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Dwell-time-based stabilization of switched linear singular systems with all unstable-mode subsystems
Accepted Manuscript
Dwell-time-based stabilization of switched singular linear systems with all unstable-mode subsystems Jinghan Li, Ruicheng Ma, Georgi M. Dimirovski, Jun Fu PII: DOI: Reference:
S0016-0032(17)30026-1 10.1016/j.jfranklin.2017.01.015 FI 2871
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
26 February 2016 14 November 2016 12 January 2017
Please cite this article as: Jinghan Li, Ruicheng Ma, Georgi M. Dimirovski, Jun Fu, Dwell-time-based stabilization of switched singular linear systems with all unstable-mode subsystems, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.01.015
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ACCEPTED MANUSCRIPT
Dwell-time-based stabilization of switched singular linear systems with all unstable-mode subsystems ? Jinghan Li a , Ruicheng Ma a , Georgi M. Dimirovski b,c Jun Fu d b c d
School of Mathematics, Liaoning University, Shenyang, 110036, China
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a
Faculty of Engineering, Dogus University of Istanbul, Istanbul, Republic of Turkey
St Cyril & St Methodius University, School FEEIT, Skopje, Republic of Macedonia
State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, 110189, China
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Abstract
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The global stabilization design of a class of switched singular linear systems via a novel dwell-time switching is investigated and solved in this work. The distinguishing feature of the proposed method is that stability of all subsystems of the switched systems is not necessarily required. A time-varying coordinate transformation is introduced first in order to convert the problem into an equivalent one of reduced-order switched conventional linear system with state jumps. Then, by constructing certain new multiple time-varying Lyapunov functions, computable sufficient conditions for the global stabilization task are proposed within the framework of dwell-time switching. Given the assumed instability of individual subsystems, the stabilization of the switched system is achieved under the condition of confining the dwell time by a certain pair of upper and lower bounds, which restrict the growth of Lyapunov function for the actively operating subsystem, thus decrease the energy of the Lyapunov function of the overall switched system at switching times. In addition, the multiple time-varying Lyapunov functions method is also used to analyze the stability analysis of a class of switched linear singular systems with stable subsystems. Two illustrative examples are presented to demonstrate the effectiveness of the proposed method.
1
Introduction
PT
ED
Key words: Switched linear singular systems; unstable subsystems; stabilization; dwell time.
AC
CE
In the recent past years, an increasing number of researchers have investigated switched systems due to their numerous applications in various fields and system engineering problems such as power electronics, flight control, network control systems, see, e.g., [14–16] and the references therein. Major efforts are devoted to problems of stability, stabilization, H∞ control, optimal control, and so on [1,11,4,9]. So far it has been well established that switching between unstable subsystems may yield stability of the overall system. As far as the stability of switched systems is concerned, most of the reported results appear confined to cases of switched systems where at least one stable subsystem within switched system exists. Thus, switching strategy is viewed as powerful control input even to stabilize a switched systems with all unstable subsystems [18] and is employed to achieve the observers design of switched linear systems with unknown inputs without posing any strong detectability requirements on subsystems of switched systems [10]. How to design such an appropriate switching became one of the most challenging problems in the study of switched systems. It is therefore that this paper is focused on the design of stabilizing switching signal for the class of switched systems with all unstable subsystems. ? This paper was not presented at any IFAC meeting. Corresponding author: Ruicheng Ma. Email addresses: [email protected] (Jinghan Li), [email protected] (Ruicheng Ma), [email protected] (Georgi M. Dimirovski), [email protected] (Jun Fu).
Preprint submitted to Journal of the Franklin Institute
1 February 2017
ACCEPTED MANUSCRIPT On the other hand, since singular linear system models not only describe the system dynamics, but also can capture algebraic constrains [3], switched singular systems have also drawn considerable attention in the recent past [2,6– 8,12,17,19–21]. As stated in [5], it is necessary to allow for solutions in switched singular systems with instantaneous state jumps, which are unavoidable even if all subsystems are regular and impulse-free. This is one of the major distinctions between switched singular systems and switched conventional systems [21]. Thus, in turn, considerable research attention has been devoted to the crucial property of stability under arbitrary switchings and under some constrained switching laws. With the help of consistency projectors, [5] proposed a common Lyapunov function for the stability analysis and design of singular systems under arbitrary switching policies.
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Work [13] developed state-dependent switching strategies for stabilization of such switched systems. On the dwelltime-based approach, the stability analysis is studied in [6,8,12,17,19–21]. These studies exploit the features of the dwell time technique: a dwell time of active subsystem can subside for potentially possible large state transients. In addition, it appeared to be a good alternative for studying electrical circuits with physical switches or cases with sudden component faults in electrical and mechanical systems. These gave the main motivation for choosing dwell time technique in the present exploration study. However, dwell-time-based methods in [6,8,12,17,19–21] require that there must be at least one stable subsystem of the switched system to be switched on for the stability analysis to be successful. Thus, naturally questions arise: Is it possible, in the framework of the dwell time technique, to achieve the stabilization design of switched singular linear systems without posing any stability requirements on subsystems of the switched systems? If possible, then under which conditions and how can we come up with a switching policy to achieve this goal? To the best of our knowledge, in the literature there are no results which provide answers to these questions. This precisely is the motivation of the present paper.
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To study the problems proposed above, a time-varying coordinate transformation is introduced first in order to convert the problem into an equivalent one of for reduced-order switched conventional linear system with state jumps. Then, by constructing certain new multiple time-varying Lyapunov functions, computable sufficient conditions for the global stabilization task are proposed within the framework of a dwell-time switching. Given the assumed instability of individual subsystems, the stabilization of the switched system is achieved under the condition of confining the dwell time by a certain pair of upper and lower bounds, which restrict the growth of Lyapunov function for the actively operating subsystem, thus decrease energy of the overall switched system at switching times. In addition, the multiple time-varying Lyapunov functions method is also used to analyze the stability analysis of a class of switched linear singular systems with stable subsystems. Two illustrative examples are presented to demonstrate the effectiveness of the proposed method.
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Further this paper is organized as follows. In Section 2, system description and necessary preliminaries are given. The main results are presented in Section 3. An illustrative example along with numerical and simulation results is provided for in Section 4. Finally, some conclusions are drawn in Section 5.
2
CE
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Notations: The notations used in this paper are fairly standard. Rn is the n dimensional Euclidean space, Rn×m denotes the set of n × m real matrices, R+ = (0, +∞), and N = {0, 1, 2, · · ·}. xT denotes the transpose of the vector x. The symbol “ ∗ ” represents arbitrary block matrix of appropriate dimensions. P > (≥, <, ≤) 0 is used to denote a positive-definite (positive-semidefinite, negative, negative-semidefinite) matrix. Denote the maximal (minimal) eigenvalues of a matrix P by λmax (P ) (λmin (P )). I and 0 represent identity matrix and zero matrix with proper dimension, respectively. System description and preliminaries
AC
Consider the following switched linear singular systems: (
Eσ(t) x(t) ˙ = Aσ(t) x(t),
(1)
x (0) = x0 ,
where x(t) ∈ Rn is the system state, x0 ∈ Rn is a vector-valued initial state, σ(t) : R+ → M = {1, 2, . . . , m} is the switching law, which is assumed to be a piecewise continuous (from the right) function of time and m > 0 is the number of modes of the switched system (i.e., subsystems). Throughout this paper, we assume that σ(t) = σ (tk ) = ik , ik ∈ M, t ∈ [tk , tk+1 ), where tk is the switching instant, this means that the ik th subsystem is activated when t ∈ [tk , tk+1 ). For every i ∈ M , Ei and Ai are constant matrices, and it is assumed that rank (Ei ) = r ≤ n. 2
ACCEPTED MANUSCRIPT For simplicity, we use (Ei , Ai ) to denote the ith subsystem. The set {tk } generated by σ(t) ∈ T [τ1 , τ2 ] denotes the switching sequences with τ1 ≤ tk − tk−1 ≤ τ2 , k ∈ N . Definition 1 [3]. For every i ∈ M , the singular system (Ei , Ai ) is said to be (i) regular if det (sEi − Ai ) is not identically zero; (ii) impulse-free if deg (det (sEi − Ai )) = rank (Ei ). Assumption 1 For every i ∈ M , the singular system (Ei , Ai ) is regular and impulse-free.
