx.com or www.polyx.cz.
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Supervisory Control and Measurement Scheduling for Discrete-Time Linear Systems Ji-Woong Lee, Member, IEEE, and Geir E. Dullerud, Fellow, IEEE
Abstract—The problem of jointly synthesizing a supervisor, a measurement scheduler, and a feedback controller for discrete-time linear systems is considered. It is shown that open-loop supervisory and scheduling laws and type are nonconservative for robust exponential stability and performance requirements, and that they can be obtained separately from the feedback controller. All the design conditions are formulated in terms of linear matrix inequalities. control, Index Terms— inequality, uncertain systems.
control, hybrid systems, linear matrix
I. INTRODUCTION Switched systems arise, among others, from supervisory control [1] and measurement scheduling [2]. In supervisory control, a supervisor chooses a controller among a finite set after sampling each measurement, and the plant state evolves accordingly. On the other hand, in measurement scheduling, a scheduler chooses a sensor among a finite set before taking each measurement, and the controller state is updated accordingly. Our goals are to treat the problems of finding supervisory and scheduling laws in a unified manner, and to relate them to the standard problem of determining a single switching sequence [3] to guarantee stability and desired performance levels. We focus on switched linear systems in the discrete time domain, and consider joint synthesis of a supervisor, a scheduler, and a feedback controller for robust extype performance ponential stabilization and guaranteed 2 and levels. Aside from supervisory control and measurement scheduling, discrete-time switched linear systems find applications in many areas such as signal processing and formal languages [4], multi-agent coordination [5], and approximate analysis and synthesis of continuous-time hybrid systems [6]. In Section II, we define and motivate the so-called “robustly” exponentially stabilizing supervisor-scheduler pairs, which ensure that the total (nonlinear) response of the “internal” switched linear system is separated into a zero-input response-like term and a zero-state response-like term. The robustness requirement guarantees that exponential stability is preserved against arbitrary state perturbations. We show that, as far as robust exponential stabilization is concerned, the problem of determining a switching law (i.e., supervisor-scheduler pair) is equivalent to that of finding a single switching sequence. This generalizes some of the results in [7] and shows the robustness requirement is crucial for sufficiency of open-loop switching. The result is in contrast to the fact that the existence of an asymptotically stabilizing switching law does not imply the existence of an asymptotically stabilizing switching sequence [8]. Moreover, whenever a supervisor, a measurement scheduler, and a path-dependent dynamic output feedback controller jointly, robustly exponentially, stabilize a switched linear plant, there exists a single switching sequence, which
H
H1
Manuscript received February 09, 2010; revised July 15, 2010; accepted November 28, 2010. Date of publication December 10, 2010; date of current version April 06, 2011. Recommended by Associate Editor J. S. Braslavsky. J.-W. Lee is with the Pennsylvania State University, University Park, PA 16802 USA (e-mail: [email protected]). G. E. Dullerud is with the University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2010.2098971 0018-9286/$26.00 © 2010 IEEE
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is robustly exponentially stabilizing and can be obtained independently of the feedback controller. type performance Next, we define in Section III our 2 and requirements. In particular, the -type performance requirement ensures that the overall system possesses the kind of small gain property that standard linear systems exhibit; that is, the overall uncertain (nonlinear) dynamics is separated into the dynamics associated with the plant and that with the uncertainty block, so that the system is robustly well-connected with sufficiently small dynamic uncertainty. We show that it suffices to look for an open-loop supervisor-scheduler pair to satisfy these performance requirements. This result is also extended to the joint synthesis of supervisor-scheduler-feedback controller triples. In particular, the result justifies the problem of jointly determining switching and feedback controllers for average output variance minimization [9], where an optimal switching sequence can be determined in an open-loop fashion in the design stage separately from the synthesis of a feedback controller. It also extends the feedback control problem for disturbance attenuation under autonomous switching [10] to joint synthesis of switching and feedback. The synthesis conditions are all formulated in terms of linear matrix inequalities [11]. An optimal switching sequence can, in principle, be determined by solving an increasing sequence of semidefinite programs indexed by the length of past switching paths that the feedback controller should recall, and then the associated controller coefficients are determined in a straightforward and nonconservative manner using the linear matrix inequality embedding technique [12], [13] or change-of-variables approaches [14], [15]. The set of real numbers is denoted by , and the set of positive (resp. 2 m2n , nonnegative) integers by (resp. 0 ). For x 2 n and denoted by kxk and k k are the Euclidean norm of x and the spectral 2 n2n , then we write < to norm of , respectively. If , 0 (resp. 0 ) is symmetric and positive definite mean that (resp. negative definite). The identity matrices, with their dimensions understood, are denoted ; similarly, the zero matrices are denoted .
H1
A Y X
A
H
H1
A X Y
XY X Y
I
0
II. SWITCHING LAWS FOR EXPONENTIAL STABILITY
A
Example 2: Consider x(t +1) = x(t), t 2 0 . There are N sensors to choose from; choosing the ith sensor at time t gives measurement y (t) = i x(t). At each time t, the measurement scheduler generates a scheduling command (t) before sampling y (t), so that a state observer xK (t +1) = xK (t)+ ( (t) xK (t) 0 y (t)), with y (t) = (t) x(t), gives rise to the state prediction error e(t) = x(t) 0 xK (t) evolving according to e(t + 1) = ( + (t) )e(t) for t 2 0 . The problem of determining a stabilizing scheduling law g = (g0 ; g1 ; . . .), such that (t) = gt (y (0); . . . ; y (t 0 1)), is then cast as the problem of finding a stabilizing switching law for S = f( + i ; i ): i = 1; . . . ; N g. In Example 1, the supervisory command (t) is computed after taking the measurement y (t). However, in Example 2, the scheduling command (t) is calculated before y (t) is sampled. To treat these situations in a unified manner, one can separate the switching sequence into a supervisory sequence = ((0); (1); . . .) 2 f1; . . . ; Nf g1 and a scheduling sequence = ( (0); (1); . . .) 2 f1; . . . ; Ng g1 , are the numbers of available supervisory and where Nf , Ng 2 scheduling commands, respectively, so that the mode of the system (1) at time t 2 0 is (t) = ((t); (t)). We will consider the case where N = Nf Ng : The set S is replaced with
C
A
C
A LC C
S = f(Aij ; Cj ): i = 1; . . . ; Nf ; j = 1; . . . ; Ng g
A Cj ) 2
where ( ij ;
2
S = f(Ai ; Ci ): i = 1; . . . ; N g
n2n 2 l2n
be an indexed set of matrix pairs. The set S defines the family of linear time-varying state-space representations of the form
x(t + 1) = A(t) x(t); t 2 y(t) = C(t) x(t); t 2
0
(1)
0
over all switching sequences = ((0); (1); . . .) 2 f1; . . . ; N g1 . The set S is called a switched linear system, or a discrete linear inclusion. Given a switching sequence 2 f1; . . . ; N g1 and an initial state x(0) 2 n , the linear time-varying system (1) generates a state sequence x = (x(0); x(1); . . .) and a measurement sequence y = (y(0); y(1); . . .). Switched systems typically arise in the context of supervisory control [1] and measurement scheduling [2]. Example 1: Consider a plant given by x(t + 1) = x(t) + u(t) and y (t) = x(t) for t 2 0 . There are N feedback gain matrices 1 ; . . . ; N to choose from; at each time t, the supervisor measures x(t) and issues a supervisory command (t) so that the control input u(t) = (t) x(t). Closing the feedback loop yields x(t + 1) = ( + (t) )x(t) for t 2 0 . The problem of determining a stabilizing supervisory law f = (f0 ; f1 ; . . .), such that (t) = ft (x(t)), is then cast as the problem of finding a stabilizing switching law for S = f( + i ; ): i = 1; . . . ; N g.
A
K
K K A BK
A BK I
B
0
(3)
Example 1 is the case with Nf = N and Ng = 1, and Example 2 has = 1 and Ng = N . The sets S and S are related by labeling the modes (i; j ) with the numbers (i 0 1)Ng + j . This relation will always be assumed; that is, throughout the technical note, we write (t) for (t) with 1j = 1 1 1 = N j = j understood for all j . Given S as in (2), let
Nf
C
C
C
f
, let
(2)
n2n 2 l2n , and the system (1) with
x(t + 1) = A(t)(t) x(t); t 2 y(t) = C(t) x(t); t 2 0 :
Definitions Given n, l, N
LC A LC
= fft: t 2 0 g;
C
C
g = fgt : t 2
0
g
be families of functions ft : f1; . . . ; Nf gt 2 f1; . . . ; Ng gt+1 2 ( l )t+1 ! f1; . . . ; Nf g and gt : f1; . . . ; Nf gt 2 f1; . . . ; Ng gt 2 ( l )t ! f1; . . . ; Ng g with the convention that f1; . . . ; Nf g0 , f1; . . . ; Ng g0 , and ( l )0 are empty sets. The family f defines a supervisory law, and g a scheduling law, for the switched linear system S such that the supervisory and scheduling sequences and , respectively, are generated according to
(t) = ft (t01 ; t ; yt );
(t) = gt (t01 ; t01 ; yt01 )
for t 2 0 . Here, s = ((0); . . . ; (s)), s = ( (0); . . . ; (s)), and y s = (y (0); . . . ; y (s)) with 01 , 01 , and y 01 being empty tuples. Note that (t01 ; t01 ; y t01 ) represents the information available to the scheduler at time t. The scheduler generates a scheduling command (t) before y (t) can be taken. Once y (t) is sampled according to the scheduling command, the information (t01 ; t ; y t ) becomes available to the supervisor at time t, and the supervisor issues a supervisory command (t) so that x(t) evolves to x(t +1) accordingly. The pair (f; g ) shall be called a switching law. Let us consider the “internal” switched linear system
x(t + 1) = A(t) x(t) + r(t);
t2
0:
(4)
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If a switching law (f; g ), with state measurements as in (3), generates a switching sequence for the internal system (4), then define
x(t; x(0)) =
x(0);
A t0
1) 1 1 1
(
0;
xt (t; r(t0 )) =
r(t0 );
0,
2
1) 1 1 1
(
so that
if t = 0 if t > 0 if t < t0 + 1 if t = t0 + 1 : if t > t0 + 1 +1) r (t0 );
(0) x(0);
A t
A t0 (
for t0 , t
A
0
t 1
x(t) = x(t; x(0)) +
for x(0); r(0); r(1); . . . 2 n , and t 2 0 . We require that, if (f; g ) is applied to the internal switched linear system, the responses x(t; x(0)) and xt (t; r(t0 )), t0 2 0 , all converge to the origin exponentially fast regardless of x(0), r(t0 ) 2 n , t0 2 0 . This ensures that the system responses to the initial state and state perturbations can be separated, and hence that the H2 and H1 type performance levels introduced in Section III are well defined. Definition 3: The switching law (f; g ) is said to be robustly exponentially stabilizing for S if there exist c > 0 and 2 (0; 1) such that, under (f; g ), the internal switched linear system (4) satisfies
xt
k
0 (t + 1; r (t0 ))k c
t t
r(t0 )k
k
(5a)
. for all t0 , t 2 0 with t t0 and for all x(0); r(0); r(1); . . . 2 Definition 4: A switching sequence = (; ) is said to be robustly exponentially stabilizing for S if there exist c > 0 and 2 (0; 1) such that
t; t0 )k ct0t
k8 ( 2
0
with t
8 (t; t0 ) =
t0 , where the state transition matrix
A t0 I; (
1) 1 1 1
A t (
);
t > t0 t = t0 .
