stabilization of switched linear systems with ... - Springer Link
J Syst Sci Complex (2012) 25: 60–70
STABILIZATION OF SWITCHED LINEAR SYSTEMS WITH CONSTRAINED INPUTS∗ Wei NI · Daizhan CHENG
DOI: 10.1007/s11424-012-9319-x Received: 11 November 2009 / Revised: 2 April 2011 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2012 Abstract The stabilization of switched linear systems with constrained inputs (SLSCI) is considered. The authors design admissible linear state feedbacks and the switching rule which has a minimal dwell time (MDT) to stabilized the system. First, for each subsystem with constrained inputs, a stabilizing linear state feedback and an invariant set of the closed-loop system are simultaneously constructed, such that the input constraints are satisfied if and only if the closed-loop system’s states lie inside this set. Then, by constructing a quadratic Lyapunov function for each closed-loop subsystem, an MDT is deduced and an MDT-based switching strategy is presented to ensure the stability of the switched system. Key words Constrained inputs, dwell time, stabilization, switched linear system.
1 Introduction Switching among different system models is a significant feature of many engineering control applications, and it can be modeled by switched systems. Roughly speaking, a switched system is an indexed family of differential/difference equations and a rule that determines the switching between them. Besides application background, the rapidly developing area of control techniques based on switching between different controllers[1−2] also motivate the study of switched systems. Research efforts on the switched systems were firstly put on stability and stabilization issues[3] , and later on, the structural properties, like reachability, controllability and observability, were also addressed[4−6]. For more details, we refer to the survey papers[3,7] and books[8−9] . For the stabilization of switched systems, most previous studies have focused on the design of controls without constraints, and very little attention has been devoted to the design of bounded stabilizers. The literatures[10−11] considered the stabilization of switched linear systems under bounded control. However, these works tackled the problem for arbitrary switching case. That is, only the control inputs are utilized to stabilize the switched systems. It’s worth noting that, Wei NI (Corresponding author) Department of Mathematics, School of Science, Nanchang University, Nanchang 330031, China. Email: [email protected]. Daizhan CHENG Key Laboratory of Systems and Control, Institute of Systems Science, Chinese Academy of Sciences, Beijing 100190, China. Email: [email protected]. ∗ This work is supported by the National Nature Science Foundation of China under Grant Nos: 60674022, 60736022, and 62821091. This paper was recommended for publication by Editor Yiguang HONG.
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besides the control inputs, the switching law can also be taken as a control tool if it is designable. The problem considered in this paper allows both the control inputs and the switching rules to be designable to ensure the exponential stability of switched systems. As for the control input, it is designed to be linear, and an invariant set is then constructed. The invariant set is designed to ensure the input constraints being satisfied. As for the design of the switching signal, each state at switching instants should lie in the invariant set of the currently active subsystem. Thus, when a subsystem is active, the corresponding trajectory stays in its invariant set, and consequently, input constraints are satisfied automatically. We will prove that MDT-based switching law meets this requirement. In short, we tackle the stabilization of SLSCI in two steps. Firstly, for each linear subsystem subject to input constraints, an admissible stabilizing state feedback is designed, and a quadratic Lyapunov function for the closed-loop subsystem is simultaneously constructed using the tool of matrix measure. Secondly, using these Lyapunov functions, we give an estimation of MDT and use the MDT-based switching strategy to ensure the stability of the switched systems. The rest of the paper is organized as follows. Section 2 contains some notations and preliminaries. Section 3 considers the stabilization of non-switched linear systems with constrained inputs, where an admissible stabilizer is designed and a quadratic Lyapunov function is constructed. The discussion then turns to stabilization of SLSCI in Section 3. An illustrative example is presented in Section 4. Section 5 is a brief conclusion.
2 Preliminaries First, we introduce some notations. For two vectors x, y ∈ Rn , x y means xi ≤ yi , i = 1, 2, · · · , n. For a real number a, a+ = max(a, 0); a− = max(−a, 0). λmin (·), λmax (·) denote the minimal and maximal eigenvalues of a symmetric matrix, respectively. ρ(·) denotes the set of eigenvalues of a matrix. R(·) denotes the range of a matrix. We use the symbol σ to denote a switching signal σ : [0, ∞) → I := {1, 2, · · · , N }, which is a piecewise right continuous map. The discontinuities t1 , t2 , · · · of σ are called switching instants. The number τ = inf{ti+1 −ti |i = 1, 2, · · · } is called dwell time. S[τ ] denotes the set of all the switching signals with dwell time no smaller than τ . For a matrix M = (mij )n×n , define M+ M− ∈ R2n×2n , M= M− M+ where M + and M − are n × n matrices whose (i, j) entries are defined as mij , if i = j, 0, if i = j, + − M (i, j) = M (i, j) = + − mij , if i = j, mij , if i = j, respectively, and define its matrix measure corresponding to a norm · as μ(M ) = lim
ε→0+
I + εM − 1 , ε
where I is an n × n identity matrix. The following lemmas will be used in the sequel. Lemma 1[12] The function V (x) = W x 2 with the k × n matrix T of rank n is a Lyapunov function of n-order linear system x˙ = Ax, if and only if there exists a matrix T ∈ Rk×k , such that W A − T W = 0, μ2 (T ) < 0.
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Lemma 2[13] Let A ∈ Rn×n and B ∈ Rm×m . The equation AX − XB = C has a unique solution X ∈ Rn×m for each C ∈ Rn×m , if and only if ρ(A) ∩ ρ(B) = ∅. Lemma 3[14] Let X be a solution of AX − XB = C. If (A, C) is controllable and R(C) ⊂ R(X), then X −1 exists.
3 Stabilization of Linear Systems with Constrained Controls Consider a linear system x˙ = Ax + Bu,
(1)
where A ∈ Rn×n , B ∈ Rn×m , rank(B) = m ≤ n, u ∈ U ⊂ Rm , U is the admissible control set defined by U = {u ∈ Rm | − umin u umax } for two given vectors umax , umin ∈ Rm with positive components. Assumption 1 The pair (A, B) is controllable. Let F ∈ Rm×n be a matrix such that A + BF is a Hurwitz matrix. The closed-loop system becomes x˙ = (A + BF )x.
(2)
D = {x ∈ Rn | − umin F x umax }.