Hi Ei Ni =
"
Ir 0 0 0
#
¯ Hi Ai Ni = =: E,
"
A11 (i) A12 (i) A21 (i) A22 (i)
#
=: A¯i .
By introducing the state transformation: x ¯1 (t) x ¯2 (t)
#
= Ni−1 x(t), t ∈ [tk , tk+1 ) ,
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x ¯(t) =
"
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Due to the fact rank (Ei ) = r ≤ n, we can find nonsingular matrices Hi and Ni (i ∈ M ), such that (2)
(3)
switched system (1) takes the following form in the new coordinates: x ¯˙ 1 (t) = A11 (σ(t)) x ¯1 (t) + A12 (σ(t)) x ¯2 (t), 0 = A21 (σ(t)) x ¯1 (t) + A22 (σ(t)) x ¯2 (t).
(4) (5)
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Note that at the switching instant tk , the system switches from (Ej , Aj ) to (Ei , Ai ). Then, considering the switching law dependent feature of the state transformation (3), we have (6)
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+ x t− ¯ t− ¯ t+ k = Nj x k , x tk = Ni x k .
According to the analysis presented in [21], we have that
Γij =
I
PT
"
(7)
0
#
Ni−1 Nj .
CE
with
x ¯ t+ ¯ t− k = Γij x k , i, j ∈ M,
−A−1 22 (i)A21 (i) 0
(8)
AC
In addition, we can obtain that +
x ¯ 1 tk
h
i
i h = I 0 Γij x ¯ t− k = I 0 Γij
"
I −A−1 22 (j)A21 (j)
#
x ¯ 1 t− ¯1 t− k = Πij x k ,
(9)
with h
Πij = I 0
i
Ni−1 Nj
"
I −A−1 22 (j)A21 (j)
#
.
(10)
3
ACCEPTED MANUSCRIPT Under Assumption 1, we know that A22 is nonsingular. Thus, by (5), we can obtain a reduced-order switched conventional linear system with state jumps (9): (
x ¯˙ 1 (t) = Aˆ (σ(t)) x ¯1 (t), + x ¯1 tk = Πij x ¯1 t− k ,
(11)
where Aˆ (σ(t)) = A11 (σ(t)) − A12 (σ(t)) A−1 22 (σ(t)) A21 (σ(t)).
3
Main results
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The objective of this work is to derive and establish sufficient conditions for the existence of the switching signal to globally asymptotically stabilize the switched singular linear system (1) under the switching law design σ(t). If one of the subsystems is asymptotically stable, then this problem is no longer a challenge. Therefore, in here none of the individual subsystems is assumed to be asymptotically stable.
In this section, we consider two classes of switched singular systems. First we consider the stabilization problem for a switched singular system with all unstable subsystems. And thereafter we study the stability analysis problem for a switched singular system with all stable subsystems for the sake of establishing a parallel.
3.1
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We are now in a position to state the main results in this study. Case a: all subsystems are unstable
θilq − λPil ≤ 0, ∀i ∈ M, l, q = 1, 2, −µPj1 ΠTij Pi2
−µi Pi2
*
#
< 0, ∀i, j ∈ M, i 6= j,
ED
"
ln µ + λτ2 < 0,
CE
ˆ + 1 (Pi1 − Pi2 ) , ∀i ∈ M, l, q = 1, 2, θilq = ϑi Pil + AˆT (i)Pil + Pil A(i) τq h
Ni−1 Nj
"
I
−A−1 22 (j)A21 (j)
(13)
#
(15)
(16)
AC
Πij = I 0
i
(12)
(14)
PT
where
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Theorem 1 Consider switched singular system (1) satisfying Assumption 1. If there exist constants τ2 ≥ τ1 > 0, λ > 0, 1 > µ > 0, µi > 1, i ∈ M , and positive definite matrices Pi1 > 0, Pi2 > 0, and any appropriate dimensional matrix Pi3 and Pi4 , such that
−1 i) ˆ with ϑi = ln(µ τ1 , A(i) = A11 (i)−A12 (i)A22 (i)A21 (i), switched singular system (1) is globally uniformly asymptotically stable under switching law σ(t) ∈ T [τ1 , τ2 ].
Proof.
For {tk } generated by σ(t) ∈ T [τ1 , τ2 ] and t ∈ [tk , tk+1 ), we define: ρ(t) =
t − tk 1 ρ(t)−1 , ρ˜(t) = 1 − ρ(t), ρ1 (t) = , φ(t) = µi . tk+1 − tk tk+1 − tk 4
(17)
ACCEPTED MANUSCRIPT When t ∈ [tk , tk+1 ), we consider the following Lyapunov function: ¯ P¯i (t)¯ Vi (t, x(t)) = φ(t)¯ xT (t)E x(t), where P¯i (t) =
"
Pi (t) 0 Pi3 Pi4
#
(18)
with Pi (t) = ρ(t)Pi1 + ρ˜(t)Pi2 , i ∈ M .
From (18), we can obtain that
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= φ(t)¯ xT1 (t)Pi (t)¯ x1 (t) = Vi (t, x ¯1 ) .
It is obvious that Vi (t, x ¯1 ) satisfies α1 (k¯ x1 k) =
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¯ P¯i (t)¯ Vi (t, x(t)) = φ(t)¯ xT (t)E x(t) " #" #" # h i I 0 P (t) 0 x ¯ (t) r i 1 = φ(t) x ¯T1 (t) x ¯T2 (t) 0 0 Pi3 Pi4 x ¯2 (t) " #" # h i P (t) 0 x ¯1 (t) i = φ(t) x ¯T1 (t) 0 Pi3 Pi4 x ¯2 (t) " # h i x ¯1 (t) = φ(t) x ¯T1 (t)Pi (t) 0 x ¯2 (t)
λ2 2 2 k¯ x1 k ≤ Vi (t, x ¯1 ) ≤ λ1 k¯ x1 k = α2 (k¯ x1 k) , ν
(19)
(20)
M
where ν = max {µi , i ∈ M }, λ1 = max {λmax (Pil ), i ∈ M, l = 1, 2}, and λ2 = min {λmin (Pil ) , i ∈ M, l = 1, 2}. When t ∈ [tk , tk+1 ), the time derivative of Vi (t, x ¯1 ) is
PT
ED
V˙ i (t, x1 (t)) = ρ(t) ˙ ln (µi ) φ(t)¯ xT1 (t)Pi (t)¯ x1 (t) + φ(t)¯ xT1 (t) ρ(t)P ˙ ˜˙ (t)Pi2 x ¯1 (t) + 2φ(t)¯ xT1 (t)Pi (t)x ¯˙ 1 (t) i1 + ρ ln (µi ) ˆ x1 (t) ≤ φ(t)¯ xT1 (t) Pi (t) + ρ1 (t) (Pi1 − Pi2 ) x ¯1 (t) + 2φ(t)¯ xT1 (t)Pi (t)A(i)¯ τ1 ln (µi ) ˆ + AˆT (i)Pi (t) + ρ1 (t) (Pi1 − Pi2 ) x = φ(t)¯ xT1 (t) Pi (t) + Pi (t)A(i) ¯1 (t). τ1
(21)
1 1 ρ2 (t) + ρ˜2 (t). τ1 τ2
(22)
AC
ρ1 (t) =
CE
We choose a function ρ2 (t) ∈ [0, 1] and ρ2 (t) = 1 − ρ˜2 (t), such that
Apparently, ρ2 (t) can be fairly easy obtained. For example, when τ2 > τ1 , we choose ρ2 (t) = such a function ρ1 (t) − τ12 / τ11 − τ12 . In addition, if τ1 = τ2 , then ρ1 (t) = τ1 = τ2 , we can easily to choose, for example, ρ2 (t) = ρ˜2 (t) = 21 , which satisfies (22).
