On stabilizability of switched positive linear systems under state-dependent switching

Applied Mathematics and Computation 307 (2017) 92–101

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On stabilizability of switched positive linear systems under state-dependent switchingR Xiuyong Ding∗, Xiu Liu School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, PR China

a r t i c l e

i n f o

Keywords: Switched systems Positive systems State-dependent switching Stabilizability

a b s t r a c t This paper addresses the stabilization of switched positive linear systems by statedependent switching. We show that if there is a Hurwitz convex (or linear) combination of the coefficient matrices, then the switched positive linear system can be exponentially stabilized by means of a single linear co-positive Lyapunov function. If there is not a stable combination of system matrices, it is shown that the exponential stabilizability is equivalent to a completeness condition on the coefficient matrices. When the switched positive systems can not be stabilized by the single Lyapunov function, we provide a unified criterion for piecewise exponential stabilizability in terms of multiple linear co-positive Lyapunov functions. © 2017 Elsevier Inc. All rights reserved.

1. Introduction As a special class of hybrid dynamical systems, switched systems have numerous applications in the control of manufacturing systems [1], traffic control [2], automotive engine control and air craft control [3], and many other fields [4]. For a discussion of various issues related to switched systems, see the survey article [4]. The typical switched system is composed of a family of continuous-time or discrete-time subsystems and a rule orchestrating the switch among them. In this paper, we consider the switched linear system:

x˙ = Aσ (t,x ) x,

(1)

where x is the state vector in the real vector space σ : [0, ∞ ) → m := {1, 2, . . . , m} is the so-called switching signal. The system matrices Ai (i ∈ m ) belong to the real n × n space Rn×n . In general, switching events can be classified, according to the switching type, into time-dependent (depending on the time t only, i.e., σ = σ (t )) and state-dependent (depending on the state x, i.e., σ = σ (x )). For a linear time-invariant (LTI) system A : x˙ = Ax, if the non-negativeness of initial condition implies that the state x is non-negative at the every t ≥ 0, then it is called positive system [5]. The switched system (1) is called a switched positive linear system (SPLS) if all its subsystems are positive systems. Recently, the importance of linear switched positive systems has been highlighted by many researchers because of their broad application in communication systems [6], formation flying [7], mathematical networked epidemiology [8,9], and other areas. The stability and stabilizability of switched positive systems, especially SPLSs, have drawn a lot of attentions in the last decade. One is the stability analysis of SPLSs under arbitrary switching (see, e.g., [10–12]); The other question is whether Rn .

× Rn

R This work was supported by the Foundation of National Nature Science of China (Grant No.61473239 and 11626196) and supported by the Fundamental Research Funds for the Central Universities (Grant No.2682015CX058 and 2682016CX117). ∗ Corresponding author. E-mail address: [email protected] (X. Ding).

http://dx.doi.org/10.1016/j.amc.2017.03.007 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.

