Exponential stability of singularly perturbed switched systems with all modes being unstable

Automatica 113 (2020) 108800

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Exponential stability of singularly perturbed switched systems with all modes being unstable✩ ∗

Wu Yang a,b,c , Yan-Wu Wang a,b , , Changyun Wen d , Jamal Daafouz e a

School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, 430074, China Key Laboratory of Image Processing and Intelligent Control, Huazhong University of Science and Technology, Ministry of Education, China c School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China d School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore e Université de Lorraine, CNRS, CRAN, F54000 Nancy, France b

article

info

Article history: Received 9 January 2019 Received in revised form 14 October 2019 Accepted 8 December 2019 Available online xxxx Keywords: Singularly perturbed switched system Exponential stability Unstable mode

a b s t r a c t In this paper, we study the exponential stability problem for singularly perturbed switched systems(SPSSs), in which subsystems with two-time-scale property are all unstable, and both the destabilizing and stabilizing switching behaviors coexist. To estimate the state divergence during each two consecutive switching instants, the general property of a two-dimensional matrix involving singular perturbation parameter is explored. The switching sequence is properly reordered to provide an appropriate way to describe different switching behaviors. In addition, multiple composite Lyapunov functions(MCLFs) are employed to derive some stability criteria for the nonlinear SPSSs. Furthermore, by using switching-time-dependent MCLFs and dwell time method, some computable stability condition is given for the linear case. The obtained results show the relationship between the ratio of the stabilizing switching behavior and the singular perturbation parameter. Besides, the obtained results are free of ill-conditioning and stiffness problems. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Singularly perturbed systems(SPSs) are dynamic systems that exhibit both slow and fast dynamics (Kokotovic, Khalil, & O’reilly, 1999) and find applications in modeling power systems, mechanical systems, chemical reaction process, etc. (Lee & Othmer, 2010; Malloci, Daafouz, Iung, Bonidal and Szczepanski, 2009; Wang, Fu, Li, Wu, & Wang, 2017; Yang, Wang, Shen and Pan, 2017; Yang, Wang, Xiao and Liu, 2017). Note that dynamic mode switching is frequently encountered in both the slow and fast modes due to instantaneous disturbances or abrupt changes at certain time. Therefore, a more comprehensive system model is required, that is, the so-called singularly perturbed switched system (SPSS) ✩ This work is supported by the National Natural Science Foundation of China under Grants 61773172, 61903147, and 51537003, the Natural Science Foundation of Hubei Province of China (2017CFA035), the Fundamental Research Funds for the Central Universities, China (2018KFYYXJJ119) and the Postdoctoral Science Foundation of China under Grant 2019M652644. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Bert Tanner under the direction of Editor Christos G. Cassandras. ∗ Corresponding author at: School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, 430074, China. E-mail addresses: [email protected] (W. Yang), [email protected] (Y.-W. Wang), [email protected] (C. Wen), [email protected] (J. Daafouz). https://doi.org/10.1016/j.automatica.2019.108800 0005-1098/© 2019 Elsevier Ltd. All rights reserved.

(Alwan & Liu, 2009; Alwan, Liu, & Ingalls, 2008; Malloci, Daafouz and Iung, 2009; Wang & Yang, 2018). As a fundamental problem, the stability problem has been studied extensively during the past few decades. For SPSs, some significant contributions can be found in Chen, Yang, Lu, and Shen (2010), Chen, Yuan, and Zheng (2013), Corless and Glielmo (1992), Fridman (2002), Teel, Moreau, and Nesic (2003) and Wang, Teel, and Nešić (2012) and references therein. Compared with the classical SPSs, the analysis and synthesis of SPSSs are usually more complicated (Malloci, Daafouz and Iung, 2009). Thus, some dedicated techniques for SPSSs are appealing, for example, singular perturbation parameter-dependent common Lyapunov function (Ma, Wang, Zhou, & Yang, 2016), singular perturbation parameter-dependent multiple Lyapunov function (Lian & Wang, 2015; Ma & Cai, 2016; Wang, Zhou, Ma, & Yang, 2018), switched Lyapunov function method (Deaecto, Daafouz, & Geromel, 2012; Malloci, Daafouz and Iung, 2009; Malloci, Daafouz, & Iung, 2010), singular perturbation technique with the minimum dwell time method (Alwan & Liu, 2009; Alwan et al., 2008; Rejeb, Morărescu, Girard, & Daafouz, 2018), etc. Note that all modes of SPSSs in Deaecto et al. (2012), Ma et al. (2016), Malloci et al. (2010) and Rejeb et al. (2018) are stable, while both the stable and unstable modes coexist in Alwan and Liu (2009), Alwan et al. (2008), Lian and Wang (2015), Ma and Cai (2016) and Wang et al. (2018). However, when considering a more severe situation with

