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Design of switching sequences for controllability realization of switched linear systems
Automatica 43 (2007) 662 – 668 www.elsevier.com/locate/automatica
Brief paper
Design of switching sequences for controllability realization of switched linear systems夡 Zhijian Ji a,∗ , Long Wang b , Xiaoxia Guo c a School of Automation Engineering, Qingdao University, 266071, China b Intelligent Control Laboratory, Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China c Department of Mathematics, Ocean University of China, 266071, China
Received 29 March 2006; received in revised form 8 October 2006; accepted 10 October 2006 Available online 19 January 2007
Abstract In this paper, we study the problem of designing switching sequences for controllability of switched linear systems. Each controllable state set of designed switching sequences coincides with the controllable subspace. Both aperiodic and periodic switching sequences are considered. For the aperiodic case, a new approach is proposed to construct switching sequences, and the number of switchings involved in each designed switching sequence is shown to be upper bounded by d(d −d1 +1). Here d is the dimension of the controllable subspace, d1 =dim m i=1 Ai |Bi , where (Ai , Bi ) are subsystems. For the periodic case, we show that the controllable subspace can be realized within d switching periods. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Switched systems; Controllability; Reachability; Switching sequence
1. Introduction As a special class of hybrid control systems, switched systems have attracted considerable attention during the last decade because of its importance from both theoretical and practical points of view (Cheng, Guo, Lin, & Wang, 2005; Sun & Ge, 2005). In spite of some existing results, the study of the stability analysis and design of switched systems is just under way and the theoretical framework is far from complete. In particular, the switching mechanism has not been fully understood, and a closely related problem is to design switching sequences with switching times as less as possible. There are excellent survey papers such as Sun and Ge (2005), Liberzon and Morse (1999) and DeCarlo, Branicky, Pettersson, and Lennartson (2000). In this paper, we are interested in designing switching sequences to achieve the controllability for switched linear systems, i.e., each controllable state set of designed switching 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor teD Iwasaki under the direction of Editor R. Tempo. ∗ Corresponding author. E-mail address: [email protected] (Z. Ji).
0005-1098/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2006.10.010
sequences coincides with the controllable subspace of switched linear systems. We call this problem a controllability realization problem. Our goal is to design switching sequences to realize the controllability with the number of switchings as small as possible. To this end, a constructive approach is proposed for the design purpose. We will show that the upper bound for the length of each designed switching sequence is d(d − d1 + 1). Hered is the dimension of controllable subspace, d1 = dim m i=1 Ai |Bi , where (Ai , Bi ) are subsystems. For periodic switching sequences, we will prove that the controllability can always be realized within d switching periods. Comparison of our results with existing ones verifies the advantage of our design approach. The controllability realization problem was formerly studied by Sun, Ge, and Lee (2002) and Xie and Wang (2003). A common feature for the switching sequences designed by them is that the number of switchings increases rapidly with the number of subsystems and dimensions. In this paper, we reduce the required number of switchings significantly to an acceptable level for both cases of aperiodic and periodic switchings. Furthermore, we will show that for the design of aperiodic switching sequences, the maximum number of switchings is only determined by the dimension of the controllable subspace.
Z. Ji et al. / Automatica 43 (2007) 662 – 668
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Definition 1. A nonzero state x ∈ Rn is controllable, if there exist a switching sequence and an input u(t) : [0, T ] → Rp such that x(0) = x, and x(T ) = 0. The system (1) is said to be (completely) controllable if any nonzero state x is controllable. We denote by C the set of all controllable states of the system (1).
Besides controllability realization, the design of switching sequences/rules has been widely employed for stabilization problems. Wicks, Peleties, and DeCarlo (1998) firstly developed an elegant construction of a stabilizing switching rule. The result therein was later used for switched controller design and robustness study by Sun (2005), Zhao and Dimirovski (2004), Ji, Wang, Xie, and Hao (2004), Ji, Wang, and Xie (2005), Zhai, Lin, and Antsaklis (2003), Bacciotti (2004), King and Shorten (2005) and Ishii, Basar, and Tempo (2005). Another stabilization method is to constrain switching sequences to satisfy a dwell or an average-dwell-time condition. This method was firstly proposed by Morse (1997), and further developed by Hespanha and Morse (1999), Hespanha (2004) and Zhai, Hu, Yasuda, and Michel (2001). Recently, Cheng et al. (2005) put forward a concept of switching frequency which can be deemed as a development of average-dwell-time method. Our proposed approach enriches constructive methods for switching sequences. Controllability and observability play a fundamental role in the design and synthesis of linear control systems. For switched linear systems, complete geometric criteria for controllability and reachability were established by Sun et al. (2002) and Xie and Wang (2003). The controllability of switched bilinear systems was investigated by Cheng (2005) using Lie algebraic technique. Other necessary and/or sufficient conditions for controllability and reachability were presented by Ezzine and Haddad (1989), Krastanov and Veliov (2005), Petreczky (2006), Yang (2002), Stikkel, Bokor, and Szabó (2004), Xie and Wang (2004), Meng and Zhang (2006), Sun (2004), Ge, Sun, and Lee (2001), Stanford and Conner (1980) and Conner and Stanford (1987), etc. In our opinion, although necessary and sufficient conditions on controllability and reachability were established, the switching behaviors for controllability has not been fully investigated. This is our main purpose in the paper. The paper is organized as follows. In Section 2, we present preliminaries including system description, definitions and supporting lemmas. Main results and comparisons with existing ones are presented in Section 3. An illustrative example is included in Section 4. Finally, Section 5 briefly concludes the work.
Definition 3 (Xie & Wang, 2003). Given two switching sequences, 1 ={(i1,0 , h1,0 ) · · · (i1,s1 , h1,s1 )}, 2 ={(i2,0 , h2,0 ) · · · (i2,s2 , h2,s2 )}, 1 ∧ 2 is defined by
2. Preliminaries
Lemma 1 (Xie & Wang, 2003). Given a switching sequence = {(i0 , h0 ) · · · (is−1 , hs−1 )}, its controllable state set is
Definition 2 (Xie & Wang, 2003). Given a switching sequence = {(i0 , h0 ) · · · (is−1 , hs−1 )}, denote T = s−1 j =0 hj . The controllable state set C() of is defined by C() = {x|there exists an input u(t), t ∈ [0, T ], such that x(0) = x and x(T ) = 0}. The reachability counterparts of Definitions 1 and 2 can be given by replacing ‘x(0) = x, and x(T ) = 0’ with ‘x(0) = 0, and x(T ) = x’.