(3)
Define
For the set D, we have the following lemma. Lemma 4[15] Domain D in (3) is positively invariant with respect to system (2), if and only if there exists a matrix H ∈ Rm×m satisfying F A + F BF = HF, 0, HU
(4) (5)
T T where U = (uT max , umin ) . For the matrix A, using Schur unitary triangularization theorem[13] , there exists an orthogonal matrix Q ∈ Rn×n such that A0 A2 := AQ , (6) QT AQ = 0 A1
where A0 has all negative real part eigenvalues, and A1 has all nonnegative real part eigenvalues. Then, taking a coordinate transformation x = Qz yields z˙ = AQ z + BQ u, where BQ := QT B. Set FQ := F Q. We assume the matrix A has r(0 ≤ r ≤ n) nonnegative real part eigenvalues. Thus, dim(A1 ) = r. For later use, split Q as Q = [(Q0 )n×(n−r) , (Q1 )n×r ]. Let umax = [Ir×r , 0r×(m−r)]umax , umin = [Ir×r , 0r×(m−r)]umin , and U = ((umax )T , (umin )T )T . We have the following theorem. Theorem 1 Consider the linear system (1) under Assumption 1 with A ∈ Rn×n , B ∈ Rn×m and u ∈ U. Suppose that A has r nonnegative real part eigenvalues. Let Q be an orthogonal matrix satisfying (6). Partition QT B and Q as (B01 )(n−r)×r ∗ T , Q = [(Q0 )n×(n−r) , (Q1 )n×r ], Q B= (B11 )r×r ∗
STABILIZATION OF SWITCHED LINEAR SYSTEMS
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where ∗ denotes the elements we do not concern. Let H1 ∈ Rr×r be a Hurwitz matrix such that 1 U 0. H
(7)
Then there exists a unique solution X ∈ Rr×r to the equation A1 X − XH1 = −B11 .
(8)
Furthermore, if R(B11 ) ⊂ R(X), the closed-loop system (2) with 0 X −1 QT F = 0 0 is asymptotically stable for all x0 ∈ D and the control u = F x is admissible. Proof Obviously, ρ(H1 ) ∩ ρ(A1 ) = ∅. According to Lemma 2, the solution of Equation (8) uniquely exists. Since (A, B) is controllable, the controllability of (A1 , B11 ) follows. This and the condition R(B11 ) ⊂ R(X), using Lemma 3, grantee that X is nonsingular. Then we can infer from (8) that A1 + B11 X −1 = XH1 X −1 , which implies ρ(A1 + B11 X −1 ) = ρ(H1 ).
(9)
Since A + BF = Q(AQ + BQ FQ )QT =Q
0 A1
=Q
A0 A2
+
B01 ∗
B11 ∗
A0 A2 + B01 X −1 0 A1 + B11 X −1
0 X −1 0
0
QT
QT ,
it can be obtained from (9) that ρ(A + BF ) = ρ(A0 ) ∪ ρ(H1 ). Therefore, A + BF is Hurwitz, and consequently, u = F x is a stabilizing feedback. We proceed to show u = F x is admissible. Define (H1 )r×r 0r×(m−r) H= . (0)(m−r)×r (0)(m−r)×(m−r) It can be easily shown that Equation (8) is equivalent to FQ AQ + FQ BQ FQ = HFQ , or equivalent to F A + F BF = HF. Noting that (5) holds due to (7), it can be inferred from Lemma 4 that D is an invariant set, and consequently, u = F x is admissible provided x0 ∈ D.
(z )
−1 , Remark 1 Considering the special form of u = F x = FQ z = X 0 z2 , where z = (z1 2n−r )r the control is admissible when the first r components of u satisfy the constraints. This can be seen in (7) by taking umax = [Ir×r , 0r×(m−r) ]umax , umin = [Ir×r , 0r×(m−r)]umin .
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Remark 2 By the above construction, since the matrix F is determined by H, it follows that the attractive set D depends on the choice of H. A challenging problem is how to choose H such that the corresponding attractive set D is as large as possible. We don’t elaborate on this question in this paper. Now, we proceed to construct a 2-norm form Lyapunov function for the closed-loop system. We need the following lemma. Lemma 5[16] Let A be a Hurwitz matrix. Then there exists a nonsingular matrix G such that μ2 (GAG−1 ) < 0. Since A0 is Hurwitz, it follows from Lemma 5 that there exists a nonsingular matrix G0 such that μ2 (G0 A0 G−1 0 ) < 0. Applying the same argument to H1 , one sees that there exists a nonsingular matrix G1 such that μ2 (G1 H1 G−1 1 ) < 0. Define I Y G0 0 QT , (10) W = 0 X −1 0 G1 where Y is a solution to the following matrix equation A0 Y − Y (A1 + B11 X −1 ) = A2 + B01 X −1 .
(11)
The existence of Y is guaranteed by the condition ρ(A0 ) ∩ ρ(H1 ) = ∅. Then, we have the following theorem. Theorem 2 The function V (x) = W x 2 is a Lyapunov function for the closed-loop system (2). Proof A straightforward calculation yields G0 0 A0 A2 + B01 X −1 + Y (A1 + B11 X −1 ) W (A + BF ) = QT . 0 G1 0 X −1 (A1 + B11 X −1 ) Setting T =
G0 0
0 G1
we have
TW =
G0 0 0 G1
A0 0 0 H1
G−1 0
0
0 G−1 1
A0 A0 Y 0 H1 X −1
,
QT .
Therefore, by (11) we have W (A + BF ) = T W . It follows from μ2 (G0 A0 G−1 0 ) < 0 and μ2 (G1 H1 G−1 ) < 0 that μ (T ) < 0. According to Lemma 1, V (x) = W x is a Lyapunov 2 2 1 function for the closed-loop system (2). Remark 3 A quadratic Lyapunov function can be obtained from the square of the 2-norm Lyapunov function in Theorem . That is, using the same notation V , we have V (x) = W x 22 = xT W T W x := xT P x.
STABILIZATION OF SWITCHED LINEAR SYSTEMS
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4 Stabilization of Switched Linear Systems with Constrained Controls We use the results in the last section, together with MDT-based switching law, to stabilize SLSCI. Consider a switched linear system x˙ = Aσ(t) x + Bσ(t) uσ(t) ,
(12)
where x ∈ Rn , σ : [0, ∞) → I = {1, 2, · · · , N } is a switching signal, Ai ∈ Rn×n , Bi ∈ Rn×mi , mi i T and ui = [u1i , u2i , · · · , um is the control inputs constrained by i ] ∈ R −umin ui umax , i i
i∈I
for pre-assigned vectors umin , umax ∈ Rmi with positive components. The ith subsystem of (12) i i is x˙ = Ai x + Bi ui ,
i ∈ I.