From (21) and (22), one has that V˙ i (t, x ¯1 )
h i ˆ + AˆT (i)Pi (t) + ρ1 (t) (Pi1 − Pi2 ) x ≤ φ(t)¯ xT1 (t) ϑi Pi (t) + Pi (t)A(i) ¯1 (t) 5
ACCEPTED MANUSCRIPT h i ˆ + AˆT (i)(ρ(t)Pi1 + ρ˜(t)Pi2 ) + ρ1 (t)(Pi1 − Pi2 ) x = φ(t)¯ xT1 (t) ϑi (ρ(t)Pi1 + ρ˜(t)Pi2 ) + (ρ(t)Pi1 + ρ˜(t)Pi2 )A(i) ¯1 (t) n o n o ˆ ˆ = φ(t)¯ xT1 (t) ρ(t) ϑi Pi1 + AˆT (i)Pi1 + Pi1 A(i) x ¯1 (t) + φ(t)¯ xT1 (t) ρ˜(t) ϑi Pi2 + AˆT (i)Pi2 + Pi2 A(i) x ¯1 (t) 1 1 +φ(t)¯ xT1 (t) ( ρ2 (t) + ρ˜2 (t))(Pi1 − Pi2 ) x ¯1 (t). (23) τ1 τ2 With the help of (12) and (23), we can obtain
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V˙ i (t, x ¯1 ) T ≤ φ(t)¯ x1 (t) {ρ(t) [ρ2 (t)θi11 + ρ˜2 (t)θi12 ]} x ¯1 (t) + φ(t)¯ xT1 (t) {˜ ρ(t) [ρ2 (t)θi21 + ρ˜2 (t)θi22 ]} x ¯1 (t) T ≤ φ(t)¯ x1 (t) {ρ(t) [ρ2 (t)λPi1 + ρ˜2 (t)λPi1 ]} x ¯1 (t) + φ(t)¯ xT1 (t) {˜ ρ(t) [ρ2 (t)λPi2 + ρ˜2 (t)λPi2 ]} x ¯1 (t) T = λφ(t)¯ x1 (t) [ρ(t)Pi1 + ρ˜(t)Pi2 ] x ¯1 (t) = λVi (t, x ¯1 ) . On the other hand, according to (13), one can find that
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Vi t+ ¯1 k ,x T + = φ t+ ¯1 tk Pi t+ ¯ 1 t+ k x k x k ρ(t+ )−1 T = µi k x ¯ 1 t+ ρ t+ ˜ t+ ¯1 t+ k k Pi1 + ρ k Pi2 x k = µ−1 ¯T1 t+ ¯ 1 t+ i x k Pi2 x k T = µ−1 ¯T1 t− ¯ 1 t− i x k Πij Pi2 Πij x k
(24)
αi (t)Vi (t, x ¯1 ), where αi (t) =
ED
For simplicity, we define V (t, x ¯1 ) =
m P
i=1
and αj (t) = 0, when t ∈ [tk , tk+1 ).
(25) (
1, if σ(t) = i, 0, otherwise,
∀i ∈ M . Assuming αi (t) = 1
PT
From (24), we can derive Vi (t, x ¯1 ) ≤ eλ(t−tk ) Vi (tk , x ¯1 ), t ∈ [tk , tk+1 ).
CE
Then, by supposing system (11) switches from subsystem j to i at switching instant tk , since σ(t) is continuous from the right, we have Vi (tk , x ¯1 ) ≤ µeλ(tk −tk−1 ) Vj (tk−1 , x ¯1 ) .
(26)
AC
Since tk − tk−1 ≤ τ2 , k = 1, 2, · · ·, which together with (14), we can obtain that µeλ(tk −tk−1 ) < 1.
(27)
Thus, we can see Vi (tk , x ¯1 ) < Vj (tk−1 , x ¯1 ) . Then, for any ε > 0, we can choose k¯ x1 (t0 )k < δ (ε) = α2−1 e−λτ2 α1 (ε) . Thus, this yields V (t0 , x ¯1 ) ≤ α2 (k¯ x1 (t0 )k) < e−λτ2 α1 (ε). Since Vi (tk , x ¯1 ) is strictly decreasing, we have Vi (tk , x ¯1 ) ≤ −λτ2 e α1 (ε). Then, we have V (t, x ¯1 ) ≤ α1 (ε). Furthermore, from (20), we can conclude k¯ x1 (t)k < ε. Obviously, for ∀δ > 0, we have k¯ x1 (t)k < ε. Due to the fact that the sequence Vi (tk , x ¯1 ) , k = 0, 1, 2, . . . is strictly decreasing, we obtained that lim k¯ x1 (t)k = 0. Therefore, we can conclude switched system (11) is globally asymptotically stabilized t→∞
under switching law σ(t) ∈ T [τ1 , τ2 ].
6
ACCEPTED MANUSCRIPT It follows from (5) that x ¯2 (t) = −A−1 ¯1 (t), thus, x ¯2 (t) also global asymptotical stable. This indicates 22 (σ(t)) A21 (σ(t)) x that system (4), or equivalently, the system (1) is globally asymptotically stabilize under switching law σ(t) ∈ T [τ1 , τ2 ]. Thus, the proof is completed. Remark 1 Theorem 1 provides a sufficient condition to achieve the stabilization for a class of switched singular systems via dwell-time-based switchings. Our method does not require stability of each subsystem, which nontrivially generalizes the result of [21] obtained under the assumption that all or part of subsystems are stable. On the other hand, when Ei = I, ∀i ∈ M , the switched singular system deduces to the switched normal system in [18]. Therefore, our obtained result extends that of the switched normal system to the switched singular system case.
3.2
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Remark 2 In our results, the common λ and µ are used to confine accordingly the upper bound of the dwell time τ2 . Since these two parameters may be dependent on individual subsystems, values of these two parameters can be different for different subsystems. It is therefore that here proposed methods appears less conservative and thus yield a better outcome design result. Case b: all subsystems are stable
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It is well known that switching among stable systems may yield instability of the overall switched system. It is therefore that is necessary also to investigate this line of design derivation for the case where all subsystems of switched singular systems are stable. In particular, we have found that if the subsystems of switched singular linear system (1) are stable, then the condition on the upper bound of the dwell-time (i.e,τ2 in Theorem 1) can be removed. Thus, we have the following theorem. Theorem 2 Consider switched singular system (1) satisfying Assumption 1. If there exist constants τ1 > 0, λ > 0, 1 > µ > 0, µi > 1, i ∈ M , and positive definite matrices Pi1 > 0, Pi2 > 0, such that θilq + λPil ≤ 0, ∀i ∈ M, l, q = 1, 2, −µPj1 ΠTij Pi2
−µi Pi2
*
#
< 0, ∀i, j ∈ M, i 6= j,
M
"
ED
Pi1 − Pi2 > 0, where
h
i
"
I
−A−1 22 (j)A21 (j)
#
,
(29)
(30)
(31)
(32)
CE
Πij = I 0 Ni−1 Nj
PT
ˆ + 1 (Pi1 − Pi2 ) , ∀i ∈ M, l, q = 1, 2, θilq = ϑi Pil + AˆT (i)Pil + Pil A(i) τ1
(28)
AC
ˆ = A11 (i) − A12 (i)A−1 with ϑi = lnτ1µi , and A(i) 22 (i)A21 (i), then switched singular system (1) is globally uniformly asymptotically stable under switching law σ(t) ∈ T [τ1 , ∞). Proof.
The proof is very similar to the proof of Theorem 1, and it can be easily derived by the methodology as above. Therefore, the proof of Theorem 2 is omitted in here. 4
Illustrative examples
In this section, we present illustrative examples along with the respective numerical and simulation results to demonstrate the effectiveness of the proposed switching design method.
7
ACCEPTED MANUSCRIPT
150 x1 x2
100
x3 x4
50
x(t)
0
-50
-100
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-150
-200 0
0.5
1 Time(s)
1.5
2
Fig. 1. The trajectories of subsystem 1.