1 ∧ 2 = {(i1,0 , h1,0 ) · · · (i1,s1 , h1,s1 ) (i2,0 , h2,0 ) · · · (i2,s2 , h2,s2 )}. Since (1 ∧ 2 ) ∧ 3 = 1 ∧ (2 ∧ 3 ), we only denote it by 1 ∧ 2 ∧ 3 . Specially, we denote ∧ ∧· · · ∧ by ∧n . n
To describe the controllable subspace, we introduce some notations. Given B ∈ Rn×p , Im B denotes the image space of B. Given a matrix A ∈ Rn×n , a linear subspace W ⊆ Rn , A|W is the minimum A-invariant subspace containing W, i.e., A|W =
(1)
where x(t) ∈ Rn is the state, u(t) ∈ Rp is the input, and (t) : [t0 , ∞) → {1, . . . , m} is the switching sequence to be designed. Moreover, (t) = i implies that the ith subsystem (Ai , Bi ) is activated. Given a switching sequence : [t0 , tf ] → {1, . . . , m}, suppose its discontinuous (jump) time instants are t0 < t1 < · · · < ts−1 . We refer to the sequence t0 , t1 , . . . , ts−1 as switching time sequence, and the sequence (t0 )=i0 , (t1 )=i1 , . . . , (ts−1 )= is−1 as switching index sequence. Let hi := ti+1 − ti , i = 0, 1, . . . , s − 1, and ts := tf . We denote a switching sequence by = {(i0 , h0 ), . . . , (is−1 , hs−1 )}, where s is the length of .
Ai−1 W.
i=1
For notational simplicity, we denote A|B = A|Im B.
Consider a switched linear control system given by x(t) ˙ = A x(t) + B u(t),
n
C() =
k s−1
e
−Aij hj
Aik |Bik ,
k=0 j =0
where
k
j =0 e
−Aij hj
= e−Ai0 h0 · · · e−Aik hk .
Given a switching sequence = {(i0 , h0 ) · · · (is−1 , hs−1 )}, −Aij hj we denote e− := s−1 . j =0 e Remark 1. Lemma 1 implies that for any two switching sequences 1 , 2 C(1 ∧ 2 ) = C(1 ) + e−1 C(2 ).
(2)
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Z. Ji et al. / Automatica 43 (2007) 662 – 668
For system (1), a nested subspace sequence is defined as follows (Sun et al., 2002; Xie & Wang, 2003). W1 =
m
Ai |Bi ,
Wj +1 =
i=1
m
consider W1 , . . . , W . By Lemma 4, they can be written as W1 =
Ai |Wj ,
(3)
i=1
where j =1, 2, . . . . Note that if dim Wj =dim Wj +1 for some j, then Wl =Wj for all l > j, and Wn =Wn+1 =· · · holds. By employing Wn , an elegant result on geometric characteristics of controllable subspace C was established as follows. Lemma 2 (Sun et al., 2002; Xie & Wang, 2003). For switched linear system (1), the controllable set is C = Wn .
m
Ai |Bi ,
W2 =
i=1
W =
m
m
e−Ai h |W1 , . . . ,
i=1
e−Ai h |W−1 .
i=1
From these equalities, we have j0 ,...,j−2 ∈{0,1,...,n−1}
W =
(e−Ai0 h )j0
i0 ,...,i−1 ∈{1,...,m}
· · · (e
−Ai−2 h j−2
)
Ai−1 |Bi−1 .
Lemma 3 (Sun et al., 2002; Xie & Wang, 2003). Both the controllable set and the reachable set are subspaces of the total space, and the two subspaces are always identical, i.e., C = V.
That is, W is the summation of m n−1 subspaces, with each
By using standard results on sampling (see, e.g., Chen, 1999), the following lemma can be easily shown.
−2 Vi00,...,i−1 = (e−Ai0 h )j0 · · · (e
Lemma 4. For matrices Ak ∈ Rn×n , there exists h > 0, such that for any linear subspace W ⊆ Rn , Ak |W = e−Ak h |W,
∀k ∈ {1, . . . , m}.
Lemma 5 (Sun et al., 2002). For any given matrices A1 , A2 ∈ Rn×n , and B1 , B2 ∈ Rn×p , inequality
j ,...,j
holds for almost all h ∈ R.
)
Ai−1 |Bi−1 .
(4)
Then there exist d linear independent vectors 1 , . . . , d , which span W . We divide these vectors into l (1 l d) groups with each group belonging to a subspace of form (4). Suppose the division is as follows: 1 , . . . , d1 ∈ V1 , d1 +1 , . . . , d2 ∈ V2 , . . . , dl−1 +1 , . . . , d ∈ Vl ,
−2 Vi00,...,i−1 = e−Ai0 h0 · · · e
j ,...,j
dim[eA2 h Im(B
−Ai−2 h j−2
where V1 , . . . , Vl are subspaces of form (4). Accordingly, W = lj =1 Vj . As Vj is in form (4), we can rewrite it as
rank[A1 eA2 h B1 , B2 ]rank[A1 B1 , B2 ]
Lemma 5 implies that dim [Im(B1 ) + Im(B2 )] .
j ,...,j
−2 subspace Vi00,...,i−1 in the form of
1)
+ Im(B2 )]
3. Design of switching sequences 3.1. Aperiodic switching sequences Let L() be the length of switching sequence . Obviously, L() indicates the number of switchings involved in . Let = min{j | dim Wj =dim Wj +1 , j =1, 2, . . .}, and d1 =dim W1 . The following result reveals the relationship between the required number of switchings and the dimension of controllable state subspace C. Theorem 1. For system (1), there exist switching sequences k , k = 1, 2, . . . , with length of each one satisfying L(k )d(d − d1 + 1), such that for each k , C(k ) = Wn = C, where d and n are the dimensions of Wn and the state, respectively. Proof. Suppose dim Wn =d. The definition of index implies that dim W = dim Wn = d, and d − d1 + 1. This, together with W ⊆ Wn , yields W = Wn . Hence, it is sufficient to
−Ai−2 h−2
Ai−1 |Bi−1 ,
(5) −A
h
where h0 := j0 h, . . . , h−2 := j−2 h. Since e i−1 −1 Ai−1 |Bi−1 = Ai−1 |Bi−1 , it follows from Lemma 1 and j ,...,j
−2 (5) that Vi00,...,i−1 is just the last term of C( ). Here is designed as follows:
= {(i0 , h0 ) · · · (i−2 , h−2 )(i−1 , h−1 )}, and h−1 > 0 is arbitrary, for example, one can set h−1 = h or 1. Concerning Vj , we denote by j, the corresponding aforementioned switching sequence j, = {(ij,0 , hj,0 ) · · · (ij,−2 , hj,−2 )(ij,−1 , hj,−1 )}, where j = 1, . . . , l. Obviously, j, is not unique. This is because the selection of hj,−1 is not unique. Then Vj ⊆
C j, , and W = lj =1 Vj ⊆ lj =1 C(j, ). On the other hand, C(j, ) ⊆ Wn . This, together with Wn = W , gives W =
l
C(j, ).
(6)
j =1
Next, we are to prove dim C(1, ∧ · · · ∧ l, ) d.
