Stability and stabilization of switched linear time-invariant systems with saddle points and switching delays

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Stability and stabilization of switched linear time-invariant systems with saddle points and switching delays Liying Zhu*

Q1

College of Mathematics, Physics, and Information Engineering, Zhejiang Normal University, Zhejiang, Jinhua 321004, China

a r t i c l e

i n f o

Article history: Received 4 May 2014 Revised 26 June 2015 Accepted 17 July 2015 Available online xxx Keywords: Switched systems Stability Stabilization Saddle points Switching delays Algorithms

1 Q2 2 3

4 5 6 7 8 9 10 11 12 13 14 15 16 17

a b s t r a c t This paper addresses global asymptotic stability (GAS) and stabilization issues of switched linear time-invariant (LTI) systems with saddle points and time-varying switching delays. Firstly, two criteria of GAS are proposed for such switched systems with all subsystems sharing a common unique saddle point under arbitrary periodic/quasi-periodic switching paths (PSP/QSP). Secondly, by using our stability results we design global asymptotic-stabilizing controls (GASC), periodic/quasi-periodic stabilizing switching paths (PSSP/QSSP), and an algorithm of GASC and PSSP/QSSP for the switched systems. Finally, we present a numerical example of a switching multi-agent system to illustrate the effectiveness and practicality of our new results. © 2015 Published by Elsevier Inc.

1. Introduction Switched systems, an important kind of hybrid systems [3,7,13], have attracted much attention in control theory [3,4,11,26] and engineering communities [1] in the past 20 years. This may be due to that switched systems can be used widely to formulate many physical systems in nature and engineering [5,8]. Till now, there are many results on system analysis and synthesis obtained for switched systems, see for example [1,6,9–12,14,16,17,21–26] and the references therein. Meanwhile, several useful methods or techniques, such as the common Lyapunov function (CLF) method [4,11], the multiple-Lyapunov functions (MLF) method [3], the multiple-storage functions (MSF) method [26], the average dwell-time approach [24,27,28], and the linear matrix inequality (LMI) technique [8,9,15,24] are well developed for analyzing or designing switched systems. Among the results in the literature are lots of system analysis and design results on switched linear time-invariant (LTI) systems [4,10,12,15,18,25,30]. By means of the LMI technique, the average dwell-time approach, and other techniques, researchers have also obtained many results on system stability analysis and synthesis for switched systems with time delays or state-dependent delays [5,7,9,15,17,19,24]. However, it is well worth pointing out the following. Many stability and stabilization results of switched systems in the literature require classical stability analysis and design techniques or ideas based on suitable Lyapunov functions. Almost all of the stability results on switched systems in the literature are unsuitable for switched systems with all subsystems unstable. The general techniques or methods of system stability analysis and design for switched systems with all subsystems stable, such as CLF, MLF, MSF, LMI, and other classical methods, seem to be useless for analyzing stability of switched systems with all subsystems unstable. Indeed, when all the subsystems are unstable, the desired Lyapunov-type functions of subsystems cannot

*

Tel.: +86 57982298188. E-mail address: [email protected], [email protected]

http://dx.doi.org/10.1016/j.ins.2015.07.042 0020-0255/© 2015 Published by Elsevier Inc.

Please cite this article as: L. Zhu, Stability and stabilization of switched linear time-invariant systems with saddle points and switching delays, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.07.042

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55

be obtained over their active state spaces. As the above techniques use Lyapunov functions or Lyapunov-type functions, they cannot be applied to switched systems with all subsystems unstable. The motivation for this paper comes from the following. Switched systems with saddle points are used to formulate some practical engineering systems, such as electric circuits [1], mobile robot systems, power systems, etc. For instance, if some emergency or system failure occurs suddenly, every subsystem of second-order switching power systems may have a saddle point (a kind of improper equilibria). Due to the existence of a saddle point, such switching power systems may be unstable, and then voltage collapses of the systems may occur. From the viewpoint of system security, it is vital to avoid voltage collapses of switching power systems with saddle points. Moreover, voltage collapses caused by the existence of saddle points are natural stability issues of switched power systems with saddle points. However, without having workable Lyapunov functions for every subsystem, the existing methods or techniques based on Lyapunov functions cannot be used to analyze stability of switched systems with saddle points. In addition, switched systems with switching delays is also more complex than switched systems without delays. It is indeed more difficult and challenging to investigate switched systems with saddle points and switching delays. Although some local asymptotic stability (LAS) results are given for two-dimensional LTI switched systems with saddle points in [18], to the author’s best knowledge, there are a few stability/stabilization results for switched LTI systems with saddle points and switching delays. This paper investigates global asymptotic stability (GAS) and stabilization of switched LTI systems with saddle points and time-varying switching delays. The main contributions of this paper are several novel results on GAS and stabilization for switched systems with all the subsystems whose state matrices are all block matrices. By means of the concept of GAS and the criterion of direction of subsystems obtained [18], we will first propose two sufficient conditions for GAS of switched LTI systems with saddle points under arbitrary periodic/quasi-periodic switching paths (PSP/QSP) with time-varying switching delays. Then, based on the new stability results obtained in this paper we will design global asymptotic-stabilizing controls (GASC), periodic/quasi-periodic stabilizing switching paths (PSSP/QSSP), and a corresponding algorithm for such switched control systems. Finally, the obtained stability and stabilization results will be applied to a switching multi-agent system (SMAS). The numerical simulations of the SMAS will then show the effectiveness and practicality of the new results. Compared with existing stability and stabilization results on switched systems in the literature, the GAS and stabilization results presented in this paper have significant advantages and novelties as follows. The hypotheses in these results are relationships among the elements of subsystems state matrices, the switching dwell times, and the switching delays. Hence, the criteria only depend on the elements of the state matrices of subsystems and switching paths. Moreover, all the results presented in this paper are obtained without resort to any type of Lyapunov functions or Lyapunov-like functions. Most important, the conditions of the main results given in this paper are explicit, easily-tested, and practical. The rest of the paper is organized as follows. Section 2 presents preliminaries including the system model, definitions, notation, and assumptions. In Section 3, several GAS and GASC-PSSP/QSSP results are proposed for switched LTI systems with saddle points and time-varying switching delays. An illustrative example of a switching multi-agent system is carried out in Section 4, which is followed by the conclusion in Section 5. Notation: With N denoting the set of natural numbers and N+ denoting the set of positive integers, [K] denotes the set {1, 2, . . . , K }, where K ∈ N+ . Also, Rn denotes the n-dimensional Euclidean space,  ·  denotes the Euclidean norm in Rn , and Rm×n denotes the m × n matrix space of real matrices, with the superscript ‘T’ denoting matrix transposition. Finally, < ·, · > denotes the inner product, and V1 ⊕V2 denotes the direct sum of two vector spaces V1 and V2 .

56

2. Preliminaries

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

57

Consider a switched LTI control system as follows

x˙ = Aσ (t ) x + Bσ (t ) uσ (t ) , x(t0 ) = x(0), 58 59 60 61 62 63 64

[(x1 )T , . . . , (xn )T ]T

65

67 68 69

(1)

∈ is the state of the system, where = ∈ for h ∈ [n]. The map σ : [t0 , +∞) → [N] is Here x = a piecewise right-continuous step function called a switching path or switching rule, where N is an integer larger than 1. The value σ (t ) = i means that subsystem i is active. The positive time delay τ i caused by the switching path σ is called the switching delay of subsystem i, which may be time-varying. The state matrix Ai = diag{Ai1 , . . . , Ain } and the input matrix Bi = diag{Bi1 , . . ., Bin } of subsystem i are 2n × 2n matrix and 2n × n matrix, respectively, where Aih = (aiwvh )2×2 ∈ R2×2 , Bih = [bih1 bih2 ]T ∈ R2×1 , for w, v ∈ {1, 2}, i ∈ [N], and h ∈ [n]. The 1 × 2n vector ui is the input of subsystem i. Particularly, when ui ≡ 0 system (1) becomes a switched LTI system as follows:

x˙ = Aσ (t ) x, x(t0 ) = x(0), 66

for t  t0 .

R2n

xh

[xh1 ,

xh2 ]T

for t  t0 .