4.1
Example 1
1 0 0 0
1 0 0 0
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Consider switched singular linear of the fourth the following system(1) with two subsystems order: 10 0 0 −1 1 0 −1 1 0 0 0 1 −2 −1 2 0 1 0 0 1 −2 −1 1 0 0 1 0 −2 −2 −1 2 E1 = , A1 = , E2 = , A2 = . 0 0 1 0 1 −3 −0.2 −0.2 0 1 0 0 −1 1 −1 −2 00 0 0 2 1 −2 −1 0 0 0 0 1 1 −1 1
ED
M
0 1 0 0 0 0 1 0 By H1 = N1 = N2 = and H2 = , we can obtain (2) with 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1
−1 1
−1
−1 −4 1
0
1 −2 −1 2
CE
PT
−1 1 −1 −2 1 −2 −1 1 ¯ ¯ A1 = H1 A1 N1 = , , A2 = H 2 A2 N2 = −2 −2 −1 2 1 −3 −0.2 −0.2 1 1 −1 1 2 1 −2 −1
−3
0
2
1 0 0
AC
ˆ and (11) with Aˆ1 = 3 −1 −3 , A2 = 1 3 −3 , Π12 = Π21 = 0 1 0 . −4 −4 1 0 0 1 0.6 −3.2 0.2 From Fig. 1 and 2, we see that both subsystem 1 and 2 are unstable with x(t0 ) = (3, 4, −2, 14)T and x(t0 ) = (3, 4, −2, 9)T , respectively. Therefore, the method in [21] cannot be applied to the studied switched system. However, since the here proposed method does not require stability of the subsystems of switched singular linear system (1), we can apply it to this example in order to stabilize asymptotically the switched system, as shown further below. Let λ = 1, µ = 0.9, µ1 = 2.1, µ2 = 2.2, τ1 = 0.08s, τ2 = 0.1s. Then by means of Theorem 1, we obtain the following
8
ACCEPTED MANUSCRIPT
1000 x1 x2 x3 x4
x(t)
500
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0
-500 0
0.5
1 Time(s)
1.5
2
Fig. 2. The trajectories of subsystem 2. 15
10
0
−5
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x(t)
5
x1 x2
−10
x3 x4
−15 0
2
4
6 Time(s)
8
10
12
matrices: 57.9402
55.2065 −21.1590
ED
M
Fig. 3. State responses of the switched singular system.
58.1763 −25.8976
56.6759
CE
PT
P11 = 55.2065 60.0027 −18.8876 , P12 = 58.1763 67.8901 −32.9528 , −25.8976 −32.9528 38.5350 −21.1590 −18.8876 28.4523 44.8649 46.5320 −9.4428 83.2191 76.1833 −47.7394 P21 = 46.5320 53.1657 −12.1667 , P22 = 76.1833 80.6093 −46.3264 . −9.4428 −12.1667 22.8090 −47.7394 −46.3264 52.6676
4.2
AC
Let x(t0 ) = (3, 4, −2, 14)T . Fig. 3 shows the state trajectories of the switched system under the dwell time switching signal σ(t) ∈ T [τ1 , τ2 ) shown in Fig. 4. Apparently Fig. 3 shows that the asymptotical stability of the switched system has been achieved rather quickly. Thus, the simulation results demonstrate pretty well the effectiveness of the proposed switching design method. Example 2
Consider the same switched singular system as in [21], where
E1 =
"
0 1 0 0
#
, A1 =
"
0 −1 1 a
#
, E2 =
"
1 1 0 0
#
, A2 =
"
−1 −1 1
9
0
#
.
The i−th subsystem
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2
1
0
2
4
6 Time(s)
8
10
12
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Fig. 4. The designed switching signal σ(t). 6
x1
5
x2
x(t)
4 3
1 0 0
2
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2
4
6 Time(s)
8
10
Fig. 5. State response of the switched singular system.
"
−1 0
0 1
#
1 0
, N2 =
"
.
0 1 1 −1
#
¯ = E1 N1 = E2 N2 = , then get E
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A¯2 = A2 N2 =
0 1
#
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As in [21], we set N1 =
"
"
1 0 0 0
, A¯1 = A1 N1 =
1 2
"
−1 0
a 1
#
,
ln (|1 − a|) = 0.5493, the
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When a = −2, it is reported in [21] that when the average dwell time is greater than system is asymptotical stability.
#
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Now, we apply our Theorem 2 to the studied system, and it has been found that, when a = −2, there exist τ1 = 0.05s, λ = 10, µ1 = µ1 = 1.002, and P11 = [11.0254], P12 = [7.0712], P21 = [3.6752], P22 = [2.3893], such that (28), (29) and (30) hold. Thus, the studied switched singular system is asymptotically stable under switching law σ(t) ∈ T [0.05, ∞). When a = −2, the state response of the switched singular system and the corresponding switching signal are illustrated in Fig. 5 and Fig. 6, respectively.
5
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Remark 3 It can be seen that the dwell time 0.05 obtained by Theorem 2 is much smaller than the one 0.5493 in [21]. Thus, our obtained stability results are less conservative than the ones given in [21].
1 2
ln (|1 − a|) =
Conclusions
This paper has investigated the global stabilization problem of a class of switched singular linear systems via dwell-time switching mechanism without requiring stability of subsystems of the switched systems. In spite of the instability of individual subsystems, the stabilization of the switched system is achieved under the condition of confining the dwell time within an upper and lower bounds representing a relevant pair of bounds, which restrict the growth of Lyapunov function of the active subsystem. At the same time such switching law decreases the energy of the Lyapunov function of the overall switched system at switching instants. In addition, the multiple time-varying
10
The i−th subsystem
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2
1
2
4
6 Time(s)
8
10
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0
Fig. 6. The corresponding switching signal.
Lyapunov functions method is also used to analyze the stability analysis of a class of switched linear singular systems with stable subsystems. A future research along the lines of the method proposed in this manuscript is envisaged in solving the finite-time on stabilization problem for the considered class of switched singular systems in order to improve the overall performance.
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Acknowledgement
This work was partially supported by the National Natural Science Foundation of China (61673198, 61304055, 61473063, 61590924), Provincial Natural Science Foundation of Liaoning Province (2015020088), the Fundamental Research Funds for the Central Universities ( N150802001), as well as supported by the Fund for Science of Dogus University and by the grant TUBITAK-RFBR-113E595 for the TurkishCRussian scientific project in Aerospace Sciences. Last but not the least, we would thank the three anonymous reviewers for their constructive comments which greatly improved the quality and presentation of this paper.
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References
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[2] Zairui Gao, Kun Yang, Yanxia Shen, and Zhicheng Ji. Input-to-state stability analysis for a class of nonlinear switched descriptor systems. International Journal of Systems Science, 46(16):2973–2981, 2015. [3] James Lam and Shengyuan Xu. Robust control and filtering of singular systems. Springer-Verlag Berlin/Heidelberg, 2006.
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[4] Shengtao Li, Xiaomei Liu, Yanyan Tan, Yanhui Ding, and Kanjian Zhang. Optimal switching time control of discrete-time switched autonomous systems. International Journal of Innovative Computing, Information and Control, 11(6):2043 – 2050, 2015. [5] Daniel Liberzon and Stephan Trenn. On stability of linear switched differential algebraic equations. In Proceedings of the 48th IEEE Conference on Decision and Control held jointly with the 28th Chinese Control Conference, pages 2156–2161, 2009.
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[6] Daniel Liberzon and Stephan Trenn. Switched nonlinear differential algebraic equations: Solution theory, lyapunov functions, and stability. Automatica, 48(5):954–963, 2012. [7] Jin-Xing Lin and Shu-Min Fei. Robust exponential admissibility of uncertain switched singular time-delay systems. Acta Automatica Sinica, 36(12):1773–1779, 2010.