(13)
Assumption 2 The pairs (Ai , Bi ), i ∈ I are controllable. Denote by ri the number of the eigenvalues of matrix Ai with nonnegative real parts. Applying Theorem 1 to (13), we can get a stabilizing feedback matrix Fi and the corresponding stability region Di = {x ∈ Rn | − umin Fi x umax }. The resulting closed-loop system is i i x˙ = (Ai + Bi Fi )x,
(14)
and the quadratic Lyapunov function for system (14) is Vi (x) = xT Pi x = xT WiT Wi x, where Wi is constructed by using Theorem 1. Next, we consider the design of stabilizing switching law. The following lemma is used. Lemma 6[17] Let x˙ = Mi x , x ∈ Rn , i ∈ I be N asymptotically stable linear systems, and Vi (x), i ∈ I be the corresponding Lyapuvov functions satisfying ai x 2 ≤ Vi (x) ≤ bi x 2 ,
i ∈ I,
(15)
dVi (x(t)) |Mi x ≤ −ci x 2 , dt
i ∈ I.
(16)
and
Denote by μi =
bi ai , λi
=
ci bi ,
and set τ > max i∈I
ln μ i
λi
.
Then for any switching signal σ ∈ S[τ ], the switched linear system x˙ = Mσ(t) x is globally exponentially stable. For each subsystem (14), its quadratic Lyapunov function is Vi (x) = xT WiT Wi x. Setting Ri = −[Pi (Ai + Bi Fi ) + (Ai + Bi Fi )T Pi ] and using Wi (Ai + Bi Fi ) = Ti Wi and Pi = WiT Wi , we can show that Ri = −WiT (Ti + TiT )Wi ,
(17)
WEI NI · DAIZHAN CHENG
66
where Ti is constructed by using Theorem 1. We need some new notations. Denote
ai = λmin WiT Wi ,
bi = λmax WiT Wi ,
ci = −λmax WiT (Ti + TiT )Wi .
(18) (19) (20)
Let τ be a positive number satisfying τ > max i∈I
ln(bi /ai ) , ci /bi
(21)
N and let Ω ⊂ i=1 Di be an Euclidian ball centered at the origin. We are then able to establish the main result. Theorem 3 For the switched system (12) under Assumption 2, let Wi and Fi be obtained as in Theorem 1, and ai , bi , ci , τ be defined as (18), (19), (20), (21). Then the switched system (12) with initial state x0 ∈ Ω can be stabilized exponentially by admissible controls ui = Fi x and any switching law σ ∈ S[τ ]. Proof For any switching signal σ ∈ S[τ ] with switching instants t1 < t2 < · · · , let x(t) be a solution of (12) with x0 ∈ Ω . We show x(ti ) ∈ Ω , i = 1, 2, · · · . Let t0 = 0 and suppose that during [ti , ti+1 ), i = 0, 1, · · · , model p(i) is active, where p(i) ∈ I. Denote λi = ci /bi , μi = bi /ai . Obviously, ai x 2 ≤ Vi (x) ≤ bi x 2 , ≤ −ci x 2 . V˙ i (x(t))
(22) (23)
Ai x
For any t > 0, there exists an integer k such that t ∈ [tk , tk+1 ). Using (22) and (23), a similar argument as in the proof of Lemma 6 yields x(t) 2 ≤ μp(k) e−λp(k) (t−tk ) x(tk ) 2 ,
t ∈ [tk , tk+1 ).
By the continuity of x(t), letting t → tk+1 , we have x(tk+1 ) 2 ≤ μp(k) e−λp(k) (tk+1 −tk ) x(tk ) 2 . Using the above inequality repeatedly, we get x(tk+1 ) 2 ≤ μp(k) μp(k−1) · · · μp(0) e−λp(k) (tk+1 −tk ) e−λp(k−1) (tk −tk−1 ) · · · e−λp(0) (t1 −t0 ) x(t0 ) 2 ≤ μp(k) μp(k−1) · · · μp(0) e−[λp(k) +···λp(0) ]τ x(t0 ) 2 ,
(24)
where the second inequality comes from that tk+1 − tk ≥ τ, k = 0, 1, · · · . What we are going to show is μp(k) · · · μp(0) e−[λp(k) +···+λp(0) ]τ < 1. To this end, denote β = max{ lnλμk k }, which means k∈I
βλk ≥ ln μk , k ∈ I. Thus, ln μ βλp(k) + · · · + βλp(0) ln μp(k) + · · · + ln μp(0) i max =β= > . i∈I λi λp(k) + · · · + λp(0) λp(k) + · · · + λp(0) Since τ > max{ lnλμk k }, we have k∈I
τ>
ln μp(k) + · · · + ln μp(0) , λp(k) + · · · + λp(0)
67
STABILIZATION OF SWITCHED LINEAR SYSTEMS
or equivalently, μp(k) · · · μp(0) e−[λp(k) +···+λp(0) ]τ < 1. Thus inequality (24) turns to x(tk+1 ) 2 ≤ x(t0 ) 2 . Since x0 ∈ Ω and Ω is an Euclidian ball, it follows that x(ti+1 ) ∈ Ω , i ∈ N. Note that t ∈ [tk , tk+1 ) and the state x(t) evolves from x(tk ) according to x(t) ˙ = (Ap(k) + Bp(k) Fp(k) )x(t). Since x(tk ) ∈ Ω ⊂ Dk and Dk is positively invariant, we have x(t) ∈ Dk . Therefore, uσ(t) (t) = Fp(k) x(t) is admissible. Since the time t is arbitrary, we conclude that the control uσ(t) (t) of the switched system is admissible. At last, according to Lemma 6, we conclude that the switched system (12) is exponentially stable for any σ ∈ S[τ ] and x0 ∈ Ω .
5 An Illustrative Example Consider a switched linear system x˙ = Aσ(t) x + Bσ(t) uσ(t) , where σ : [0, +∞) → {1, 2}, 0.2210 −0.1190 A1 = , 0.3400 −0.1870
B1 =
1 1
,
A2 =
2 0.8 3
6
,
B2 =
1 0
2 2
,
= 0.9, umax = 1, umin = (7, 6)T , umax = (6, 7)T . The eigenvalues of subsystem and umin 1 1 2 2 matrices are ρ(A1 ) = {0.0510, −0.0170},
ρ(A2 ) = {1.4702, 6.5298},
respectively. The Schur decomposition of matrix A1 and A2 are given by −0.0170 0.4590 1.4702 2.2000 T T Q1 A1 Q1 = , Q2 A2 Q2 = , 0 0.0510 0 6.5298 where
Q1 =
−0.4472 −0.8944 −0.8944
0.4472
,
Q2 =
−0.8337 0.5522 0.5522
0.8337
.