R2

(2)

Corresponding to the parts Aih of all the matrices Ai of its subsystems, for system (2) there are nN second-order systems as follows

x˙ h = Aih xh , xh (t0 ) = xh (0), where xh

[xh1 ,

xh2 ]T

for t  t0 , i ∈ [N],

and h ∈ [n],

(3)

R2 . Note that systems (3) are in fact n second-order LTI switched systems scheduled by the same switching

= ∈ path σ of systems (1) and (2). When the parts Aih of the matrices Ai are all block-triangular for h ∈ [n], we let

[N]uh = {i ∈ [N] : ai11h ai22h < 0, ai12h = 0, ai21h = 0}

(4)

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3

denote the index subset of the hth block-upper-triangular parts Aih . Similarly, we let

[N]lh = {i ∈ [N] : ai11h ai22h < 0, ai12h = 0, ai21h = 0} 71

be the index subset of the hth block-lower-triangular parts

(5) Aih .

Then, the index set [N] can be divided into

[N]uh

and

[N]lh .

[N] = [N]uh ∪ [N]lh , h ∈ [n]. 72 73 74 75 76 77 78 79

hi = xh and E hi = ai xh + (ai hi and l hi as follows: (i) For i ∈ [N]uh and h ∈ [n], let Ea1 − ai22h )xh1 , we define asymptotic planes la1 2 a2 a2 12h 2 11h hi hi hi hi la1 := {x ∈ R2n : Ea1 = 0} and la2 := {x ∈ R2n : Ea2 = 0 }.

(ii) For i ∈

82

83

(6)

Furthermore, from the eigenvalue equation det (sI − Aih ) = s2 − (ai11h + ai22h )s + (ai11h ai22h − ai12h ai21h ) = 0 of a block part Aih of the state matrix Ai , it can be shown for h ∈ [n] that there are both a positive eigenvalue and a negative eigenvalue if and only if ai11h ai22h − ai12h ai21h < 0. Therefore, the inequality ai11h ai22h < 0 in (4) and (5) implies that the origin x = 0 is a common unique saddle point of all the subsystems of systems (3). In this case, the origin x = 0 is also called, throughout this paper, the common unique saddle point of all subsystems of systems (1) and (2). Every subsystem i of system (2) has 2n “asymptotic planes”, which hi and l hi of subsystem are planes the state can not cross. Similar to the discussion of asymptotes in [17], the asymptotic planes la1 a2 i can then be defined below.

80

81

[N]lh

and h ∈ [n], let

hi Ea1

=

ai21h xh1

− (ai11h

− ai22h )xh2

and

hi Ea2

=

(7) xh1 ,

we define asymptotic planes

hi hi hi hi la2 := {x ∈ R2n : Ea2 = 0} and la1 := {x ∈ R2n : Ea1 = 0 }.

Then, the asymptotic planes

hi la1

and

hi la2

85 86 87 88 89 90 91 92 93

hi la1

and

hi la2

as follows: (8)

define the following domains:

hi hi hi hi +ahi := {x ∈ R2n : Ea1 Ea2 > 0} and − := {x ∈ R2n : Ea1 Ea2 < 0}, ahi

84

That is,

for i ∈ [N], h ∈ [n].

(9)

To facilitate the analysis, we let x(t) := x(t; t0 , x(0), σ (t)) denote the trajectory or state of system (1) or system (2) under a switching path σ . The path σ is expressed as σ (t ) = im+1 ∈ [N], for t ∈ [tm + τim , tm+1 + τim+1 ) and m ∈ N, where im is m

m+1

the index of subsystem im , and τim is the time-varying switching delay. We let x(m) = x(tm ) denote the switching states at the m switching times tm for m ∈ N+ . Let {im }+∞ , {tm }+∞ , and {x(m)}+∞ denote the switching index sequence, the switching time m=1 m=0 m=0 sequence, and the switching state sequence of the path σ , respectively. We assume throughout this paper that each switching path σ satisfies: (i) the states of systems (1)–(3) under the path σ do not jump at the switching instants tm , (ii) the path σ has a finite number of switchings on any finite time intervals, and (iii) the path σ is a periodical/quasi-periodical switching path, which is defined below.

Definition 1. (Periodical/quasi-periodical switching path (PSP/QSP)). A switching path of systems (1) and (2) is said to be a periodic switching path or a quasi-periodic switching path if the switching sequences {im }+∞ and {tm }+∞ satisfy m=1 m=0

iNk+l = il ∈ [N], tNk+l − tNk+l−1 = T > 0, il = il+1 , or 94

iNk+l = il ∈ [N], tNk+l − tNk+l−1 = T il > 0, T il = T il+1 , il = il+1 , T il

for

l ∈ [N] and k ∈ N,

95

where T and

are all positive constants.

96

3. Main results

97 98

This section propose a GAS result of system (1) in Section 3.1 and a global asymptotic stabilization result in Section 3.2 for system (2).

99

3.1. Stability analysis

101

We first recall the sets − and + (for i ∈ [N] and h ∈ [n]) as defined in (9) and a criterion of the direction of systems from ahi ahi [18] as follows.

102

Lemma 1 ([18]). For systems (3), the following statements hold for i ∈ [N] and h ∈ [n].

100

103 104 105 106 107 108

(i) If ai12h > 0 (resp. ai12h < 0), then the direction of system h of systems (3) is clockwise (resp. anti-clockwise) in − , and antiahi clockwise (resp. clockwise) in + . ahi

(ii) If ai21h < 0 (resp. ai21h > 0), then the direction of system h of systems (3) is clockwise (resp. anti-clockwise) in − , and antiahi clockwise (resp. clockwise) in + . ahi Lemma 2. If [N] = [N]lh ∪ [N]uh , then for i ∈ [N] and h ∈ [n] the state xh (t) of system h of systems (3) over any time interval [tu , tv ], t0 ≤ tu < tv , satisfies

xh (t )2  max{xh (tu )2 , xh (tv )2 }, for t ∈ [tu , tv ],

(10)

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109

where [N]uh and [N]lh are the nonempty sets as defined in (4) and (5), respectively.

110

Proof. Without loss of generality, we just need to show the case that ai11h > 0, ai21h > 0, and ai22h < 0, for i ∈ [N]lh and h ∈ [n]. The other cases are similar. Differentiating the function

111 112

V (xh ) = xh 2 = (xh1 )2 + (xh2 )2 113

along the state

xh (t)

V˙ (x (t )) = h

114 115 116

(11) [N]lh ,

of system h of systems (3) over [tu , tv ] yields that i ∈

2xh1

(t ) (t ) + x˙ h1

2xh2

(t ) (t ) = x˙ h2

2ai11h

( (t )) + xh1

2

2ai21h xh1

t ∈ [tu , tv ],

(t ) (t ) + 2ai22h (xh2 (t ))2 . xh2

V˙ (xh (t ))

It is obvious from (12) that there must exist the following three cases:(1) V˙ (xh (t ))  0 nor V˙ (xh (t )) < 0 always holds in [tu , tv ]. For Cases (1) and (2), one obtains from (12) that

V (xh (tv ))  V (xh (t ))  V (xh (tu ))orV (xh (tv ))  V (xh (t ))  V (xh (tu )), 117 118

120

 0, and (3) neither

for t ∈ [tu , tv ],

(13)

 (ai21h )2 − 4ai11h ai22h h x2 = 0 

(14)

which together with (11) implies that (10) holds. As for Case (3), it follows from (12), ai11h > 0, ai21h > 0, and ai22h < 0 that



 V˙ (xh (t )) 119

< 0, (2)

(12) V˙ (xh (t ))

(3)

= 0 ⇒ ai11h (xh1 )2 + ai21h xh1 xh2 + ai22h (xh2 )2 = 0

and then

 

ai11h xh1

ai21h +

+



(ai21h )2 − 4ai11h ai22h h x2 





2 ai11h

ai11h xh1

+



ai21h −

2 ai11h

which implies there must exist a instant t¯ ∈ (tu , tv ) such that V˙ (xh (t¯))|(3) = 0. In this case, one of the lines 





li1 ={xh ∈R2 :



− ahi

121

(ai11h )−1/2 [ai21h + (ai21h )2 −4ai11h ai22h ]xh2 =0} and li2 ={xh ∈R2 : ai11h xh1 +(ai11h )−1/2 [ai21h − (ai21h )2 −4ai11h ai22h ]xh2 =0} must be contained in

122

¯ and crossed by trajectory xh (t) at the time t. ¯ tv ], and hence It now follows from (12) and (14) that V˙ (xh (t )) is always negative or always positive over [tu , t¯) and (t,

123

124

¯ for t ∈ [tu , t],

(15)

V (xh (tv ))  V (xh (t ))  V (xh (t¯))orV (xh (tv ))  V (xh (t ))  V (xh (t¯)),

¯ tv ]. for t ∈ [t,

(16)

It follows from (11), (15), and (16) that (10) holds for Case (3). 