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[8] Jinxing Lin, Shumin Fei, and Zhifeng Gao. Stabilization of discrete-time switched singular time-delay systems under asynchronous switching. Journal of the Franklin Institute, 349(5):1808–1827, 2012. [9] Xiaomei Liu, Kanjian Zhang, Shengtao Li, and Haikun Wei. Optimal control for switched delay systems. International Journal of Innovative Computing, Information and Control, 10(4):1555 – 1566, 2014. [10] Ruicheng Ma and Jun Fu. Dwell-time-based observers design for unknown inputs switched linear systems without requiring strong detectability of subsystems. IEEE Transactions on Automatic Control, Revised. [11] Ruicheng Ma and Jun Zhao. Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings. Automatica, 46(11):1819–1823, 2010. [12] Xiaowu Mu, Jumei Wei, and Rui Ma. Stability of linear switched differential algebraic equations with stable and unstable subsystems. International Journal of Systems Science, 44(10):1879–1884, 2013. [13] Jamila Raouf and Hannah Michalska. Exponential stabilization of singular systems by controlled switching. In Proceedings of the 49th IEEE Conference on Decision and Control, pages 414–419, 2010.
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[14] P. Shi, X. Su, and F. Li. Dissipativity-based filtering for fuzzy switched systems with stochastic perturbation. IEEE Transactions on Automatic Control, 61(6):1694–1699, 2016. [15] X. Su, P. Shi, L. Wu, and Y. D. Song. Fault detection filtering for nonlinear switched stochastic systems. IEEE Transactions on Automatic Control, 61(5):1310–1315, 2016. [16] X. Su, L. Wu, P. Shi, and C. L. P. Chen. Model approximation for fuzzy switched systems with stochastic perturbation. IEEE Transactions on Fuzzy Systems, 23(5):1458–1473, 2015. [17] Yijing Wang, Yanchao Zou, Yuhua Liu, Xiaomeng Shi, and Zhiqiang Zuo. Average dwell time approach to finite-time stabilization of switched singular linear systems. Journal of the Franklin Institute, 352(7):2920–2933, 2015. [18] Weiming Xiang and Jian Xiao. Stabilization of switched continuous-time systems with all modes unstable via dwell time switching. Automatica, 50(3):940–945, 2014. [19] Iman Zamani and Masoud Shafiee. On the stability issues of switched singular time-delay systems with slow switching based on average dwell-time. International Journal of Robust and Nonlinear Control, 24(4):595–624, 2014.
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[20] Iman Zamani, Masoud Shafiee, and Asier Ibeas. Stability analysis of hybrid switched nonlinear singular time-delay systems with stable and unstable subsystems. International Journal of Systems Science, 45(5):1128–1144, 2014.
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[21] Lei Zhou, Daniel WC Ho, and Guisheng Zhai. Stability analysis of switched linear singular systems. Automatica, 49(5):1481–1487, 2013.
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Dwell-time-based stabilization of switched singular linear systems with all unstable-mode subsystems Jinghan Li, Ruicheng Ma, Georgi M. Dimirovski, Jun Fu PII: DOI: Reference:
S0016-0032(17)30026-1 10.1016/j.jfranklin.2017.01.015 FI 2871
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
26 February 2016 14 November 2016 12 January 2017
Please cite this article as: Jinghan Li, Ruicheng Ma, Georgi M. Dimirovski, Jun Fu, Dwell-time-based stabilization of switched singular linear systems with all unstable-mode subsystems, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.01.015
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Dwell-time-based stabilization of switched singular linear systems with all unstable-mode subsystems ? Jinghan Li a , Ruicheng Ma a , Georgi M. Dimirovski b,c Jun Fu d b c d
School of Mathematics, Liaoning University, Shenyang, 110036, China
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a
Faculty of Engineering, Dogus University of Istanbul, Istanbul, Republic of Turkey
St Cyril & St Methodius University, School FEEIT, Skopje, Republic of Macedonia
State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, 110189, China
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Abstract
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The global stabilization design of a class of switched singular linear systems via a novel dwell-time switching is investigated and solved in this work. The distinguishing feature of the proposed method is that stability of all subsystems of the switched systems is not necessarily required. A time-varying coordinate transformation is introduced first in order to convert the problem into an equivalent one of reduced-order switched conventional linear system with state jumps. Then, by constructing certain new multiple time-varying Lyapunov functions, computable sufficient conditions for the global stabilization task are proposed within the framework of dwell-time switching. Given the assumed instability of individual subsystems, the stabilization of the switched system is achieved under the condition of confining the dwell time by a certain pair of upper and lower bounds, which restrict the growth of Lyapunov function for the actively operating subsystem, thus decrease the energy of the Lyapunov function of the overall switched system at switching times. In addition, the multiple time-varying Lyapunov functions method is also used to analyze the stability analysis of a class of switched linear singular systems with stable subsystems. Two illustrative examples are presented to demonstrate the effectiveness of the proposed method.
1
Introduction
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Key words: Switched linear singular systems; unstable subsystems; stabilization; dwell time.
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In the recent past years, an increasing number of researchers have investigated switched systems due to their numerous applications in various fields and system engineering problems such as power electronics, flight control, network control systems, see, e.g., [14–16] and the references therein. Major efforts are devoted to problems of stability, stabilization, H∞ control, optimal control, and so on [1,11,4,9]. So far it has been well established that switching between unstable subsystems may yield stability of the overall system. As far as the stability of switched systems is concerned, most of the reported results appear confined to cases of switched systems where at least one stable subsystem within switched system exists. Thus, switching strategy is viewed as powerful control input even to stabilize a switched systems with all unstable subsystems [18] and is employed to achieve the observers design of switched linear systems with unknown inputs without posing any strong detectability requirements on subsystems of switched systems [10]. How to design such an appropriate switching became one of the most challenging problems in the study of switched systems. It is therefore that this paper is focused on the design of stabilizing switching signal for the class of switched systems with all unstable subsystems. ? This paper was not presented at any IFAC meeting. Corresponding author: Ruicheng Ma. Email addresses: [email protected] (Jinghan Li), [email protected] (Ruicheng Ma), [email protected] (Georgi M. Dimirovski), [email protected] (Jun Fu).
Preprint submitted to Journal of the Franklin Institute
1 February 2017
ACCEPTED MANUSCRIPT On the other hand, since singular linear system models not only describe the system dynamics, but also can capture algebraic constrains [3], switched singular systems have also drawn considerable attention in the recent past [2,6– 8,12,17,19–21]. As stated in [5], it is necessary to allow for solutions in switched singular systems with instantaneous state jumps, which are unavoidable even if all subsystems are regular and impulse-free. This is one of the major distinctions between switched singular systems and switched conventional systems [21]. Thus, in turn, considerable research attention has been devoted to the crucial property of stability under arbitrary switchings and under some constrained switching laws. With the help of consistency projectors, [5] proposed a common Lyapunov function for the stability analysis and design of singular systems under arbitrary switching policies.
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Work [13] developed state-dependent switching strategies for stabilization of such switched systems. On the dwelltime-based approach, the stability analysis is studied in [6,8,12,17,19–21]. These studies exploit the features of the dwell time technique: a dwell time of active subsystem can subside for potentially possible large state transients. In addition, it appeared to be a good alternative for studying electrical circuits with physical switches or cases with sudden component faults in electrical and mechanical systems. These gave the main motivation for choosing dwell time technique in the present exploration study. However, dwell-time-based methods in [6,8,12,17,19–21] require that there must be at least one stable subsystem of the switched system to be switched on for the stability analysis to be successful. Thus, naturally questions arise: Is it possible, in the framework of the dwell time technique, to achieve the stabilization design of switched singular linear systems without posing any stability requirements on subsystems of the switched systems? If possible, then under which conditions and how can we come up with a switching policy to achieve this goal? To the best of our knowledge, in the literature there are no results which provide answers to these questions. This precisely is the motivation of the present paper.
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To study the problems proposed above, a time-varying coordinate transformation is introduced first in order to convert the problem into an equivalent one of for reduced-order switched conventional linear system with state jumps. Then, by constructing certain new multiple time-varying Lyapunov functions, computable sufficient conditions for the global stabilization task are proposed within the framework of a dwell-time switching. Given the assumed instability of individual subsystems, the stabilization of the switched system is achieved under the condition of confining the dwell time by a certain pair of upper and lower bounds, which restrict the growth of Lyapunov function for the actively operating subsystem, thus decrease energy of the overall switched system at switching times. In addition, the multiple time-varying Lyapunov functions method is also used to analyze the stability analysis of a class of switched linear singular systems with stable subsystems. Two illustrative examples are presented to demonstrate the effectiveness of the proposed method.