(7)
Z. Ji et al. / Automatica 43 (2007) 662 – 668
It follows from (2) that
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3.2. Periodic switching sequences
C(1, ∧ · · · ∧ l, ) = C(1, ) + e−Ai1,0 h1,0
Since {1 , . . . , d1 } ⊆ V1 ⊆ C(1, ), repeatedly applying Lemma 5 to the right-hand-side of the above equality yields that
Periodic switching sequences are interesting from the viewpoint of implementation. Let h0 ,...,hm−1 ={(1, h0 ) · · · (m, hm−1 )} . The switching sequence ∧d h0 ,...,hm−1 possesses periodic switching index and periodic switching time sequences, and there are d switching periods associated with it. Let R+ be the set of all positive real numbers. We have the following result.
dim C(1, ∧ · · · ∧ l, ) dim{span{1 , . . . , d1 } + C(2, ∧ · · · ∧ l, )},
Theorem 2. For the switched linear system (1), there exists a periodic switching sequence ∧d h0 ,...,hm−1 , such that
holds for almost all h1,j ∈ R, j =0, 1, . . . , −1. Repeating the same reasoning on the right-hand-side of the above inequality, one can finally get
C(∧d h0 ,...,hm−1 ) = C,
···e
−Ai1,−1 h1,−1
C(2, ∧ · · · ∧ l, ).
dim C(1, ∧ · · · ∧ l, ) dim{span{1 , . . . , d1 } + · · · + span{dl−1 +1 , . . . , d }}. Then, (7) follows from the independence of 1 , . . . , d . Since C(1, ∧ · · · ∧ l, ) ⊆ Wn = W , and dim W = d, we have
where h0 , . . . , hm−1 ∈ R+ . Proof. Let us consider W1 , . . . , Wd with d, the dimension of the controllable subspace C. It can be seen that Wd = Wn = C. By (3), Wd can be written as j0 ,...,jd−2 ∈{0,1,...,n−1} i0 ,...,id−1 ∈{1,...,m}
C(1, ∧ · · · ∧ l, ) = W . This means that = 1, ∧ · · · ∧ l, is always a switching sequence to be required. Because j, , j = 1, . . . , l, is not unique, is not unique as well. We denote all these switching sequences by k , k = 1, 2, . . . . Clearly, the length of each k satisfies L(k ) l d(d − d1 + 1).
Wd =
j
j
d−2 Ai00 · · · Aid−2 Aid−1 |Bid−1 .
(8)
In the following, we prove C(∧d h0 ,...,hm−1 ) ⊇ Wd .
(9)
It follows from direct expansion of C(∧k h0 ,...,hm−1 ) that (e−A1 h0 · · · e−Am hm−1 )k−1 e−A1 h0 · · · e−Ai hi−1 Ai |Bi ⊆ C(∧k h0 ,...,hm−1 ),
∀i ∈ {1, . . . , m}; k = 1, 2, . . . .
Finally, we show Wn = C. Suppose is an arbitrary switching sequence, then C = ∀ C(). It follows that Wn = C(k ) ⊆ C. On the other hand, it follows from Lemma 1 and definition of Wn that for ∀, C() ⊆ Wn . We have C ⊆ Wn . Hence, C = Wn .
Denote
Remark 2. From the proof, we see that if l < d, L(k ) < d(d− d1 + 1). That is, L(k ) is always smaller than d(d − d1 + 1) whenever the division number is smaller than the dimension of controllable subspace C. Moreover, the smaller the number l, the smaller the length L(k ). An extreme case is l = 1. In this case, L(k )d − d1 + 1. We also see that the smaller the dimension of controllable subspace, the smaller the number of switchings required.
where id−1 plays the same role as i in the foregoing inclusion associated with C(∧k h0 ,...,hm−1 ). Then, for k = d, and any given i0 , . . . , id−1 ∈{1, . . . , m},
Remark 3. The proof of Theorem 1 provides a new proof for Lemma 2, and Lemma 2 is a direct consequence of Theorem 1.
i0 ,...,id−1 := (e−A1 h0 · · · e−Ai0 hi0 −1 · · · e−Am hm−1 )
· · · (e−A1 h0 · · · e−Aid−2 hid−2 −1 · · · e−Am hm−1 ) × e−A1 h0 · · · e−Aid−1 hid−1 −1 Aid−1 |Bid−1 ,
i0 ,...,id−1 ⊆ C(∧d h0 ,...,hm−1 ). It follows that C(∧d h0 ,...,hm−1 ) ⊇ =
i0 ,...,id−1
id−1 ∈{1,...,m}
i0 ,...,id−1 ,
(10)
i0 ,...,id−1 ∈{1,...,m}
Remark 4. The switching sequences constructed in Theorem 1 are not unique since h is not unique, and V1 , . . . , Vl are not unique as well. Actually, there are infinitely many h’s. Hence, the number of k s is infinite as well.
where the equality is due to the fact that the terms
It should be noted that a reachability counterpart of Theorem 1 can also be established by Lemma 3.
are actually the same regardless of what the values of i0 , . . . , id−2 are, and that id−1 is independent of i0 , . . . , id−2 .
(e−A1 h0 · · · e−Ai0 hi0 −1 · · · e−Am hm−1 ) · · · (e−A1 h0 · · · e−Aid−2 hid−2 −1 · · · e−Am hm−1 )
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Z. Ji et al. / Automatica 43 (2007) 662 – 668
Next, we are to show that for any given i0 , . . . , id−1 ∈ {1, . . . , m}, jd−2 j Aid−1 |Bid−1 , (11) i0 ,...,id−1 ⊇ Ai00 · · · Aid−2 where ∀j0 , . . . , jd−2 ∈ {0, 1, . . . , n − 1}. This can be verified using the power expansion of the matrix exponential. By Cayley–Hamilton theorem, e−As hs−1 = 0 (hs−1 )I + 1 (hs−1 )As + · · · + n−1 (hs−1 )An−1 , s where s = 1, . . . , m. Substituting e−As hs−1 in the form of the power expansion into i0 ,...,id−1 implies that j
[0 (h0 )I · · · 0 (hi0 −2 )I · j0 (hi0 −1 )Ai00 · 0 (hi0 )I · · · 0 (hm−1 )I ] · · · j
d−2 [0 (h0 )I · · · 0 (hid−2 −2 )I · jd−2 (hid−2 −1 )Aid−2
· 0 (hid−2 )I · · · 0 (hm−1 )I ]0 (h0 )I · · · 0 (hid−1 −1 )I Aid−1 |Bid−1 is one of the adding terms that constitute i0 ,...,id−1 . Observing that 0 (hs−1 ), . . . , n−1 (hs−1 ) are all scalars for ∀s ∈ {1, . . . , m}, and Aid−1 |Bid−1 is a subspace, we claim that (11) holds. Combining this with (10) and (8) gives (9). On the other hand, since C(∧d h0 ,···,hm−1 ) ⊆ Wn = Wd , it follows that C(∧d h0 ,...,hm−1 ) = Wn = C.
Let = min{j | dim Wj = dim Wj +1 , j = 1, 2, . . .}. Corollary 1. For the switched linear system (1), there ex∧ ists a periodic switching sequence h0 ,...,hm−1 , such that ∧
C(h0 ,...,hm−1 ) = C, where h0 , . . . , hm−1 ∈ R+ . Furthermore, for any k > , C(∧k h0 ,...,hm−1 ) = C holds as well.