126

Let

 αh1 (i, j) := 1,

127

T

j a12h

· (e(a22h −a11h )T − 1) j

j j a22h − a11h



i i i 1 − e(a22h −a11h )T ,

α (i, j) : = 2 h

+ 128

j a12h j j a22h − a11h

 α (i, j) : = 3 h

j

ai11h − ai22h ai21h

−

,

(17)



j a12h j a22h

−

e(a22h −a11h )T +(a22h −a11h )T i

j a11h

i



129

α (i, j) := 4 h

i

ai21h ai11h − ai22h

i

i

j

i

j

T j j i i i j (e(a22h −a11h )T + e(a22h −a11h )T − 1) ,

j j a22h − a11h j a12h

j

−

ai11h − ai22h j

j

ai21h

e(a11h −a22h )T +(a11h −a22h )T + i

i

i

j

j

j

ai11h − ai22h

T j

j

j

j

(18)



ai21h

·(e(a11h −a22h )T + e(a11h −a22h )T − 1), 1 − e(a11h −a22h )T

j

,

(19)

T i i i (e(a11h −a22h )T − 1), 1

,

(20)

for i ∈ [N]lh , j ∈ [N]uh , and h ∈ [n], where Ti > 0 and Tj > 0 are constants. We then define j

j

j

j

q1h (i, j) := max{αh1 (i, j), |ai21h (ai11h − ai22h )−1 |αh2 (i, j)}ea11h T +a11h T , i

131

or + ahi

V (xh (tu ))  V (xh (t ))  V (xh (t¯))orV (xh (tu ))  V (xh (t ))  V (xh (t¯)), 125

130

xh + 11h 1

ai

j j j q2h (i, j) := max{|a12h (a22h − a11h )−1 |αh3 (i, j),

i

αh4 (i, j)}ea22h T +a22h T , i

i

(21) (22)

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5 i

132

for h ∈ [n], i ∈ [N]lh , and j ∈ [N]uh . Also, let T i2l−1 := tN1 k+2l−1 − tN1 k+2l−2 , T i2l := tN1 k+2l − tN1 k+2l−1 , TN2l−1 := tN1 k+2l−1 − k+2l−1

133

tN1 k+2l−2 + τN2l−1 − τN2l−2 , and TN2lk+2l := tN1 k+2l − tN1 k+2l−1 + τN2l k+2l − τN2l−1 , for k ∈ N and l ∈ [N1 ], where N1 is an k+2l−1 k+2l−2 k+2l−1

i

i

1

i

1

1

1

i

i

1

1

135

integer satisfying 2N1 ∈ [N]. Using Lemmas 1 and 2, and Eqs. (21) and (22), we obtain the following stability result.

136

Theorem 1. Consider system (2) with state matrices being block-triangular. Assume that

134

137 138 139 140 141

(i) [N] = [N]l1 ∪ [N]u1 ; (ii) either [N]lh = [N]l1 and [N]uh = [N]u1 , or [N]uh = [N]l1 and [N]lh = [N]u1 hold for h ∈ [n];

(iii) all the conditions of (1) ai21h a12h < 0, (2) ai21h a12h = (ai11h − ai22h )(a22h − a11h ), and (3) q1h (i, j) < 1 and q2h (i, j) < 1 hold for j

j

j

i ∈ [N]lh , j ∈ [N]uh , and h ∈ [n].

Then system (2) is GAS under arbitrary PSP/QSP σ expressed by either



σ1 (t ) =

i2l−1 ∈ [N]lh , t ∈ [tN1 k+2l−2 + τN2l−2 ,t + τN2l−1 ) k+2l−2 N1 k+2l−1 k+2l−1 i

i

1

1

i2l ∈ [N]uh , t ∈ [tN1 k+21−1 + τN2l−1 ,t + τN2l k+2l ) k+2l−1 N1 k+2l i

i

1



142

σ2 (t ) = 143

i

i

1

i2l ∈

∈ [tN1 k+21−1 + τ

[N]lh , t

1

i2l−1 , tN1 k+2l N1 k+2l−1

+τ

i

i2l N1 k+2l

i

1

i2l−1 N1 k+2l−1


and 0  τ

i2l−1

i2l N1 k+2l

1

i2l


(25)

Here [N]lh and [N]uh are the nonempty sets as defined in (4) and (5). Also, q1h (i, j) and q2h (i, j) are defined in (21) and (22). All the i

i

146

Proof. We will first show that

1

lim xh (t ) = 0,

150 151 152 153 154 155

1

for h ∈ [n].

t→+∞

149

(24)

)

time intervals of T i2l−1 , T i2l , TN2l−1 , and TN2lk+2l satisfy Condition (3) of (iii). k+2l−1

148

(23)

i2l−1 ∈ [N]uh , t ∈ [tN1 k+2l−2 + τN2l−2 , tN1 k+2l−1 + τN2l−1 ) k+2l−2 k+2l−1

145

147

or

1

and τN2l k+2l satisfying for k ∈ N and l ∈ [N1 ], with time-varying switching delays τN2l−1 k+2l−1

0τ 144

j

(26)

Let xh (0) be any initial state in R2 . Without loss of generality, we consider any switching path σ 1 as expressed in (23). For the path σ 2 as expressed in (24), the arguments are parallel. By Lemma 1 we know from (7), (8), and Condition (i) that the following observations hold. (1) For h ∈ [n], i ∈ [N]lh , and j ∈ [N]uh , hj

hj

hi and l hi and l the asymptotes lak (for k ∈ {1, 2}) of subsystems i and j of systems (3) are different from each other. (2) la1 cannot a2 ak − + − be contained simultaneously in any of + ,  ,  and  . (3) In the above domains all the directions of subsystems i and ahi ahi ah j ah j

j are the same. Therefore, every trajectory of subsystem i (or subsystem j) crosses through one of the asymptotes of subsystem j (or subsystem i). i

i

It is obvious that system (3) can be expressed as follows. When t ∈ [tN1 k+2l−2 + τN2l−2 ,t + τN2l−1 ) for k ∈ N and k+2l−2 N1 k+2l−1 k+2l−1 1

l ∈ [N1 ],

1



i2l−1 h i2l−1 h i2l−1 h x1 (t ), x˙h2 (t ) = a21h x2 (t ) + a22h x2 (t ), x˙h1 (t ) = a11h

(27)

xh (tN1 k+2l−2 + τN2l−2 ) = [xh1 (tN1 k+2l−2 + τN2l−2 ), xh2 (tN1 k+2l−2 + τN2l−2 )]T , k+2l−2 k+2l−2 k+2l−2 i

i

1

156

i

1

i

1

i

,t + τN2l k+2l ) for k ∈ N and l ∈ [N1 ], and when t ∈ [tN1 k+2l−1 + τN2l−1 k+2l−1 N1 k+2l 1



1

i2l i2l i2l x˙h1 (t ) = a11h xh1 (t ) + a12h xh2 (t ), x˙h2 (t ) = a22h xh2 (t ),

(28)

xh (tN1 k+2l−1 + τN2l−1 ) = [xh1 (tN1 k+2l−1 + τN2l−1 ), xh2 (tN1 k+2l−1 + τN2l−1 )]T , k+2l−1 k+2l−1 k+2l−1 i

i

1

157 158

i

i

1

1

i

and τN2l k+2l are time-varying switching delays of subsystems i2l−1 and i2l , respectively. where τN2l−1 k+2l−1 1

1

Solving ODEs (27), (28) and together with (25) and xh (m) = xh (tm ) yields

⎧ i2l−1 i2l−1 ⎪ ⎨xh1 (N1 k + 2l − 1) = xh1 (N1 k + 2l − 2)ea11h TN1 k+2l−1

2l−1 2l−1 ⎪ ⎩xh2 (N1 k + 2l − 1) = xh2 (N1 k + 2l − 2)ea22h TN1 k+2l−1 + i

i

i

2l−1 h a21h x1 (N1 k+2l−2) i2l−1 i2l−1 a11h −a22h

· (e

i

i

2l−1 2l−1 a11h TN k+2l−1 1

i

−e

i

2l−1 2l−1 a22h TN k+2l−1 1

)

(29)

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L. Zhu / Information Sciences xxx (2015) xxx–xxx

⎧ i2l i2l ⎨xh1 (N1 k + 2l ) = xh1 (N1 k + 2l − 1)ea11h TN1 k+2l + ⎩

xh2

160

(N1 k + 2l ) =

xh2

(N1 k + 2l − 1)e

i

2l xh (N k+2l−1) a12h 1 2 i2l i2l a22h −a11h

i2l

i2l

i2l

i2l

(ea22h TN1 k+2l − ea11h TN1 k+2l )

1

.