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the SPLS is stable or stabilizable when there are some restrictions on the switching signals. These restrictions may be either time-dependent or state-dependent. The stability of SPLSs under time-dependent switching captured wide attention and has more abundant achievements (see, e.g., [13–16] and some references therein). On the other hand, state-dependent stabilizability of SPLSs is a topic only partially explored [17–21]. In [17], the authors discuss the existence of a Hurwitz convex combination of the sysytem (Metzler) matrices. In [17,18], the authors show that some special SPLSs, such as second order systems, two mode systems with rank one difference, can be exponentially stabilized. Similarly, for discrete-time SPLSs, the existence of a Schur convex combination of the system matrices implies state-dependent stabilizability [19]. The necessity is only for some special cases (for example, the second order case [19], cyclic monomial matrix and circulant matrix cases [20]). It should be emphasized that all these results are essentially based on some special restrictions, such as the low dimension, the existence of a Hurwitz convex combination, etc., thus leading to deeper insights into the state-dependent stabilization problem of SPLSs must be endowed with. This paper will focus on the stabilization issue of SPLSs under state-dependent switching. We aim at exploiting some constructive switching strategies which are applicable to several broad classes of SPLSs. The layout of the paper is as follows. Section 2 states some preliminary results. In Section 3, we provide some stabilization strategies via the single Lyapunov function methods. We first show that the existence of a Hurwitz convex combination of the coefficient matrices implies state-dependent stabilizability, and that the converse is true only for the two mode systems. When the assumption of a Hurwitz combination is not possible, it is shown that exponential stabilizability is equivalent to a completeness condition on the coefficient matrices. Section 4 focuses on piecewise stabilizability. When the stabilization can not be carried out with the help of a single Lyapunov function, we prove exponential stabilizability by using multiple Lyapunov functions. It is worth noting that the proposed results are unified criteria for SPLSs which are without any a priori restriction, in contrast to others in the literature. Finally, Section 5 concludes this paper. 2. Preliminaries Throughout, for matrix A, B or vector x, y, A  B(AB) or xy(xy) means that all elements of matrix A − B or vector x − y are non-negative (positive). Similarly, A B(A≺B) or x y(x≺y) means that all elements of matrix A − B or vector x − y are non-positive (negative). AT represents the transpose of matrix A. I denotes the identity matrix. x = 0 means that there  exists at least one non-zero entry in vector x. x = nk=1 |xk |, where xk is the kth element of x ∈ Rn . 1 is the vector of all ones. Before proceeding, we recall some facts which are relevant for this paper. The following S-procedure for linear version is presented in [22,23]. Theorem 2.1 [22,23]. (“S-procedure for linear forms”) Let σk (y ) = yT sk + rk , where vectors sk , y ∈ Rn and rk ∈ R, k = 0, 1, . . . , N. If σ k (y) is regular (i.e., there is one y∗ such that σ k (y∗ ) > 0 for k = 1, . . . , N), then the following statements are equivalent for any finite number of constraints N: (i) σ 0 (y) ≥ 0 for y ∈ Rn whenever σ k (y) ≥ 0, k = 1, . . . , N.  (ii) There exists constants τ k ≥ 0 (k = 1, . . . , N) such that σ0 (y ) − N k=1 τk σk (y ) ≥ 0. The following lemma which is straightforward from Theorem 2.1 will play a key role in deriving the results of this paper. Lemma 2.1. Let vectors w0 , and w1 be in Rn . If there exists a constant τ ≥ 0 such that w0 − τ w1  0, then for every x0 in Rn , xT w0 ≥ 0 whenever xT w1 ≥ 0. A matrix is a Metzler matrix if its off-diagonal entries are non-negative. A classic result is that an LTI system  A is positive if and only if its system matrix A is a Metzler matrix [5]. A matrix is a Hurwitz matrix if and only if all its eigenvalues lie in the open left half of the complex plane. There are a number of equivalent conditions for a Metzler–Hurwitz matrix and the stability of positive LTI system  A . Some conditions which are relevant for the work of this paper are collected here. Theorem 2.2 [10,24]. Let A ∈ Rn×n be a Metzler matrix, the following statements are equivalent. (i) The LTI system  A is asymptotically stable, i.e., x → 0 as t → ∞. (ii) A ∈ Rn×n is a Hurwitz matrix. (iii) There exists a vector v  0 in Rn such that AT v ≺ 0. Related to statement (iii), we recall a common definition which is a powerful research tool for positive systems. The function V (x ) = xT v is call a linear co-positive Lyapunov function (LCLF) [10] of the positive LTI system  A if V(x) > 0 and V˙ (x ) = xT AT v < 0 for all non-zero x  0. Definition 2.1. The SPLS (1) is exponentially stable if there exist constants γ > 0 and δ > 0 such that the system solution x satisfies x ≤ γ e−δ (t−t0 ) x0 for the initial x0 0 and t ≥ t0 . In addition, The SPLS (1) is called exponentially stabilizable if there exist switching signals σ such that the SPLS (1) is exponentially stable. In this paper, we will design some effective state-dependent switching strategies σ (x) to exponentially stabilize SPLS (1). Notice that if at least one of the individual is asymptotically stable, or equivalently, there exists at least one system matrix

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Ai (i ∈ m ) is Hurwitz, this problem is trivial (just keep to activate the stable subsystem). We thus, in the sequel, steadily make a assumption that none of the individual subsystems of SPLS (1) is asymptotically stable. 3. Exponential stabilization by single LCLF method In this section, we deal with the stabilization problem by using a single LCLF V (x ) = xT v. We will prove that, if the existence of a Hurwitz convex (or, linear) combination of system matrices is a sufficient condition for state-dependent stabilizability. The necessity is true only for the case of two subsystems. Conversely, the system matrices may not possess a Hurwitz combination, in which case we also establish an equivalent condition between the exponential stabilizability and the strict completeness [25] of the set {ATi v : i ∈ m}. 3.1. Stable convex combinations First of all we make a classic assumption that lead to some elegant constructions of stabilizing switching signals.  Assumption 3.1. There exist some constants α i (i ∈ m ) satisfying 0 < α i < 1 and m i=1 αi = 1 such that the convex combim nation A := i=1 αi Ai is Hurwitz. Now we provide a stabilization result involving Assumption 3.1 Theorem 3.1. Consider the SPLS (1) with Metzler matrices Ai ∈ Rn×n (i ∈ m ). If Assumption 3.1 holds, then there exist some statedependent switching signals with the form