Proof. Since W =Wd , the first equality follows by replacing d with in the proof of Theorem 2. As for the second equality, by Lemma 1, for any k > , ∧
C(∧k h0 ,...,hm−1 ) ⊇ C(h0 ,...,hm−1 ). ∧
Observe that C(h0 ,...,hm−1 )=C=Wn , and Wn ⊇C(∧k h0 ,...,hm−1 ). ∧k We get C(h0 ,...,hm−1 ) = Wn = C. Remark 5. Theorem 2 implies that the controllable subspace can always be realized via a periodic switching sequence within d periods, and hence the number of switchings involved is not greater than md. The switching sequence possesses not only the periodic switching index sequence but also the periodic time sequence. Remark 6. Sun (2004) proved that the controllability can be realized by a periodic and synchronous switching sequence, where synchronization means that there exist a base rate and a sequence of natural numbers {1 , 2 , . . .}, such that the switching time sequence is {0, 1 , 2 , . . .}. Whether or not
the switching times involved this switching path are finite is not answered therein. Hence, the difference between our periodic switching sequence and the one used by Sun (2004) lies in the switching time sequence and the number of switchings. Remark 7. Theorem 2 holds for switching index sequence of ∧d h0 ,...,hm−1 not only for the order {1, . . . , m}, but also for any other permutation of {1, . . . , m}. h0 , . . . , hm−1 are arbitrary positive real numbers. Hence, the number of periodic switching sequences is infinite from the viewpoint of switching time sequence. The reachability counterpart of the above results can be established in the same manner. Let us compare the results with existing ones. Xie and Wang (2003) put forward another method for constructing an aperiodic switching sequence for the controllability realization problem. By computation, it can be seen that the upperj bound for switching sequences designed therein is mn d−1 j =0 n , which is much greater than d(d −d1 +1)( n2 ) given in Theorem 1. Sun et al. (2002) solved the same problem by employing a switching sequence switching index sequence, with only periodic k where L() n−1 m(mn) − 1. It can be seen that the minik=0 mum number of switchings given by them is much greater than the maximum number of switchings given in Theorems 1 and 2. These comparisons verify that the constructive approaches presented in Theorems 1 and 2 significantly reduce the required number of switchings for the controllability realization problem. 4. An illustrative example Example 1. Consider the switched system (1) with m = n = 3, and 0 1 0 A1 = 0 0 0 , b1 =03×1 ; A2 =03×3 , b2 =[0, 1, 0]T , 0 0 0 A3 = 03×3 , b3 = [0, 0, 1]T . It can be verified that d1 = dim W1 = 2, W2 = W3 = R3 . Hence, the controllable subspace C is R3 , = 2, and d = 3. After some computations, we have 0 0 −h W2 = span ∀h > 0 1 , 0 , 1 0 1 0 = span{1 , 2 , 3 ; ∀h > 0}. Since 1 ∈ A2 |b2 ⊆ C(1 ), 2 ∈ A3 |b3 ⊆ C(2 ),
1 = {(2, h)}, 2 = {(3, h)},
3 ∈ e−A1 h A2 |b2 ⊆ C(3 ),
3 = {(1, h)(2, h)},
it follows from the proof of Theorem 1 that the desired switching sequence is (h) = 1 ∧ 2 ∧ 3 = {(2, h)(3, h)(1, h)(2, h)}.
Z. Ji et al. / Automatica 43 (2007) 662 – 668
Some simple calculations show that C((h)) =R3 =C, ∀h > 0. Hence, (h) is a desired aperiodic switching sequence for controllability realization. Due to another basis expression for W2 , similar arguments show that (h) = {(1, h)(2, h)(1, h)(3, h)(1, 2h)(3, h)} is also a desired switching sequence. The following are some observations, which reveal varieties inherent in both switching time and switching index sequences for the design of aperiodic switching sequences. • L((h))=4 < d(d −d1 +1)=6, and L((h))=d(d −d1 +1), as is in accordance with Theorem 1. • C((h)) = C holds for ∀ h > 0, as is also for (h). • Denote (h)={(1, h)(2, h)(1, h)(3, h)}. It can be calculated that C( (h)) = C((h)) = C. This means there may exist redundant switchings involved in the designed switching sequences. How to reduce redundant switchings is a problem worthy of a further investigation. Next, let us consider periodic switchings. Let h0 , h1 , h2 > 0 be any given positive real numbers, and h0 ,h1 ,h2 = {(1, h0 )(2, h1 )(3, h2 )} . By computation, we have ∧3 3 ∧2 h0 ,h1 ,h2 = h0 ,h1 ,h2 = R = C.
Denote i0 ,h0 ;i1 ,h1 ;i2 ,h2 = {(i0 , h0 )(i1 , h1 )(i2 , h2 )} with {i0 , i1 , i2 } being any permutation of {1, 2, 3}. It can be seen that ∧2 i0 ,h0 ;i1 ,h1 ;i2 ,h2 = C. By computation, we also see that for this example, the controllability cannot be realized through one switching period. This means that, for this example, = 2 is the minimum number of switching periods required for the controllability realization problem. 5. Conclusion An approach is proposed in this paper to design aperiodic switching sequences with much less switchings than the existing approaches in literature. The result shows a relationship between the number of switchings and the dimension of the controllable subspace. We also show that it is sufficient to employ periodic switchings to realize controllability within d switching periods, where d is the dimension of the controllable subspace. The present results bring a deep understanding on switching mechanism when controllability is considered. The application of the proposed approach to stabilization control problems will be studied in the future. Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 60604032, 60674050, 10601050, 60528007) and National 973 Program (2002CB312200). The authors would like to thank the three reviewers for their constructive and insightful suggestions for further improving the quality and presentation of this paper.
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Xie, G., & Wang, L. (2003). Controllability and stabilizability of switched linear-systems. Systems & Control Letters, 48(2), 135–155. Xie, G., & Wang, L. (2004). Necessary and sufficient conditions for controllability and observability of switched impulsive control systems. IEEE Transactions on Automatic Control, 49(6), 960–966. Yang, Z. (2002). An algebraic approach towards the controllability of controlled switching linear hybrid systems. Automatica, 38(7), 1221–1228. Zhai, G., Hu, B., Yasuda, K., & Michel, A. N. (2001). Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach. International Journal of Systems Science, 32(8), 1055–1061. Zhai, G., Lin, H., & Antsaklis, P. J. (2003). Quadratic stabilizability of switched linear systems with polytopic uncertainties. International Journal of Control, 76(7), 747–753. Zhao, J., & Dimirovski, G. M. (2004). Quadratic stability of a class of switched non-linear systems. IEEE Transactions on Automatic Control, 49(4), 574–578. Chen, C. T. (1999). Linear system theory and design. (3rd ed)., Oxford: Oxford University Press. Zhijian Ji received the M.S. degree in Applied Mathematics from Ocean University of China in 1998, and the Ph.D. degree in system theory from Intelligent Control Laboratory, Center for Systems and Control, Department of Mechanics and Engineering Science, Peking University, Beijing, China in 2005. He is currently an Associate Professor with School of Automation Engineering, Qingdao University. His current research interests are in the fields of nonlinear control systems, switched and hybrid systems, networked systems, and swarm dynamics. He was the winner of the First-class Scholarship of China Petrol in 2003, the Academic Innovation Award and the May Fourth Scholarship of Peking University in 2004.