⎧ i1 i1 ⎨xh1 (N1 k + 1) = xh1 (N1 k)ea11h TN1 k+1 2

2

i

i

1 xh (0) a21h 1 i1 i1 a11h −a22h

i1

i1

i1

(31)

i1

(ea11h TN1 k+1 − ea22h TN1 k+1 ),

⎧ i i i i i i i i i a 2l−2 xh2 (N1 k+2l−3) a 2l−2 T 2l−2 +a 2l−1 T 2l−1 a 2l−1 T 2l−1 +a 2l−2 T 2l−2 ⎪ xh1 (N1 k + 2l − 1) = xh1(N1 k+2l−3) e 11h N1 k+2l−2 11h N1 k+2l−1 + 12h i2l−2 · (e 11h N1 k+2l−1 22h N1 k+2l−2 ⎪ i2l−2 ⎪ a −a ⎪ 22h 11h ⎪ ⎪ ⎪ i2l−2 i2l−2 i2l−1 i2l−1 ⎪ a11h TN k+2l−2 +a11h TN k+2l−1 ⎪ 1 1 ⎪ −e ) ⎪ ⎪ ⎪ i i i i i i i i i ⎪ ⎨ h a 2l−1 a 2l−2 T 2l−2 +a 2l−1 T 2l−1 a 2l−2 T 2l−2 +a 2l−1 T 2l−1 x2 (N1 k + 2l − 1) = xh1 (N1 k + 2l − 3) i2l−121h i2l−1 [e 11h N1 k+2l−2 11h N1 k+2l−1 − e 11h N1 k+2l−2 22h N1 k+2l−1 ] a11h −a22h ⎪  ⎪ i2l−1 i2l−2 i i i i i i i i ⎪ ⎪ a21h a12h a 2l−1 T 2l−1 a 2l−1 T 2l−1 a 2l−2 T 2l−2 a 2l−2 T 2l−2 ⎪ + xh2 (N1 k + 2l − 3) i2l−1 i2l−1 · (e 11h N1 k+2l−1 − e 22h N1 k+2l−1 )(e 22h N1 k+2l−2 − e 11h N1 k+2l−2 ) ⎪ i2l−2 i2l−2 ⎪ (a11h −a22h )(a22h −a11h ) ⎪ ⎪ ⎪  ⎪ ⎪ i2l−2 i2l−2 i2l−1 i2l−1 ⎪ ⎪ ⎩ + ea22h TN1 k+2l−2 +a22h TN1 k+2l−1 , for l ∈ {2, . . . , N1 }, ⎧ i2l−1 i2l−1 i2l−1 i2l−1 i i2l i2l i2l i2l a 2l xh (N k+2l−2) ⎪ xh1 (N1 k + 2l ) = 12h 2i2l 1 i2l (ea22h TN1 k+2l−1 +a22h TN1 k+2l − ea22h TN1 k+2l−1 +a11h TN1 k+2l ) + xh1 (N1 k + 2l − 2) ⎪ ⎪ a22h −a11h ⎪ ⎪ ⎪ ⎪ i2l−1 i i2l−1 i2l−1 i i i i i2l i2l i i i i 2l ⎪ a12h a T +a a 2l−1 T 2l−1 a 2l−1 T 2l−1 a 2l T 2l a 2l T 2l ⎪ ⎨ · [e 11h N1 k+2l−1 11h TN1 k+2l + i2l−1 ai21h · (e 11h N1 k+2l−1 − e 22h N1 k+2l−1 )(e 22h N1 k+2l − e 11h N1 k+2l )] i2l i2l 2l−1 (a11h −a22h )(a22h −a11h )

i i2l−1 i2l−1 i2l−1 i2l−1 i2l i2l i2l i2l ⎪ ⎪ a 2l−1 xh1 (N1 k+2l−2) ⎪ (ea11h TN1 k+2l−1 +a22h TN1 k+2l − ea22h TN1 k+2l−1 +a22h TN1 k+2l ) xh2 (N1 k + 2l ) = 21h i2l−1 ⎪ i2l−1 ⎪ a −a ⎪ 11h 22h ⎪ ⎪ ⎪ i2l−1 i2l−1 i i ⎩ a T +a 2l T 2l + xh2 (N1 k + 2l − 2)e 22h N1 k+2l−1 22h N1 k+2l , for l ∈ [N1 ].

163

⎧ i2l−2 i2l−2 i2l−1 i2l−1 ⎪ ⎨xh1 (N1 k + 2l − 1) =< xh (N1 k + 2l − 3), ea11h TN1 k+2l−2 +a11h TN1 k+2l−1 αh1 (i2l−2 , i2l−1 ) >,

xh2

165 167

i2l−1 a21h i2l−1 i2l−1 a11h −a22h

i

e

i

i

i

2l−2 2l−2 2l−1 2l−1 a11h TN k+2l−2 +a11h TN k+2l−1 1

1

αh2 (i2l−2 , i2l−1 ) >,

⎧ i2l−1 i2l−1 i2l i2l i2l 3 ⎪ ⎨xh1 (N1 k + 2l ) =< xh (N1 k + 2l − 2), a12h αi2lh (i2l−1i2l ,i2l ) ea22h TN1 k+2l−1 +a22h TN1 k+2l >, ⎪ ⎩

166

(32)

(33)

For k ∈ N and l ∈ [N1 ], it can be obtained from (17)–(20), (32), and (33) that

⎪ ⎩xh2 (N1 k + 2l − 1) =< xh (N1 + 2l − 3), 164

(30)

i

It follows from (29) and (30) that for k ∈ N,

i

162

i

2l T 2l a22h N k+2l

1 T 1 ⎩xh (N1 k + 1) = xh (N1 k)ea22h N1 k+1 +

161

[m3Gsc;July 29, 2015;7:35]

a22h −a11h

(N1 k + 2l ) =< x (N1 k + 2l − 2), e h

i2l−1 i2l−1 i2l i a22h TN k+2l−1 +a22h TN2lk+2l 1 1

(34)

(35)

α (i2l−1 , i2l ) >, 4 h

with αh1 , αh2 , αh3 , and αh4 as defined in (17)–(19), respectively. One obtains from (34) and (35) the following two conclusions. (1) When k = 0,

xh (2l − 1) < q1h (i2(l−1) , i2(l−1)+1 )xh (2(l − 1) − 1) < q1h (i2(l−1) , i2(l−1)+1 )q1h (i2(l−2) , i2(l−2)+1 )xh (2(l − 2) − 1) < · · · <

l−1 

q1h (i2(l− j) , i2(l− j)+1 )xh (1),

for

l ∈ {2, . . . , N1 },

(36)

j=1

168

xh (2l ) < q1h (i2(l−1)+1 , i2(l−1)+2 )xh (2(l − 1)) < q1h (i2(l−1)+1 , i2(l−1)+2 )q1h (i2(l−2)+1 , i2(l−2)+2 )xh (2(l − 2)) < · · · <

l 

q1h (i2(l− j)+1 , i2(l− j)+2 )xh (0),

for l ∈ [N1 ].