2

W. Yang, Y.-W. Wang, C. Wen et al. / Automatica 113 (2020) 108800

all the subsystems being unstable, the trade-off strategy between the stable and the unstable modes used in Alwan and Liu (2009), Alwan et al. (2008), Lian and Wang (2015), Ma and Cai (2016) and Wang et al. (2018) fails. Although some achievements for switched systems with fully unstable subsystems in a single time scale are made, see, e.g., Wang, Karimi, and Wu (2019) and Xiang and Xiao (2014), the methods used there are not directly applicable to our case due to numerical ill-conditioning and stiffness problems (Kokotovic et al., 1999). Moreover, in our case, the question related to how to accurately estimate the bound of the divergent state is challenging since the fast and slow subsystems have different divergence rates, and they are also affected by the singular perturbation parameter. Lastly, even reduced to the case of regularly switched systems, the switching behavior in Wang et al. (2019) and Xiang and Xiao (2014) is a special case of our model as explained in the sequel, and considering a more general switching behavior brings additional difficulties in our study. Motivated by these observations, we will study the exponential stability of SPSSs with all modes being unstable for the first time. By using the so-called multiple composite Lyapunov functions(MCLFs) and by exploring the stabilizing effect of switching behavior, the exponential stability conditions for nonlinear SPSSs are derived firstly. Furthermore, with the help of switching-timedependent MCLFs method, a linear matrix inequalities(LMIs) criterion is proposed for the exponential stability of the linear case. The contributions and the related approaches of this paper can be summarized as follows: (1) Each subsystem with two-time-scale feature is unstable. Moreover, both the destabilizing and stabilizing switching behaviors coexist, which further brings challenges here and broadens the application of the proposed SPSSs as well. (2) The state divergence bound during two successive switching instants is estimated by exploring the property of an ε -dependent two-dimensional matrix, while both the destabilizing and stabilizing switching behaviors are described properly via reordering the switching sequence. By employing MCLFs, some Lyapunov-like conditions are established for the exponential stability of the nonlinear SPSSs. Several well-conditioned stability criteria are further deduced for linear SPSSs by using the switching time-dependent MCLFs. Notations: Rn denotes the real n dimensional vectors. N and N+ are the sets of nonnegative integer and positive integer, respectively. For a set S , the cardinality of S is denoted as |S |. For a matrix P, the expression P < 0 means that P is real symmetric negative definite. λmax (A) and λmin (A) denote the maximum and minimum eigenvalues of A, respectively. He{A} stands for A + AT . The symbol ∗ within the matrix means the symmetric item in block matrices. The symbol {1 → 2 3→1  →  } means repeat the

tk , then, the switching sequence can be defined as {tk }k∈N with 0 ≤ t0 < t1 < · · · < tk . . . , limk→∞ tk = ∞, and 0 < τ¯1 ≤ τk = tk − tk−1 ≤ τ¯2 < ∞, k ∈ N+ , where τ¯1 and τ¯2 are positive constants. Assumption 1. The origin (x = 0, z = 0) is an isolated unstable equilibrium for each mode i ∈ M. Remark 1. All or parts of modes are stable in Alwan and Liu (2009), Alwan et al. (2008), Deaecto et al. (2012), Lian and Wang (2015), Ma and Cai (2016), Ma et al. (2016), Malloci et al. (2010), Rejeb et al. (2018) and Wang et al. (2018), while Assumption 1 shows that all modes are unstable. From this point, the approaches in these existing works cannot be applied to the case here. In view of this, it is reasonable to impose the aforementioned condition on τk . The upper bound τ¯2 ensures that the unstable modes will only diverge in finite time, and thus the divergence will not be too large to be compensated. The lower bound τ¯1 ensures the exclusive of Zeno behavior. In what follows, let N (τ¯1 , τ¯2 ) := {{tj }j∈N ; 0 < τ¯1 ≤ τj ≤ τ¯2 < ∞, j ∈ N+ } be the set of admissible switching sequence that can stabilize system (1). The two-time-scale property of system (1) gives the opportunity to investigate the stability of the full-order system through the stability of the reduced-order subsystems in separate time scales. Using the fast-slow decomposition technique, the slow and fast subsystems are presented as follows, respectively, slow subsystem : x˙ (t) = fi (x(t), hi (x(t))) ,