σ (x ) = arg min xT ATi v, v  0

(2)

i∈m

such that the SPLS (1) is exponentially stabilizable. Proof. Note that the endpoints αi = 1(i ∈ m ) are excluded because Ai are not Hurwitz. Now for Metzler matrices Ai (i ∈  T m), if Assumption 3.1 holds, there exists, according to Theorem 2.2, a vector v  0 such that AT v = m i=1 αi Ai v ≺ 0. Which m leads, for any non-zero x0, to i=1 αi xT ATi v < 0. This implies that for every non-zero x there exists at least one i such that xT ATi v < 0. Now we set AT v := −w with w  0, and further construct regions i as follows

i = {x ∈ Rn+ |xT ATi v ≤ −xT w}, i ∈ m.

(3)

Rn+ /{0}.

Obviously, these regions together cover Indeed, assume that there is a non-zero vector x0 0 which does not belong  m T T T T T T T to any i , that is, xT0 ATi v > −xT0 w for all i ∈ m. It follows that xT0 m i=1 αi Ai v = x0 A v > − i=1 αi x0 w = −x0 w = x0 A v. A contradiction thus occurs. ˜ i , such that each i is a To prevent a sliding motion (see [26] for more details), we pick a set of overlapping regions,  ˜ i . We thus define proper subset of 

1

˜ i = {x ∈ Rn+ |xT AT v ≤ − xT w}, ϑ > 1, i ∈ m.  i ϑ

(4)

Now by the switching signals (2), a stabilizing switching strategy can be described as follows. First, according to the switch˜ i , keep σ (x ) = i. In addition, if the state reaches ing signals (2), choose the initial active mode σ (x0 ) = i0 . If the state x ∈  ˜ i , determine the jth mode according to the switching signals (2) and switch to the jth mode. the boundary of  Based on the above discussion, we now show the exponential stability of (1) under the switching signals (2). Consider the single LCLF

V (x(t )) = xT (t )v. ˜ i for t ∈ [tk , tk+1 ), the derivative of V Suppose that the ith mode is active on [tk , tk+1 ) for some integer k > 0, i.e., x(t ) ∈  along the solution of (1) is

V˙ (x(t )) = xT (t )ATi v ≤ −

1

ϑ

xT (t )w ≤ −δ xT (t )v = −δV (x(t )),

T

where δ = mini∈m minx∈˜ ϑxV (wx ) . This implies that V (x(t )) ≤ exp(−δ (t − tk ))V (x(tk )). Furthermore, observe that the vali ues of the Lyapunov function certainly matches on the switching instants, it therefore follows that V (x(t )) ≤ exp(−δ (t − t0 ))V (x(t0 )), which implies that the system (1) converges exponentially to zero. This completes the proof.  Remark 3.1. In Assumption 3.1, if we remove the restriction of convex, namely, we just assume that there exist some con stants ζ i > 0 (i ∈ m ) such that the linear combination A¯ := m i=1 ζi Ai is Hurwitz. Notice that this assumption (called as Hurwitz linear combination) is equivalent to Assumption 3.1. In fact, Assumption 3.1 obviously implies Hurwitz linear com ζ m i bination. In contrast, if the Hurwitz linear combination is satisfied, it immediately follows that m1 ζ A¯ = m A is i=1 ζ i Hurwitz. Denoting αi :=