(37)

j=1

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169

(2) When k ∈ N+ ,



xh (N1 k + 2l − 1) <

N 1 −1

k q1h (i2(l− j) , i2(l− j)+1 )

l−1 

·

j=1

 x (N1 k + 2l ) < h

N1 

for l ∈ {2, . . . , N1 },

k q1h

(i2(l− j)+1 , i2(l− j)+2 )

l 

·

j=1

where

(k, l ) =

N 1 −1

q1h

(i2(l− j) , i2(l− j)+1 )

·

l−1 

j=1



172

Qh2

(k, l ) =

N1 

174

k q1h

(i2(l− j)+1 , i2(l− j)+2 )

176 177

·

q1h (i2(l− j)+1 , i2(l− j)+2 ),

l ∈ [N1 ], k ∈ N,

(41)

for

i ∈ [2], k ∈ N,

for

and l ∈ [N1 ].

(42) − 1)}+∞ k=0

{xh (N1 k + 2l )}+∞ k=0

xh (N1 k + 2l − 1) = 0, lim xh (N1 k + 2l ) = 0, for l ∈ [N1 ] and h ∈ [n], k→+∞

lim

xh (m) = 0, for h ∈ [n].

(44)

Let

xh (t ) = 0 ⇐⇒ m→+∞ lim xh (m) = 0, for h ∈ [n].

   i j i j αh := max ea11h T , ea22h T +  i, j∈l

ai21h ai11h − ai22h

(45)

   ai T j (e 11h − eai22h T j ) , h ∈ [n]. 

By Lemma 2 we also obtain from (38),(39),(42), and (31) that



x (t )  max e h

i

i

1 T 1 a11h 1

,e

i

i

1 T 1 a22h 1

+

  max e

ai11h T ji

i, j∈l



,e

ai22h T ji

+

1 ai21h 1 1 ai11h − ai22h

ai21h ai11h − ai22h

(e

(e

i

i

1 T 1 a11h 1

ai11h T ji

−e

−e

i

i

1 T 1 a22h 1

ai22h T ji

 ) xh (0)

 ) xh (0)

αh · xh (0), for t  t0 and h ∈ [n].

For any ε > 0, there must exist a positive constant δ :=

αh−1 ε

(46) > 0 such that

xh (0) < δ ⇒ xh (t ) < ε , for t  t0 and h ∈ [n], 183 184

(47)

which implies that (26) holds. On the other hand, let V = R2n and Vh = R2 for h ∈ [n]. Since each Ai is block-triangular,

V = V1 ⊕ V2 ⊕ · · · ⊕ Vn . 185

(48)

Based on the orthogonality of Vh and the orthogonal decomposition of the space V =

lim

t→+∞

186

(43)

By Lemma 2 we obtain t→+∞

182

l 

which implies

lim

181

(40)

and are both One knows from (36)–(39) that for l ∈ [N1 ] and h ∈ [n], the sequences monotonically strictly decreasing and bounded below by zero. By the bounded monotonic sequential theory, we then obtain from (36)–(39) and (42) that

k→+∞

180

l ∈ {2, . . . , N1 }, k ∈ N,

{xh (N1 k + 2l

lim

179

for

with q1h and q2h defined as in (21) and (22), respectively. It follows from Condition (iii), (21), (22), (40), and (41) that

k→+∞

178

q1h (i2(l− j) , i2(l− j)+1 ),

j=1

0 < Qhi (k, l ) < 1, 175

(39)

j=1

j=1

173

q1h (i2(l− j)+1 , i2(l− j)+2 ) · xh (0)

for l ∈ [N1 ],

k

 Qh1

(38)

j=1

= Qh2 (k, l ) · xh (0), 171

q1h (i2(l− j) , i2(l− j)+1 ) · xh (1)

j=1

= Qh1 (k, l ) · xh (1), 170

7

x(t ) = 0 ⇐⇒ lim x (t ) = 0 for h ∈ [n]. h

t→+∞

R2n ,

one obtains from (48) that (49)

It then follows from (26) and (49) that the statement of Theorem 1 holds.  Please cite this article as: L. Zhu, Stability and stabilization of switched linear time-invariant systems with saddle points and switching delays, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.07.042

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187 188 189 190 191 192 193 194

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Remark 1. If Condition (iii) in Theorem 1 is replaced by Qhi (k, l )  1, for i ∈ {1, 2}, k ∈ N, and l ∈ [N1 ], with Qh1 and Qh2 as defined in (40) and (41), respectively, then the GAS guaranteed in Theorem 1 can be replaced by the GS of system (2) under certain PSP/ QSP σ expressed in (23) or (24). Remark 2. Condition (2) of (iii) in Theorem 1 implies all the asymptotes of any pair of subsystems i and j (for i ∈ [N]lh and j ∈ [N]uh ) of systems (3) are different from each other. Remark 3. It can be seen from (17)–(22) that q1h and q2h depend only on the elements of the matrices Aih (for h ∈ [n] and i ∈ [N]) and/or the time intervals Ti of a path σ . They can then be obtained easily via (21) and (22). Therefore, all the conditions of Theorem 1 do not involve any subsidiary and extrinsic tools that come from outside the system.

197

Remark 4. For the general case that the matrices Ai (for i ∈ [N]) are not all block-triangular, if there is a nonsingular matrix P such that all the matrices PAi P −1 (for i ∈ [N]) are block-triangular and satisfy all the conditions of Theorem 1, then it follows from Theorem 1 that system (2) is GAS under any PSP/QSP σ as expressed in (23) or (24).

198

Corollary 1. Consider a stochastic switched LTI system described as the following stochastic differential equation (SDE) [22]

195 196

dx(t ) = Aσ (t ) x(t )dt + Dσ (t ) x(t )dw(t ), x(t0 ) = x(0), 199 200 201 202 203 204 205 206

for t  0.

(50)

Here x(t ) ∈ R2n is the state of the system, which is a random variable of t. The map σ : [t0 , +∞) → [N] is a switching path. The matrices i , d i }, Ai and Bi are the same state matrices in system (1) being block-matrix. The matrices Di = diag{Di1 , . . . , Din } and Dih = diag{dh1 h2 i i 2n where dh1 and dh2 are constants, for h ∈ [n] and i ∈ [N]. Also, w(t ) ∈ R is a stochastic variable of Brownian motion. If for h ∈ [n] and i and d i satisfy i ∈ [N] the constants dh1 h2





(1) min |aivvh − 0.5(dhi v )2 | : v ∈ {1, 2}, h ∈ [n], i ∈ [N] > |dhi v |, for v ∈ {1, 2}, and (2) Aih − 12 (Dih )2 are all block-triangular and satisfy all the conditions of Theorem 1, then system (50) is GAS with probability 1 under arbitrary PSP/QSP σ , i.e., for any nonzero x(0) ∈ R2n , the solution x(t) of system (50) satisfies

lim x(t ) = 0, withprobability1.

(51)

t→+∞

207

Proof. One obtains from SDE (50) that when σ (t ) = im the exact solution of the system is

x(t ) = x(m − 1) exp 208 209 210 211 212 213 214 215 216 217 218

1 2 D 2 im



 (t − tm−1 ) + Dim [W (t ) − W (tm−1 )] ,

T

Theorem 2. Suppose that for system (2), the state is denoted by x = [(x1 )T , (xd )T ] ∈ Rn , and the state matrices of subsystems are as follows: Ai = diag{Ai1 , Aid }, where xd = [x3 , . . . , xn ]T ∈ Rn−2 , Ai1 = (aijk1 )2×2 ∈ R2×2 , Aid = diag{λi1 , λi2 ,. . . , λin−2 } ∈ R(n−2)×(n−2) , and λiν < 0, for i ∈ [N], ν ∈ [n − 2] and j, k ∈ {1, 2}. If [N] = [N]l1 ∪ [N]u1 , and Condition (iii) of Theorem 1 hold for h = 1, then system (2) is GAS under any switching path σ as expressed in (23) or (24), where [N]l1 and [N]u1 are the nonempty sets as defined in (4) and (5).

Proof. Let V = Rn , V1 = R2 , and V2 = Rn−2 . We know from the fact that the state matrices Ai are all block-triangular and diagonal that V = V1 ⊕ V2 , which implies that

x(t ) = 0 ⇐⇒ lim x1 (t ) = 0 and t→+∞

lim

t→+∞

i

i

1

i

(53)

i

2l−1 2l−1 λh−2 TN k+2l−1 1

1

xdh (tN1 k+2l + τN2l k+2l ) = xdh (tN1 k+2l−1 + τN2l−1 )e k+2l−1 i

i

1

222

xd (t ) = 0.