Assign the closed-loop eigenvalues as ρ(A1 + B1 F1 ) = {−0.1000, −3.6041},
ρ(A2 + B2 F2 ) = {−0.0120 − 0.0110},
respectively. For simplicity, we choose diagonal matrices (H1 )1 and (H2 )1 as −0.012 0 (H1 )1 = −0.1, (H2 )1 = 0 −0.011
(25)
WEI NI · DAIZHAN CHENG
68
and the corresponding augmented matrices are H1 = (H1 )1 and H2 = (H2 )1 which verify 1 U1 0 and H 2 U2 0. Choosing G01 = G11 = 1 and G12 = I2 (we don’t need G02 since A02 H does not appear in the Schur decomposition (25)), it can be verified that the resulting matrices T1 , T2 satisfy μ2 (T1 ) < 0 and μ2 (T2 ) < 0. The feedback matrices can be calculated as −2.0119 −0.8004
. F1 = −0.3020, 0.1510 , F2 = 0.5118 −2.2052 According to (10), we have −0.5119 −0.8621 , W1 = −0.3020 0.1510 Thus
P1 =
W1T W1
=
0.3532 0.3957
W2 =
−2.0119 −0.8004 0.5118
0.3957 0.7660
,
P2 =
W2T W2
=
−2.2052
.
4.7764 0.3139 0.3139 5.7330
.
Using (18)–(21), we can compute a1 = 0.9906, b1 = 1.0000, a2 = 4.6826, b2 = 5.8268, c1 = 0.0842, c2 = 0.2682. τ = max(0.1117, 4.7495) = 4.7495. The invariant sets D1 , D2 of the two subsystems and the domain of attraction Ω are plotted in Figure 1. For simulation, choosing the initial condition x0 = (−2.4; −1.1)T which lies in Ω and letting the switching law be such that the two subsystems are active alternatively with dwell time 5, the trajectories of x1 (t) and x2 (t) are depicted in Figure 2. In order to see clearly the trajectories at switching instants, we extract a locally exemplified graph in Figure 2. At the same time, we plot the time responses u1 ∈ R and u2 ∈ R2 of the inputs in Figure 3, from which one sees that the saturation is actually avoided.
3 2 1
x2
r =2.6458 0 −1 −2 −3 −4 −4
−3
−2
−1
0 x1
1
2
3
4
Figure 1 Estimation of the domain of attraction
69
STABILIZATION OF SWITCHED LINEAR SYSTEMS 0
The component x 1 The component x2
Two components of the state x
−0.5
−1 −1.15 −1.5
−1.2 −1.25 −1.3
−2
−2.5 0
5
50
10
15 20 25 30
100
150
200 250 Time (sec)
300
350
400
Time responses of inputs u1to subsystem 1 and u2 to subsystem 2
Figure 2 Time response of x(t) with initial state x0 = (−2.4, −1.1)T 5
One dimensional input u 1 to subsystem 1 Two dimensional input u 2 to subsystem 2
4
5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0
3 2 1
0 5 10 15 20 25 30 35 40 45 50
0
−1
−2 −3
0
50
100
150
200 250 Time (sec)
300
350
400
Figure 3 Time response of u1 (t) and u2 (t) for the switched system
6 Conclusions The stabilization of switched linear systems with constrained inputs was considered in this paper. For each linear subsystem, a stabilizing state feedback control subject to input constraints was designed and a quadratic Lyapunov function for the closed-loop subsystem was simultaneously constructed. Then, using these Lyapunov functions, an estimation of MDT was obtained and an MDT-based switching strategy was used to ensure the stability of the switched systems. The validity of the results was shown in the simulation. References [1] S. R. Kulkarni and P. J. Ramadge, Model and controller selection policies based on output predictionerrors, IEEE Transactions on Automatic Control, 1996, 41(11): 1594–1604. [2] K. S. Narendra and J. Balakrishnan, Adaptive control using multiple models, IEEE Transactions on Automatic Control, 1997, 42(2): 171–187. [3] D. Liberzon and A. S. Morse, Basic problems in stbility and design of switched systems, IEEE Contr. Sys. Mag., 1999, 19: 59–70.
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[4] D. Cheng, Controllability of switched bilinear systems, IEEE Trans. Aut. Contr., 2005, 50(4): 511–515. [5] D. Cheng, Y. Lin, and Y. Wang, Accessibility of switched linear systems, IEEE Trans. Aut. Contr., 2006, 51(9): 1486–1491. [6] Z. Sun and D. Zheng, On reachability and stabilization of switched linear systems, IEEE Trans. Aut. Contr., 2001, 46(2): 291–295. [7] R. Shorten, et al., Stability criteria for switched and hybrid systems, SIAM Review, 2007, 49(4): 545–592. [8] D. Liberzon, Switching in Systems and Control, Birkhauser, Boston, 2003. [9] Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design, Springer, New York, 2005. [10] A. Benzaouia, L. Saydy, and O. Akhrif, Stability and control synthesis of switched systems subject to actuator saturation, American Control Conference, 2004, 6(30): 5818–5823. [11] Y. Song, et al., Control of switched systems with actuator saturation, Journal of Control Theory and Applications, 2006, 1: 38–43. [12] H. Kiendl, J. Adamy, and P. Stelzner, Vector norms as Lyapunov functions for linear systems, IEEE Transactions on Automatic Control, 1992, 37(6): 839–842. [13] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1986. [14] J. Z. Hearon, Nonsingular solutions of T A + BT = C, Linear Algebra and its Applications, 1977, 16(1): 57–63. [15] A. Benzaouia, M. A. Rami, and S. E. Faiz, Stabilization of linear system with saturation: A Sylvester equation approach, IMA Journal of Mathematical Control and Information, 2004, 21: 247–259. [16] Z. Zahreddine, Matrix measure and application to stability of matrices and interval dynamical systems, International Journal of Mathematics and Mathematical Sciences, 2003(2): 75–85. [17] W. Ni and D. Cheng, Control of switched linear systems with input saturation, Internal Journal of Systems Science, 2010, 41(9): 1057–1065.