Long Wang was born in Xian, China on Feb. 13, 1964. He received his Bachelor’s, Master’s, and Doctor’s degrees in Dynamics and Control from Tsinghua University and Peking University in 1986, 1989, and 1992, respectively. He has held research positions at the University of Toronto, Canada, and the German Aerospace Center, Munich, Germany. He is currently Cheung Kong Chair Professor of Dynamics and Control and Director of Center for Systems and Control of Peking University. He is also Vice-Director of National Key Laboratory of Complex Systems and Turbulence. He is a panel member of the Division of Information Science, National Natural Science Foundation of China. He is on the editorial boards of Progress in Natural Science, Acta Automatica Sinica, Journal of Control Theory and Applications, Control and Decision, Information and Control, etc. His research interests are in the fields of networked systems, hybrid systems, swarm dynamics, cognitive science, collective intelligence, and bio-mimetic robotics. Xiaoxia Guo received her M.S. degree in Applied Mathematics from Ocean University of China in 1997, and the Ph.D. degree in Computational Mathematics from Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Science in 2005. She is now with Department of Mathematics, Ocean University of China. Her research interests are in the fields of Numerical Computation, and matrix equations in control theory.
Brief paper
Design of switching sequences for controllability realization of switched linear systems夡 Zhijian Ji a,∗ , Long Wang b , Xiaoxia Guo c a School of Automation Engineering, Qingdao University, 266071, China b Intelligent Control Laboratory, Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China c Department of Mathematics, Ocean University of China, 266071, China
Received 29 March 2006; received in revised form 8 October 2006; accepted 10 October 2006 Available online 19 January 2007
Abstract In this paper, we study the problem of designing switching sequences for controllability of switched linear systems. Each controllable state set of designed switching sequences coincides with the controllable subspace. Both aperiodic and periodic switching sequences are considered. For the aperiodic case, a new approach is proposed to construct switching sequences, and the number of switchings involved in each designed switching sequence is shown to be upper bounded by d(d −d1 +1). Here d is the dimension of the controllable subspace, d1 =dim m i=1 Ai |Bi , where (Ai , Bi ) are subsystems. For the periodic case, we show that the controllable subspace can be realized within d switching periods. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Switched systems; Controllability; Reachability; Switching sequence
1. Introduction As a special class of hybrid control systems, switched systems have attracted considerable attention during the last decade because of its importance from both theoretical and practical points of view (Cheng, Guo, Lin, & Wang, 2005; Sun & Ge, 2005). In spite of some existing results, the study of the stability analysis and design of switched systems is just under way and the theoretical framework is far from complete. In particular, the switching mechanism has not been fully understood, and a closely related problem is to design switching sequences with switching times as less as possible. There are excellent survey papers such as Sun and Ge (2005), Liberzon and Morse (1999) and DeCarlo, Branicky, Pettersson, and Lennartson (2000). In this paper, we are interested in designing switching sequences to achieve the controllability for switched linear systems, i.e., each controllable state set of designed switching 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor teD Iwasaki under the direction of Editor R. Tempo. ∗ Corresponding author. E-mail address: [email protected] (Z. Ji).
0005-1098/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2006.10.010
sequences coincides with the controllable subspace of switched linear systems. We call this problem a controllability realization problem. Our goal is to design switching sequences to realize the controllability with the number of switchings as small as possible. To this end, a constructive approach is proposed for the design purpose. We will show that the upper bound for the length of each designed switching sequence is d(d − d1 + 1). Hered is the dimension of controllable subspace, d1 = dim m i=1 Ai |Bi , where (Ai , Bi ) are subsystems. For periodic switching sequences, we will prove that the controllability can always be realized within d switching periods. Comparison of our results with existing ones verifies the advantage of our design approach. The controllability realization problem was formerly studied by Sun, Ge, and Lee (2002) and Xie and Wang (2003). A common feature for the switching sequences designed by them is that the number of switchings increases rapidly with the number of subsystems and dimensions. In this paper, we reduce the required number of switchings significantly to an acceptable level for both cases of aperiodic and periodic switchings. Furthermore, we will show that for the design of aperiodic switching sequences, the maximum number of switchings is only determined by the dimension of the controllable subspace.
Z. Ji et al. / Automatica 43 (2007) 662 – 668
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Definition 1. A nonzero state x ∈ Rn is controllable, if there exist a switching sequence and an input u(t) : [0, T ] → Rp such that x(0) = x, and x(T ) = 0. The system (1) is said to be (completely) controllable if any nonzero state x is controllable. We denote by C the set of all controllable states of the system (1).
Besides controllability realization, the design of switching sequences/rules has been widely employed for stabilization problems. Wicks, Peleties, and DeCarlo (1998) firstly developed an elegant construction of a stabilizing switching rule. The result therein was later used for switched controller design and robustness study by Sun (2005), Zhao and Dimirovski (2004), Ji, Wang, Xie, and Hao (2004), Ji, Wang, and Xie (2005), Zhai, Lin, and Antsaklis (2003), Bacciotti (2004), King and Shorten (2005) and Ishii, Basar, and Tempo (2005). Another stabilization method is to constrain switching sequences to satisfy a dwell or an average-dwell-time condition. This method was firstly proposed by Morse (1997), and further developed by Hespanha and Morse (1999), Hespanha (2004) and Zhai, Hu, Yasuda, and Michel (2001). Recently, Cheng et al. (2005) put forward a concept of switching frequency which can be deemed as a development of average-dwell-time method. Our proposed approach enriches constructive methods for switching sequences. Controllability and observability play a fundamental role in the design and synthesis of linear control systems. For switched linear systems, complete geometric criteria for controllability and reachability were established by Sun et al. (2002) and Xie and Wang (2003). The controllability of switched bilinear systems was investigated by Cheng (2005) using Lie algebraic technique. Other necessary and/or sufficient conditions for controllability and reachability were presented by Ezzine and Haddad (1989), Krastanov and Veliov (2005), Petreczky (2006), Yang (2002), Stikkel, Bokor, and Szabó (2004), Xie and Wang (2004), Meng and Zhang (2006), Sun (2004), Ge, Sun, and Lee (2001), Stanford and Conner (1980) and Conner and Stanford (1987), etc. In our opinion, although necessary and sufficient conditions on controllability and reachability were established, the switching behaviors for controllability has not been fully investigated. This is our main purpose in the paper. The paper is organized as follows. In Section 2, we present preliminaries including system description, definitions and supporting lemmas. Main results and comparisons with existing ones are presented in Section 3. An illustrative example is included in Section 4. Finally, Section 5 briefly concludes the work.
Definition 3 (Xie & Wang, 2003). Given two switching sequences, 1 ={(i1,0 , h1,0 ) · · · (i1,s1 , h1,s1 )}, 2 ={(i2,0 , h2,0 ) · · · (i2,s2 , h2,s2 )}, 1 ∧ 2 is defined by
2. Preliminaries
Lemma 1 (Xie & Wang, 2003). Given a switching sequence = {(i0 , h0 ) · · · (is−1 , hs−1 )}, its controllable state set is
Definition 2 (Xie & Wang, 2003). Given a switching sequence = {(i0 , h0 ) · · · (is−1 , hs−1 )}, denote T = s−1 j =0 hj . The controllable state set C() of is defined by C() = {x|there exists an input u(t), t ∈ [0, T ], such that x(0) = x and x(T ) = 0}. The reachability counterparts of Definitions 1 and 2 can be given by replacing ‘x(0) = x, and x(T ) = 0’ with ‘x(0) = 0, and x(T ) = x’.