It is immediate that, if a switching sequence (; ) is robustly exponentially stabilizing for S , then the switching law (f; g ) with ft and gt identically equal to (t) and (t), respectively, for all t is robustly exponentially stabilizing for S . Our goals are to show that the converse also holds true and to establish a convex characterization of the existence of such a switching sequence. To do so requires adapting some of the notions introduced for S in [7] to S . Definition 5: Let M 2 0 . Each element of f1; . . . ; Nf gM +1 (resp. f1; . . . ; Ng gM +1 ) is called a supervisory path (resp. scheduling path) of length M . If (i0 ; . . . ; iM ) is a supervisory path and if (j0 ; . . . ; jM ) is a scheduling path, then ((i0 ; j0 ); . . . ; (iM ; jM )) is called a switching path of length M . A nonempty set N of switching paths of length M is said to be admissible if, for each ((i0 ; j0 ); . . . ; (iM ; jM )) 2 N , there exist a K 2 , with K > M , and a switching path ((iM +1 ; jM +1 ); . . . ; (iK ; jK )) such that ((iK 0M ; jK 0M ); . . . ; (iK ; jK )) = ((i0 ; j0 ); . . . ; (iM ; jM )) and ((it ; jt ); . . . ; (it+M ; jt+M )) 2 N for all t 2 f0; . . . ; K 0 M g. An admissible set N of switching paths is called minimal if none of the proper subsets of N is admissible. In a sense, an admissible set of switching paths is a set of switching paths that lead to themselves via the other switching paths in the same set. Stability properties of a switching sequence are solely determined by the switching paths that occur infinitely many times in , and the set of such switching paths is admissible [16, Lemma 8]. Moreover, since an admissible set of switching paths is a finite union of minimal sets of switching paths, one only needs to consider minimal sets to look for
t2
0 g:
(6)
Such a is periodic and is unique up to a time shift [7]. A. Result on Robust Exponential Stabilization The following theorem characterizes the existence of robustly exponentially stabilizing switching laws. Theorem 6: Let S be as in (2). The following are equivalent: (a) There exists a robustly exponentially stabilizing switching law for S . (b) There exists a robustly exponentially stabilizing switching sequence for S . (c) There exist an M 2 0 , a minimal set N of switching paths of n2n such that length M , and matrices (i ;...;i ) 2
Y
(5b) n
for all t0 , t
= f(i0 ; . . . ; iM ): (i0 ; . . . ; iM ) = ( (t); . . . ; (t + M ));
t =0
x(t; x(0))k ct kx(0))k
stabilizing switching sequences. Each minimal set of switching paths of length M corresponds to a unique elementary cycle [17] in a directed graph whose nodes are switching paths of length M 0 1 [18]. To simplify notation, we will often write switching paths as (i0 ; . . . ; iM ) with ik = (ik ; jk ) for all k . Also, we use the convention that (i0 ; . . . ; iM 01 ) = 0 if M = 0; otherwise, (i0 ; . . . ; iM 01 ) is a switching path of length M 0 1. It is readily seen that, if N is a minimal set of switching paths of length M , then there is a switching sequence = (; ), with (t) = ((t); (t)) for all t, such that N
xt (t; r(t0 ))
k
875
(
Ai Y i ; (
Y i ; ;i Ai Y i ; ;i
...;i
)
T
0
...
(
...
) )
0
(7a) (7b)
for all (i0 ; . . . ; iM ) 2 N . Moreover, if condition (c) holds, then any periodic switching sequence satisfying (6) is robustly exponentially stabilizing for S . Proof: Suppose a switching law (f; g ) is robustly exponentially stabilizing for S , so that the internal system (4) satisfies (5) under (f; g ) for t t0 and for x(0); r(0); r(1); . . . 2 n . For k = 1; . . . ; n, denote the k th column of the n-by-n identity matrix by ek . Let M 1 be no less than the maximum of k ij k over all i and j ; let T 2 0 be p such that ncT 0n M n < 1. To show that (a) implies (b), choose x(0) and r for the internal system (4) as follows: Set x(0) = 0 and obtain (0) under (f; g ) (or more precisely, obtain (0) under g and obtain (0) from y (0) = (0) x(0) under f to form (0) = ((0); (0))); apply r(0) = (0) e1 to the internal system at time t = 0 and obtain (1) under (f; g ); apply r(1) = (1) (0) e2 to the internal system at time t = 1 and obtain (2) under (f; g ); proceed in this manner to obtain r(0); . . . ; r(n 0 1) and (0); . . . ; (n 0 1); and apply r(t) = 0 to the internal system at all time t n to obtain (n); (n + 1); . . .. Now, for any x0 = [ 1 1 1 1 n ]T 2 n with kx0 k = 1, we have
A
A
C
A A
n
T; 0)x0 k
8 (k; 0)ek k k jk8 (T; k)8
k8 (
j
k=1 n
=
k jkxk01 (T ; r(k 0 1))k
j
k=1 n
k jcT 0k kr(k 0 1)k
j
k=1 n
=
k jcT 0k k8(k; 0)ek k
j
k=1
p
ncT 0n M n kx0 k
which implies k8 (T; 0)k < 1. Thus, the periodic switching sequence with period T , where the switching path ((0); . . . ; (T 0 1)) repeats, is robustly exponentially stabilizing. This establishes that (a) implies (b). Thus, (a) and (b) are equivalent.
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It remains to show (b) Theorem 2] to our case, 2 existence of an paths of length , and that T ( ;...; )
is equivalent to (c). By adapting [7, we see that (b) is equivalent to the 0 , an admissible set N of switching such matrices ( ;...; ) 0 ( ;...; holds for all )
M Xi i > 0 M Ai X i i Ai Xi i < 0 (i0 ; . . . ;ii ) 2 N . A Schur complement argument yields that this inA 0 X(0i1 ... i ) < 0 equality is equivalent to Ai X(0i1 ... i ) i holding true for (i0 ; . . . ;ii ) 2 N . Choose a minimal set N N 01 and put Y(i ... i for (i0 ; . . . ;ii ) 2 N . Then, ) = X(i ... i ) since N is admissible, conditions (b) and (c) are equivalent. As N is minimal, there is a periodic switching sequence satisfying (6). Such a is robustly exponentially stabilizing for S .
xt xtx x tr
T
;
;
;
;
;
;
;
If a robustly exponentially stabilizing switching law is known and one wishes to obtain a robustly exponentially stabilizing switching sequence, then an alternative to solving (7) is provided in the first half of the proof of Theorem 6, where a periodic switching sequence with a period 2 0 such that k8 ( 0)k 1 is constructed. This period is likely to be much larger than the path length in condition (c) of the theorem; a minimal set of switching paths of length can have up to ( f g )M switching paths and can lead to a switching sequence of period ( f g )M . A counterexample in [8] shows that the existence of an asymptotically stabilizing supervisory law does not imply the existence of an asymptotically stabilizing supervisory sequence. According to Theorem 6, however, adding a robustness requirement to the stability notion makes the situation different. Moreover, this theorem clarifies and generalizes the claim in [7, Section III] that open-loop switching laws suffice. The following examples illustrate these points. Example 7: Consider the supervisory control problem to robustly exponentially stabilize the system S with f = 2, g = 1, and
T
T
T;
2jx2j;
with j 1 j
otherwise.
On this other hand, as is shown in [8], there is no robustly exponentially stabilizing supervisory sequence for this case. Therefore, we conclude by Theorem 6 that none of the asymptotically stabilizing supervisory laws, including the one given above, is robustlypexponentially stabilizing. Indeed, if (0) = [2 2]T , (0) = [0 1 0 3]T , and (1) = (2) = 1 1 1 = 0, then the internal system (4) under the supervisory law above yields the supervisory sequence = (2 1 1 . . .),
r
r
f
x
r
N
N
1
;;;
xtx
; CC12 == [[ 11
p12 p12 0 p12 p12
; A12 =
0
x^ (y1;y2) = P( 2) [ y1 y2 ] i
=
for
T
i;
p2 3 2 0 1
1]
01 ] :
0
i 2 f1; 2g and y1 , y2 2
P(
i;j )
=
0
y1 ; if i = 1 y2 y1 ; if i = 2 y2
0 3p1 2 0 p12
p12 0 p12
, where
A1 A1 [ C A1 C ] CCA1 j
T i
i
T i
i
T j
j
01
[
i
C A1 C ] : T i
T i
T j
g g g ;y (t) = g ( 01;y 01) 1 if (t 0 1) = 2 and x ^ ( 02) y (t 0 2);y (t 0 1) = = [^ x1 x^2] with jx^1j > jx^2j 2; otherwise for t 2. Since x ^ ( 02) y (t 0 2);y (t 0 1) = x(t) whenever t 2 and (t 0 1) = 2, a 1 is always followed by a 2 in and no more than three consecutive 2’s can appear in . It turns out that (7) is feasible over the scheduling paths of length M = 3 in the admissible, but not minimal, set f(1; 2; 1; 2), (1; 2; 2; 1), (1; 2; 2; 2), (2; 1; 2; 1), (2; 1; 2; 2), (2; 2; 1; 2), (2; 2; 2; 1)g of scheduling paths. Since ( (t); . . . ; (t + 3)) must belong to this set for all x(0) and t under g, we invoke [10, Corollary 3.4] and conclude that the scheduling law g is robustly exponentially stabilizing. Moreover, as Theorem 6 implies, there exists a robustly exponentially stabilizing scheduling sequence. Such a (periodic) scheduling sequence is given by = (1; 2; 1; 2; . . .), in which 1 and 2 alternate. Consider the measurement scheduling law given by (0) = 0 ( ) = 2, (1) = 1 ( (0) (0)) = 1, and t
t
t
t
T
t
B. Joint Synthesis of Supervisor, Scheduler, and Feedback Controller In view of Theorem 6 and [20, Theorem 20], joint synthesis of supervisor-scheduler-controller triples is possible. Let 2 n2n 2 n2m and j 2 l2n be given for ( ij ij ) = 1 ... = 1 ... g . Consider the controlled plant f and given by
i
A ;B
; ;N
j
; ;N
C
x(t + 1) = A ( ) ( )x(t) + B ( ) ( ) u(t); t 2 0 y(t) = C ( )x(t); t 2 0: (8) Assume that both (t) and (t), as well as y (t), become available to the controller at each time t. Consider all dynamic output feedback controllers that generate the control input u(t), t 2 0 , according to x (t +1) =A ( ( 0 ) ... ( ))x (t)+ B ( ( 0 ) ... ( ))y(t) u(t) = C ( ( 0 ) ... ( ))x (t)+D ( ( 0 ) ... ( ))y(t): Here, L 2 0 is the number of past modes that the controller recalls and (t) = ((t); (t)) for t 2 0 . The closed-loop system is x~(t +1) = A0( ) 00 + 0I B0( ) K(t) C0( ) 0I x~(t) (9) t t
It is shown in [19] that there exist asymptotically stabilizing supervisory laws for this S . Such a law is given by
(t) = f ( 01;x ) 1 if x(t) = [x1 x2 ] =
t> t
Define
M
;
1 2 0
A11 =
M
M
t x tr
where (0) = 2 and ( ) = 1 for 0; in this case, while ( ) = ( ; (0)) + 0 ( ; (0)) ! 0 as ! 1, both ( ; (0)) and 0 ( ; (0)) diverge. Example 9: Consider the measurement scheduling problem to stabilize the system S with f = 1, g = 2, and
t t
t
K
K; t
L ;
; t
K
K; t
L ;
; t
K
t
K; t
K; t
L ;
L ;
t
t
; t
; t
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where x ~(t) = [x(t)T xK (t)]T and
K(t) = K t0L ; ; t AK; t0L ; ; t BK; t0L ; = CK; t0L ; ; t DK; t0L ; ( (
) ...