ζ

m i

gives that matrix i=1 ζi

i=1 i

i=1 i

A¯ is a Hurwitz convex combination. In this case, we can also in i=1 ζi

m1

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Theorem 3.1 remove the convex restriction, or equivalently, replace the condition Assumption 3.1 by the Hurwitz linear combination assumption. It should be emphasized that the Hurwitz convex (or, linear) combination is only a sufficient condition for exponential stabilizability. The necessity, unfortunately, is not true in general (a counter-example can be found in [17]). However, if one restricts the attention to the case of two modes, necessity can be achieved. Theorem 3.2. Assume that the SPLS (1) is composed of two subsystems (i.e., m = {1, 2}), then the following are equivalent. (i) Assumption 3.1 holds.  (ii) There exist some constants ζ i > 0 (i ∈ m ) such that the linear combination A¯ := m i=1 ζi Ai is Hurwitz. (iii) There exists a switching strategy with the form (2) such that the SPLS (1) is exponentially stabilizable. Proof. Remark 3.1 shows the equivalence between (i) and (ii). Hence we only need to show that (i) is equivalent to (iii). Sufficiency follows from Theorem 3.1. Now if there exists a switching strategy with the form (2) such that the SPLS (1) is exponentially stabilizable, then there exists a LCLF V (x ) = xT v whose derivative along the system solutions satisfies V˙ (x ) = xT ATi v < − x for some ε > 0 and i ∈ {1, 2}. Since the switching is state-dependent, this implies that for every nonzero x we must have either xT AT1 v < − x or xT AT2 v < − x . We can restate this as follows:

xT (AT1 v + 1 ) < 0 whenever xT (AT2 v + 1 ) ≥ 0,

(5)

xT (AT2 v + 1 ) < 0 whenever xT (AT1 v + 1 ) ≥ 0.

(6)

xT AT1 v

Rn+ ,

If < 0 holds for all non-zero x ∈ from Theorem 2.2, the matrix A1 is thus Metzler Hurwitz and the proof is trivial. Similarly, if for all nonzero x, xT AT2 v < 0 holds, then A2 is also Metzler–Hurwitz. Discarding these trivial cases, now from one of the conditions (5) and (6) we can apply Theorem 2.1 to derive that there exist some τ > 0 such that

xT (AT1 v + 1 ) + τ xT (AT2 v + 1 ) ≤ 0,

∀x ∈ Rn+ .

This is,

xT

( A1 + τ A2 )T v ≤ − xT 1 < 0, ∀x ∈ Rn+ . 1+τ

As the matrix A1 + τ A2 is also Metzler, this implies that the LTI positive system x˙ = the LCLF V (x ) = By Theorem 2.2, we thus know A2 , and so Assumption 3.1 holds.  x T v.

A1 +τ A2 1+τ

A1 +τ A2 1+τ x

is asymptotically stable with

is Hurwitz, which is a Hurwitz convex combination of A1 and

Example 1. Consider the switched linear system (1) with two 2 × 2 Metzler matrices



A1 =

2 0



2 , −8



A2 =

−6 0 2 2



Obviously, both subsystems are positive and unstable. Now by choosing α1 = α2 = 0.5, a Hurwitz convex combination is given by A = α1 A1 + α2 A2 = [

−2 1

1 ]. Hence, for a positive vector w = [1 2]T , there exists a associated positive vec−3

tor v = [1 1]T such that AT v = −w = −[1 2]T . From (3) this can give i = {x ∈ Rn+ |xT ATi v ≤ −xT w}. To overcome the sliding ˜ i = {x ∈ Rn+ |xT AT v ≤ − 1 xT w} with some overlapping regions. Specifmotion, by selecting ϑ = 2 we set up according to (4)  i ϑ 1 T n T T T ˜ ˜ ically, 1 = {x ∈ R+ |x [2 − 6] ≤ − x [1 2] } and 2 = {x ∈ Rn+ |xT [−4 2]T ≤ − 1 xT [1 2]T }. The switching rule is illustrated 2

2

in Fig. 1. Then it follows from Theorem 3.1 that the SPLS is exponentially stabilizable under the switching strategy (2). This is confirmed by the numerical simulation shown in Fig. 1. 3.2. Strict completeness In this subsection, we will establish a relation between the exponential stabilizability and a strict completeness [25] condition on the matrices Ai (i ∈ m ) when the coefficient matrices do not possess a Hurwitz combination.

Theorem 3.3. There exist some switching signals with the form σ (x ) = arg mini∈m {xT ATi v}, v  0 such that the SPLS (1) is exponentially stabilizable if and only if there is a vector v  0 such that the set {ATi v : i ∈ m} is strictly complete in Rn+ , i.e., for any non-zero x0, there exists i ∈ m such that xT ATi v < 0. Proof. First, if SPLS (1) is exponentially stabilizable under the switching signal σ (x ) = mini∈m {xT ATi v}, then for any x ∈ Rn+ /{0}, there must exist a matrix Ai such that the derivative of LCLF V (x ) = xT v along the trajectories of SPLS (1) satisfies

V˙ (x ) = xT ATi v < 0. This condition is equivalent to the strict completeness of the set {ATi v : i ∈ m}.