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Further this paper is organized as follows. In Section 2, system description and necessary preliminaries are given. The main results are presented in Section 3. An illustrative example along with numerical and simulation results is provided for in Section 4. Finally, some conclusions are drawn in Section 5.
2
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Notations: The notations used in this paper are fairly standard. Rn is the n dimensional Euclidean space, Rn×m denotes the set of n × m real matrices, R+ = (0, +∞), and N = {0, 1, 2, · · ·}. xT denotes the transpose of the vector x. The symbol “ ∗ ” represents arbitrary block matrix of appropriate dimensions. P > (≥, <, ≤) 0 is used to denote a positive-definite (positive-semidefinite, negative, negative-semidefinite) matrix. Denote the maximal (minimal) eigenvalues of a matrix P by λmax (P ) (λmin (P )). I and 0 represent identity matrix and zero matrix with proper dimension, respectively. System description and preliminaries
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Consider the following switched linear singular systems: (
Eσ(t) x(t) ˙ = Aσ(t) x(t),
(1)
x (0) = x0 ,
where x(t) ∈ Rn is the system state, x0 ∈ Rn is a vector-valued initial state, σ(t) : R+ → M = {1, 2, . . . , m} is the switching law, which is assumed to be a piecewise continuous (from the right) function of time and m > 0 is the number of modes of the switched system (i.e., subsystems). Throughout this paper, we assume that σ(t) = σ (tk ) = ik , ik ∈ M, t ∈ [tk , tk+1 ), where tk is the switching instant, this means that the ik th subsystem is activated when t ∈ [tk , tk+1 ). For every i ∈ M , Ei and Ai are constant matrices, and it is assumed that rank (Ei ) = r ≤ n. 2
ACCEPTED MANUSCRIPT For simplicity, we use (Ei , Ai ) to denote the ith subsystem. The set {tk } generated by σ(t) ∈ T [τ1 , τ2 ] denotes the switching sequences with τ1 ≤ tk − tk−1 ≤ τ2 , k ∈ N . Definition 1 [3]. For every i ∈ M , the singular system (Ei , Ai ) is said to be (i) regular if det (sEi − Ai ) is not identically zero; (ii) impulse-free if deg (det (sEi − Ai )) = rank (Ei ). Assumption 1 For every i ∈ M , the singular system (Ei , Ai ) is regular and impulse-free.
Hi Ei Ni =
"
Ir 0 0 0
#
¯ Hi Ai Ni = =: E,
"
A11 (i) A12 (i) A21 (i) A22 (i)
#
=: A¯i .
By introducing the state transformation: x ¯1 (t) x ¯2 (t)
#
= Ni−1 x(t), t ∈ [tk , tk+1 ) ,
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x ¯(t) =
"
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Due to the fact rank (Ei ) = r ≤ n, we can find nonsingular matrices Hi and Ni (i ∈ M ), such that (2)
(3)
switched system (1) takes the following form in the new coordinates: x ¯˙ 1 (t) = A11 (σ(t)) x ¯1 (t) + A12 (σ(t)) x ¯2 (t), 0 = A21 (σ(t)) x ¯1 (t) + A22 (σ(t)) x ¯2 (t).
(4) (5)
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Note that at the switching instant tk , the system switches from (Ej , Aj ) to (Ei , Ai ). Then, considering the switching law dependent feature of the state transformation (3), we have (6)
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+ x t− ¯ t− ¯ t+ k = Nj x k , x tk = Ni x k .
According to the analysis presented in [21], we have that
Γij =
I
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"
(7)
0
#
Ni−1 Nj .
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with
x ¯ t+ ¯ t− k = Γij x k , i, j ∈ M,
−A−1 22 (i)A21 (i) 0
(8)
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In addition, we can obtain that +
x ¯ 1 tk
h
i
i h = I 0 Γij x ¯ t− k = I 0 Γij
"
I −A−1 22 (j)A21 (j)
#
x ¯ 1 t− ¯1 t− k = Πij x k ,
(9)
with h
Πij = I 0
i
Ni−1 Nj
"
I −A−1 22 (j)A21 (j)
#
.
(10)
3
ACCEPTED MANUSCRIPT Under Assumption 1, we know that A22 is nonsingular. Thus, by (5), we can obtain a reduced-order switched conventional linear system with state jumps (9): (
x ¯˙ 1 (t) = Aˆ (σ(t)) x ¯1 (t), + x ¯1 tk = Πij x ¯1 t− k ,
(11)
where Aˆ (σ(t)) = A11 (σ(t)) − A12 (σ(t)) A−1 22 (σ(t)) A21 (σ(t)).
3
Main results
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The objective of this work is to derive and establish sufficient conditions for the existence of the switching signal to globally asymptotically stabilize the switched singular linear system (1) under the switching law design σ(t). If one of the subsystems is asymptotically stable, then this problem is no longer a challenge. Therefore, in here none of the individual subsystems is assumed to be asymptotically stable.
In this section, we consider two classes of switched singular systems. First we consider the stabilization problem for a switched singular system with all unstable subsystems. And thereafter we study the stability analysis problem for a switched singular system with all stable subsystems for the sake of establishing a parallel.
3.1
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We are now in a position to state the main results in this study. Case a: all subsystems are unstable
θilq − λPil ≤ 0, ∀i ∈ M, l, q = 1, 2, −µPj1 ΠTij Pi2
−µi Pi2
*
#
< 0, ∀i, j ∈ M, i 6= j,
ED
"
ln µ + λτ2 < 0,
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ˆ + 1 (Pi1 − Pi2 ) , ∀i ∈ M, l, q = 1, 2, θilq = ϑi Pil + AˆT (i)Pil + Pil A(i) τq h
Ni−1 Nj
"
I
−A−1 22 (j)A21 (j)
(13)
#
(15)
(16)
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Πij = I 0
i
(12)
(14)
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where
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Theorem 1 Consider switched singular system (1) satisfying Assumption 1. If there exist constants τ2 ≥ τ1 > 0, λ > 0, 1 > µ > 0, µi > 1, i ∈ M , and positive definite matrices Pi1 > 0, Pi2 > 0, and any appropriate dimensional matrix Pi3 and Pi4 , such that
−1 i) ˆ with ϑi = ln(µ τ1 , A(i) = A11 (i)−A12 (i)A22 (i)A21 (i), switched singular system (1) is globally uniformly asymptotically stable under switching law σ(t) ∈ T [τ1 , τ2 ].
Proof.
For {tk } generated by σ(t) ∈ T [τ1 , τ2 ] and t ∈ [tk , tk+1 ), we define: ρ(t) =
t − tk 1 ρ(t)−1 , ρ˜(t) = 1 − ρ(t), ρ1 (t) = , φ(t) = µi . tk+1 − tk tk+1 − tk 4
(17)
ACCEPTED MANUSCRIPT When t ∈ [tk , tk+1 ), we consider the following Lyapunov function: ¯ P¯i (t)¯ Vi (t, x(t)) = φ(t)¯ xT (t)E x(t), where P¯i (t) =
"
Pi (t) 0 Pi3 Pi4
#
(18)
with Pi (t) = ρ(t)Pi1 + ρ˜(t)Pi2 , i ∈ M .
From (18), we can obtain that
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= φ(t)¯ xT1 (t)Pi (t)¯ x1 (t) = Vi (t, x ¯1 ) .