( ))
( (
) ...
( ))
( (
) ...; (t))
( (
) ...
( ))
( (
) ...; (t))
(10)
Fig. 1. Given each switching sequence generated by (f; g ), the uncertain system is well-connected over all sufficiently small linear uncertainties 1.
for t 2 0 . We are to determine the number of past modes L to recall, the controller matrices (i ;...;i ) 2 (n+m)2(n+l) for all switching paths (i0 ; . . . ; iL ) of length L, and the switching law (f; g ) that robustly exponentially stabilizes the closed-loop system (9). Corollary 10: Given a plant (8), the following are equivalent: (a) There exist a switching law (f; g ) and a family of controller matrices (10), for some L 2 0 , such that (f; g ) is robustly exponentially stabilizing for the closed-loop system (9). (b) There exist a switching sequence = (; ) and a family of controller matrices (10), for some L 2 0 , such that the closedloop system (9) is robustly exponentially stable. (c) There exist an M 2 0 , a minimal set N of switching paths of , (i ;...;i 2 n2n such that length M , and (i ;...;i ) )
We will consider two performance measures. The first is of H2 type, and gives the square root of the average output variance (per unit time) of (12) under white Gaussian disturbance sequence w . This performance measure is well-defined provided that the switching law (f; g ) is robustly exponentially stabilizing for T (i.e., for S with ij , j replaced by ij , 2;j in T ). Definition 11: A switching law (f; g ) is said to achieve output regulation level > 0 for the system T if it is robustly exponentially stabilizing for T and if there exists a ~ 2 (0; ) such that, whenever x(0) = 0, the state-space equations (12) satisfy
K
R
B N (Ci
S
Ai R i ; )T AiT S i ;
N ( iT )T
i
(
...;
(
...;
i
Ai 0 R i ; ;i B Ai 0 S i ; ;i N (Ci R i ; ;i I I S i ; ;i )
)
T
(
(
(
...
)
...
...
)
(
...
)
for all (i0 ; . . . ; iM ) 2 N , where N ( ) is any full-column-rank matrix whose columns span the null space of . Moreover, if condition (c) holds, then the closed-loop system (9) is robustly exponentially stable under a periodic switching sequence satisfying (6) and feedback controller matrices (10) with L = M . A guaranteed, nonconservative procedure for obtaining a stabilizing set of controller matrices (10) is based on the well-known linear matrix inequality embedding technique [12], [13], and it is the same as [20, Algorithm 1] adapted to our case.
M
III. SWITCHING LAWS FOR GUARANTEED PERFORMANCE
A B C D C D ;j ):
= f( ij ; ij ; 1;ij ; 1;ij ; 2;j ; i = 1; . . . ; Nf ; j = 1; . . . ; Ng g
n2n 2
n2m 2 l
2n 2
l
l
2n 2
l
2m (11)
defines a family of linear time-varying state-space equations
A C C
B D D
x(t + 1) = (t)(t) x(t) + (t)(t) w(t); t 2 0 z (t) = 1;(t)(t) x(t) + 1;(t)(t) w(t); t 2 y(t) = 2;(t) x(t) + 2;(t) w(t); t 2 0 over all 2 f1; . . . ; Nf g1
I; 0;
t=s t 6= s.
Definition 12: A switching sequence = (; ) is said to achieve output regulation level > 0 for the system T if it is robustly exponentially stabilizing for T and if T
1 tr t!1 T + 1 t=0 ~; where (t0 ; t) lim
Y
t01
s=t
0;
C ; t Y (t ; t)CT; t + D ; t DT; t 1
( )
0
1
( )
1
( )
1
( )
B B
8 (t; s + 1) (s) T(s)8 (t; s + 1)T ; t > t0 ;
t = t0 .
The second performance measure is of H1 type, where (f; g ) is required to satisfy the following small gain property [21]: If is a switching sequence for T generated under (f; g ), the system (12) (or (1) in Fig. 1) is robustly well-connected for all uncertainties 1 whose `2 -induced gain is sufficiently small. This property in a sense guarantees that one can deal with the dynamics associated with the plant and uncertainty separately. This performance measure is well-defined if (f; g ) is robustly exponentially stabilizing for T . Definition 13: A switching law (f; g ) is said to achieve robust disturbance attenuation level > 0 for the system T if it is robustly exponentially stabilizing for T and if there exists a ~ 2 (0; ) such that, whenever x(0) = 0, the state-space equations (12), denoted (1) , is such that the uncertain system in Fig. 1 has z = 0 as the unique response to w = 0 over all linear operators 1 satisfying
G
2
2m 2
E w(t)w(s)T =
E[w(t)] = 0;
A. Problem Statement
T
kz(t)k2 ~2
where E[1] denotes the expectation with respect to w with
= We will now consider the problems of guaranteeing infinite-horizon performance measures. Given n, m, l1 , l2 , Nf , Ng 2 , the set
T
1 E T !1 T + 1 t=0 lim
N ( iT )
0 for T if it is robustly exponentially stabilizing for T and if there is a ~ 2 (0; ) such that
M (t; t )k ~
k for all t0 , t 2
M (t; t ) = 0
with t0 t, where
0
D
C C
B
1;(t )
1;(t +1)
.. .
8(
1;(t) t; t0
D
(t )
0
+ 1)B(t )
0
0 0
111 111
1;(t +1)
.. .
..
:
.. .
.