1 ∧ 2 = {(i1,0 , h1,0 ) · · · (i1,s1 , h1,s1 ) (i2,0 , h2,0 ) · · · (i2,s2 , h2,s2 )}. Since (1 ∧ 2 ) ∧ 3 = 1 ∧ (2 ∧ 3 ), we only denote it by 1 ∧ 2 ∧ 3 . Specially, we denote ∧ ∧· · · ∧ by ∧n . n
To describe the controllable subspace, we introduce some notations. Given B ∈ Rn×p , Im B denotes the image space of B. Given a matrix A ∈ Rn×n , a linear subspace W ⊆ Rn , A|W is the minimum A-invariant subspace containing W, i.e., A|W =
(1)
where x(t) ∈ Rn is the state, u(t) ∈ Rp is the input, and (t) : [t0 , ∞) → {1, . . . , m} is the switching sequence to be designed. Moreover, (t) = i implies that the ith subsystem (Ai , Bi ) is activated. Given a switching sequence : [t0 , tf ] → {1, . . . , m}, suppose its discontinuous (jump) time instants are t0 < t1 < · · · < ts−1 . We refer to the sequence t0 , t1 , . . . , ts−1 as switching time sequence, and the sequence (t0 )=i0 , (t1 )=i1 , . . . , (ts−1 )= is−1 as switching index sequence. Let hi := ti+1 − ti , i = 0, 1, . . . , s − 1, and ts := tf . We denote a switching sequence by = {(i0 , h0 ), . . . , (is−1 , hs−1 )}, where s is the length of .
Ai−1 W.
i=1
For notational simplicity, we denote A|B = A|Im B.
Consider a switched linear control system given by x(t) ˙ = A x(t) + B u(t),
n
C() =
k s−1
e
−Aij hj
Aik |Bik ,
k=0 j =0
where
k
j =0 e
−Aij hj
= e−Ai0 h0 · · · e−Aik hk .
Given a switching sequence = {(i0 , h0 ) · · · (is−1 , hs−1 )}, −Aij hj we denote e− := s−1 . j =0 e Remark 1. Lemma 1 implies that for any two switching sequences 1 , 2 C(1 ∧ 2 ) = C(1 ) + e−1 C(2 ).
(2)
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For system (1), a nested subspace sequence is defined as follows (Sun et al., 2002; Xie & Wang, 2003). W1 =
m
Ai |Bi ,
Wj +1 =
i=1
m
consider W1 , . . . , W . By Lemma 4, they can be written as W1 =
Ai |Wj ,
(3)
i=1
where j =1, 2, . . . . Note that if dim Wj =dim Wj +1 for some j, then Wl =Wj for all l > j, and Wn =Wn+1 =· · · holds. By employing Wn , an elegant result on geometric characteristics of controllable subspace C was established as follows. Lemma 2 (Sun et al., 2002; Xie & Wang, 2003). For switched linear system (1), the controllable set is C = Wn .
m
Ai |Bi ,
W2 =
i=1
W =
m
m
e−Ai h |W1 , . . . ,
i=1
e−Ai h |W−1 .
i=1
From these equalities, we have j0 ,...,j−2 ∈{0,1,...,n−1}
W =
(e−Ai0 h )j0
i0 ,...,i−1 ∈{1,...,m}
· · · (e
−Ai−2 h j−2
)
Ai−1 |Bi−1 .
Lemma 3 (Sun et al., 2002; Xie & Wang, 2003). Both the controllable set and the reachable set are subspaces of the total space, and the two subspaces are always identical, i.e., C = V.
That is, W is the summation of m n−1 subspaces, with each
By using standard results on sampling (see, e.g., Chen, 1999), the following lemma can be easily shown.
−2 Vi00,...,i−1 = (e−Ai0 h )j0 · · · (e
Lemma 4. For matrices Ak ∈ Rn×n , there exists h > 0, such that for any linear subspace W ⊆ Rn , Ak |W = e−Ak h |W,
∀k ∈ {1, . . . , m}.
Lemma 5 (Sun et al., 2002). For any given matrices A1 , A2 ∈ Rn×n , and B1 , B2 ∈ Rn×p , inequality
j ,...,j
holds for almost all h ∈ R.
)
Ai−1 |Bi−1 .
(4)
Then there exist d linear independent vectors 1 , . . . , d , which span W . We divide these vectors into l (1 l d) groups with each group belonging to a subspace of form (4). Suppose the division is as follows: 1 , . . . , d1 ∈ V1 , d1 +1 , . . . , d2 ∈ V2 , . . . , dl−1 +1 , . . . , d ∈ Vl ,
−2 Vi00,...,i−1 = e−Ai0 h0 · · · e
j ,...,j
dim[eA2 h Im(B
−Ai−2 h j−2
where V1 , . . . , Vl are subspaces of form (4). Accordingly, W = lj =1 Vj . As Vj is in form (4), we can rewrite it as
rank[A1 eA2 h B1 , B2 ]rank[A1 B1 , B2 ]
Lemma 5 implies that dim [Im(B1 ) + Im(B2 )] .
j ,...,j
−2 subspace Vi00,...,i−1 in the form of
1)
+ Im(B2 )]
3. Design of switching sequences 3.1. Aperiodic switching sequences Let L() be the length of switching sequence . Obviously, L() indicates the number of switchings involved in . Let = min{j | dim Wj =dim Wj +1 , j =1, 2, . . .}, and d1 =dim W1 . The following result reveals the relationship between the required number of switchings and the dimension of controllable state subspace C. Theorem 1. For system (1), there exist switching sequences k , k = 1, 2, . . . , with length of each one satisfying L(k )d(d − d1 + 1), such that for each k , C(k ) = Wn = C, where d and n are the dimensions of Wn and the state, respectively. Proof. Suppose dim Wn =d. The definition of index implies that dim W = dim Wn = d, and d − d1 + 1. This, together with W ⊆ Wn , yields W = Wn . Hence, it is sufficient to
−Ai−2 h−2
Ai−1 |Bi−1 ,
(5) −A
h
where h0 := j0 h, . . . , h−2 := j−2 h. Since e i−1 −1 Ai−1 |Bi−1 = Ai−1 |Bi−1 , it follows from Lemma 1 and j ,...,j
−2 (5) that Vi00,...,i−1 is just the last term of C( ). Here is designed as follows:
= {(i0 , h0 ) · · · (i−2 , h−2 )(i−1 , h−1 )}, and h−1 > 0 is arbitrary, for example, one can set h−1 = h or 1. Concerning Vj , we denote by j, the corresponding aforementioned switching sequence j, = {(ij,0 , hj,0 ) · · · (ij,−2 , hj,−2 )(ij,−1 , hj,−1 )}, where j = 1, . . . , l. Obviously, j, is not unique. This is because the selection of hj,−1 is not unique. Then Vj ⊆
C j, , and W = lj =1 Vj ⊆ lj =1 C(j, ). On the other hand, C(j, ) ⊆ Wn . This, together with Wn = W , gives W =
l
C(j, ).