Next, without loss of generality, we consider system (2) under any switching path σ as expressed in (23). By solving ODE (2), we obtain the following two equalities for k ∈ N, l ∈ [N1 ], and 3  h  n,

xdh (tN1 k+2l−1 + τN2l−1 ) = xdh (tN1 k+2l−2 + τN2l−2 )e k+2l−1 k+2l−2 221

(52)

It should be pointed out that Theorem 1 and Corollary 1 can be extended to other type of state matrices of subsystems, such as another GAS result on system (2) as proposed below.

lim

219

A im −

for t ∈ [tm−1 , tm ) and m ∈ N+ . Note that the solution x(t) is a stochastic variable in R2n . Similar to the proof of Theorem 1, we obtain from (52), Conditions (1) and (2) that (51) is satisfied for system (50). This completes the proof of Corollary 1. 

t→+∞

220



1

d(m)

which together with (54) and xh

i

,

(54)

i

2l T 2l λh−2 N k+2l 1

,

(55)

= xdh (tm ) yields that for k ∈ N, 3  h  n, and l ∈ [N1 ],

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xdh(N1 k+2l−1) = xdh(N1 k+2l−2) e = xdh(N1 k) e = xdh(0) e 223

l−1 j=0

k

τ =0

= xdh(N1 k) e = xdh(0) e

k

i

i2(l− j)−1 i2(l− j)−1 TN k+2(l− j)−1 h−2 1

λ

ij

j=1

ij

λh−2 TN

1τ+j

i2l i T 2l h−2 N1 k+2l

λ i

j=1

j=0

i

i

i

i

2l−1 2l−1 2l−2 2l−2 λh−2 TN k+2l−1 +λh−2 TN k+2l−2 1

1

= ···

= ··· i

i

2(l− j)−1 2(l− j)−1 λh−2 TN k+2(l− j)−1 1

,

(56)

i2l−1 i2l−1 i2l i T 2l + h−2 TN k+2l−1 h−2 N1 k+2l 1

λ

λ

= ···

i

1

N1

+

l−1

= xdh(N1 k+2l−2) e

2(l− j) 2(l− j) TN k+2(l− j) j=0 λh−2

τ =0

= xdh(N1 k+2l−3) e

1

N1

xdh(N1 k+2l ) = xdh(N1 k+2l−1) e l−1

i

2l−1 2l−1 λh−2 TN k+2l−1

9

ij

ij

λh−2 TN

1τ+j

+

= ···

l−1 j=0

i

i

2(l− j) 2(l− j) λh−2 TN k+2(l− j) 1

.

(57)

Let λ = min {|λih |}. It follows from the fact that λiv < 0 holds for i ∈ [N] and μ ∈ [n − 2], (56), and (57) that for k ∈ N, 3  h

224

3hN1

225

 n, and l ∈ [N1 ],

|xdh(N1 k+2l−1) |  |xdh(0) |e−λ(tN1 k+2l−1 +τN1 k+2l−1 −t0 ) , |xdh(N1 k+2l) |  |xdh(0) |e−λ(tN1 k+2l +τN1 k+2l −t0 ) , 226

(58)

which implies that

lim |xdh(N1 k+2l−1) | = 0

k→+∞

and

lim |xdh(N1 k+2l ) | = 0,

k→+∞

l ∈ [N1 ]

for

and 3  h  n.

(59)

228

In view of xd = [xd3 , . . . , xdn ]T , we know from (59) that limt→+∞ xd (t ) = 0. Similar to the proof of Theorem 1, we obtain from Conditions of Theorem 2 that limt→+∞ x1 (t ) = 0. Then, it follows from (53) that Theorem 2 holds. 

229

3.2. Stabilization

227

230

To investigate stabilization of system (1), we first recall a completely controllability criterion for system (1) in [29] as follows.

231

Lemma 3 ([29] Necessary and sufficient condition for complete controllability). For h ∈ [n], the following switched control system

x˙ h = Aih xh + Bih uih , xh (t0 ) = xh (0), 232

233

235 236 237 238 239 240 241 242

and i ∈ [N],

(60)

is completely controllable under arbitrary switching paths if and only if

Rank (Bih 234

for t  t0 ,

.. . Aih Bih ) = 2,

i ∈ [N] and

for

h ∈ [n].

(61)

By means of the property of the orthogonal decomposition of the state space R2n , we then generalize Lemma 3 to system (1) and obtain the following result. The proof is obvious and omitted here. Lemma 4 (Necessary and sufficient condition for complete controllability). If every state matrix of subsystems of system (1) is block-matrix, then system (1) is completely controllable under arbitrary switching paths, if and only if Condition (61) of Lemma 3 holds. Let a¯ i11h := ai11h − bih1 kih1 , a¯ i22h := ai22h − bih2 kih2 , a¯ i12h := ai12h − bih1 kih2 , a¯ i21h := ai21h − bih2 kih1 , for i ∈ [N] and h ∈ [n]. In the definitions of q1h (i, j) and q2h (i, j) (see (17)–(22)), replace ai11h , ai22h , ai12h , and ai21h by a¯ i11h , a¯ i22h , a¯ i12h , and a¯ i21h , respectively, we obtain the definitions of q¯ 1h (i, j) and q¯ 2h (i, j) for h ∈ [n]. Then, based on Theorem 1 and Lemma 4, we obtain another main result of this paper as follows. Theorem 3. Consider the switched control system (1). If Condition (61) of Lemma 3 holds, and there are constants kih1 , kih2 ∈ R and







243

j j j j Ti > 0 such that all the sets [N]lh = i ∈ [N] : a¯ i11h a¯ i22h < 0, a¯ i21h = 0, a¯ i12h = 0 and [N]uh = j ∈ [N] : a¯ 11h a¯ 22h < 0, a¯ 12h = 0, a¯ 21h =

244

0 are nonempty and satisfying [N] = [N]lh ∪ [N]uh , and all the following conditions of

245 246 247 248



(i) either [N]lh = [N]l1 and [N]uh = [N]u1 , or [N]uh = [N]l1 and [N]lh = [N]u1 ,

j j j j (ii) a¯ i21h a¯ 12h < 0, a¯ i21h a¯ 12h =(a¯ i11h − a¯ i22h )(a¯ 22h − a¯ 11h ), 1 2 (iii) q¯ h (i, j) < 1 and q¯ h (i, j) < 1,

hold for i ∈ [N]l1 , j ∈ [N]u1 , and h ∈ [n], then system (1) can be globally asymptotically stabilized by the following GASC

uσ (t ) = −Bσ (t ) Kσ (t ) x(t ),

for t  t0 ,

(62) i

i

249

and any PSSP/QSSP σ as expressed in (23) or (24) of Theorem 1 with time-varying switching delays τN2l−1 and τN2l k+2l . Here k+2l−1

250

Ki = diag{K1i , . . . , Kni } ∈ Rn×2n , where Khi = [kih1 kih2 ]T ∈ R1×2 , for i ∈ [N] and h ∈ [n]. The delays τN2l−1 and τN2l k+2l are any positive k+2l−1

251

constants satisfying Condition (iii), 0 < τN2l−1  T i2l−1 , and 0 < τN2l k+2l  T i2l , for i2l , i2l−1 ∈ [N], l ∈ [N1 ], and k ∈ N. k+2l−1