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x2(t)

3

2

system mode

4 3x2=3.5x1

Switched signals 3 2 1 0 0

2

4 6 Sample Time

Overlapping region

8

∼ Ω1

2x =x 2

1

1 ∼ Ω2 0 0

1

2 x (t)

3

4

1

Fig. 1. The trajectory, switching regions, and switching signals of the SPLS in Example 1, where the initial condition is [2 4]T . Switching rule: First, σ (x0 ) = 1 ˜ 1 ’s boundary 2x2 = x1 , then switch σ (x ) = 2. As is for x ∈  ˜ 2. ˜ 1 , keep σ (x ) = 1. As far as the state reaches the  ˜ 1 . When x ∈  as [2 4]T ∈ 

On the other hand, if the set {ATi v : i ∈ m} is strictly complete, then it follows that for every x ∈ i there exists i ∈ m such that xT AT v < 0. Furthermore, there must exist a constant  i > 0 such that xT AT v < −i x , or, equivalently, V˙ (x ) < 0 i

i

along the trajectories of the ith subsystem by choosing the LCLF V (x ) = xT v. Now it remain to consider the points on the switching boundary i, j or j, i . From Filippov [27] we have

V˙ (x ) = sup xT (α Ai + (1 − α )A j )T v α ∈[0,1]

= sup

{α xT ATi v + (1 − α )xT ATj v}.

α ∈[0,1]

Since the set is strictly complete, it follows by the construction of the switching signal that on each side of the switching boundary that the subsystem is chosen such that xT ATl v < −l x with  l > 0 for l = i, j. Hence, for any nonzero x ∈ Rn+ , V˙ (x ) < − x with  = min{i , i ∈ m}, and the exponential stabilizability is thus achieved.  Remark 3.2. It should be emphasized that checking the strict completeness is a nontrivial task and NP hard in general. However, it is easily checked that the stable convex combination (or equivalently, stable linear combination) implies strict completeness. Especially, the converse is true only for the case of two subsystems (see, e.g., [28]). 4. Exponential stabilization by multiple LCLF method In the previous section, the stabilization analysis was carried out with the help of a single LCLF. When this is impossible, one can try to find a stabilizing switching signal and prove exponential stability by using multiple LCLFs. It should be emphasized that, different from the earlier proposed in the literature, the purpose of this section is to give constructive switching synthesis results that can be applied to general SPLSs without any a priori restriction. In general, state-dependent stabilization implies the different subsystems will only be active in parts of the state space, specified by switching regions i , i ∈ m. However, since we do not make any assumption on SPLSs in advance, these state space partitions i are not known a priori, and need to be designed elaborately. The following lemma ensures this problem to be fulfilled. Lemma 4.1. For any x ∈ Rn+ , consider regions taking the forms

i = {x ∈ Rn+ |xT wi ≥ 0} with wi ∈ m 

Rn , i

(7)

∈ m. If there exist constants hi ≥ 0 (i ∈ m ) such that

hi xT w i ≥ 0 ,

(8)

i=1

then ∪m  = Rn+ . i=1 i Proof. We shall argue by contradiction. Assume that there is a non-zero state x0 ∈ Rn+ which does not belongs to any i .  T This implies that xT0 wi < 0 for any i ∈ m. It further follows m i=1 hi x0 wi < 0, a contradiction. 

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Notice that, in (7) each partition region i is purposely given by the linear forms because this can facilitate the search for proper LCLFs, each region i is associated with a LCLF V (x ) = xT vi , i ∈ m. Now we should design a switching condition which can make sure that: (i) The ith subsystem is active only when x ∈ i , i ∈ m. (ii) A switch from subsystem i to j occurs only for states where the regions i and j are adjacent, denoted as i, j , ¯ j , where  ¯ i and  ¯ j denote the closure of i and j , respectively. This merely ensures that the switches ¯i∩ i.e., i, j ⊆  only occur on i, j , none of the other regions exist switching phenomena. As for which regions will intersect, it is a design result and impossible to be known in advance. To accomplish this aim, we define the following switching strategy