(2)

dy(τ )

= gi (x, y + hi (x)), (3) dτ where τ = (t − t0 )/ε is an expanded time variable, x is treated as a fixed parameter in Eq. (3), and a new state vector y(τ ) is defined as y(τ ) = z(t) − hi (x(t)). fast subsystem :

Definition 1. For a given switching sequence {tk }k∈N ∈ N (τ¯1 , τ¯2 ) and a given positive scalar ε ∗ , the equilibrium (x, z) = (0, 0) of system (1) is said to be globally uniformly exponentially stable (GUES) for ε ∈ (0, ε ∗ ), if there exist positive constants κ and λ such that

∥x (t )∥ + ∥z (t )∥ ≤ κ (∥x0 ∥ + ∥z0 ∥) e−λ(t −t0 ) ,

(4)

for any t ≥ t0 , any initial condition (x0 , z0 ) and any ε ∈ (0, ε ∗ ). Moreover, λ is called the exponential convergence rate. For given positive constants ai , i = 1 [ ], 2, 3, 4, define a a1 a2 with ε > 0 and 2 × 2 symmetric matrix A(ε ) = ∗ a3 − a4 /ε Lemma 1.

its largest eigenvalue λmax (A(ε )). Then, the following facts hold. (1) λmax (A(ε )) is monotonically increasing with respect to ε , dλ (A(ε )) i.e., maxdε > 0, for any ε > 0. (2) lim λ (A( ε )) = a1 and ]limε→∞ λmax (A(ε )) = 12 ε→ 0 max [

sequence 2 → 3 → 1 in the following time.

(a1 + a3 ) +

√

(a1 − a3 )2 + 4a22 .

2. Problem description Proof. [

Consider the following nonlinear SPSSs: x˙ (t) = fσ (t ) (x(t), z(t)) , x(t0 ) = x0 , ε˙z (t) = gσ (t ) (x(t), z(t)) , z(t0 ) = z0 ,

{

(1)

where x(t) ∈ Rnx is the slow state vector, z(t) ∈ Rnz is the fast state vector, ε is a small parameter that indicates the degree of the fast and slow dynamics separation. The piecewise right continuous σ (t) : [0, ∞) → M := {1, 2, . . . , m} is the switching signal and m is the number of modes. fi : Rnx × Rnz → Rnx and gi : Rnx × Rnz → Rnz are smooth functions with fi (0, 0) = 0, gi (0, 0) = 0, ∀i ∈ M. The kth switching instant is denoted by

1 2

It follows from ]some computation that λmax (A(ε )) = √

ϕ1 (ε ) +

ϕ22 (ε ) + 4a22 and the second equation in assertion

(2) holds, where ϕ1 (ε ) = a1 + a3 − a4 /ε and ϕ2 (ε ) = a1 − (a3 − a4 /ε ). Moreover, the derivative of λmax (A(ε[)) with respect ]to ε is

[

dλmax (A(ε )) dε

=

1 2

a4

ε2

+

2ϕ2 (ε )

−a4 √ 2 2 ϕ22 (ε )+4a22 ε

]

a4

=

for any ε > 0, and limε→0 λmax (A(ε )) =

= a1 . The proof is thus completed.

■

1 2

√

ϕ22 (ε )+4a22 −ϕ2 (ε) √ 2ε 2 ϕ22 (ε )+4a22

> 0,

ϕ 2 (ε )+4a22 −ϕ12 (ε)

limε→0 √ 2

ϕ22 (ε )+4a22 −ϕ1 (ε)

W. Yang, Y.-W. Wang, C. Wen et al. / Automatica 113 (2020) 108800

3

Now, with all these preliminary results, the main result on the GUES of system (1) is ready to be presented. Theorem 1. Consider system (1) with switching signal σ (t). Suppose that Assumptions 1 and 2 are fulfilled, if for each i ∈ M, the following conditions hold,

Fig. 1. Switching sequence {tlk }k∈N , where the notation ↑ denotes that the switching behavior at this instant is destabilizing, while the notation ↓ means that the switching behavior at this instant is stabilizing.