(6)
j =1
Next, we are to prove dim C(1, ∧ · · · ∧ l, ) d.
(7)
Z. Ji et al. / Automatica 43 (2007) 662 – 668
It follows from (2) that
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3.2. Periodic switching sequences
C(1, ∧ · · · ∧ l, ) = C(1, ) + e−Ai1,0 h1,0
Since {1 , . . . , d1 } ⊆ V1 ⊆ C(1, ), repeatedly applying Lemma 5 to the right-hand-side of the above equality yields that
Periodic switching sequences are interesting from the viewpoint of implementation. Let h0 ,...,hm−1 ={(1, h0 ) · · · (m, hm−1 )} . The switching sequence ∧d h0 ,...,hm−1 possesses periodic switching index and periodic switching time sequences, and there are d switching periods associated with it. Let R+ be the set of all positive real numbers. We have the following result.
dim C(1, ∧ · · · ∧ l, ) dim{span{1 , . . . , d1 } + C(2, ∧ · · · ∧ l, )},
Theorem 2. For the switched linear system (1), there exists a periodic switching sequence ∧d h0 ,...,hm−1 , such that
holds for almost all h1,j ∈ R, j =0, 1, . . . , −1. Repeating the same reasoning on the right-hand-side of the above inequality, one can finally get
C(∧d h0 ,...,hm−1 ) = C,
···e
−Ai1,−1 h1,−1
C(2, ∧ · · · ∧ l, ).
dim C(1, ∧ · · · ∧ l, ) dim{span{1 , . . . , d1 } + · · · + span{dl−1 +1 , . . . , d }}. Then, (7) follows from the independence of 1 , . . . , d . Since C(1, ∧ · · · ∧ l, ) ⊆ Wn = W , and dim W = d, we have
where h0 , . . . , hm−1 ∈ R+ . Proof. Let us consider W1 , . . . , Wd with d, the dimension of the controllable subspace C. It can be seen that Wd = Wn = C. By (3), Wd can be written as j0 ,...,jd−2 ∈{0,1,...,n−1} i0 ,...,id−1 ∈{1,...,m}
C(1, ∧ · · · ∧ l, ) = W . This means that = 1, ∧ · · · ∧ l, is always a switching sequence to be required. Because j, , j = 1, . . . , l, is not unique, is not unique as well. We denote all these switching sequences by k , k = 1, 2, . . . . Clearly, the length of each k satisfies L(k ) l d(d − d1 + 1).
Wd =
j
j
d−2 Ai00 · · · Aid−2 Aid−1 |Bid−1 .
(8)
In the following, we prove C(∧d h0 ,...,hm−1 ) ⊇ Wd .
(9)
It follows from direct expansion of C(∧k h0 ,...,hm−1 ) that (e−A1 h0 · · · e−Am hm−1 )k−1 e−A1 h0 · · · e−Ai hi−1 Ai |Bi ⊆ C(∧k h0 ,...,hm−1 ),
∀i ∈ {1, . . . , m}; k = 1, 2, . . . .
Finally, we show Wn = C. Suppose is an arbitrary switching sequence, then C = ∀ C(). It follows that Wn = C(k ) ⊆ C. On the other hand, it follows from Lemma 1 and definition of Wn that for ∀, C() ⊆ Wn . We have C ⊆ Wn . Hence, C = Wn .
Denote
Remark 2. From the proof, we see that if l < d, L(k ) < d(d− d1 + 1). That is, L(k ) is always smaller than d(d − d1 + 1) whenever the division number is smaller than the dimension of controllable subspace C. Moreover, the smaller the number l, the smaller the length L(k ). An extreme case is l = 1. In this case, L(k )d − d1 + 1. We also see that the smaller the dimension of controllable subspace, the smaller the number of switchings required.
where id−1 plays the same role as i in the foregoing inclusion associated with C(∧k h0 ,...,hm−1 ). Then, for k = d, and any given i0 , . . . , id−1 ∈{1, . . . , m},
Remark 3. The proof of Theorem 1 provides a new proof for Lemma 2, and Lemma 2 is a direct consequence of Theorem 1.
i0 ,...,id−1 := (e−A1 h0 · · · e−Ai0 hi0 −1 · · · e−Am hm−1 )
· · · (e−A1 h0 · · · e−Aid−2 hid−2 −1 · · · e−Am hm−1 ) × e−A1 h0 · · · e−Aid−1 hid−1 −1 Aid−1 |Bid−1 ,
i0 ,...,id−1 ⊆ C(∧d h0 ,...,hm−1 ). It follows that C(∧d h0 ,...,hm−1 ) ⊇ =
i0 ,...,id−1
id−1 ∈{1,...,m}
i0 ,...,id−1 ,
(10)
i0 ,...,id−1 ∈{1,...,m}
Remark 4. The switching sequences constructed in Theorem 1 are not unique since h is not unique, and V1 , . . . , Vl are not unique as well. Actually, there are infinitely many h’s. Hence, the number of k s is infinite as well.
where the equality is due to the fact that the terms
It should be noted that a reachability counterpart of Theorem 1 can also be established by Lemma 3.
are actually the same regardless of what the values of i0 , . . . , id−2 are, and that id−1 is independent of i0 , . . . , id−2 .
(e−A1 h0 · · · e−Ai0 hi0 −1 · · · e−Am hm−1 ) · · · (e−A1 h0 · · · e−Aid−2 hid−2 −1 · · · e−Am hm−1 )
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Next, we are to show that for any given i0 , . . . , id−1 ∈ {1, . . . , m}, jd−2 j Aid−1 |Bid−1 , (11) i0 ,...,id−1 ⊇ Ai00 · · · Aid−2 where ∀j0 , . . . , jd−2 ∈ {0, 1, . . . , n − 1}. This can be verified using the power expansion of the matrix exponential. By Cayley–Hamilton theorem, e−As hs−1 = 0 (hs−1 )I + 1 (hs−1 )As + · · · + n−1 (hs−1 )An−1 , s where s = 1, . . . , m. Substituting e−As hs−1 in the form of the power expansion into i0 ,...,id−1 implies that j
[0 (h0 )I · · · 0 (hi0 −2 )I · j0 (hi0 −1 )Ai00 · 0 (hi0 )I · · · 0 (hm−1 )I ] · · · j
d−2 [0 (h0 )I · · · 0 (hid−2 −2 )I · jd−2 (hid−2 −1 )Aid−2
· 0 (hid−2 )I · · · 0 (hm−1 )I ]0 (h0 )I · · · 0 (hid−1 −1 )I Aid−1 |Bid−1 is one of the adding terms that constitute i0 ,...,id−1 . Observing that 0 (hs−1 ), . . . , n−1 (hs−1 ) are all scalars for ∀s ∈ {1, . . . , m}, and Aid−1 |Bid−1 is a subspace, we claim that (11) holds. Combining this with (10) and (8) gives (9). On the other hand, since C(∧d h0 ,···,hm−1 ) ⊆ Wn = Wd , it follows that C(∧d h0 ,...,hm−1 ) = Wn = C.