+ 2)B(t +1) 1 1 1 D1;(t) Whenever x(0) = 0, we have z (0) = D1;(0) w(0) and
C
8(
Y
1;(t) t; t0
( )=
z t
0
t 1
s=0
C
for t 2 . Thus, if there exists a switching sequence (; ) that achieves an output regulation level, or a robust disturbance attenuation level, then the switching law (f; g ) with (ft ; gt ) identically equal to ((t); (t)) for all t achieves the same performance level. Our goal in this section is to establish that the converse also holds true. B. Results on Guaranteed Performance The following theorems characterize the existence of switching laws that achieve guaranteed performance levels. Theorem 15: Let T be as in (11); let > 0. The following are equivalent. (a) There exists a switching law that achieves output regulation level
for T . (b) There exists a switching sequence that achieves output regulation level for T . (c) There exist an M 2 0 , a minimal set N of switching paths of n2n such that length M , and matrices (i ;...;i ) 2
Y
( ;...;
Ai Y i
( ;...;i
1
jN j
Yi i Ai 0 Y i i
)
( ;...;
for all (i0 ; . . . ; iM ) 2 N , and
tr C1;i
(i ;...;i )2N
Yi
( ;...;i
)
0 < Bi Bi
) > )
(14a)
T
(14b)
T 1;
T 1;
1;
< 2
(14c)
where jN j denotes the cardinality of N . Moreover, if condition (c) holds, then any periodic switching sequence satisfying (6) achieves output regulation level for T . Proof: The proof is similar to those of [22, Theorem 3.7] and [9, Theorem 1], so we will only sketch it. Suppose (f; g ) achieves an output regulation level > 0 for T . Let be any realization of the switching sequence under (f; g ) with x(0) = 0. Then it can be shown that there are t 2 n2n , t 2 0 , such that
Y
I Y I A Y A 0 Y 0 B B 0 I 1 0 tr C Y C + D D lim !1 T
(t)
T
T t (t) T 1 t=0
t
t+1
1;(t)
(t)
t
T 1;(t)
T (t)
1;(t)
T 1;(t)
(15a) (15b)
~2 (15c)
( ;...;
( ;...;
1;
)
(16a) T
)
1;
1;
( ;...;
)
1;
2
(16b)
for all (i0 ; . . . ; iM ) 2 N . Moreover, if condition (c) holds, then any periodic switching sequence satisfying (6) achieves disturbance attenuation level for T . Proof: Suppose (f; g ) achieves a robust disturbance attenuation level > 0 for T . Set x(0) = 0 and choose w(0); w(1); . . . 2 m to generate a switching sequence for T under (f; g ). Suppose k (t; t0 )k for some t, t0 2 0 with t t0 . Let 1 in Fig. 1 be such that w(s) = 0 whenever s < t0 or s > t and such that
M
[ w(t0 )T
1 1 1 w(t)T ]T
T (t; t0 ) = kM M (t; t0)k2 [ z(t0)T
1 1 1 z (t)T ]T :
This 1 satisfies (13) for any ~ 2 (0; ), and yet, because the matrix 0 (t; t0 ) (t; t0 )T =k (t; t0 )k2 is singular, z = 0 is not the unique solution to the equation z = (1)1 z , which governs the uncertain system in Fig. 1. This contradicts the assumption on (f; g ), and hence (a) and (b) are equivalent. To show (b) is equivalent to (c), suppose a switching sequence achieves a robust disturbance attenuation level > 0. Then by applying [23, Theorem 11] to the scaled version of T given by ( ij ; 01=2 ij ; 01=2 1;ij ; 01 n21n;ij ; 2;j ; 2;j ) for all i and j , satisfying we deduce that there exist t 2
I M
M
M
A
C i +D i D i
Y i i >0 B Y i i 0 Ai Bi D ii 0 I Ci Di Y 0 i i 0 0 I , and 2 ; . Then, due to [22, Lemma 2.3], this achieves the output regulation level for T . This establishes that (a) implies (b), and hence that (a) and (b) are equivalent. To show (b) and (c) are equivalent, suppose a switching sequence achieves an output regulation level > . Then it can be shown by adapting the proof of [9, Theorem 1] to our case that there are a path length M 2 0 and a minimal set N of switching paths of length M such that (14) holds true. The converse implication follows because, choosing a satisfying (6), we recover (15) from (14). Theorem 16: Let T be as in (11); let > . The following are equivalent. (a) There exists a switching law that achieves robust disturbance attenuation level for T . (b) There exists a switching sequence that achieves robust disturbance attenuation level for T . (c) There exist an M 2 0 , a minimal set N of switching paths of n2n such that length M , and matrices (i ;...;i ) 2
B
A C
(t)
B D
(t)
1;(t)
for all t 2
C
0
T
1;(t)
G
D C D
X
I X I X 00 A 0 I C X 0 0 I 0 0 I t
t+1
(t)
1
1;(t)
t
and for some , > 0, 2
B D
(17a)
(t)
1;(t)
(17b)
(0; ). Define
WM () = f(i0 ; . . . ; iM ):
(i0 ; . . . ; iM ) = ((t); . . . ; (t + M )); t 2
0g
for M 2 0 . Then, by [10, Theorem 3.3] and the Schur complement formular, there exist an M 2 0 and matrices (i ;...;i ) such that (16) holds for all (i0 ; . . . ; iM ) 2 WM (). There exists at least one minimal set N of switching paths of length M such that N WM (), so we obtain (c). Conversely, assuming (c) holds, reversing the above
X
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 4, APRIL 2011
argument gives that (17) holds for all t 2 0 , where 2 (0; ) is sufficiently close to , and where satisfies (6). Therefore, (b) and (c) are equivalent. C. Joint Synthesis of Supervisor, Scheduler, and Feedback Controller Joint synthesis of supervisor-scheduler-controller triples is also possible for achieving guaranteed disturbance attenuation performance in view of Theorem 15 and [10, Theorem 3.7] and for achieving guaranteed robust output regulation performance in view of Theorem 16 and [22, Theorem 4.2]. The technical development is similar to that of Section II-C, so the details are omitted. In particular, adapting [9, Theorem 1] to our case will result in a nonconservative joint synthesis condition for guaranteed output regulation. IV. CONCLUSION Problems of robust exponential stabilization, output regulation, and robust disturbance attenuation have been considered for discrete-time switched linear systems. It is shown that supervisor-scheduler pairs can be determined open-loop in the design stage, and that open-loop supervisory-scheduling laws suffice in the implementation stage. Determining the best supervisory and scheduling sequences requires solving an increasing sequence of semidefinite programs. As one moves from one such program to the next, the computational complexity is likely to grow exponentially. Although this is due to the “semi-decidable” problem nature [24] and the nonconvexity of the associated Lyapunov functions [25], the presented results are nonconservative and expected to be useful for small problems. Moreover, depending on one’s tradeoff between performance and computational efficiency, one has the option to either stop at a point along the sequence of semidefinite programs and settle for the best performance achievable at that point or move further down the sequence by paying more computational cost. Even though one can always find open-loop solutions, it is desirable to use closed-loop solutions in practice whenever there is uncertainty in the system coefficients or the system is required to be fault-tolerant. In some cases, fault-tolerance can be ensured by using a set-valued switching law as suggested in [7]. However, the problem of determining a closed-loop switching law to guarantee robustness against uncertainty in the system coefficients remains open.
REFERENCES [1] A. S. Morse, “Supervisory control of families of linear set-point controllers—Part 1: Exact matching,” IEEE Trans. Autom. Control, vol. 41, no. 10, pp. 1413–1431, Oct. 1996. [2] L. Meier, III, J. Peschon, and R. M. Dressler, “Optimal control of measurement subsystems,” IEEE Trans. Autom. Control, vol. AC-12, no. 5, pp. 528–536, Oct. 1967. [3] D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Syst. Mag., vol. 19, no. 5, pp. 59–70, Oct. 1999. [4] R. Jungers, The Joint Spectral Radius: Theory and Applications. Berlin, Germany: Springer, 2009. [5] A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules,” IEEE Trans. Autom. Control, vol. 48, no. 6, pp. 988–1001, Jun. 2003. [6] A. Girard, G. Pola, and P. Tabuada, “Approximate bisimilar symbolic models for incrementally stable switched systems,” IEEE Trans. Autom. Control, vol. 55, no. 1, pp. 116–126, Jan. 2010. [7] J.-W. Lee and G. E. Dullerud, “Uniformly stabilizing sets of switching sequences for switched linear systems,” IEEE Trans. Autom. Control, vol. 52, no. 5, pp. 868–874, May 2007. [8] D. P. Stanford and J. M. Urbano, “Some convergence properties of matrix sets,” SIAM J. Matrix Anal. Appl., vol. 15, no. 4, pp. 1132–1140, 1994.
879
[9] J.-W. Lee, “Infinite-horizon joint LQG synthesis of switching and feedback in discrete time,” IEEE Trans. Autom. Control, vol. 54, no. 8, pp. 1945–1951, Aug. 2009. [10] J.-W. Lee and G. E. Dullerud, “Optimal disturbance attenuation for discrete-time switched and Markovian jump linear systems,” SIAM J. Control Optimiz., vol. 45, no. 4, pp. 1329–1358, 2006. [11] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [12] A. Packard, “Gain scheduling via linear fractional transformations,” Syst. Control Lett., vol. 22, no. 2, pp. 79–92, 1994. [13] P. Gahinet and P. Apkarian, “A linear matrix inequality approach to control,” Int. J. Robust Nonlin. Control, vol. 4, no. 4, pp. 421–448, 1994. [14] C. Scherer, P. Gahinet, and M. Chilali, “Multiobjective output-feedback control via LMI optimization,” IEEE Trans. Autom. Control, vol. 42, no. 7, pp. 896–911, Jul. 1997. [15] I. Masubuchi, A. Ohara, and N. Suda, “LMI-based controller synthesis: A unified formulation and solution,” Int. J. Robust Nonlin. Control, vol. 8, no. 8, pp. 669–686, 1998. [16] J.-W. Lee, “Uniform consensus among self-driven particles,” in Hybrid Systems: Computation and Control, R. Majumdar and P. Tabuada, Eds. Berlin, Germany: Springer-Verlag, 2009, vol. 5469, Lecture Notes in Computer Science, pp. 252–261. [17] D. B. Johnson, “Finding all the elementary circuits of a directed graph,” SIAM J. Comput., vol. 4, no. 1, pp. 77–84, 1975. [18] S. A. Krishnamurthy and J.-W. Lee, “A computational stability analysis of discrete-time piecewise linear systems,” in Proc. 48th IEEE Conf. Decision Control, 2009, pp. 1106–1111. [19] D. P. Stanford, “Stability for a multi-rate sampled-data system,” SIAM J. Control Optimiz., vol. 17, no. 3, pp. 390–399, 1979. [20] J.-W. Lee and G. E. Dullerud, “Uniform stabilization of discrete-time switched and Markovian jump linear systems,” Automatica, vol. 42, no. 2, pp. 205–218, 2006. [21] G. E. Dullerud and F. Paganini, A Course in Robust Control Theory: A Convex Approach. New York: Springer-Verlag, 2000. [22] J.-W. Lee and P. P. Khargonekar, “Optimal output regulation for discrete-time switched and Markovian jump linear systems,” SIAM J. Control Optimiz., vol. 47, no. 1, pp. 40–72, 2008. [23] G. E. Dullerud and S. Lall, “A new approach for analysis and synthesis of time-varying systems,” IEEE Trans. Autom. Control, vol. 44, no. 8, pp. 1486–1497, Aug. 1999. [24] P.-A. Bliman and G. Ferrari-Trecate, “Stability analysis of discrete-time switched systems through Lyapunov functions with nonminimal state,” in Proc. IFAC Conf. Anal. Design Hybrid Syst., 2003, pp. 325–330. [25] F. Blanchini and C. Savorgnan, “Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions,” Automatica, vol. 44, no. 4, pp. 1166–1170, 2008.
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D
[9] D. Henrion, O. Bachelier, and M. Sebek, “ -stability of polynomial matrices,” Int. J. Contr., vol. 74, no. 8, pp. 845–856, 2001. [10] D. Henrion, D. Arzelier, D. Peaucelle, and M. Sebek, “An LMI condition for robust stability of polynomial matrix polytopes,” Automatica, vol. 37, no. 3, pp. 461–468, 2001. [11] D. Henrion, D. Arzelier, and D. Peaucelle, “Positive polynomial matrices and improved LMI robustness conditions,” Automatica, vol. 39, no. 8, pp. 1479–1485, 2003. [12] Y. Ebigara, K. Maeda, and T. Hagiwara, “Robust -stability analysis of uncertain polynomial matrices via polynomial-type multipliers,” in Proc. 16th IFAC World Congr., Prague, Czechoslovakia, 2005. [13] D. C. W. Ramos and P. L. D. Peres, “An LMI condition for the robust stability of uncertain continuous-time linear systems,” IEEE Trans. Autom. Control, vol. 47, no. 4, pp. 675–678, Apr. 2002. [14] R. C. L. F. Oliveira and P. L. D. Peres, “Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations,” IEEE Trans. Autom. Control, vol. 52, no. 7, pp. 1334–1340, Jul. 2007. [15] P. A. Parrilo, “Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization,” Ph.D. dissertation, California Institute of Technology, Pasadena, CA, 2000. [16] P.-A. Bliman, “A convex approach to robust stability for linear systems with uncertain scalar parameters,” SIAM J. Contr. Optimiz., vol. 42, no. 6, pp. 2016–2042, 2004. [17] G. Chesi, A. Garulli, A. Tesi, and A. Vicino, “Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: An LMI approach,” IEEE Trans. Autom. Control, vol. 50, no. 3, pp. 65–370, Mar. 2005. [18] Y. Y. Cao and Z. Lin, “A discriptor approach to robust stability analysis and controller synthesis,” IEEE Trans. Autom. Control, vol. 49, no. 11, pp. 2081–2084, Nov. 2004. [19] R. C. L. F. Oliveira, M. C. de Oliveira, and P. L. D. Peres, “Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions,” Syst. Control Lett., vol. 57, no. 8, pp. 680–689, 2008. [20] T. M. Guerra, A. Kruszewski, and M. Bernal, “Control law proposition for the stabilization of discrete Takagi-Sugeno models,” IEEE Trans. Fuzzy Syst., vol. 17, no. 3, pp. 724–731, Jun. 2009. [21] R. E. Skelton, T. Iwasaki, and K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design. New York: Taylor & Francis, 1998. [22] D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, “Vector spaces of lizarizations for matrix polynomials,” SIAM J. Matrix Anal. Appl., vol. 28, no. 4, pp. 971–1004. [23] I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials. New York: Academic, 1982. [24] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985. [25] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox. Natick, MA: MathWorks, 1995. [26] Polyx, Ltd., The Polynomial Toolbox for Matlab Release 2.5.0. [Online]. Available: www.polyx.com or www.polyx.cz.