Remark 2. Note that the fast and slow subsystems have different divergence rates, they are also affected by singular perturbation parameter. Therefore, how to accurately estimate the bound of the divergent state of system (1) is challenging. Lemma 1 is helpful in the following to deduce the state bound of system (1) and to obtain the upper bound ε ∗ . 3. Stability analysis

(i) there exist function Vi of x and positive constants αi , β1 , i = ∂V 1, 2, such that α1 ∥x∥2 ≤ Vi ≤ α2 ∥x∥2 and ∂ xi fi (x, hi (x)) ≤ 2 nx β1 ∥x∥ for all x ∈ R ; (ii) there exist function Wi of x and y, and positive constants αi , β2 , i = 3, 4, such that α3 ∥y∥2 ≤ Wi ≤ α4 ∥y∥2 and ∂ Wi g x, y + hi (x)) ≤ −β2 ∥y∥2 for all x ∈ Rnx , y ∈ Rnz ; ∂y i ( ∂ Wi ∂x

y + hi (x)) − fi (x, hi (x))] ≤ 2γ1 ∥x∥∥y∥ and

−

(x, y + hi (x)) ≤ 2γ2 ∥x∥∥y∥ + γ3 ∥y∥ ; (iv) there ( exist ) two constants ( ) 0 < ν < 1 and µ (≥ 1 such ) that 2

ϑi tl+k +p ( ) tl−+p k

↑

≤ µϑj tl−k +p , if p ∈ Sk , and ϑi tl+k +p

To analyze the GUES of system (1), we propose the so-called MCLFs method. The key idea of the MCLFs method is to compensate the divergence of the corresponding Lyapunov function by using the stabilization effect of some switching behaviors. In order to provide a proper description of the switching behavior and further to explore the stabilizing effect of some switching, motivated by Wang et al. (2019), the switching sequence is reordered as a new switching sequence {tlk }k∈N , which is depicted in Fig. 1. In the new switching sequence {tlk }k∈N , set tl0 = t1 and lk+1 − lk = s, ∀lk , k ∈ N, where s ∈ N+ is a predefined positive integer. The time interval [tlk , tlk+1 ) is denoted as the (k + 1)th segment. Then, any switching instant within the (k+1)th segment can be described as tlk +p , ∀p ∈ S := {0, 1, . . . , s − 1}. Motivated by Wang et al. (2019), which focuses on the normal switched ↑ ↓ systems, we extend those results to SPSSs, and let Sk and Sk be the index sets referring to the destabilizing and stabilizing switching behaviors within the kth segment, respectively, then, the ratios of the destabilizing and stabilizing switching behaviors ↑ ↓ are |Sk |/s and |Sk |/s, respectively. Assumption 2. There exist two positive integers s1 and s2 with s1 + s2 = s such that the numbers of the destabilizing and stabilizing switching behaviors within each segment satisfy ↑ ↓ |Sk | ≤ s1 and |Sk | ≥ s2 , respectively. Assumption 2 introduces an upper (lower) bound for the number of the destabilizing (stabilizing) switching behaviors within each segment, and is necessary for analyzing the GUES of system (1). Remark 3. The switching sequence considered here is a more general one. Specifically, only destabilizing switching behavior is considered in Xiang and Xiao (2014), although both the destabilizing and stabilizing switching behaviors are considered in Wang et al. (2019), the stabilizing (destabilizing) switching behaviors are assumed to occur at the first s1 (the remainder s2 ) switching behaviors. Here, we assume that the numbers of the destabilizing and stabilizing switching behaviors within each segment satisfy ↑ ↓ |Sk | ≤ s1 and |Sk | ≥ s2 , respectively. Obviously, the switching sequences in Wang et al. (2019) and Xiang and Xiao (2014) are special cases of our model. Therefore, the stability problem here is much more challenging.