STABILIZATION OF SWITCHED LINEAR SYSTEMS WITH CONSTRAINED INPUTS∗ Wei NI · Daizhan CHENG
DOI: 10.1007/s11424-012-9319-x Received: 11 November 2009 / Revised: 2 April 2011 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2012 Abstract The stabilization of switched linear systems with constrained inputs (SLSCI) is considered. The authors design admissible linear state feedbacks and the switching rule which has a minimal dwell time (MDT) to stabilized the system. First, for each subsystem with constrained inputs, a stabilizing linear state feedback and an invariant set of the closed-loop system are simultaneously constructed, such that the input constraints are satisfied if and only if the closed-loop system’s states lie inside this set. Then, by constructing a quadratic Lyapunov function for each closed-loop subsystem, an MDT is deduced and an MDT-based switching strategy is presented to ensure the stability of the switched system. Key words Constrained inputs, dwell time, stabilization, switched linear system.
1 Introduction Switching among different system models is a significant feature of many engineering control applications, and it can be modeled by switched systems. Roughly speaking, a switched system is an indexed family of differential/difference equations and a rule that determines the switching between them. Besides application background, the rapidly developing area of control techniques based on switching between different controllers[1−2] also motivate the study of switched systems. Research efforts on the switched systems were firstly put on stability and stabilization issues[3] , and later on, the structural properties, like reachability, controllability and observability, were also addressed[4−6]. For more details, we refer to the survey papers[3,7] and books[8−9] . For the stabilization of switched systems, most previous studies have focused on the design of controls without constraints, and very little attention has been devoted to the design of bounded stabilizers. The literatures[10−11] considered the stabilization of switched linear systems under bounded control. However, these works tackled the problem for arbitrary switching case. That is, only the control inputs are utilized to stabilize the switched systems. It’s worth noting that, Wei NI (Corresponding author) Department of Mathematics, School of Science, Nanchang University, Nanchang 330031, China. Email: [email protected]. Daizhan CHENG Key Laboratory of Systems and Control, Institute of Systems Science, Chinese Academy of Sciences, Beijing 100190, China. Email: [email protected]. ∗ This work is supported by the National Nature Science Foundation of China under Grant Nos: 60674022, 60736022, and 62821091. This paper was recommended for publication by Editor Yiguang HONG.
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61
besides the control inputs, the switching law can also be taken as a control tool if it is designable. The problem considered in this paper allows both the control inputs and the switching rules to be designable to ensure the exponential stability of switched systems. As for the control input, it is designed to be linear, and an invariant set is then constructed. The invariant set is designed to ensure the input constraints being satisfied. As for the design of the switching signal, each state at switching instants should lie in the invariant set of the currently active subsystem. Thus, when a subsystem is active, the corresponding trajectory stays in its invariant set, and consequently, input constraints are satisfied automatically. We will prove that MDT-based switching law meets this requirement. In short, we tackle the stabilization of SLSCI in two steps. Firstly, for each linear subsystem subject to input constraints, an admissible stabilizing state feedback is designed, and a quadratic Lyapunov function for the closed-loop subsystem is simultaneously constructed using the tool of matrix measure. Secondly, using these Lyapunov functions, we give an estimation of MDT and use the MDT-based switching strategy to ensure the stability of the switched systems. The rest of the paper is organized as follows. Section 2 contains some notations and preliminaries. Section 3 considers the stabilization of non-switched linear systems with constrained inputs, where an admissible stabilizer is designed and a quadratic Lyapunov function is constructed. The discussion then turns to stabilization of SLSCI in Section 3. An illustrative example is presented in Section 4. Section 5 is a brief conclusion.
2 Preliminaries First, we introduce some notations. For two vectors x, y ∈ Rn , x y means xi ≤ yi , i = 1, 2, · · · , n. For a real number a, a+ = max(a, 0); a− = max(−a, 0). λmin (·), λmax (·) denote the minimal and maximal eigenvalues of a symmetric matrix, respectively. ρ(·) denotes the set of eigenvalues of a matrix. R(·) denotes the range of a matrix. We use the symbol σ to denote a switching signal σ : [0, ∞) → I := {1, 2, · · · , N }, which is a piecewise right continuous map. The discontinuities t1 , t2 , · · · of σ are called switching instants. The number τ = inf{ti+1 −ti |i = 1, 2, · · · } is called dwell time. S[τ ] denotes the set of all the switching signals with dwell time no smaller than τ . For a matrix M = (mij )n×n , define M+ M− ∈ R2n×2n , M= M− M+ where M + and M − are n × n matrices whose (i, j) entries are defined as mij , if i = j, 0, if i = j, + − M (i, j) = M (i, j) = + − mij , if i = j, mij , if i = j, respectively, and define its matrix measure corresponding to a norm · as μ(M ) = lim
ε→0+
I + εM − 1 , ε
where I is an n × n identity matrix. The following lemmas will be used in the sequel. Lemma 1[12] The function V (x) = W x 2 with the k × n matrix T of rank n is a Lyapunov function of n-order linear system x˙ = Ax, if and only if there exists a matrix T ∈ Rk×k , such that W A − T W = 0, μ2 (T ) < 0.
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Lemma 2[13] Let A ∈ Rn×n and B ∈ Rm×m . The equation AX − XB = C has a unique solution X ∈ Rn×m for each C ∈ Rn×m , if and only if ρ(A) ∩ ρ(B) = ∅. Lemma 3[14] Let X be a solution of AX − XB = C. If (A, C) is controllable and R(C) ⊂ R(X), then X −1 exists.
3 Stabilization of Linear Systems with Constrained Controls Consider a linear system x˙ = Ax + Bu,
(1)
where A ∈ Rn×n , B ∈ Rn×m , rank(B) = m ≤ n, u ∈ U ⊂ Rm , U is the admissible control set defined by U = {u ∈ Rm | − umin u umax } for two given vectors umax , umin ∈ Rm with positive components. Assumption 1 The pair (A, B) is controllable. Let F ∈ Rm×n be a matrix such that A + BF is a Hurwitz matrix. The closed-loop system becomes x˙ = (A + BF )x.
(2)
D = {x ∈ Rn | − umin F x umax }.