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Supervisory Control and Measurement Scheduling for Discrete-Time Linear Systems Ji-Woong Lee, Member, IEEE, and Geir E. Dullerud, Fellow, IEEE
Abstract—The problem of jointly synthesizing a supervisor, a measurement scheduler, and a feedback controller for discrete-time linear systems is considered. It is shown that open-loop supervisory and scheduling laws and type are nonconservative for robust exponential stability and performance requirements, and that they can be obtained separately from the feedback controller. All the design conditions are formulated in terms of linear matrix inequalities. control, Index Terms— inequality, uncertain systems.
control, hybrid systems, linear matrix
I. INTRODUCTION Switched systems arise, among others, from supervisory control [1] and measurement scheduling [2]. In supervisory control, a supervisor chooses a controller among a finite set after sampling each measurement, and the plant state evolves accordingly. On the other hand, in measurement scheduling, a scheduler chooses a sensor among a finite set before taking each measurement, and the controller state is updated accordingly. Our goals are to treat the problems of finding supervisory and scheduling laws in a unified manner, and to relate them to the standard problem of determining a single switching sequence [3] to guarantee stability and desired performance levels. We focus on switched linear systems in the discrete time domain, and consider joint synthesis of a supervisor, a scheduler, and a feedback controller for robust extype performance ponential stabilization and guaranteed 2 and levels. Aside from supervisory control and measurement scheduling, discrete-time switched linear systems find applications in many areas such as signal processing and formal languages [4], multi-agent coordination [5], and approximate analysis and synthesis of continuous-time hybrid systems [6]. In Section II, we define and motivate the so-called “robustly” exponentially stabilizing supervisor-scheduler pairs, which ensure that the total (nonlinear) response of the “internal” switched linear system is separated into a zero-input response-like term and a zero-state response-like term. The robustness requirement guarantees that exponential stability is preserved against arbitrary state perturbations. We show that, as far as robust exponential stabilization is concerned, the problem of determining a switching law (i.e., supervisor-scheduler pair) is equivalent to that of finding a single switching sequence. This generalizes some of the results in [7] and shows the robustness requirement is crucial for sufficiency of open-loop switching. The result is in contrast to the fact that the existence of an asymptotically stabilizing switching law does not imply the existence of an asymptotically stabilizing switching sequence [8]. Moreover, whenever a supervisor, a measurement scheduler, and a path-dependent dynamic output feedback controller jointly, robustly exponentially, stabilize a switched linear plant, there exists a single switching sequence, which
H
H1
Manuscript received February 09, 2010; revised July 15, 2010; accepted November 28, 2010. Date of publication December 10, 2010; date of current version April 06, 2011. Recommended by Associate Editor J. S. Braslavsky. J.-W. Lee is with the Pennsylvania State University, University Park, PA 16802 USA (e-mail: [email protected]). G. E. Dullerud is with the University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2010.2098971 0018-9286/$26.00 © 2010 IEEE
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is robustly exponentially stabilizing and can be obtained independently of the feedback controller. type performance Next, we define in Section III our 2 and requirements. In particular, the -type performance requirement ensures that the overall system possesses the kind of small gain property that standard linear systems exhibit; that is, the overall uncertain (nonlinear) dynamics is separated into the dynamics associated with the plant and that with the uncertainty block, so that the system is robustly well-connected with sufficiently small dynamic uncertainty. We show that it suffices to look for an open-loop supervisor-scheduler pair to satisfy these performance requirements. This result is also extended to the joint synthesis of supervisor-scheduler-feedback controller triples. In particular, the result justifies the problem of jointly determining switching and feedback controllers for average output variance minimization [9], where an optimal switching sequence can be determined in an open-loop fashion in the design stage separately from the synthesis of a feedback controller. It also extends the feedback control problem for disturbance attenuation under autonomous switching [10] to joint synthesis of switching and feedback. The synthesis conditions are all formulated in terms of linear matrix inequalities [11]. An optimal switching sequence can, in principle, be determined by solving an increasing sequence of semidefinite programs indexed by the length of past switching paths that the feedback controller should recall, and then the associated controller coefficients are determined in a straightforward and nonconservative manner using the linear matrix inequality embedding technique [12], [13] or change-of-variables approaches [14], [15]. The set of real numbers is denoted by , and the set of positive (resp. 2 m2n , nonnegative) integers by (resp. 0 ). For x 2 n and denoted by kxk and k k are the Euclidean norm of x and the spectral 2 n2n , then we write < to norm of , respectively. If , 0 (resp. 0 ) is symmetric and positive definite mean that (resp. negative definite). The identity matrices, with their dimensions understood, are denoted ; similarly, the zero matrices are denoted .
H1
A Y X
A
H
H1
A X Y
XY X Y
I
0
II. SWITCHING LAWS FOR EXPONENTIAL STABILITY
A
Example 2: Consider x(t +1) = x(t), t 2 0 . There are N sensors to choose from; choosing the ith sensor at time t gives measurement y (t) = i x(t). At each time t, the measurement scheduler generates a scheduling command (t) before sampling y (t), so that a state observer xK (t +1) = xK (t)+ ( (t) xK (t) 0 y (t)), with y (t) = (t) x(t), gives rise to the state prediction error e(t) = x(t) 0 xK (t) evolving according to e(t + 1) = ( + (t) )e(t) for t 2 0 . The problem of determining a stabilizing scheduling law g = (g0 ; g1 ; . . .), such that (t) = gt (y (0); . . . ; y (t 0 1)), is then cast as the problem of finding a stabilizing switching law for S = f( + i ; i ): i = 1; . . . ; N g. In Example 1, the supervisory command (t) is computed after taking the measurement y (t). However, in Example 2, the scheduling command (t) is calculated before y (t) is sampled. To treat these situations in a unified manner, one can separate the switching sequence into a supervisory sequence = ((0); (1); . . .) 2 f1; . . . ; Nf g1 and a scheduling sequence = ( (0); (1); . . .) 2 f1; . . . ; Ng g1 , are the numbers of available supervisory and where Nf , Ng 2 scheduling commands, respectively, so that the mode of the system (1) at time t 2 0 is (t) = ((t); (t)). We will consider the case where N = Nf Ng : The set S is replaced with
C
A
C
A LC C
S = f(Aij ; Cj ): i = 1; . . . ; Nf ; j = 1; . . . ; Ng g
A Cj ) 2
where ( ij ;
2
S = f(Ai ; Ci ): i = 1; . . . ; N g
n2n 2 l2n
be an indexed set of matrix pairs. The set S defines the family of linear time-varying state-space representations of the form
x(t + 1) = A(t) x(t); t 2 y(t) = C(t) x(t); t 2
0
(1)
0
over all switching sequences = ((0); (1); . . .) 2 f1; . . . ; N g1 . The set S is called a switched linear system, or a discrete linear inclusion. Given a switching sequence 2 f1; . . . ; N g1 and an initial state x(0) 2 n , the linear time-varying system (1) generates a state sequence x = (x(0); x(1); . . .) and a measurement sequence y = (y(0); y(1); . . .). Switched systems typically arise in the context of supervisory control [1] and measurement scheduling [2]. Example 1: Consider a plant given by x(t + 1) = x(t) + u(t) and y (t) = x(t) for t 2 0 . There are N feedback gain matrices 1 ; . . . ; N to choose from; at each time t, the supervisor measures x(t) and issues a supervisory command (t) so that the control input u(t) = (t) x(t). Closing the feedback loop yields x(t + 1) = ( + (t) )x(t) for t 2 0 . The problem of determining a stabilizing supervisory law f = (f0 ; f1 ; . . .), such that (t) = ft (x(t)), is then cast as the problem of finding a stabilizing switching law for S = f( + i ; ): i = 1; . . . ; N g.