≤ νϑj

↓

, if p ∈ Sk , for all j ∈ M, j ̸ = i, k ∈ N, ϑ ∈ {V , W };

(v) there exists a positive constant ε ∗ such that s2 ln ν + s1 ln µ+ maxi∈M {λmax (Λi (ε ))} τ¯ θ (ε ∗ ) < 0 with θ1 (ε) = min , Λi (ε ) = [2 1 ] i∈M {(1−di )α1 ,di α3 } (1 − di )β1 (1 − di )γ1 + di γ2 , and di ∈ (0, 1) is the β ∗ di (γ3 − ε2 ) s

3.1. Stability of nonlinear SPSSs

∂ Vi [f (x, ∂ x i] ∂ Wi ∂ hi fi ∂y ∂x

(iii) there exist positive constants γi , i = 1, 2, 3, such [ that

s

selected parameter; then the origin of system (1) is GUES for any ε ∈ (0, ε ∗ ). Proof. Motivated by Chapter 7 of Kokotovic et al. (1999), choose the following MCLFs candidate vi := (1 − di )Vi + di Wi , where Vi , Wi are defined as above, and di ∈ (0, 1) is an adjustable parameter. For t = tlk +p , k ∈ N, p ∈ S , it follows from condition (iv) of Theorem 1 that

vσ (tlk +p )

(

tl++p k

)

⎧ ( ) ↑ ⎨µvσ (t tl−+p , p ∈ Sk , lk +p−1 ) k ( ) ≤ ↓ ⎩νvσ (t tl−+p , p ∈ Sk . lk +p−1 ) k

(5)

For t ∈ [tk , tk+1 ), k ∈ N, by using conditions (i)-(iii) of Theorem 1, the derivative of vi along the trajectories of system (1) in the coordinates (x, y) is given by

∂ Vi di ∂ Wi fi (x, hi (x)) + gi (x, y + hi (x)) ∂x ε ∂y ∂ Vi + (1 − di ) [fi (x, y + hi (x)) − fi (x, hi (x))] ∂x ] [ ∂ Wi ∂ hi ∂ Wi + di − fi (x, y + hi (x)) ∂x ∂y ∂x d i β2 ≤ (1 − di )β1 ∥x∥2 − ∥y∥2 + di γ3 ∥y∥2 ε + 2((1 − di )γ1 + di γ2 )∥x∥∥y∥ [ ]T [ ] ∥ x∥ ∥ x∥ = Λi (ε ) . ∥y∥ ∥y∥

v˙ i = (1 − di )

(6)

According to Lemma 1, it follows that λmax (Λi (ε )) is monotonically increasing with respect to ε and limε→0 λmax (Λi (ε )) = (1 − di )β1 > 0, then θ1 (ε ) is also monotonically increasing with respect to ε and is positive as well. Therefore, it follows from (6) that v˙ i (t) ≤ θ1 (ε )vi (t), ∀t ∈ [tlk +p , tlk +p+1 ), k ∈ N, p ∈ S , which implies that for any t ∈ [tlk +p , tlk +p+1 ), k ∈ N, p ∈ S ,

vσ (tlk +p ) (t) ≤ exp{θ1 (ε )(t − tlk +p )}vσ (tlk +p ) (tl+k +p ).

(7)

Note that it follows from condition (v) of Theorem 1 that there exists sufficient small positive constant ϱ such that s2 ln ν + s1 ln µ + sτ¯2 θ1 (ε ) + ϱ < 0 holds for any ε ∈ (0, ε ∗ ).

4

W. Yang, Y.-W. Wang, C. Wen et al. / Automatica 113 (2020) 108800

Combining (5) and (7) together, it can be obtained that for any t ∈ [tlk +p , tlk +p+1 ), k ∈ N, p ∈ S ,

vσ (tlk +p ) (t) ≤ µs1 (ν s2 µs1 )k exp{θ1 (ε )(t − t0 )}vσ (t0 ) (t0 ) = µs1 exp{k(s2 ln ν + s1 ln µ)} × exp{θ1 (ε )(t − t0 )}vσ (t0 ) (t0 ) ϱ < κ¯ exp{− (t − t0 )}vσ (t0 ) (t0 ), sτ¯2

(8)

where κ¯ = µ−s1 /s ν −s2 (1+1/s) and the fact that k + 1 ≥

t −t0 −τ¯2 sτ¯2

, ∀t ∈

[tlk +p , tlk +p+1 ), k ∈ N, p ∈ S is used. Which shows that the zero solution of system (1) is GUES with respect to the switching sequence σ (t) for any ε ∈ (0, ε ∗ ). This ends the proof. ■ Remark 4. Several explanations on the conditions of Theorem 1 are given as follows:

• Condition (iv) indicates that switching behaviors can be roughly divided into two types: the destabilizing switching behavior, and µ (ν ) is regarded as the destabilizing (stabilizing) switching strength. Note that the switching behavior is treated as the destabilizing one in Alwan and Liu (2009), Alwan et al. (2008), Deaecto et al. (2012), Lian and Wang (2015), Ma and Cai (2016), Ma et al. (2016), Malloci et al. (2010), Rejeb et al. (2018) and Wang et al. (2018) and all the switching behavior is treated as the stabilizing one in Xiang and Xiao (2014). In this sense, this work provides a more general view of switching behaviors. • Condition (v) imposes the requirement on s2 and s, and reveals that the stability of system (1) can be ensured as long as the ratio of the stabilizing switching behavior is large enough with properly chosen µ, ν and τk . Moreover, the larger the ratio of the stabilizing switching behavior is, the larger upper bound ε will be obtained. In addition, if the parameters τ¯1 , µ, ν, s1 , and s2 are fixed, τ¯2 is negatively correlated with the singular perturbation parameter. All these relations will be illustrated in the simulation example. Remark 5. When conditions (i)-(iv) are satisfied, by using Lemma 1 and condition (v), √ if the following inequality holds, s2 s

ln ν +

s1 s

ln µ +

τ¯2 [a1 +a3 + (a1 −a3 )2 +4a22 ] 2min{(1−di )α1 ,di α3 }

< 0, with a1 = (1 −

di )β1 , a2 = (1−di )γ1 +di γ2 , a3 = di γ3 , then system (1) can achieve the GUES for all ε > 0. Remark 6. It is very difficult to obtain the exact upper bound ε ∗ even for continuous-time linear time-invariant systems, see Chen and Lin (1990) and Sen and Datta (1993). The situation is more complex for nonlinear systems or dynamical systems with switching behaviors. Here, we provide a procedure to better describe how to obtain numerically an upper bound ε ∗ in our case: (1) For given SPSSs (1), determine the parameters αi (i = 1, 2, 3, 4), βj (j = 1, 2) and γk (k = 1, 2, 3), by choosing proper Lyapunov functions Vi and Wi ; (2) For given constants ν and µ, select the parameters s and s2 such that the switching sequence {tk }k∈N satisfies Assumption 2 and the condition ss2 ln ν + ss1 ln µ+ τ¯2 θ1 (0) < 0 holds, then the global exponentially stability of the full-order system can be guaranteed for small enough ε > 0; (3) By using a simple one-dimensional search over ε ∗ > 0, one obtains the upper bound ε ∗ satisfying the condition (v) of Theorem 1. When there is no destabilizing switching behavior, that is, ↑ Sk = ∅, based on Theorem 1, the following corollary can be achieved immediately.

Corollary 1. Suppose that Assumption 1 is fulfilled. The origin of system (1) is GUES with respect to switching sequence σ (t) for any ε ∈ (0, ε ∗ ), if there exist continuous non-negative functions Vi , Wi and positive constants αi , i = 1, 2, 3, 4, βi , i = 1, 2 such that conditions (i)-(iii) and the following conditions hold. (iv)′ there exists constant 0 < ν < 1 such that for all k ∈ N and ∀p ∈ S , ϑi (tl++p ) ≤ νϑj (tl−+p ), ϑ ∈ {V , W }, where k k i ̸ = j, i, j ∈ M. (v)′ there exists a positive constant ε ∗ such that ln ν + τ¯2 θ1 (ε ∗ ) < 0, where θ1 (ε ) is defined in Theorem 1. How to verify Theorem 1 or Corollary 1 is generally nontrivial since Lyapunov function design for nonlinear system is challenging. In the following subsection, a computable result will be presented for the linear SPSSs. 3.2. Stability of linear SPSSs Consider the following linear SPSSs:

{

12 x˙ (t) = A11 σ (t ) x(t) + Aσ (t ) z(t), x(t0 ) = x0 ,

(9)

22 ε˙z (t) = A21 σ (t ) x(t) + Aσ (t ) z(t), z(t0 ) = z0 ,

where x(t), z(t), σ (t) and ε are defined the same as in system 0 (1). Assume that A22 i is Hurwitz matrix for any i ∈ M. Let Ai =

(

22 12 A11 i − Ai Hi , where Hi = Ai

)−1

A21 i . According to Assumption 1,

A0i

is not Hurwitz stable for each mode i ∈ M. Using the fast-slow decomposition, the slow subsystem and the fast subsystem can be given as follows, respectively, 12 x˙ (t) = (A11 σ (t) − Aσ (t) Hσ (t) )x(t), x(t0 ) = x0 ,