(3)
Define
For the set D, we have the following lemma. Lemma 4[15] Domain D in (3) is positively invariant with respect to system (2), if and only if there exists a matrix H ∈ Rm×m satisfying F A + F BF = HF, 0, HU
(4) (5)
T T where U = (uT max , umin ) . For the matrix A, using Schur unitary triangularization theorem[13] , there exists an orthogonal matrix Q ∈ Rn×n such that A0 A2 := AQ , (6) QT AQ = 0 A1
where A0 has all negative real part eigenvalues, and A1 has all nonnegative real part eigenvalues. Then, taking a coordinate transformation x = Qz yields z˙ = AQ z + BQ u, where BQ := QT B. Set FQ := F Q. We assume the matrix A has r(0 ≤ r ≤ n) nonnegative real part eigenvalues. Thus, dim(A1 ) = r. For later use, split Q as Q = [(Q0 )n×(n−r) , (Q1 )n×r ]. Let umax = [Ir×r , 0r×(m−r)]umax , umin = [Ir×r , 0r×(m−r)]umin , and U = ((umax )T , (umin )T )T . We have the following theorem. Theorem 1 Consider the linear system (1) under Assumption 1 with A ∈ Rn×n , B ∈ Rn×m and u ∈ U. Suppose that A has r nonnegative real part eigenvalues. Let Q be an orthogonal matrix satisfying (6). Partition QT B and Q as (B01 )(n−r)×r ∗ T , Q = [(Q0 )n×(n−r) , (Q1 )n×r ], Q B= (B11 )r×r ∗
STABILIZATION OF SWITCHED LINEAR SYSTEMS
63
where ∗ denotes the elements we do not concern. Let H1 ∈ Rr×r be a Hurwitz matrix such that 1 U 0. H
(7)
Then there exists a unique solution X ∈ Rr×r to the equation A1 X − XH1 = −B11 .
(8)
Furthermore, if R(B11 ) ⊂ R(X), the closed-loop system (2) with 0 X −1 QT F = 0 0 is asymptotically stable for all x0 ∈ D and the control u = F x is admissible. Proof Obviously, ρ(H1 ) ∩ ρ(A1 ) = ∅. According to Lemma 2, the solution of Equation (8) uniquely exists. Since (A, B) is controllable, the controllability of (A1 , B11 ) follows. This and the condition R(B11 ) ⊂ R(X), using Lemma 3, grantee that X is nonsingular. Then we can infer from (8) that A1 + B11 X −1 = XH1 X −1 , which implies ρ(A1 + B11 X −1 ) = ρ(H1 ).
(9)
Since A + BF = Q(AQ + BQ FQ )QT =Q
0 A1
=Q
A0 A2
+
B01 ∗
B11 ∗
A0 A2 + B01 X −1 0 A1 + B11 X −1
0 X −1 0
0
QT
QT ,
it can be obtained from (9) that ρ(A + BF ) = ρ(A0 ) ∪ ρ(H1 ). Therefore, A + BF is Hurwitz, and consequently, u = F x is a stabilizing feedback. We proceed to show u = F x is admissible. Define (H1 )r×r 0r×(m−r) H= . (0)(m−r)×r (0)(m−r)×(m−r) It can be easily shown that Equation (8) is equivalent to FQ AQ + FQ BQ FQ = HFQ , or equivalent to F A + F BF = HF. Noting that (5) holds due to (7), it can be inferred from Lemma 4 that D is an invariant set, and consequently, u = F x is admissible provided x0 ∈ D.
(z )
−1 , Remark 1 Considering the special form of u = F x = FQ z = X 0 z2 , where z = (z1 2n−r )r the control is admissible when the first r components of u satisfy the constraints. This can be seen in (7) by taking umax = [Ir×r , 0r×(m−r) ]umax , umin = [Ir×r , 0r×(m−r)]umin .
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Remark 2 By the above construction, since the matrix F is determined by H, it follows that the attractive set D depends on the choice of H. A challenging problem is how to choose H such that the corresponding attractive set D is as large as possible. We don’t elaborate on this question in this paper. Now, we proceed to construct a 2-norm form Lyapunov function for the closed-loop system. We need the following lemma. Lemma 5[16] Let A be a Hurwitz matrix. Then there exists a nonsingular matrix G such that μ2 (GAG−1 ) < 0. Since A0 is Hurwitz, it follows from Lemma 5 that there exists a nonsingular matrix G0 such that μ2 (G0 A0 G−1 0 ) < 0. Applying the same argument to H1 , one sees that there exists a nonsingular matrix G1 such that μ2 (G1 H1 G−1 1 ) < 0. Define I Y G0 0 QT , (10) W = 0 X −1 0 G1 where Y is a solution to the following matrix equation A0 Y − Y (A1 + B11 X −1 ) = A2 + B01 X −1 .
(11)
The existence of Y is guaranteed by the condition ρ(A0 ) ∩ ρ(H1 ) = ∅. Then, we have the following theorem. Theorem 2 The function V (x) = W x 2 is a Lyapunov function for the closed-loop system (2). Proof A straightforward calculation yields G0 0 A0 A2 + B01 X −1 + Y (A1 + B11 X −1 ) W (A + BF ) = QT . 0 G1 0 X −1 (A1 + B11 X −1 ) Setting T =
G0 0
0 G1
we have
TW =
G0 0 0 G1
A0 0 0 H1
G−1 0
0
0 G−1 1
A0 A0 Y 0 H1 X −1
,
QT .
Therefore, by (11) we have W (A + BF ) = T W . It follows from μ2 (G0 A0 G−1 0 ) < 0 and μ2 (G1 H1 G−1 ) < 0 that μ (T ) < 0. According to Lemma 1, V (x) = W x is a Lyapunov 2 2 1 function for the closed-loop system (2). Remark 3 A quadratic Lyapunov function can be obtained from the square of the 2-norm Lyapunov function in Theorem . That is, using the same notation V , we have V (x) = W x 22 = xT W T W x := xT P x.
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65
4 Stabilization of Switched Linear Systems with Constrained Controls We use the results in the last section, together with MDT-based switching law, to stabilize SLSCI. Consider a switched linear system x˙ = Aσ(t) x + Bσ(t) uσ(t) ,
(12)
where x ∈ Rn , σ : [0, ∞) → I = {1, 2, · · · , N } is a switching signal, Ai ∈ Rn×n , Bi ∈ Rn×mi , mi i T and ui = [u1i , u2i , · · · , um is the control inputs constrained by i ] ∈ R −umin ui umax , i i
i∈I
for pre-assigned vectors umin , umax ∈ Rmi with positive components. The ith subsystem of (12) i i is x˙ = Ai x + Bi ui ,
i ∈ I.