It is obvious that Vi (t, x ¯1 ) satisfies α1 (k¯ x1 k) =
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¯ P¯i (t)¯ Vi (t, x(t)) = φ(t)¯ xT (t)E x(t) " #" #" # h i I 0 P (t) 0 x ¯ (t) r i 1 = φ(t) x ¯T1 (t) x ¯T2 (t) 0 0 Pi3 Pi4 x ¯2 (t) " #" # h i P (t) 0 x ¯1 (t) i = φ(t) x ¯T1 (t) 0 Pi3 Pi4 x ¯2 (t) " # h i x ¯1 (t) = φ(t) x ¯T1 (t)Pi (t) 0 x ¯2 (t)
λ2 2 2 k¯ x1 k ≤ Vi (t, x ¯1 ) ≤ λ1 k¯ x1 k = α2 (k¯ x1 k) , ν
(19)
(20)
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where ν = max {µi , i ∈ M }, λ1 = max {λmax (Pil ), i ∈ M, l = 1, 2}, and λ2 = min {λmin (Pil ) , i ∈ M, l = 1, 2}. When t ∈ [tk , tk+1 ), the time derivative of Vi (t, x ¯1 ) is
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V˙ i (t, x1 (t)) = ρ(t) ˙ ln (µi ) φ(t)¯ xT1 (t)Pi (t)¯ x1 (t) + φ(t)¯ xT1 (t) ρ(t)P ˙ ˜˙ (t)Pi2 x ¯1 (t) + 2φ(t)¯ xT1 (t)Pi (t)x ¯˙ 1 (t) i1 + ρ ln (µi ) ˆ x1 (t) ≤ φ(t)¯ xT1 (t) Pi (t) + ρ1 (t) (Pi1 − Pi2 ) x ¯1 (t) + 2φ(t)¯ xT1 (t)Pi (t)A(i)¯ τ1 ln (µi ) ˆ + AˆT (i)Pi (t) + ρ1 (t) (Pi1 − Pi2 ) x = φ(t)¯ xT1 (t) Pi (t) + Pi (t)A(i) ¯1 (t). τ1
(21)
1 1 ρ2 (t) + ρ˜2 (t). τ1 τ2
(22)
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ρ1 (t) =
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We choose a function ρ2 (t) ∈ [0, 1] and ρ2 (t) = 1 − ρ˜2 (t), such that
Apparently, ρ2 (t) can be fairly easy obtained. For example, when τ2 > τ1 , we choose ρ2 (t) = such a function ρ1 (t) − τ12 / τ11 − τ12 . In addition, if τ1 = τ2 , then ρ1 (t) = τ1 = τ2 , we can easily to choose, for example, ρ2 (t) = ρ˜2 (t) = 21 , which satisfies (22).
From (21) and (22), one has that V˙ i (t, x ¯1 )
h i ˆ + AˆT (i)Pi (t) + ρ1 (t) (Pi1 − Pi2 ) x ≤ φ(t)¯ xT1 (t) ϑi Pi (t) + Pi (t)A(i) ¯1 (t) 5
ACCEPTED MANUSCRIPT h i ˆ + AˆT (i)(ρ(t)Pi1 + ρ˜(t)Pi2 ) + ρ1 (t)(Pi1 − Pi2 ) x = φ(t)¯ xT1 (t) ϑi (ρ(t)Pi1 + ρ˜(t)Pi2 ) + (ρ(t)Pi1 + ρ˜(t)Pi2 )A(i) ¯1 (t) n o n o ˆ ˆ = φ(t)¯ xT1 (t) ρ(t) ϑi Pi1 + AˆT (i)Pi1 + Pi1 A(i) x ¯1 (t) + φ(t)¯ xT1 (t) ρ˜(t) ϑi Pi2 + AˆT (i)Pi2 + Pi2 A(i) x ¯1 (t) 1 1 +φ(t)¯ xT1 (t) ( ρ2 (t) + ρ˜2 (t))(Pi1 − Pi2 ) x ¯1 (t). (23) τ1 τ2 With the help of (12) and (23), we can obtain
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V˙ i (t, x ¯1 ) T ≤ φ(t)¯ x1 (t) {ρ(t) [ρ2 (t)θi11 + ρ˜2 (t)θi12 ]} x ¯1 (t) + φ(t)¯ xT1 (t) {˜ ρ(t) [ρ2 (t)θi21 + ρ˜2 (t)θi22 ]} x ¯1 (t) T ≤ φ(t)¯ x1 (t) {ρ(t) [ρ2 (t)λPi1 + ρ˜2 (t)λPi1 ]} x ¯1 (t) + φ(t)¯ xT1 (t) {˜ ρ(t) [ρ2 (t)λPi2 + ρ˜2 (t)λPi2 ]} x ¯1 (t) T = λφ(t)¯ x1 (t) [ρ(t)Pi1 + ρ˜(t)Pi2 ] x ¯1 (t) = λVi (t, x ¯1 ) . On the other hand, according to (13), one can find that
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Vi t+ ¯1 k ,x T + = φ t+ ¯1 tk Pi t+ ¯ 1 t+ k x k x k ρ(t+ )−1 T = µi k x ¯ 1 t+ ρ t+ ˜ t+ ¯1 t+ k k Pi1 + ρ k Pi2 x k = µ−1 ¯T1 t+ ¯ 1 t+ i x k Pi2 x k T = µ−1 ¯T1 t− ¯ 1 t− i x k Πij Pi2 Πij x k
(24)
αi (t)Vi (t, x ¯1 ), where αi (t) =
ED
For simplicity, we define V (t, x ¯1 ) =
m P
i=1
and αj (t) = 0, when t ∈ [tk , tk+1 ).
(25) (
1, if σ(t) = i, 0, otherwise,
∀i ∈ M . Assuming αi (t) = 1
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From (24), we can derive Vi (t, x ¯1 ) ≤ eλ(t−tk ) Vi (tk , x ¯1 ), t ∈ [tk , tk+1 ).
CE
Then, by supposing system (11) switches from subsystem j to i at switching instant tk , since σ(t) is continuous from the right, we have Vi (tk , x ¯1 ) ≤ µeλ(tk −tk−1 ) Vj (tk−1 , x ¯1 ) .
(26)
AC
Since tk − tk−1 ≤ τ2 , k = 1, 2, · · ·, which together with (14), we can obtain that µeλ(tk −tk−1 ) < 1.
(27)
Thus, we can see Vi (tk , x ¯1 ) < Vj (tk−1 , x ¯1 ) . Then, for any ε > 0, we can choose k¯ x1 (t0 )k < δ (ε) = α2−1 e−λτ2 α1 (ε) . Thus, this yields V (t0 , x ¯1 ) ≤ α2 (k¯ x1 (t0 )k) < e−λτ2 α1 (ε). Since Vi (tk , x ¯1 ) is strictly decreasing, we have Vi (tk , x ¯1 ) ≤ −λτ2 e α1 (ε). Then, we have V (t, x ¯1 ) ≤ α1 (ε). Furthermore, from (20), we can conclude k¯ x1 (t)k < ε. Obviously, for ∀δ > 0, we have k¯ x1 (t)k < ε. Due to the fact that the sequence Vi (tk , x ¯1 ) , k = 0, 1, 2, . . . is strictly decreasing, we obtained that lim k¯ x1 (t)k = 0. Therefore, we can conclude switched system (11) is globally asymptotically stabilized t→∞
under switching law σ(t) ∈ T [τ1 , τ2 ].
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ACCEPTED MANUSCRIPT It follows from (5) that x ¯2 (t) = −A−1 ¯1 (t), thus, x ¯2 (t) also global asymptotical stable. This indicates 22 (σ(t)) A21 (σ(t)) x that system (4), or equivalently, the system (1) is globally asymptotically stabilize under switching law σ(t) ∈ T [τ1 , τ2 ]. Thus, the proof is completed. Remark 1 Theorem 1 provides a sufficient condition to achieve the stabilization for a class of switched singular systems via dwell-time-based switchings. Our method does not require stability of each subsystem, which nontrivially generalizes the result of [21] obtained under the assumption that all or part of subsystems are stable. On the other hand, when Ei = I, ∀i ∈ M , the switched singular system deduces to the switched normal system in [18]. Therefore, our obtained result extends that of the switched normal system to the switched singular system case.