(10)

ε˙y(t) = Aσ (t ) y(t), y(t0 ) = z0 + Hσ (t) x0 . 22

(11)

As pointed out in Rejeb et al. (2018), the state y(t) will occur instantaneous jumps, i.e., the dynamics expressed in (x, y) ˜ be the set of subsystem variable is a discontinuous one. Let M pairs associated with the stabilizing switching behavior, that is, ˜ := {(i, j); σ (t) = i, t ∈ [tlk +p , tlk +p+1 ), σ (t) = j, t ∈ M ↓ [tlk +p+1 , tlk +p+2 ), p + 1 ∈ Sk , ∃k ∈ N}. In order to reveal both the stabilizing and destabilizing switching behaviors, motivated by Xiang and Xiao (2014) and Wang et al. (2019), the switching-time-dependent MCLFs are designed as Vi (t) = xT [ρ (t)Pi0 + (1 − ρ (t))Pi1 ]x and Wi (t) = yT [ρ (t)Qi0 + (1 − ρ (t))Qi1 ]y, where i ∈ M, ρ (t) = (tlk +p+1 − t)/τk , t ∈ [tlk +p , tlk +p+1 ), p ∈ S , and Pij , Qij , i ∈ M, j ∈ {0, 1}, are positive definite matrices. Theorem 2. Suppose that Assumption 2 is fulfilled. Given positive scalars τ¯1 and τ¯2 satisfying τ¯1 ≤ τ¯2 , µ ≥ 1, and ν ∈ (0, 1), consider ˜ ̸= ∅ and system (9) with switching signal σ (t) ∈ N (τ¯1 , τ¯2 ). If M the following conditions hold, j

(I) there exist positive definite matrices χi , χ ∈ {P , Q }, j ∈ {0, 1}, and positive constants β1 , β2 , γ3 such that Ξlij <

{

ij

j

}

0, l = 1, 2, 3, where Ξ1 = He Pi A0i +

{

j

}

j

ij

{

j

Pi1 −Pi0

}

He Qi A22 +β2 Qi , Ξ3 = He Qi Hi A12 + i i

τ¯k

− β1 Pij , Ξ2ij =

Qi1 −Qi0

τ¯k

1, 2; ˜ , otherwise, χi0 ≤ µχj1 ; (II) χi0 ≤ νχj1 , if (j, i) ∈ M (III) there exists a positive constant ε ∗ such that

τ¯2 θ2 (ε ∗ ) < 0, where θ2 (ε ) = defined in Theorem 1;

−γ3 Qij , k =

ν + ss1 ln µ+

s2 ln s maxi∈M {λmax (Λi (ε ))} mini∈M {(1−di ),di }

with Λi (ε )

then the origin of system (9) is GUES for any ε ∈ (0, ε ∗ ).

W. Yang, Y.-W. Wang, C. Wen et al. / Automatica 113 (2020) 108800

5

Table 1 The upper bound of ε with different s and s2 .

ε∗ s 2

3

4

5

6

1

×

×

×

×

×

2

0.00459

0.000318

×

×

×

3

−

0.00459

0.00138

×

×

4

−

−

0.00459

0.00202

0.000318

5

−

−

−

0.00459

0.00245

6

−

−

−

−

0.00459

s2

Table 2 The upper bound of ε with different τ¯2 . Fig. 2. Simulation results of Example 1 when ε = 0.0005.

τ¯2 ε∗

1.7 1.08 × 10−3

1.8 0.906 × 10−3

1.9 0.703 × 10−3

τ¯2 ε∗

2.0 0.572 × 10−3

2.1 0.451 × 10−3

2.2 0.342 × 10−3

In line with the proof of Theorem 1, it can be proved that system (9) is GUES for any ε ∈ (0, ε ∗ ) by combining the convex combination technique and LMI method, and we omit the proofs. Remark 7. The parameter β1 in Theorems 1–2 is not necessarily positive, which means that the obtained results can also be applicable for system with stable modes (Deaecto et al., 2012; Malloci, Daafouz and Iung, 2009), this point will be demonstrated via Example 2. Remark 8. To estimate τ¯2 and τ¯1 , the following strategy can be used: (1) given a τ¯1 (or τ¯2 ), choose an initial τ¯2 (or τ¯1 ) such that all conditions of Theorem 2 are satisfied; (2) choose ∆τ > 0 and update τ¯2 with τ¯2 + ∆τ (or τ¯1 − ∆τ ), check the conditions of Theorem 2; (3) repeat (2) until Theorem 2 is infeasible. 4. Simulation Example 1. This example is to illustrate the stability result for the SPSS with all modes being unstable. Consider SPSSs (9) with the following parameters: A11 1 =