(13)
Assumption 2 The pairs (Ai , Bi ), i ∈ I are controllable. Denote by ri the number of the eigenvalues of matrix Ai with nonnegative real parts. Applying Theorem 1 to (13), we can get a stabilizing feedback matrix Fi and the corresponding stability region Di = {x ∈ Rn | − umin Fi x umax }. The resulting closed-loop system is i i x˙ = (Ai + Bi Fi )x,
(14)
and the quadratic Lyapunov function for system (14) is Vi (x) = xT Pi x = xT WiT Wi x, where Wi is constructed by using Theorem 1. Next, we consider the design of stabilizing switching law. The following lemma is used. Lemma 6[17] Let x˙ = Mi x , x ∈ Rn , i ∈ I be N asymptotically stable linear systems, and Vi (x), i ∈ I be the corresponding Lyapuvov functions satisfying ai x 2 ≤ Vi (x) ≤ bi x 2 ,
i ∈ I,
(15)
dVi (x(t)) |Mi x ≤ −ci x 2 , dt
i ∈ I.
(16)
and
Denote by μi =
bi ai , λi
=
ci bi ,
and set τ > max i∈I
ln μ i
λi
.
Then for any switching signal σ ∈ S[τ ], the switched linear system x˙ = Mσ(t) x is globally exponentially stable. For each subsystem (14), its quadratic Lyapunov function is Vi (x) = xT WiT Wi x. Setting Ri = −[Pi (Ai + Bi Fi ) + (Ai + Bi Fi )T Pi ] and using Wi (Ai + Bi Fi ) = Ti Wi and Pi = WiT Wi , we can show that Ri = −WiT (Ti + TiT )Wi ,
(17)
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66
where Ti is constructed by using Theorem 1. We need some new notations. Denote
ai = λmin WiT Wi ,
bi = λmax WiT Wi ,
ci = −λmax WiT (Ti + TiT )Wi .
(18) (19) (20)
Let τ be a positive number satisfying τ > max i∈I
ln(bi /ai ) , ci /bi
(21)
N and let Ω ⊂ i=1 Di be an Euclidian ball centered at the origin. We are then able to establish the main result. Theorem 3 For the switched system (12) under Assumption 2, let Wi and Fi be obtained as in Theorem 1, and ai , bi , ci , τ be defined as (18), (19), (20), (21). Then the switched system (12) with initial state x0 ∈ Ω can be stabilized exponentially by admissible controls ui = Fi x and any switching law σ ∈ S[τ ]. Proof For any switching signal σ ∈ S[τ ] with switching instants t1 < t2 < · · · , let x(t) be a solution of (12) with x0 ∈ Ω . We show x(ti ) ∈ Ω , i = 1, 2, · · · . Let t0 = 0 and suppose that during [ti , ti+1 ), i = 0, 1, · · · , model p(i) is active, where p(i) ∈ I. Denote λi = ci /bi , μi = bi /ai . Obviously, ai x 2 ≤ Vi (x) ≤ bi x 2 , ≤ −ci x 2 . V˙ i (x(t))
(22) (23)
Ai x
For any t > 0, there exists an integer k such that t ∈ [tk , tk+1 ). Using (22) and (23), a similar argument as in the proof of Lemma 6 yields x(t) 2 ≤ μp(k) e−λp(k) (t−tk ) x(tk ) 2 ,
t ∈ [tk , tk+1 ).
By the continuity of x(t), letting t → tk+1 , we have x(tk+1 ) 2 ≤ μp(k) e−λp(k) (tk+1 −tk ) x(tk ) 2 . Using the above inequality repeatedly, we get x(tk+1 ) 2 ≤ μp(k) μp(k−1) · · · μp(0) e−λp(k) (tk+1 −tk ) e−λp(k−1) (tk −tk−1 ) · · · e−λp(0) (t1 −t0 ) x(t0 ) 2 ≤ μp(k) μp(k−1) · · · μp(0) e−[λp(k) +···λp(0) ]τ x(t0 ) 2 ,
(24)
where the second inequality comes from that tk+1 − tk ≥ τ, k = 0, 1, · · · . What we are going to show is μp(k) · · · μp(0) e−[λp(k) +···+λp(0) ]τ < 1. To this end, denote β = max{ lnλμk k }, which means k∈I
βλk ≥ ln μk , k ∈ I. Thus, ln μ βλp(k) + · · · + βλp(0) ln μp(k) + · · · + ln μp(0) i max =β= > . i∈I λi λp(k) + · · · + λp(0) λp(k) + · · · + λp(0) Since τ > max{ lnλμk k }, we have k∈I
τ>
ln μp(k) + · · · + ln μp(0) , λp(k) + · · · + λp(0)
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STABILIZATION OF SWITCHED LINEAR SYSTEMS
or equivalently, μp(k) · · · μp(0) e−[λp(k) +···+λp(0) ]τ < 1. Thus inequality (24) turns to x(tk+1 ) 2 ≤ x(t0 ) 2 . Since x0 ∈ Ω and Ω is an Euclidian ball, it follows that x(ti+1 ) ∈ Ω , i ∈ N. Note that t ∈ [tk , tk+1 ) and the state x(t) evolves from x(tk ) according to x(t) ˙ = (Ap(k) + Bp(k) Fp(k) )x(t). Since x(tk ) ∈ Ω ⊂ Dk and Dk is positively invariant, we have x(t) ∈ Dk . Therefore, uσ(t) (t) = Fp(k) x(t) is admissible. Since the time t is arbitrary, we conclude that the control uσ(t) (t) of the switched system is admissible. At last, according to Lemma 6, we conclude that the switched system (12) is exponentially stable for any σ ∈ S[τ ] and x0 ∈ Ω .
5 An Illustrative Example Consider a switched linear system x˙ = Aσ(t) x + Bσ(t) uσ(t) , where σ : [0, +∞) → {1, 2}, 0.2210 −0.1190 A1 = , 0.3400 −0.1870
B1 =
1 1
,
A2 =
2 0.8 3
6
,
B2 =
1 0
2 2
,
= 0.9, umax = 1, umin = (7, 6)T , umax = (6, 7)T . The eigenvalues of subsystem and umin 1 1 2 2 matrices are ρ(A1 ) = {0.0510, −0.0170},
ρ(A2 ) = {1.4702, 6.5298},
respectively. The Schur decomposition of matrix A1 and A2 are given by −0.0170 0.4590 1.4702 2.2000 T T Q1 A1 Q1 = , Q2 A2 Q2 = , 0 0.0510 0 6.5298 where
Q1 =
−0.4472 −0.8944 −0.8944
0.4472
,
Q2 =
−0.8337 0.5522 0.5522
0.8337
.