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Remark 2 In our results, the common λ and µ are used to confine accordingly the upper bound of the dwell time τ2 . Since these two parameters may be dependent on individual subsystems, values of these two parameters can be different for different subsystems. It is therefore that here proposed methods appears less conservative and thus yield a better outcome design result. Case b: all subsystems are stable
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It is well known that switching among stable systems may yield instability of the overall switched system. It is therefore that is necessary also to investigate this line of design derivation for the case where all subsystems of switched singular systems are stable. In particular, we have found that if the subsystems of switched singular linear system (1) are stable, then the condition on the upper bound of the dwell-time (i.e,τ2 in Theorem 1) can be removed. Thus, we have the following theorem. Theorem 2 Consider switched singular system (1) satisfying Assumption 1. If there exist constants τ1 > 0, λ > 0, 1 > µ > 0, µi > 1, i ∈ M , and positive definite matrices Pi1 > 0, Pi2 > 0, such that θilq + λPil ≤ 0, ∀i ∈ M, l, q = 1, 2, −µPj1 ΠTij Pi2
−µi Pi2
*
#
< 0, ∀i, j ∈ M, i 6= j,
M
"
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Pi1 − Pi2 > 0, where
h
i
"
I
−A−1 22 (j)A21 (j)
#
,
(29)
(30)
(31)
(32)
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Πij = I 0 Ni−1 Nj
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ˆ + 1 (Pi1 − Pi2 ) , ∀i ∈ M, l, q = 1, 2, θilq = ϑi Pil + AˆT (i)Pil + Pil A(i) τ1
(28)
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ˆ = A11 (i) − A12 (i)A−1 with ϑi = lnτ1µi , and A(i) 22 (i)A21 (i), then switched singular system (1) is globally uniformly asymptotically stable under switching law σ(t) ∈ T [τ1 , ∞). Proof.
The proof is very similar to the proof of Theorem 1, and it can be easily derived by the methodology as above. Therefore, the proof of Theorem 2 is omitted in here. 4
Illustrative examples
In this section, we present illustrative examples along with the respective numerical and simulation results to demonstrate the effectiveness of the proposed switching design method.
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150 x1 x2
100
x3 x4
50
x(t)
0
-50
-100
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-150
-200 0
0.5
1 Time(s)
1.5
2
Fig. 1. The trajectories of subsystem 1.
4.1
Example 1
1 0 0 0
1 0 0 0
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Consider switched singular linear of the fourth the following system(1) with two subsystems order: 10 0 0 −1 1 0 −1 1 0 0 0 1 −2 −1 2 0 1 0 0 1 −2 −1 1 0 0 1 0 −2 −2 −1 2 E1 = , A1 = , E2 = , A2 = . 0 0 1 0 1 −3 −0.2 −0.2 0 1 0 0 −1 1 −1 −2 00 0 0 2 1 −2 −1 0 0 0 0 1 1 −1 1
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0 1 0 0 0 0 1 0 By H1 = N1 = N2 = and H2 = , we can obtain (2) with 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1
−1 1
−1
−1 −4 1
0
1 −2 −1 2
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−1 1 −1 −2 1 −2 −1 1 ¯ ¯ A1 = H1 A1 N1 = , , A2 = H 2 A2 N2 = −2 −2 −1 2 1 −3 −0.2 −0.2 1 1 −1 1 2 1 −2 −1
−3
0
2
1 0 0
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ˆ and (11) with Aˆ1 = 3 −1 −3 , A2 = 1 3 −3 , Π12 = Π21 = 0 1 0 . −4 −4 1 0 0 1 0.6 −3.2 0.2 From Fig. 1 and 2, we see that both subsystem 1 and 2 are unstable with x(t0 ) = (3, 4, −2, 14)T and x(t0 ) = (3, 4, −2, 9)T , respectively. Therefore, the method in [21] cannot be applied to the studied switched system. However, since the here proposed method does not require stability of the subsystems of switched singular linear system (1), we can apply it to this example in order to stabilize asymptotically the switched system, as shown further below. Let λ = 1, µ = 0.9, µ1 = 2.1, µ2 = 2.2, τ1 = 0.08s, τ2 = 0.1s. Then by means of Theorem 1, we obtain the following
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1000 x1 x2 x3 x4
x(t)
500
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0
-500 0
0.5
1 Time(s)
1.5
2
Fig. 2. The trajectories of subsystem 2. 15
10
0
−5
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x(t)
5
x1 x2
−10
x3 x4
−15 0
2
4
6 Time(s)
8
10
12
matrices: 57.9402
55.2065 −21.1590
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Fig. 3. State responses of the switched singular system.
58.1763 −25.8976
56.6759
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P11 = 55.2065 60.0027 −18.8876 , P12 = 58.1763 67.8901 −32.9528 , −25.8976 −32.9528 38.5350 −21.1590 −18.8876 28.4523 44.8649 46.5320 −9.4428 83.2191 76.1833 −47.7394 P21 = 46.5320 53.1657 −12.1667 , P22 = 76.1833 80.6093 −46.3264 . −9.4428 −12.1667 22.8090 −47.7394 −46.3264 52.6676
4.2
AC
Let x(t0 ) = (3, 4, −2, 14)T . Fig. 3 shows the state trajectories of the switched system under the dwell time switching signal σ(t) ∈ T [τ1 , τ2 ) shown in Fig. 4. Apparently Fig. 3 shows that the asymptotical stability of the switched system has been achieved rather quickly. Thus, the simulation results demonstrate pretty well the effectiveness of the proposed switching design method. Example 2
Consider the same switched singular system as in [21], where
E1 =
"
0 1 0 0
#
, A1 =
"
0 −1 1 a
#
, E2 =
"
1 1 0 0
#
, A2 =
"
−1 −1 1
9
0
#
.
The i−th subsystem
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2
1
0
2
4
6 Time(s)
8
10
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Fig. 4. The designed switching signal σ(t). 6
x1
5
x2
x(t)
4 3
1 0 0
2
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2
4
6 Time(s)
8
10
Fig. 5. State response of the switched singular system.
"
−1 0
0 1
#
1 0
, N2 =
"
.
0 1 1 −1
#
¯ = E1 N1 = E2 N2 = , then get E
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A¯2 = A2 N2 =
0 1
#
M
As in [21], we set N1 =
"
"
1 0 0 0
, A¯1 = A1 N1 =
1 2
"
−1 0
a 1
#
,
ln (|1 − a|) = 0.5493, the
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When a = −2, it is reported in [21] that when the average dwell time is greater than system is asymptotical stability.
#
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Now, we apply our Theorem 2 to the studied system, and it has been found that, when a = −2, there exist τ1 = 0.05s, λ = 10, µ1 = µ1 = 1.002, and P11 = [11.0254], P12 = [7.0712], P21 = [3.6752], P22 = [2.3893], such that (28), (29) and (30) hold. Thus, the studied switched singular system is asymptotically stable under switching law σ(t) ∈ T [0.05, ∞). When a = −2, the state response of the switched singular system and the corresponding switching signal are illustrated in Fig. 5 and Fig. 6, respectively.
5
AC
Remark 3 It can be seen that the dwell time 0.05 obtained by Theorem 2 is much smaller than the one 0.5493 in [21]. Thus, our obtained stability results are less conservative than the ones given in [21].
1 2
ln (|1 − a|) =
Conclusions
This paper has investigated the global stabilization problem of a class of switched singular linear systems via dwell-time switching mechanism without requiring stability of subsystems of the switched systems. In spite of the instability of individual subsystems, the stabilization of the switched system is achieved under the condition of confining the dwell time within an upper and lower bounds representing a relevant pair of bounds, which restrict the growth of Lyapunov function of the active subsystem. At the same time such switching law decreases the energy of the Lyapunov function of the overall switched system at switching instants. In addition, the multiple time-varying
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The i−th subsystem
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2
1
2
4
6 Time(s)
8
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0
Fig. 6. The corresponding switching signal.
Lyapunov functions method is also used to analyze the stability analysis of a class of switched linear singular systems with stable subsystems. A future research along the lines of the method proposed in this manuscript is envisaged in solving the finite-time on stabilization problem for the considered class of switched singular systems in order to improve the overall performance.
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Acknowledgement
This work was partially supported by the National Natural Science Foundation of China (61673198, 61304055, 61473063, 61590924), Provincial Natural Science Foundation of Liaoning Province (2015020088), the Fundamental Research Funds for the Central Universities ( N150802001), as well as supported by the Fund for Science of Dogus University and by the grant TUBITAK-RFBR-113E595 for the TurkishCRussian scientific project in Aerospace Sciences. Last but not the least, we would thank the three anonymous reviewers for their constructive comments which greatly improved the quality and presentation of this paper.
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