Let = min{j | dim Wj = dim Wj +1 , j = 1, 2, . . .}. Corollary 1. For the switched linear system (1), there ex∧ ists a periodic switching sequence h0 ,...,hm−1 , such that ∧
C(h0 ,...,hm−1 ) = C, where h0 , . . . , hm−1 ∈ R+ . Furthermore, for any k > , C(∧k h0 ,...,hm−1 ) = C holds as well.
Proof. Since W =Wd , the first equality follows by replacing d with in the proof of Theorem 2. As for the second equality, by Lemma 1, for any k > , ∧
C(∧k h0 ,...,hm−1 ) ⊇ C(h0 ,...,hm−1 ). ∧
Observe that C(h0 ,...,hm−1 )=C=Wn , and Wn ⊇C(∧k h0 ,...,hm−1 ). ∧k We get C(h0 ,...,hm−1 ) = Wn = C. Remark 5. Theorem 2 implies that the controllable subspace can always be realized via a periodic switching sequence within d periods, and hence the number of switchings involved is not greater than md. The switching sequence possesses not only the periodic switching index sequence but also the periodic time sequence. Remark 6. Sun (2004) proved that the controllability can be realized by a periodic and synchronous switching sequence, where synchronization means that there exist a base rate and a sequence of natural numbers {1 , 2 , . . .}, such that the switching time sequence is {0, 1 , 2 , . . .}. Whether or not
the switching times involved this switching path are finite is not answered therein. Hence, the difference between our periodic switching sequence and the one used by Sun (2004) lies in the switching time sequence and the number of switchings. Remark 7. Theorem 2 holds for switching index sequence of ∧d h0 ,...,hm−1 not only for the order {1, . . . , m}, but also for any other permutation of {1, . . . , m}. h0 , . . . , hm−1 are arbitrary positive real numbers. Hence, the number of periodic switching sequences is infinite from the viewpoint of switching time sequence. The reachability counterpart of the above results can be established in the same manner. Let us compare the results with existing ones. Xie and Wang (2003) put forward another method for constructing an aperiodic switching sequence for the controllability realization problem. By computation, it can be seen that the upperj bound for switching sequences designed therein is mn d−1 j =0 n , which is much greater than d(d −d1 +1)( n2 ) given in Theorem 1. Sun et al. (2002) solved the same problem by employing a switching sequence switching index sequence, with only periodic k where L() n−1 m(mn) − 1. It can be seen that the minik=0 mum number of switchings given by them is much greater than the maximum number of switchings given in Theorems 1 and 2. These comparisons verify that the constructive approaches presented in Theorems 1 and 2 significantly reduce the required number of switchings for the controllability realization problem. 4. An illustrative example Example 1. Consider the switched system (1) with m = n = 3, and 0 1 0 A1 = 0 0 0 , b1 =03×1 ; A2 =03×3 , b2 =[0, 1, 0]T , 0 0 0 A3 = 03×3 , b3 = [0, 0, 1]T . It can be verified that d1 = dim W1 = 2, W2 = W3 = R3 . Hence, the controllable subspace C is R3 , = 2, and d = 3. After some computations, we have 0 0 −h W2 = span ∀h > 0 1 , 0 , 1 0 1 0 = span{1 , 2 , 3 ; ∀h > 0}. Since 1 ∈ A2 |b2 ⊆ C(1 ), 2 ∈ A3 |b3 ⊆ C(2 ),
1 = {(2, h)}, 2 = {(3, h)},
3 ∈ e−A1 h A2 |b2 ⊆ C(3 ),
3 = {(1, h)(2, h)},
it follows from the proof of Theorem 1 that the desired switching sequence is (h) = 1 ∧ 2 ∧ 3 = {(2, h)(3, h)(1, h)(2, h)}.
Z. Ji et al. / Automatica 43 (2007) 662 – 668
Some simple calculations show that C((h)) =R3 =C, ∀h > 0. Hence, (h) is a desired aperiodic switching sequence for controllability realization. Due to another basis expression for W2 , similar arguments show that (h) = {(1, h)(2, h)(1, h)(3, h)(1, 2h)(3, h)} is also a desired switching sequence. The following are some observations, which reveal varieties inherent in both switching time and switching index sequences for the design of aperiodic switching sequences. • L((h))=4 < d(d −d1 +1)=6, and L((h))=d(d −d1 +1), as is in accordance with Theorem 1. • C((h)) = C holds for ∀ h > 0, as is also for (h). • Denote (h)={(1, h)(2, h)(1, h)(3, h)}. It can be calculated that C( (h)) = C((h)) = C. This means there may exist redundant switchings involved in the designed switching sequences. How to reduce redundant switchings is a problem worthy of a further investigation. Next, let us consider periodic switchings. Let h0 , h1 , h2 > 0 be any given positive real numbers, and h0 ,h1 ,h2 = {(1, h0 )(2, h1 )(3, h2 )} . By computation, we have ∧3 3 ∧2 h0 ,h1 ,h2 = h0 ,h1 ,h2 = R = C.
Denote i0 ,h0 ;i1 ,h1 ;i2 ,h2 = {(i0 , h0 )(i1 , h1 )(i2 , h2 )} with {i0 , i1 , i2 } being any permutation of {1, 2, 3}. It can be seen that ∧2 i0 ,h0 ;i1 ,h1 ;i2 ,h2 = C. By computation, we also see that for this example, the controllability cannot be realized through one switching period. This means that, for this example, = 2 is the minimum number of switching periods required for the controllability realization problem. 5. Conclusion An approach is proposed in this paper to design aperiodic switching sequences with much less switchings than the existing approaches in literature. The result shows a relationship between the number of switchings and the dimension of the controllable subspace. We also show that it is sufficient to employ periodic switchings to realize controllability within d switching periods, where d is the dimension of the controllable subspace. The present results bring a deep understanding on switching mechanism when controllability is considered. The application of the proposed approach to stabilization control problems will be studied in the future. Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 60604032, 60674050, 10601050, 60528007) and National 973 Program (2002CB312200). The authors would like to thank the three reviewers for their constructive and insightful suggestions for further improving the quality and presentation of this paper.
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Long Wang was born in Xian, China on Feb. 13, 1964. He received his Bachelor’s, Master’s, and Doctor’s degrees in Dynamics and Control from Tsinghua University and Peking University in 1986, 1989, and 1992, respectively. He has held research positions at the University of Toronto, Canada, and the German Aerospace Center, Munich, Germany. He is currently Cheung Kong Chair Professor of Dynamics and Control and Director of Center for Systems and Control of Peking University. He is also Vice-Director of National Key Laboratory of Complex Systems and Turbulence. He is a panel member of the Division of Information Science, National Natural Science Foundation of China. He is on the editorial boards of Progress in Natural Science, Acta Automatica Sinica, Journal of Control Theory and Applications, Control and Decision, Information and Control, etc. His research interests are in the fields of networked systems, hybrid systems, swarm dynamics, cognitive science, collective intelligence, and bio-mimetic robotics. Xiaoxia Guo received her M.S. degree in Applied Mathematics from Ocean University of China in 1997, and the Ph.D. degree in Computational Mathematics from Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Science in 2005. She is now with Department of Mathematics, Ocean University of China. Her research interests are in the fields of Numerical Computation, and matrix equations in control theory.