Assign the closed-loop eigenvalues as ρ(A1 + B1 F1 ) = {−0.1000, −3.6041},
ρ(A2 + B2 F2 ) = {−0.0120 − 0.0110},
respectively. For simplicity, we choose diagonal matrices (H1 )1 and (H2 )1 as −0.012 0 (H1 )1 = −0.1, (H2 )1 = 0 −0.011
(25)
WEI NI · DAIZHAN CHENG
68
and the corresponding augmented matrices are H1 = (H1 )1 and H2 = (H2 )1 which verify 1 U1 0 and H 2 U2 0. Choosing G01 = G11 = 1 and G12 = I2 (we don’t need G02 since A02 H does not appear in the Schur decomposition (25)), it can be verified that the resulting matrices T1 , T2 satisfy μ2 (T1 ) < 0 and μ2 (T2 ) < 0. The feedback matrices can be calculated as −2.0119 −0.8004
. F1 = −0.3020, 0.1510 , F2 = 0.5118 −2.2052 According to (10), we have −0.5119 −0.8621 , W1 = −0.3020 0.1510 Thus
P1 =
W1T W1
=
0.3532 0.3957
W2 =
−2.0119 −0.8004 0.5118
0.3957 0.7660
,
P2 =
W2T W2
=
−2.2052
.
4.7764 0.3139 0.3139 5.7330
.
Using (18)–(21), we can compute a1 = 0.9906, b1 = 1.0000, a2 = 4.6826, b2 = 5.8268, c1 = 0.0842, c2 = 0.2682. τ = max(0.1117, 4.7495) = 4.7495. The invariant sets D1 , D2 of the two subsystems and the domain of attraction Ω are plotted in Figure 1. For simulation, choosing the initial condition x0 = (−2.4; −1.1)T which lies in Ω and letting the switching law be such that the two subsystems are active alternatively with dwell time 5, the trajectories of x1 (t) and x2 (t) are depicted in Figure 2. In order to see clearly the trajectories at switching instants, we extract a locally exemplified graph in Figure 2. At the same time, we plot the time responses u1 ∈ R and u2 ∈ R2 of the inputs in Figure 3, from which one sees that the saturation is actually avoided.
3 2 1
x2
r =2.6458 0 −1 −2 −3 −4 −4
−3
−2
−1
0 x1
1
2
3
4
Figure 1 Estimation of the domain of attraction
69
STABILIZATION OF SWITCHED LINEAR SYSTEMS 0
The component x 1 The component x2
Two components of the state x
−0.5
−1 −1.15 −1.5
−1.2 −1.25 −1.3
−2
−2.5 0
5
50
10
15 20 25 30
100
150
200 250 Time (sec)
300
350
400
Time responses of inputs u1to subsystem 1 and u2 to subsystem 2
Figure 2 Time response of x(t) with initial state x0 = (−2.4, −1.1)T 5
One dimensional input u 1 to subsystem 1 Two dimensional input u 2 to subsystem 2
4
5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0
3 2 1
0 5 10 15 20 25 30 35 40 45 50
0
−1
−2 −3
0
50
100
150
200 250 Time (sec)
300
350
400
Figure 3 Time response of u1 (t) and u2 (t) for the switched system
6 Conclusions The stabilization of switched linear systems with constrained inputs was considered in this paper. For each linear subsystem, a stabilizing state feedback control subject to input constraints was designed and a quadratic Lyapunov function for the closed-loop subsystem was simultaneously constructed. Then, using these Lyapunov functions, an estimation of MDT was obtained and an MDT-based switching strategy was used to ensure the stability of the switched systems. The validity of the results was shown in the simulation. References [1] S. R. Kulkarni and P. J. Ramadge, Model and controller selection policies based on output predictionerrors, IEEE Transactions on Automatic Control, 1996, 41(11): 1594–1604. [2] K. S. Narendra and J. Balakrishnan, Adaptive control using multiple models, IEEE Transactions on Automatic Control, 1997, 42(2): 171–187. [3] D. Liberzon and A. S. Morse, Basic problems in stbility and design of switched systems, IEEE Contr. Sys. Mag., 1999, 19: 59–70.
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[4] D. Cheng, Controllability of switched bilinear systems, IEEE Trans. Aut. Contr., 2005, 50(4): 511–515. [5] D. Cheng, Y. Lin, and Y. Wang, Accessibility of switched linear systems, IEEE Trans. Aut. Contr., 2006, 51(9): 1486–1491. [6] Z. Sun and D. Zheng, On reachability and stabilization of switched linear systems, IEEE Trans. Aut. Contr., 2001, 46(2): 291–295. [7] R. Shorten, et al., Stability criteria for switched and hybrid systems, SIAM Review, 2007, 49(4): 545–592. [8] D. Liberzon, Switching in Systems and Control, Birkhauser, Boston, 2003. [9] Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design, Springer, New York, 2005. [10] A. Benzaouia, L. Saydy, and O. Akhrif, Stability and control synthesis of switched systems subject to actuator saturation, American Control Conference, 2004, 6(30): 5818–5823. [11] Y. Song, et al., Control of switched systems with actuator saturation, Journal of Control Theory and Applications, 2006, 1: 38–43. [12] H. Kiendl, J. Adamy, and P. Stelzner, Vector norms as Lyapunov functions for linear systems, IEEE Transactions on Automatic Control, 1992, 37(6): 839–842. [13] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1986. [14] J. Z. Hearon, Nonsingular solutions of T A + BT = C, Linear Algebra and its Applications, 1977, 16(1): 57–63. [15] A. Benzaouia, M. A. Rami, and S. E. Faiz, Stabilization of linear system with saturation: A Sylvester equation approach, IMA Journal of Mathematical Control and Information, 2004, 21: 247–259. [16] Z. Zahreddine, Matrix measure and application to stability of matrices and interval dynamical systems, International Journal of Mathematics and Mathematical Sciences, 2003(2): 75–85. [17] W. Ni and D. Cheng, Control of switched linear systems with input saturation, Internal Journal of Systems Science, 2010, 41(9): 1057–1065.