σ (x ) = arg max xT wi

(9)

i∈m

Applying this strategy, the switching condition is fulfilled if the covering property (8) is satisfied. Indeed, from (8), based on the strategy (9), the state with the current active mode i satisfies xT wi ≥ 0. This implies that x ∈ i , i.e., the ith subsystem is active. Moreover, the boundary between i and j is given by i, j = {x ∈ Rn+ |xT wi = xT w j }. According to the strategy (9), we easily know that the switchings only occur at the adjacent regions i, j since i and j given by (7) are both closed. The following critical question is sliding motions [26] which may occur on the switching surfaces and will destroy system stability. Since the directions of the ith subsystem’s vector field Ai x and the jth subsystem’s vector field Aj x are unknown in advance, the same intersecting region may be i, j (or j, i ), this is decided by the entrance manners of Ai x and Aj x. Precisely, i, j implies that Ai x points toward the switching surface and Aj x departs from it. In contrast for j, i . Therefore the switching manners are uncertain at the switching surfaces, sliding motions will occur when both Ai x and Aj x point toward the switching surfaces. To overcome sliding motions, we define the switching surfaces as the form

{x ∈ Rn+ |xT wi = xT w j }.

(10)

Notice that no sliding motion occurs if and only if both vector fields Ai x and Aj x, at the corresponding point x in surface, point in the same direction. This implies that the non-existence of a sliding motion is equivalent to [27] either

xT ATi (wi − w j ) ≥ 0

whenever

xT ATj (wi − w j ) ≥ 0.

(11)

xT ATi (wi − w j ) ≤ 0

whenever

xT ATj (wi − w j ) ≤ 0.

(12)

or

holds for the states x with xT wi = xT w j . Now we are in position to state our main result. Theorem 4.1. For i, j ∈ m, if there exist vectors vi , wi ∈ Rn and real numbers κ 1 > 0, κ 2 > 0, hi > 0, ai ≥ 0, bi ≥ 0, ci ≥ 0, ei ≥ 0, ri ≥ 0, and di, j such that the following conditions are satisfied: m 

hi w i  0 ,

(13)

i=1

(ATi − ri ATj + ei I )(wi − w j ) 0,

(14)

κ11 + ai wi vi κ21 − bi wi ,

(15)

ATi vi + ci wi −1,

(16)

v j + di, j (wi − w j ) = vi ,

(17)

where di, j = d j,i . Then the SPLS (1) is exponentially stabilizable (with decay rate δ = 1/κ2 ) under the switching strategy (9).  T Proof. First of all, condition (13) gives m i=1 hi x wi  0 for any non-zero x0. By Lemma 4.1 we can define switching regions as the form of (7) whose union covers the whole positive orthant Rn+ . Furthermore we define switching strategy (9) which ensures the switching condition is satisfied. Now we choose the following piecewise Lyapunov function:

V (x(t )) = Vσ (x ) (x(t )) = xT (t )vσ (x ) ,

σ : Rn+ → m.

(18)

We first consider the case of x(t) ∈ i , i ∈ m, i.e, the ith subsystem is active, and suppose that the acting time t ∈ [tk , tk+1 ) for some integer k > 0. From (15) we have, for any non-zero x(t)0,

κ1 xT (t )1 + ai xT (t )wi ≤ xT (t )vi ≤ κ2 xT (t )1 − bi xT (t )wi . By applying Lemma 2.1 we further get

κ1 xT (t )1 ≤ xT (t )vi ≤ κ2 xT (t )1 whenever x(t ) ∈ i .

(19)

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This implies that the Lyapunov function (18) is positive definite in the corresponding switching region i . On the other hand, for any non-zero x(t)0, we premultiply the inequality (16) by x(t) and obtain

xT (t )ATi vi + ci xT (t )wi ≤ −xT (t )1. Also using Lemma 2.1, it follows that

xT (t )ATi vi ≤ −xT (t )1

x(t ) ∈ i .

whenever

Together this with (19), we can get that

1 1 d Vi (x(t )) ≤ −xT (t )1 ≤ − xT (t )vi = − Vi (x(t )), t ∈ [tk , tk+1 ), dt κ2 κ2 which yields

V (x(t )) = Vσ (x ) (x(t )) ≤ exp(−

1

κ2

(t − tk ))V (x(tk )).

(20)

Now we should consider the case x(t) ∈ i, j , i, j ∈ m, where switching surface i, j is as the form of (10) and implies that the SPLS switches from the ith mode to the jth. Firstly, from condition (14), it follows that, for any non-zero x(t)0,

xT (t )(ATi − ri ATj + ei I )(wi − w j ) ≤ 0. Since x(t) ∈ i, j , we can obtain

xT (t )(ATi − ri ATj )(wi − w j ) ≤ 0. By Lemma 2.1, we have for x(t) ∈ i, j

xT (t )ATi (wi − w j ) ≤ 0

whenever

xT (t )ATj (wi − w j ) ≤ 0.