1

i

1

i

1

i

1

i

1

1

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Proof. It follows from (61) and Lemma 4 that system (1) is completely controllable under arbitrary switching paths. One then knows that system (1) is stabilizable under arbitrary switching paths. It is easy to see from Conditions (i)–(iii) that all the conditions of Theorem 1 are satisfied. Then, it follows from Theorem 1 that Theorem 3 holds.  Remark 5. Note that if kih1 = k1 and kih2 = k2 hold in Theorem 3 for h ∈ [n] and i ∈ [N], then there is a common stabilizing controller u = −Kx for every subsystem of system (1). Remark 6. If all the conditions of Theorem 3 are satisfied, then there must exist a domain of kih1 and kih2 of the matrix Khi of the state feedback control input ui = −Ki x of subsystem i so that all the conditions relation to kih1 and kih2 of Theorem 3 are satisfied. Based on Theorem 3, we design an algorithm of GASC-PSSP/QSSP for system (1) as follows. GASC-PSSP/QSSP algorithm: Step 1. Check Condition (61) in Theorem 3. If yes, go to Step 2. If no, go to Step 8. Step 2. Choose kih1 and kih2 from R. Calculate aiwvh − bihw kihv and α¯ h1 − α¯ h4 , for w, v ∈ {1, 2}, i ∈ [N], and h ∈ [n]. Step 3. Check Conditions (1)–(3) of Theorem 3. If yes, go to Step 4. If no, go back to Step 2. Step 4. Verify Conditions (4) and (5) of Theorem 3. If yes, go to Step 5. If no, go back to Step 2 . Step 5. Choose Ti > 0 and Tj > 0, and then calculate q¯ 1h (i, j) and q¯ 2h (i, j), for i ∈ [N]l1 , j ∈ [N]u1 , and h ∈ [n]. Step 6. Check q¯ 1h (i, j) < 1 and q¯ 2h (i, j) < 1 hold for i ∈ [N]l1 , j ∈ [N]u1 , and h ∈ [n]. If yes, go to Step 7. If no, go back to Step 5.

267

Step 7. The desired GASC of subsystems are expressed as ui = −Ki x, where Ki = diag{K1i , . . . , Kni }, and Khi = [kih1 , kih2 ]T , for i ∈

268

[N] and h ∈ [n]. The desired PSSP/QSSP σ can be expressed as in (23) or (24) of Theorem 1 with switching delays TN2l−1 k+2l−1

269

and TN2l−1 , where (TN2l−1 , T 2l−1 ) are contained in (τ i , τ j ) ∈ R2 :, q¯ 1h (i, j) < 1, q¯ 2h (i, j) < 1, 0 < τ i  T i , 0 < τ j  k+2l−1 k+2l−1 N k+2l−1

270 271

i

i

i

1

T j , for Step 8. End.



i

1



1

i ∈ [N]l1 , j ∈ [N]u1 ,

and

1

h ∈ [n] .

272

Remark 7. The feasibility of the global ASC-PSSP/QSSP algorithm follows from Remark 6.

273

Remark 8. Based on the above algorithm, we know that if both kih1 = k111 and kih2 = k112 hold for i ∈ [N] and h ∈ [n], there is a common stabilizing controller u = −Kx for all subsystem, where K = diag{K 1 , . . . , K 1 } and K 1 = [k111 k112 ].

274

277

Remark 9. Based on (61) of Lemma 3 and Conditions (i)–(iii) of Theorem 3 one can calculate straightforwardly and obtain the maximum time complexity of the GASC-PSSP/QSSP algorithm as follows: f (m) = 50m3 + 248m2 + 548m + 372, where m = max{N, n}, which implies that the algorithm is polynomial.

278

4. Application to switching multi-agent systems

279

This section applies the results obtained in Section 3 to aggregation issues of SMASs and then shows the effectiveness and practicality of our new results. Consider a SMAS modeled by a directed star [20] that is denoted by K1,4 = G (V, E ). Here V = {0, 1, 2, 3, 4} is a vertex set composed of five agents, which may be mobile robots, or cells, or soldiers, or missiles, etc. The set E = {(0, h) : h ∈ [4]} is an arc set, where each arc (0, h) denotes a directed channel from agent 0 to agent h so that agent h can receive information directly from agent 0. In this SMAS, agent 0 is a leader that has the information of the target and sends orders/ information to other agents. Agents 1 through 4 are subordinate to agent 0 and do not exchange any information among them. The state (or target point) of agent 0 is a static and fixed point denoted by xe = [xe1 , xe2 ]T ∈ R2 , which is undesired and difficult to reach in general, such as a saddle point. Each agent h is equipped with four modes and four controls. As there is a random interference signal, such as a Brownian motion, the dynamic of every mode of agents can be formulated as a two-dimensional LTI SDE with the saddle point xe . The order given by agent 0 is a piecewise and right-continuous step function σ with switching delays that determines which one mode and one control law of every agent should be chosen from its four modes and four controls during different time intervals. The main aim of the five agents is as follow: agents 1 through 4 will amass simultaneously to the target xe leaded by agent 0. Therefore, the dynamic of SMAS K1,4 can be formulated by a two-dimensional stochastic switched LTI system as follows:

275 276

280 281 282 283 284 285 286 287 288 289 290 291 292

d(xh (t ) − xe ) = Aσh (t ) (xh (t ) − xe ) + Bσh (t ) uσh (t ) + Dσh (xh (t ) − xe )dw(t ), 293 294 295

for t  0 and

h ∈ [4],

(63)

T

with the initial conditions xh (0) and w(0) = 0. Here the state xh = [xh1 , xh2 ] is contained in R2 . The Brownian motion w(t) in R2 denotes the same random interference signal of all agents. The switching path σ maps [0, +∞) to [4] and denotes an order sent from agent 0 to other agents. The matrices Aih of four modes of agent h are respectively as follows.



A11 =



296

A12 =

1 9 0.2 −3



−4 , −7



A21 =



1 , −2

−1 2

 A22 =

−1 2



1 , −8



A31 =



−2 , −5

−1 1

 A32 =

0.2 0.5





−3 , 1

A41 =



2.7 , −5

3.3 −1.5



A42 =

−2.2 −3



−2.8 , −4



0.25 , 0.2

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297

A13



298

A14 299 300 301 302 303 304 305

−3.5 = 1





2 , 2.6

A23



A24



1 , 4



−6 , −0.6

3.5 = −2



−2 = −2

A33



−9.5 = 4



−6 = 4

3 , −1.5



8 , 0.7

A34



A43



−1.5 , −10

1.2 = 2

11



−1 , −5

2 = −0.8



A44



2 = 5

6 . 4

The control matrices Bih relation to all the modes of agents 1 through 4 are as follows. B11 = [1, 2]T , B21 = [−1, 1]T , B31 = [−1, 0]T , B41 = [−1, 0.5]T , B12 = [0, 1]T , B22 = [1, 3]T , B32 = [−2, 1]T , B42 = [0.5, 0]T , B13 = [1, 1]T , B23 = [2, 1]T , B33 = [−3, 2]T , B43 = [0, 0.4]T , B14 = [1, −1]T , B24 = [2, 0]T , B34 = [0.5, 1]T , and B44 = [−2, −1]T . The matrices Dih = diag{0.01, 0.01} for i ∈ [N] and h ∈ [4]. The control inputs uih = −Khi x and Khi = [kih1 , kih2 ], for i, h ∈ [4], are contained in R1×2 , where kih1 and kih2 are undetermined constants. Moreover, the dynamic of SMAS K1,4 can also be formulated by the following eight-dimensional stochastic switched LTI system

d(x(t ) − x(e)) = Aσ (t ) (x(t ) − x(e))dt + Bσ (t ) uσ (t ) (t ) + Dσ (t ) (x(t ) − x(e))dW (t ), 306 307 308 309 310 311 312 313 314

T

The equilibrium point x(e) = [(xe )T , (xe )T , (xe )T , (xe )T ] ∈ R8 is a common unique saddle point of all subsystems. The switching path σ maps [0, +∞) to [4] , σ (t ) = i means that mode i of each agent h is active at time t. The matrices Ai = diag{Ai1 , Ai2 , Ai3 , Ai4 }, Bi = diag{Bi1 , Bi2 , Bi3 , Bi4 }, and Di = diag{Di1 , Di2 , Di3 , Di4 }. The controls ui = [ui1 , ui2 , ui3 , ui4 ]T ∈ R4 , where Aih , Bih , Dih , and uih are the same as mentioned before. The Brownian motion W (t ) = [wT (t ), wT (t ), wT (t ), wT (t )]T ∈ R8 denotes the random interference signal for all the agents. The time-varying switching delays, denoted by τki , of subsystem i are arbitrary positive constants satisfying 0  τki  τci for k ∈ N and i ∈ [4], where the maximum delays τci are positive constants contained in R. According to the GASC-PSSP/QSSP algorithm designed in Section 5, we obtain the following. (1) The GASCs of system (64) can be expressed as

315 316 317

(2)

 

319

0.2 A¯ 12 = 0



320

−2.5 A¯ 13 = 2



321

1.5 A¯ 14 = 0

323 324 325 326

328 329 330 331



(65)

in which K11 = [3, −4], = [−1, 2], K32 = [−2, 1],



−2 , −5

1 A¯ 21 = 0



1 , −4





−2 A¯ 22 = −1

0 , 0.6



−1 , −5.6

= [2, = [1,

−3], K13 = [−1, 3], K14 = K21 = [−3, 2], −1], K34 = [−2, 2.5], K41 = [2, −5], K42 =

K22

= [1, −2], [−3, 4], K43 =

h, i ∈ [4] of the closed-loop system (63) are as follows.