A
K
K K A BK
A BK I
B
0
(3)
Example 1 is the case with Nf = N and Ng = 1, and Example 2 has = 1 and Ng = N . The sets S and S are related by labeling the modes (i; j ) with the numbers (i 0 1)Ng + j . This relation will always be assumed; that is, throughout the technical note, we write (t) for (t) with 1j = 1 1 1 = N j = j understood for all j . Given S as in (2), let
Nf
C
C
C
f
, let
(2)
n2n 2 l2n , and the system (1) with
x(t + 1) = A(t)(t) x(t); t 2 y(t) = C(t) x(t); t 2 0 :
Definitions Given n, l, N
LC A LC
= fft: t 2 0 g;
C
C
g = fgt : t 2
0
g
be families of functions ft : f1; . . . ; Nf gt 2 f1; . . . ; Ng gt+1 2 ( l )t+1 ! f1; . . . ; Nf g and gt : f1; . . . ; Nf gt 2 f1; . . . ; Ng gt 2 ( l )t ! f1; . . . ; Ng g with the convention that f1; . . . ; Nf g0 , f1; . . . ; Ng g0 , and ( l )0 are empty sets. The family f defines a supervisory law, and g a scheduling law, for the switched linear system S such that the supervisory and scheduling sequences and , respectively, are generated according to
(t) = ft (t01 ; t ; yt );
(t) = gt (t01 ; t01 ; yt01 )
for t 2 0 . Here, s = ((0); . . . ; (s)), s = ( (0); . . . ; (s)), and y s = (y (0); . . . ; y (s)) with 01 , 01 , and y 01 being empty tuples. Note that (t01 ; t01 ; y t01 ) represents the information available to the scheduler at time t. The scheduler generates a scheduling command (t) before y (t) can be taken. Once y (t) is sampled according to the scheduling command, the information (t01 ; t ; y t ) becomes available to the supervisor at time t, and the supervisor issues a supervisory command (t) so that x(t) evolves to x(t +1) accordingly. The pair (f; g ) shall be called a switching law. Let us consider the “internal” switched linear system
x(t + 1) = A(t) x(t) + r(t);
t2
0:
(4)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 4, APRIL 2011
If a switching law (f; g ), with state measurements as in (3), generates a switching sequence for the internal system (4), then define
x(t; x(0)) =
x(0);
A t0
1) 1 1 1
(
0;
xt (t; r(t0 )) =
r(t0 );
0,
2
1) 1 1 1
(
so that
if t = 0 if t > 0 if t < t0 + 1 if t = t0 + 1 : if t > t0 + 1 +1) r (t0 );
(0) x(0);
A t
A t0 (
for t0 , t
A
0
t 1
x(t) = x(t; x(0)) +
for x(0); r(0); r(1); . . . 2 n , and t 2 0 . We require that, if (f; g ) is applied to the internal switched linear system, the responses x(t; x(0)) and xt (t; r(t0 )), t0 2 0 , all converge to the origin exponentially fast regardless of x(0), r(t0 ) 2 n , t0 2 0 . This ensures that the system responses to the initial state and state perturbations can be separated, and hence that the H2 and H1 type performance levels introduced in Section III are well defined. Definition 3: The switching law (f; g ) is said to be robustly exponentially stabilizing for S if there exist c > 0 and 2 (0; 1) such that, under (f; g ), the internal switched linear system (4) satisfies
xt
k
0 (t + 1; r (t0 ))k c
t t
r(t0 )k
k
(5a)
. for all t0 , t 2 0 with t t0 and for all x(0); r(0); r(1); . . . 2 Definition 4: A switching sequence = (; ) is said to be robustly exponentially stabilizing for S if there exist c > 0 and 2 (0; 1) such that
t; t0 )k ct0t
k8 ( 2
0
with t
8 (t; t0 ) =
t0 , where the state transition matrix
A t0 I; (
1) 1 1 1
A t (
);
t > t0 t = t0 .
It is immediate that, if a switching sequence (; ) is robustly exponentially stabilizing for S , then the switching law (f; g ) with ft and gt identically equal to (t) and (t), respectively, for all t is robustly exponentially stabilizing for S . Our goals are to show that the converse also holds true and to establish a convex characterization of the existence of such a switching sequence. To do so requires adapting some of the notions introduced for S in [7] to S . Definition 5: Let M 2 0 . Each element of f1; . . . ; Nf gM +1 (resp. f1; . . . ; Ng gM +1 ) is called a supervisory path (resp. scheduling path) of length M . If (i0 ; . . . ; iM ) is a supervisory path and if (j0 ; . . . ; jM ) is a scheduling path, then ((i0 ; j0 ); . . . ; (iM ; jM )) is called a switching path of length M . A nonempty set N of switching paths of length M is said to be admissible if, for each ((i0 ; j0 ); . . . ; (iM ; jM )) 2 N , there exist a K 2 , with K > M , and a switching path ((iM +1 ; jM +1 ); . . . ; (iK ; jK )) such that ((iK 0M ; jK 0M ); . . . ; (iK ; jK )) = ((i0 ; j0 ); . . . ; (iM ; jM )) and ((it ; jt ); . . . ; (it+M ; jt+M )) 2 N for all t 2 f0; . . . ; K 0 M g. An admissible set N of switching paths is called minimal if none of the proper subsets of N is admissible. In a sense, an admissible set of switching paths is a set of switching paths that lead to themselves via the other switching paths in the same set. Stability properties of a switching sequence are solely determined by the switching paths that occur infinitely many times in , and the set of such switching paths is admissible [16, Lemma 8]. Moreover, since an admissible set of switching paths is a finite union of minimal sets of switching paths, one only needs to consider minimal sets to look for
t2
0 g:
(6)
Such a is periodic and is unique up to a time shift [7]. A. Result on Robust Exponential Stabilization The following theorem characterizes the existence of robustly exponentially stabilizing switching laws. Theorem 6: Let S be as in (2). The following are equivalent: (a) There exists a robustly exponentially stabilizing switching law for S . (b) There exists a robustly exponentially stabilizing switching sequence for S . (c) There exist an M 2 0 , a minimal set N of switching paths of n2n such that length M , and matrices (i ;...;i ) 2
Y
(5b) n
for all t0 , t
= f(i0 ; . . . ; iM ): (i0 ; . . . ; iM ) = ( (t); . . . ; (t + M ));
t =0
x(t; x(0))k ct kx(0))k
stabilizing switching sequences. Each minimal set of switching paths of length M corresponds to a unique elementary cycle [17] in a directed graph whose nodes are switching paths of length M 0 1 [18]. To simplify notation, we will often write switching paths as (i0 ; . . . ; iM ) with ik = (ik ; jk ) for all k . Also, we use the convention that (i0 ; . . . ; iM 01 ) = 0 if M = 0; otherwise, (i0 ; . . . ; iM 01 ) is a switching path of length M 0 1. It is readily seen that, if N is a minimal set of switching paths of length M , then there is a switching sequence = (; ), with (t) = ((t); (t)) for all t, such that N
xt (t; r(t0 ))
k
875
(
Ai Y i ; (
Y i ; ;i Ai Y i ; ;i
...;i
)
T
0
...
(
...
) )
0
(7a) (7b)
for all (i0 ; . . . ; iM ) 2 N . Moreover, if condition (c) holds, then any periodic switching sequence satisfying (6) is robustly exponentially stabilizing for S . Proof: Suppose a switching law (f; g ) is robustly exponentially stabilizing for S , so that the internal system (4) satisfies (5) under (f; g ) for t t0 and for x(0); r(0); r(1); . . . 2 n . For k = 1; . . . ; n, denote the k th column of the n-by-n identity matrix by ek . Let M 1 be no less than the maximum of k ij k over all i and j ; let T 2 0 be p such that ncT 0n M n < 1. To show that (a) implies (b), choose x(0) and r for the internal system (4) as follows: Set x(0) = 0 and obtain (0) under (f; g ) (or more precisely, obtain (0) under g and obtain (0) from y (0) = (0) x(0) under f to form (0) = ((0); (0))); apply r(0) = (0) e1 to the internal system at time t = 0 and obtain (1) under (f; g ); apply r(1) = (1) (0) e2 to the internal system at time t = 1 and obtain (2) under (f; g ); proceed in this manner to obtain r(0); . . . ; r(n 0 1) and (0); . . . ; (n 0 1); and apply r(t) = 0 to the internal system at all time t n to obtain (n); (n + 1); . . .. Now, for any x0 = [ 1 1 1 1 n ]T 2 n with kx0 k = 1, we have
A
A
C
A A
n
T; 0)x0 k
8 (k; 0)ek k k jk8 (T; k)8
k8 (
j
k=1 n
=
k jkxk01 (T ; r(k 0 1))k
j
k=1 n
k jcT 0k kr(k 0 1)k
j
k=1 n
=
k jcT 0k k8(k; 0)ek k
j
k=1
p
ncT 0n M n kx0 k
which implies k8 (T; 0)k < 1. Thus, the periodic switching sequence with period T , where the switching path ((0); . . . ; (T 0 1)) repeats, is robustly exponentially stabilizing. This establishes that (a) implies (b). Thus, (a) and (b) are equivalent.
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 4, APRIL 2011
It remains to show (b) Theorem 2] to our case, 2 existence of an paths of length , and that T ( ;...; )
is equivalent to (c). By adapting [7, we see that (b) is equivalent to the 0 , an admissible set N of switching such matrices ( ;...; ) 0 ( ;...; holds for all )
M Xi i > 0 M Ai X i i Ai Xi i < 0 (i0 ; . . . ;ii ) 2 N . A Schur complement argument yields that this inA 0 X(0i1 ... i ) < 0 equality is equivalent to Ai X(0i1 ... i ) i holding true for (i0 ; . . . ;ii ) 2 N . Choose a minimal set N N 01 and put Y(i ... i for (i0 ; . . . ;ii ) 2 N . Then, ) = X(i ... i ) since N is admissible, conditions (b) and (c) are equivalent. As N is minimal, there is a periodic switching sequence satisfying (6). Such a is robustly exponentially stabilizing for S .
xt xtx x tr
T
;
;
;
;
;
;
;
If a robustly exponentially stabilizing switching law is known and one wishes to obtain a robustly exponentially stabilizing switching sequence, then an alternative to solving (7) is provided in the first half of the proof of Theorem 6, where a periodic switching sequence with a period 2 0 such that k8 ( 0)k 1 is constructed. This period is likely to be much larger than the path length in condition (c) of the theorem; a minimal set of switching paths of length can have up to ( f g )M switching paths and can lead to a switching sequence of period ( f g )M . A counterexample in [8] shows that the existence of an asymptotically stabilizing supervisory law does not imply the existence of an asymptotically stabilizing supervisory sequence. According to Theorem 6, however, adding a robustness requirement to the stability notion makes the situation different. Moreover, this theorem clarifies and generalizes the claim in [7, Section III] that open-loop switching laws suffice. The following examples illustrate these points. Example 7: Consider the supervisory control problem to robustly exponentially stabilize the system S with f = 2, g = 1, and
T
T
T;
2jx2j;
with j 1 j
otherwise.