This implies from (12) that there is no sliding motion on the switching surface i, j . In addition, we can premultiply the condition (17) by x(t) and obtain

xT (t )v j + di, j xT (t )(wi − w j ) = xT (t )vi , we can further get

xT (t )v j = xT (t )vi

xT (t ) ∈ i, j .

whenever

This can be rewritten as

Vi (x(t )) = V j (x(t ))

xT (t ) ∈ i, j .

whenever

(21)

Namely, the values of the Lyapunov function are certainly matches on the switching surface. Therefore, for any initial condition x0 = x(t0 ), we can deduce from (20) and (21) that

V (x(t )) = Vσ (x ) (x(t )) ≤ exp(−

1

κ2

(t − t0 ))V (x(t0 )),

which implies, by (19), that

x(t ) ≤

1 κ2 exp(− (t − t0 )) x(t0 ) → 0, t → +∞. κ1 κ2

Hence the solution x(t) of (1) exponentially converges to zero with decay rate 1/κ 2 . This completes the proof.



Remark 4.1. According to (12), the condition (14) can be replaced by

(ATi − ri ATj + ei I )(wi − w j )  0. Remark 4.2. For condition (17), if di, j = 0, then v1 = v2 = · · · = vm = v. This implies that all the subsystems share a single LCLF. In this case, multiplying condition (16) by hi /ci (where ci = 0) and summing up gives m m m    hi T hi  Ai v + hi w i − 1, ci ci i=1

i=1

i=1

Under condition (13), it follows that m m   hi T hi  Ai v − 1, ci ci i=1

i=1

which implies, by Theorem 2.2, that

m

hi T i=1 ci Ai

is a stable linear combination, and thus the SPLS (1) by Theorem 3.1 is  exponentially stabilizable. In addition, setting m i=1 hi /ci = 1, we can get a Hurwitz convex combination of system matrices, and then Theorem 4.1 is reduced to Theorem 3.1.

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Consider the case when the SPLS (1) is composed with two subsystems. Let w1 = −w2 = w and h1 = h2 = 1 satisfy condition (13). In this case, by denoting d1,2 = d2,1 = d, (17) can be rewritten as v1 = v2 + 2d, we can further get v1 = v + 2d by setting v2 = v. Finally a corollary naturally follows from Theorem 4.1. Corollary 4.1. Consider the SPLS (1) composed of two subsystems, for i ∈ {1, 2}, if there exist vectors v, w ∈ Rn and real numbers κ 1 > 0, κ 2 > 0, ai ≥ 0, bi ≥ 0, ci ≥ 0, ei ≥ 0, ri ≥ 0, and d such that the following conditions are satisfied:

(AT1 − r1 AT2 + e1 I )w 0,

(22)

−(AT2 − r2 AT1 + e2 I )w 0,

(23)

κ11 + (a1 − 2d )w v κ21 − (b1 + 2d )w,

(24)

κ11 − a2 w v κ21 + b2 w,

(25)

AT1 v + 2dAT1 w + c1 w −1,

(26)

AT2 v − c2 w −1.

(27)

Then the SPLS (1) is exponentially stabilizable (with decay rate 1/κ 2 ) under the switching strategy (9). Again, consider the two subsystems case, in Theorem 4.1, by setting w1 = v1 − v2 = −w2 , a1 = a2 = b1 = b2 = 0, d1,2 = d2,1 = 1/2, we can derive the following result. Corollary 4.2. Consider the SPLS (1) composed of two subsystems, for i ∈ {1, 2}, if there exist a vector v ∈ Rn and real numbers κ 1 > 0, κ 2 > 0, and ci ≥ 0 such that the following conditions are satisfied:

κ11 vi κ21,

(28)

AT1 v1 + c1 (v2 − v1 ) −1.

(29)

AT2 v2 + c2 (v1 − v2 ) −1.

(30)

Then the SPLS (1) is exponentially stabilizable (with decay rate 1/κ 2 ), where the switching strategy can be defined by σ (x ) = arg mini∈{1,2} {xT vi }. Remark 4.3. Note that, in Corollary 4.2, we do not introduce the conditions (22) and (23). This implies that such two conditions are always satisfied, or the sliding motions are allowed to occur. In fact, on the switching surface, if there is a sliding motion, which can be characterized by the inequalities

xT AT1 (v1 − v2 ) ≥ 0

and xT AT2 (v1 − v2 ) ≤ 0

for x ∈ S := {x ∈ RT+ |xT v1 = xT v2 }. If a sliding motion occurs on S, then σ (x) is not uniquely defined, so we let σ (x ) = 1 without loss of generality. Let us show that V1 (x) decreases along the corresponding Filippov solution [27]. For every α ∈ (0, 1), we have

V˙ 1 (x ) = sup xT (α A1 + (1 − α )A2 )T v1 α ∈[0,1]

= sup

{α xT AT1 v1 + (1 − α )xT AT2 v1 } < 0.