0 , 1

K12 K33



2 A¯ 23 = 0





−2 A¯ 31 = 1





0 , 1



0.7 , −4



−3 A¯ 33 = 2



0 , 0.7







0.3 A¯ 41 = 0



1.2 A¯ 32 = 0

−1 , −5

−3.5 A¯ 24 = 4



0 , 1



0 , 0.5

0.2 A¯ 34 = 0



−0.8 , −5

−3.2 A¯ 42 = −3



2 A¯ 43 = 0



−1 , −9





0 , 0.2



−1 , −6

−2 A¯ 44 = 3



0 . 1

(3) Letting [4]l1 = {1, 3} and [4]u1 = {2, 4}, the index set [4] = [4]l1 ∪ [4]u . Condition (iii) of Theorem 3 is satisfied for system (64). It is obvious that the target point xe ∈ R2 is the common unique saddle point of all the subsystems of system (63). The fixed point x(e) ∈ R8 is then the common unique saddle point of all subsystems of system (64). Two kinds of PSSP/QSSPs relation to the same time interval T > 0 are considered here as follows:

σ1 (t ) =

327

for all i ∈ [4],

where Ki = diag{K1i , K2i , K3i , K4i }, K23 = [0.5, −1], K24 = [2, 0.5], K31 [2, −1], and K44 = [−2, −3]. The matrices A¯ ih := Aih + Bih uih for

−2 A¯ 11 = 3

322

(64)

with the initial conditions x(0) and W (0) = 0. Here the state x = [(x1 )T , . . . , (x4 )T ]T ∈R8 , where xh = [xh1 , xh2 ]T ∈ R2 for h ∈ [4].

ui = −Ki x,

318

for t  0,

⎧ i , t ∈ [t4k , t4k+1 ) ⎪ ⎪1 ⎨

⎧ j , t ∈ [t4k , t4k+1 ) ⎪ ⎪ 1 ⎨

j1 , t ∈ [t4k+1 , t4k+2 ) i1 , t ∈ [t4k+1 , t4k+2 ) and σ2 (t ) = , i , t ∈ [t , t ) j2 , t ∈ [t4k+2 , t4k+3 ) ⎪ ⎪ 2 4k+2 4k+3 ⎪ ⎪ ⎩ ⎩ j2 , t ∈ [t4k+3 , t4k+4 ) i2 , t ∈ [t4k+3 , t4k+4 )

(66)

for k ∈ N, i1 = i2 ∈ [4]l1 , and j1 = j2 ∈ [4]u1 , which are accompanied with time-varying switching delays τki that are any constants i between 0 and the maximum delays τci , where t4k+i − t4k+i−1 = T + τ4k+i , for i ∈ [4] and k ∈ N. Based on the processes of the GASC-PSSP/QSSP algorithm, we know that all the conditions of Corollary 1 and Theorem 3 are satisfied for system (64). Then it follows from Theorem 3 and Corollary 1 that system (64) can be globally asymptotically stabilized by the control inputs ui = −Ki x in (65) and any of PSSP/QSSP σ in (66). Please cite this article as: L. Zhu, Stability and stabilization of switched linear time-invariant systems with saddle points and switching delays, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.07.042

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Fig. 4.1. Trajectories of system (63) under the paths σ 1 and σ 2 , respectively.

332 333 334 335 336 337

Numerical simulations have been carried out with the following choices. The static state of agent 0: xe = [5, 10]T . Four initial states of agents 1 through 4 of system (63): x1 (0) = [5, 40]T , x2 (0) = [25, −20]T , x3 (0) = [−20, 10]T , and x4 (0) = [30, 20]T . hi = {xh ∈ R2 : xh = 10} for h = 2, 4 and i = 1, 3, where l hi are respectively the asympNote that x3 (0) is contained in la1 2 a1 totic planes of subsystems 1 and 3 of system (64) relation to agents 1 and 3. So the initial state of system (64) is x(0) = [5, 40, 25, −20, −20, 10, 30, 20]T . The Brownian motion W(t) is a 8 × 1 standardized normal random variable denoted by randn(8, 1). Taking T = 0.2315 and τci = 0.072, two different switching paths are chosen as follows:

σ1 (t ) =

338 339 340 341 342 343 344 345 346 Q4 347

⎧ 2, t ∈ [t4m , t4m+1 ) ⎪ ⎪ ⎨

⎧ 4, t ∈ [t4m , t4m+1 ) ⎪ ⎪ ⎨

3, t ∈ [t4m+1 , t4m+2 ) 1, t ∈ [t4m+1 , t4m+2 ) and σ2 (t ) = , 4, t ∈ [t , t ) 2, t ∈ [t4m+2 , t4m+3 ) ⎪ ⎪ 4m+2 4m+3 ⎪ ⎪ ⎩ ⎩ 1, t ∈ [t4m+3 , t4m+4 ) 3, t ∈ [t4m+3 , t4m+4 )

(67)

which are accompanied by the time-varying switching delays τki satisfying 0  τki  0.072. Here t4m+i − t4m+i−1 = 0.2315s + (0.072 × rand)s, for m ∈ N and i ∈ [4]. Also, “rand” denotes a random number between 0 and 1. Figs. 4.1 and 4.2 are respectively the responses of the trajectories of systems (63) and (64) under the paths σ 1 and σ 2 starting respectively from the initial states xh (0) of agents 1 through 4 and x(0) of system (64). Fig 4.3 denotes the switching paths σ 1 and σ 2 . The solid curves and dotted curves in Figs 4.1 and 4.2 are the trajectories of system (63) under the paths σ 1 and σ 2 with or without switching delays. The solid curves and dotted curves in Fig 4.3 are the QSSPs/PSSPs σ 1 and σ 2 with or without

switching delays. From Fig. 4.1 it can be seen that all the trajectories of system (63) converge to the saddle point xe = [5, 10]T quickly under the two paths, i.e., agents 1 through 4 amass quickly to the static state xe of agent 0 simultaneously. Fig 4.3 shows that the switching paths σ 1 and σ 2 are QSPs or PSPs. From Figs. 4.1–4.3 it can be seen that Theorems 1 and 3, Corollary 1, and the GASC-PSSP/QSSP algorithm designed in Section 3 are effective and practical.

348

5. Conclusion

349

We have studied stability and stabilization of switched LTI systems with saddle points and time-varying switching delays. The main contributions of this paper are some explicit and practical sufficient conditions for globally asymptotical stability and stabilization results on such switched systems. The new system stability analysis and stabilization ideas introduced in this paper have shown the following. (i) The standard system stability analysis and control design methods or ideas based on Lyapunov functions, or Lyapunov-like functions, or generalized Lyapunov functions are not always suitable, necessary, and appropriative techniques of system analysis and synthesis for all kinds of switched systems, especially not for switched systems with all subsystems unstable. (ii) To stabilize a switched control system with unstable subsystems, it is not always necessary to first design the distributed controllers so that every unstable subsystem is stabilized. Therefore, the pole assignment of switched LTI systems may be very different from that of non-switching LTI systems. A future research topic is to investigate the pole assignment issues of switched LTI systems.

350 351 352 353 354 355 356 357 358

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Fig. 4.2. Trajectories of system (64) under the paths σ 1 and σ 2 , respectively.

Fig. 4.3. Signals of the switching paths σ 1 and σ 2 .

359 Q5 360

Uncited reference [2].

361

Acknowledgments

362

The author would like to thank the Editor, the Associate Editor, and the anonymous referees for their constructive comments and kind suggestions on the original manuscript.

363

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