On this other hand, as is shown in [8], there is no robustly exponentially stabilizing supervisory sequence for this case. Therefore, we conclude by Theorem 6 that none of the asymptotically stabilizing supervisory laws, including the one given above, is robustlypexponentially stabilizing. Indeed, if (0) = [2 2]T , (0) = [0 1 0 3]T , and (1) = (2) = 1 1 1 = 0, then the internal system (4) under the supervisory law above yields the supervisory sequence = (2 1 1 . . .),
r
r
f
x
r
N
N
1
;;;
xtx
; CC12 == [[ 11
p12 p12 0 p12 p12
; A12 =
0
x^ (y1;y2) = P( 2) [ y1 y2 ] i
=
for
T
i;
p2 3 2 0 1
1]
01 ] :
0
i 2 f1; 2g and y1 , y2 2
P(
i;j )
=
0
y1 ; if i = 1 y2 y1 ; if i = 2 y2
0 3p1 2 0 p12
p12 0 p12
, where
A1 A1 [ C A1 C ] CCA1 j
T i
i
T i
i
T j
j
01
[
i
C A1 C ] : T i
T i
T j
g g g ;y (t) = g ( 01;y 01) 1 if (t 0 1) = 2 and x ^ ( 02) y (t 0 2);y (t 0 1) = = [^ x1 x^2] with jx^1j > jx^2j 2; otherwise for t 2. Since x ^ ( 02) y (t 0 2);y (t 0 1) = x(t) whenever t 2 and (t 0 1) = 2, a 1 is always followed by a 2 in and no more than three consecutive 2’s can appear in . It turns out that (7) is feasible over the scheduling paths of length M = 3 in the admissible, but not minimal, set f(1; 2; 1; 2), (1; 2; 2; 1), (1; 2; 2; 2), (2; 1; 2; 1), (2; 1; 2; 2), (2; 2; 1; 2), (2; 2; 2; 1)g of scheduling paths. Since ( (t); . . . ; (t + 3)) must belong to this set for all x(0) and t under g, we invoke [10, Corollary 3.4] and conclude that the scheduling law g is robustly exponentially stabilizing. Moreover, as Theorem 6 implies, there exists a robustly exponentially stabilizing scheduling sequence. Such a (periodic) scheduling sequence is given by = (1; 2; 1; 2; . . .), in which 1 and 2 alternate. Consider the measurement scheduling law given by (0) = 0 ( ) = 2, (1) = 1 ( (0) (0)) = 1, and t
t
t
t
T
t
B. Joint Synthesis of Supervisor, Scheduler, and Feedback Controller In view of Theorem 6 and [20, Theorem 20], joint synthesis of supervisor-scheduler-controller triples is possible. Let 2 n2n 2 n2m and j 2 l2n be given for ( ij ij ) = 1 ... = 1 ... g . Consider the controlled plant f and given by
i
A ;B
; ;N
j
; ;N
C
x(t + 1) = A ( ) ( )x(t) + B ( ) ( ) u(t); t 2 0 y(t) = C ( )x(t); t 2 0: (8) Assume that both (t) and (t), as well as y (t), become available to the controller at each time t. Consider all dynamic output feedback controllers that generate the control input u(t), t 2 0 , according to x (t +1) =A ( ( 0 ) ... ( ))x (t)+ B ( ( 0 ) ... ( ))y(t) u(t) = C ( ( 0 ) ... ( ))x (t)+D ( ( 0 ) ... ( ))y(t): Here, L 2 0 is the number of past modes that the controller recalls and (t) = ((t); (t)) for t 2 0 . The closed-loop system is x~(t +1) = A0( ) 00 + 0I B0( ) K(t) C0( ) 0I x~(t) (9) t t
It is shown in [19] that there exist asymptotically stabilizing supervisory laws for this S . Such a law is given by
(t) = f ( 01;x ) 1 if x(t) = [x1 x2 ] =
t> t
Define
M
;
1 2 0
A11 =
M
M
t x tr
where (0) = 2 and ( ) = 1 for 0; in this case, while ( ) = ( ; (0)) + 0 ( ; (0)) ! 0 as ! 1, both ( ; (0)) and 0 ( ; (0)) diverge. Example 9: Consider the measurement scheduling problem to stabilize the system S with f = 1, g = 2, and
t t
t
K
K; t
L ;
; t
K
K; t
L ;
; t
K
t
K; t
K; t
L ;
L ;
t
t
; t
; t
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 4, APRIL 2011
877
where x ~(t) = [x(t)T xK (t)]T and
K(t) = K t0L ; ; t AK; t0L ; ; t BK; t0L ; = CK; t0L ; ; t DK; t0L ; ( (
) ...
( ))
( (
) ...
( ))
( (
) ...; (t))
( (
) ...
( ))
( (
) ...; (t))
(10)
Fig. 1. Given each switching sequence generated by (f; g ), the uncertain system is well-connected over all sufficiently small linear uncertainties 1.
for t 2 0 . We are to determine the number of past modes L to recall, the controller matrices (i ;...;i ) 2 (n+m)2(n+l) for all switching paths (i0 ; . . . ; iL ) of length L, and the switching law (f; g ) that robustly exponentially stabilizes the closed-loop system (9). Corollary 10: Given a plant (8), the following are equivalent: (a) There exist a switching law (f; g ) and a family of controller matrices (10), for some L 2 0 , such that (f; g ) is robustly exponentially stabilizing for the closed-loop system (9). (b) There exist a switching sequence = (; ) and a family of controller matrices (10), for some L 2 0 , such that the closedloop system (9) is robustly exponentially stable. (c) There exist an M 2 0 , a minimal set N of switching paths of , (i ;...;i 2 n2n such that length M , and (i ;...;i ) )
We will consider two performance measures. The first is of H2 type, and gives the square root of the average output variance (per unit time) of (12) under white Gaussian disturbance sequence w . This performance measure is well-defined provided that the switching law (f; g ) is robustly exponentially stabilizing for T (i.e., for S with ij , j replaced by ij , 2;j in T ). Definition 11: A switching law (f; g ) is said to achieve output regulation level > 0 for the system T if it is robustly exponentially stabilizing for T and if there exists a ~ 2 (0; ) such that, whenever x(0) = 0, the state-space equations (12) satisfy
K
R
B N (Ci
S
Ai R i ; )T AiT S i ;
N ( iT )T
i
(
...;
(
...;
i
Ai 0 R i ; ;i B Ai 0 S i ; ;i N (Ci R i ; ;i I I S i ; ;i )
)
T
(
(
(
...
)
...
...
)
(
...
)
for all (i0 ; . . . ; iM ) 2 N , where N ( ) is any full-column-rank matrix whose columns span the null space of . Moreover, if condition (c) holds, then the closed-loop system (9) is robustly exponentially stable under a periodic switching sequence satisfying (6) and feedback controller matrices (10) with L = M . A guaranteed, nonconservative procedure for obtaining a stabilizing set of controller matrices (10) is based on the well-known linear matrix inequality embedding technique [12], [13], and it is the same as [20, Algorithm 1] adapted to our case.
M
III. SWITCHING LAWS FOR GUARANTEED PERFORMANCE
A B C D C D ;j ):
= f( ij ; ij ; 1;ij ; 1;ij ; 2;j ; i = 1; . . . ; Nf ; j = 1; . . . ; Ng g
n2n 2
n2m 2 l
2n 2
l
l
2n 2
l
2m (11)
defines a family of linear time-varying state-space equations
A C C
B D D
x(t + 1) = (t)(t) x(t) + (t)(t) w(t); t 2 0 z (t) = 1;(t)(t) x(t) + 1;(t)(t) w(t); t 2 y(t) = 2;(t) x(t) + 2;(t) w(t); t 2 0 over all 2 f1; . . . ; Nf g1
I; 0;
t=s t 6= s.
Definition 12: A switching sequence = (; ) is said to achieve output regulation level > 0 for the system T if it is robustly exponentially stabilizing for T and if T
1 tr t!1 T + 1 t=0 ~; where (t0 ; t) lim
Y
t01
s=t
0;
C ; t Y (t ; t)CT; t + D ; t DT; t 1
( )
0
1
( )
1
( )
1
( )
B B
8 (t; s + 1) (s) T(s)8 (t; s + 1)T ; t > t0 ;
t = t0 .
The second performance measure is of H1 type, where (f; g ) is required to satisfy the following small gain property [21]: If is a switching sequence for T generated under (f; g ), the system (12) (or (1) in Fig. 1) is robustly well-connected for all uncertainties 1 whose `2 -induced gain is sufficiently small. This property in a sense guarantees that one can deal with the dynamics associated with the plant and uncertainty separately. This performance measure is well-defined if (f; g ) is robustly exponentially stabilizing for T . Definition 13: A switching law (f; g ) is said to achieve robust disturbance attenuation level > 0 for the system T if it is robustly exponentially stabilizing for T and if there exists a ~ 2 (0; ) such that, whenever x(0) = 0, the state-space equations (12), denoted (1) , is such that the uncertain system in Fig. 1 has z = 0 as the unique response to w = 0 over all linear operators 1 satisfying
G
2
2m 2
E w(t)w(s)T =
E[w(t)] = 0;
A. Problem Statement
T
kz(t)k2 ~2
where E[1] denotes the expectation with respect to w with
= We will now consider the problems of guaranteeing infinite-horizon performance measures. Given n, m, l1 , l2 , Nf , Ng 2 , the set
T
1 E T !1 T + 1 t=0 lim
N ( iT )
0 for T if it is robustly exponentially stabilizing for T and if there is a ~ 2 (0; ) such that
M (t; t )k ~
k for all t0 , t 2
M (t; t ) = 0
with t0 t, where
0
D
C C
B
1;(t )
1;(t +1)
.. .
8(
1;(t) t; t0
D
(t )
0
+ 1)B(t )
0
0 0
111 111
1;(t +1)
.. .
..
:
.. .
.