α ∈[0,1]

Therefore, the energy on S is decreasing and the sliding motions do not destroy the stability. Example 2. Consider the SPLS (1) with two 2 × 2 Metzler matrices



A1 =



1.2 0.9 , 0.2 −a



A2 =



−2.9 0.8 0.5 1.1

Obviously, both subsystems are unstable when a > 0. Fixing ai = bi = d = 0.1, c1 = 2, c2 = 3.5, ei = ri = 1 and solving (22)–(27) in Corollary 4.1 by linear programming toolbox of MATLAB, we can find the lower bound of a is 2.240. To show the stability, let us choose a = 2.9, a feasible solution provides κ1 = 0.4627, κ2 = 6.2720, v = [2.0734 5.0414]T , and w = [−1.2461 4.1021]T . This gives 1 = {(x1 x2 )T |4.1021x2 ≥ 1.2461x1 } and 2 = {(x1 x2 )T |4.1021x2 ≤ 1.2461x1 }. The switching regions and system trajectory are shown in Fig. 2 (notice that we do not need to deal with the sliding motions since the feasible solution results from no sliding motion conditions

100

X. Ding, X. Liu / Applied Mathematics and Computation 307 (2017) 92–101

system mode

3.5 3 2.5

4.1021x2=1.2461x1 1000

2 1 1000

2

x (t)

2

switching signals 3 2 1 0 0

1.5

2000 Sample Time

3000

4000

1050

Ω1

1 0.5

Ω2

0 0

2

4

6

8

10

x (t) 1

Fig. 2. The trajectory, switching regions, and switching signals obtained by Corollary 4.1 with a = 2.9, where the initial condition [10 1]T and the step is 0.02. Switching rule: First, σ (x0 ) = 1 since [10 1]T ∈ 2 . When x ∈ 2 , σ (x ) = 2. When x ∈ 1 , σ (x ) = 1.

3

switching signals

2.5

2

x (t)

2

1.5

system mode

3 2 1 0 0

2000

3 2 1 0 2000

4000 6000 Sample Time

2020

8000

10000

4.1021x =1.2461x

2040

2

1

1

Ω1 0.5

Ω2 0 0

0.5

1

1.5

2

2.5

3

x1(t) Fig. 3. The trajectory, switching regions, and switching signals of resulted from Corollary 4.2 with a = 8.99, where the initial condition [10 1]T and the step is 0.02. Switching rule: First, σ (x0 ) = 1 since [2 3]T ∈ 1 . When x ∈ 1 , σ (x ) = 1. When x ∈ 2 , σ (x ) = 2.

(22) and (23) which can guarantee the states to pass through the switching surface successfully), which confirms that the SPLS is exponentially stabilizable under the switching strategy (9) with exponential decay rate 1/6.2720. Also choosing c1 = 2, c2 = 3.5, one can alternatively apply Corollary 4.2 to obtain the lower bound of a is 8.473. Specially, choosing a = 8.99 and solving the linear programming problem (28)–(30), results in a feasible solution κ1 = 7.1826, κ2 = 73.3195, v1 = [37.3680 23.3186]T , and v2 = [22.9379 73.3195]T . This gives 1 = {(x1 x2 )T |50.0 0 09x2 ≥ 14.4301x1 } and 2 = {(x1 x2 )T |50.0 0 09x2 ≤ 14.4301x1 }. The simulation is shown in Fig. 3, from which one can see that the SPLS is exponentially stabilizable with exponential decay rate 1/73.3195. By the above comparison, one can easily conclude that Corollary 4.1 is less conservative than Corollary 4.2. This exactly agrees with the fact the latter has stricter demand with parameters. 5. Conclusions In this paper, we have presented some constructive switching strategies for SPLSs with state-dependent switching. The key idea is based on the use of single or multiple LCLF methods. These delicately constructed switching strategies, different from the earlier proposed results in the literature, can exponentially stabilize several broader classes of SPLSs. Future work will extend these results to switched positive systems with time-delays.

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