+ 2)B(t +1) 1 1 1 D1;(t) Whenever x(0) = 0, we have z (0) = D1;(0) w(0) and
C
8(
Y
1;(t) t; t0
( )=
z t
0
t 1
s=0
C
for t 2 . Thus, if there exists a switching sequence (; ) that achieves an output regulation level, or a robust disturbance attenuation level, then the switching law (f; g ) with (ft ; gt ) identically equal to ((t); (t)) for all t achieves the same performance level. Our goal in this section is to establish that the converse also holds true. B. Results on Guaranteed Performance The following theorems characterize the existence of switching laws that achieve guaranteed performance levels. Theorem 15: Let T be as in (11); let > 0. The following are equivalent. (a) There exists a switching law that achieves output regulation level
for T . (b) There exists a switching sequence that achieves output regulation level for T . (c) There exist an M 2 0 , a minimal set N of switching paths of n2n such that length M , and matrices (i ;...;i ) 2
Y
( ;...;
Ai Y i
( ;...;i
1
jN j
Yi i Ai 0 Y i i
)
( ;...;
for all (i0 ; . . . ; iM ) 2 N , and
tr C1;i
(i ;...;i )2N
Yi
( ;...;i
)
0 < Bi Bi
) > )
(14a)
T
(14b)
T 1;
T 1;
1;
< 2
(14c)
where jN j denotes the cardinality of N . Moreover, if condition (c) holds, then any periodic switching sequence satisfying (6) achieves output regulation level for T . Proof: The proof is similar to those of [22, Theorem 3.7] and [9, Theorem 1], so we will only sketch it. Suppose (f; g ) achieves an output regulation level > 0 for T . Let be any realization of the switching sequence under (f; g ) with x(0) = 0. Then it can be shown that there are t 2 n2n , t 2 0 , such that
Y
I Y I A Y A 0 Y 0 B B 0 I 1 0 tr C Y C + D D lim !1 T
(t)
T
T t (t) T 1 t=0
t
t+1
1;(t)
(t)
t
T 1;(t)
T (t)
1;(t)
T 1;(t)
(15a) (15b)
~2 (15c)
( ;...;
( ;...;
1;
)
(16a) T
)
1;
1;
( ;...;
)
1;
2
(16b)
for all (i0 ; . . . ; iM ) 2 N . Moreover, if condition (c) holds, then any periodic switching sequence satisfying (6) achieves disturbance attenuation level for T . Proof: Suppose (f; g ) achieves a robust disturbance attenuation level > 0 for T . Set x(0) = 0 and choose w(0); w(1); . . . 2 m to generate a switching sequence for T under (f; g ). Suppose k (t; t0 )k for some t, t0 2 0 with t t0 . Let 1 in Fig. 1 be such that w(s) = 0 whenever s < t0 or s > t and such that
M
[ w(t0 )T
1 1 1 w(t)T ]T
T (t; t0 ) = kM M (t; t0)k2 [ z(t0)T
1 1 1 z (t)T ]T :
This 1 satisfies (13) for any ~ 2 (0; ), and yet, because the matrix 0 (t; t0 ) (t; t0 )T =k (t; t0 )k2 is singular, z = 0 is not the unique solution to the equation z = (1)1 z , which governs the uncertain system in Fig. 1. This contradicts the assumption on (f; g ), and hence (a) and (b) are equivalent. To show (b) is equivalent to (c), suppose a switching sequence achieves a robust disturbance attenuation level > 0. Then by applying [23, Theorem 11] to the scaled version of T given by ( ij ; 01=2 ij ; 01=2 1;ij ; 01 n21n;ij ; 2;j ; 2;j ) for all i and j , satisfying we deduce that there exist t 2
I M
M
M
A
C i +D i D i
Y i i >0 B Y i i 0 Ai Bi D ii 0 I Ci Di Y 0 i i 0 0 I , and 2 ; . Then, due to [22, Lemma 2.3], this achieves the output regulation level for T . This establishes that (a) implies (b), and hence that (a) and (b) are equivalent. To show (b) and (c) are equivalent, suppose a switching sequence achieves an output regulation level > . Then it can be shown by adapting the proof of [9, Theorem 1] to our case that there are a path length M 2 0 and a minimal set N of switching paths of length M such that (14) holds true. The converse implication follows because, choosing a satisfying (6), we recover (15) from (14). Theorem 16: Let T be as in (11); let > . The following are equivalent. (a) There exists a switching law that achieves robust disturbance attenuation level for T . (b) There exists a switching sequence that achieves robust disturbance attenuation level for T . (c) There exist an M 2 0 , a minimal set N of switching paths of n2n such that length M , and matrices (i ;...;i ) 2
B
A C
(t)
B D
(t)
1;(t)
for all t 2
C
0
T
1;(t)
G
D C D
X
I X I X 00 A 0 I C X 0 0 I 0 0 I t
t+1
(t)
1
1;(t)
t
and for some , > 0, 2
B D
(17a)
(t)
1;(t)
(17b)
(0; ). Define
WM () = f(i0 ; . . . ; iM ):
(i0 ; . . . ; iM ) = ((t); . . . ; (t + M )); t 2
0g
for M 2 0 . Then, by [10, Theorem 3.3] and the Schur complement formular, there exist an M 2 0 and matrices (i ;...;i ) such that (16) holds for all (i0 ; . . . ; iM ) 2 WM (). There exists at least one minimal set N of switching paths of length M such that N WM (), so we obtain (c). Conversely, assuming (c) holds, reversing the above
X
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 4, APRIL 2011
argument gives that (17) holds for all t 2 0 , where 2 (0; ) is sufficiently close to , and where satisfies (6). Therefore, (b) and (c) are equivalent. C. Joint Synthesis of Supervisor, Scheduler, and Feedback Controller Joint synthesis of supervisor-scheduler-controller triples is also possible for achieving guaranteed disturbance attenuation performance in view of Theorem 15 and [10, Theorem 3.7] and for achieving guaranteed robust output regulation performance in view of Theorem 16 and [22, Theorem 4.2]. The technical development is similar to that of Section II-C, so the details are omitted. In particular, adapting [9, Theorem 1] to our case will result in a nonconservative joint synthesis condition for guaranteed output regulation. IV. CONCLUSION Problems of robust exponential stabilization, output regulation, and robust disturbance attenuation have been considered for discrete-time switched linear systems. It is shown that supervisor-scheduler pairs can be determined open-loop in the design stage, and that open-loop supervisory-scheduling laws suffice in the implementation stage. Determining the best supervisory and scheduling sequences requires solving an increasing sequence of semidefinite programs. As one moves from one such program to the next, the computational complexity is likely to grow exponentially. Although this is due to the “semi-decidable” problem nature [24] and the nonconvexity of the associated Lyapunov functions [25], the presented results are nonconservative and expected to be useful for small problems. Moreover, depending on one’s tradeoff between performance and computational efficiency, one has the option to either stop at a point along the sequence of semidefinite programs and settle for the best performance achievable at that point or move further down the sequence by paying more computational cost. Even though one can always find open-loop solutions, it is desirable to use closed-loop solutions in practice whenever there is uncertainty in the system coefficients or the system is required to be fault-tolerant. In some cases, fault-tolerance can be ensured by using a set-valued switching law as suggested in [7]. However, the problem of determining a closed-loop switching law to guarantee robustness against uncertainty in the system coefficients remains open.
REFERENCES [1] A. S. Morse, “Supervisory control of families of linear set-point controllers—Part 1: Exact matching,” IEEE Trans. Autom. Control, vol. 41, no. 10, pp. 1413–1431, Oct. 1996. [2] L. Meier, III, J. Peschon, and R. M. Dressler, “Optimal control of measurement subsystems,” IEEE Trans. Autom. Control, vol. AC-12, no. 5, pp. 528–536, Oct. 1967. [3] D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Syst. Mag., vol. 19, no. 5, pp. 59–70, Oct. 1999. [4] R. Jungers, The Joint Spectral Radius: Theory and Applications. Berlin, Germany: Springer, 2009. [5] A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules,” IEEE Trans. Autom. Control, vol. 48, no. 6, pp. 988–1001, Jun. 2003. [6] A. Girard, G. Pola, and P. Tabuada, “Approximate bisimilar symbolic models for incrementally stable switched systems,” IEEE Trans. Autom. Control, vol. 55, no. 1, pp. 116–126, Jan. 2010. [7] J.-W. Lee and G. E. Dullerud, “Uniformly stabilizing sets of switching sequences for switched linear systems,” IEEE Trans. Autom. Control, vol. 52, no. 5, pp. 868–874, May 2007. [8] D. P. Stanford and J. M. Urbano, “Some convergence properties of matrix sets,” SIAM J. Matrix Anal. Appl., vol. 15, no. 4, pp. 1132–1140, 1994.
879
[9] J.-W. Lee, “Infinite-horizon joint LQG synthesis of switching and feedback in discrete time,” IEEE Trans. Autom. Control, vol. 54, no. 8, pp. 1945–1951, Aug. 2009. [10] J.-W. Lee and G. E. Dullerud, “Optimal disturbance attenuation for discrete-time switched and Markovian jump linear systems,” SIAM J. Control Optimiz., vol. 45, no. 4, pp. 1329–1358, 2006. [11] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [12] A. Packard, “Gain scheduling via linear fractional transformations,” Syst. Control Lett., vol. 22, no. 2, pp. 79–92, 1994. [13] P. Gahinet and P. Apkarian, “A linear matrix inequality approach to control,” Int. J. Robust Nonlin. Control, vol. 4, no. 4, pp. 421–448, 1994. [14] C. Scherer, P. Gahinet, and M. Chilali, “Multiobjective output-feedback control via LMI optimization,” IEEE Trans. Autom. Control, vol. 42, no. 7, pp. 896–911, Jul. 1997. [15] I. Masubuchi, A. Ohara, and N. Suda, “LMI-based controller synthesis: A unified formulation and solution,” Int. J. Robust Nonlin. Control, vol. 8, no. 8, pp. 669–686, 1998. [16] J.-W. Lee, “Uniform consensus among self-driven particles,” in Hybrid Systems: Computation and Control, R. Majumdar and P. Tabuada, Eds. Berlin, Germany: Springer-Verlag, 2009, vol. 5469, Lecture Notes in Computer Science, pp. 252–261. [17] D. B. Johnson, “Finding all the elementary circuits of a directed graph,” SIAM J. Comput., vol. 4, no. 1, pp. 77–84, 1975. [18] S. A. Krishnamurthy and J.-W. Lee, “A computational stability analysis of discrete-time piecewise linear systems,” in Proc. 48th IEEE Conf. Decision Control, 2009, pp. 1106–1111. [19] D. P. Stanford, “Stability for a multi-rate sampled-data system,” SIAM J. Control Optimiz., vol. 17, no. 3, pp. 390–399, 1979. [20] J.-W. Lee and G. E. Dullerud, “Uniform stabilization of discrete-time switched and Markovian jump linear systems,” Automatica, vol. 42, no. 2, pp. 205–218, 2006. [21] G. E. Dullerud and F. Paganini, A Course in Robust Control Theory: A Convex Approach. New York: Springer-Verlag, 2000. [22] J.-W. Lee and P. P. Khargonekar, “Optimal output regulation for discrete-time switched and Markovian jump linear systems,” SIAM J. Control Optimiz., vol. 47, no. 1, pp. 40–72, 2008. [23] G. E. Dullerud and S. Lall, “A new approach for analysis and synthesis of time-varying systems,” IEEE Trans. Autom. Control, vol. 44, no. 8, pp. 1486–1497, Aug. 1999. [24] P.-A. Bliman and G. Ferrari-Trecate, “Stability analysis of discrete-time switched systems through Lyapunov functions with nonminimal state,” in Proc. IFAC Conf. Anal. Design Hybrid Syst., 2003, pp. 325–330. [25] F. Blanchini and C. Savorgnan, “Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions,” Automatica, vol. 44, no. 4, pp. 1166–1170, 2008.
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