[ −0.9

]T

0

0.3 0.2 0 , A12 , A21 1 = 1 = 1.1 0.01 0

0.02 0

0 .2 0.6 0 , A12 , A21 2 = 2 = 0.8 −1.2 0

0.05 0.02

0.15 0.05 0.01 21 , A12 3 = −0.1 , A3 = −0.1 −1

]

[

]

[

,

hold with ε ∗ = 0.000572. Thus, according to Theorem 2, the system in this example can achieve GUES for any ε∈(0, 0.000572). Moreover, the ratio of the stabilizing switching behavior in this case equals 75%. In the simulation, select the order of the switched modes as {1 → 2 → 3 → 1 → 3 →  1 → 3 → 2 → 1}. It follows from the

˜ that the ratio of the stabilizing switching behavior notation M is 75% and {tk }k∈N \ {t8k−6 , t8k−1 }k∈N+ is the stabilizing switching instant, the corresponding switching sequence is depicted [ ] in Fig. 2(b). Moreover, set the initial value 8 10 −2 and ε = 0.0005, the state evolution is given in Fig. 2(a), which shows that the trajectories of the system in Example 1 converge to zero. In addition, we emphasize that the feasible switching sequences are not unique, for example, the switching sequence {3 → 1 → 2 → 3 → 1 →  3 → 2 → 1 → 3} is also feasible. 4.2. The relationship between ε ∗ and the switching behavior

4.1. The effectiveness of Theorem 2

To show the relationship between the upper bound ε ∗ and the ↓ ratio of stabilizing switching behavior |Sk |/s, let τ¯1 = τ¯2 = 1.6, that is, the switch is periodic. Choose µ = 1.02 and ν = 0.48, the ↓ relationship between ε ∗ and |Sk |/s is shown in Table 1 by choosing different s and s2 . In Table 1, × means that the conditions in Theorem 2 do not hold, − denotes that such situation does not exist. Table 1 shows that the larger the ratio of stabilizing switching behaviors is, the larger ε ∗ will be obtained. Also note that for the cases of nonoccurrence of the destabilizing switching behavior(s = s2 = 2, 3, 4, 5, 6 in Table 1), the upper bound ε ∗ equals 0.00459. To show the relationship between the singular perturbation parameter and the upper bound τ¯2 , let τ¯1 = 1.6, µ = 1.02, ν = 0.45, and τ¯2 can be found according to Remark 8. To ensure the stability, let s = 4 and s2 = 3, then the relationship between ε ∗ and τ¯2 is illustrated in Table 2, which confirms that ε ∗ is negatively correlated with τ¯2 .

˜ = {(1, 2), (1, 3), (2, 1), (3, 1)}, let τ¯1 = 1.6, τ¯2 = Choose M 2.0, i.e., the switching sequence belongs to N (1.6, 2.0), fix µ = 1.02, ν = 0.45, β1 = 0.23, β2 = 1.39, γ3 = 0.80, s1 = 1, s2 = 3 and di = 0.5, it can be found that the conditions of Theorem 2

Example 2. This example is to illustrate the application of the obtained result to the SPSS with stable modes and to show the advantage of the obtained result over the result in Malloci, Daafouz and Iung (2009). Consider SPSSs (9) with the following

A11 2 = A11 3 =

[ [

]

]

[

[

]

]

[

[

]T

,

]T

,

22 22 A22 1 = − 1.2, A2 = −0.7, A3 = −3.

Note that each mode is unstable. Therefore, the methods employed in Alwan and Liu (2009), Alwan et al. (2008), Deaecto et al. (2012), Lian and Wang (2015), Ma and Cai (2016), Ma et al. (2016), Malloci et al. (2010), Rejeb et al. (2018) and Wang et al. (2018) are not applicable. Besides, due to the two-time-scale feature, the approaches given in Xiang and Xiao (2014) and Wang et al. (2019) are also inapplicable.

6

W. Yang, Y.-W. Wang, C. Wen et al. / Automatica 113 (2020) 108800

been proposed for a general nonlinear SPSSs to achieve the sufficient stability criteria. Then, by using switching-time-dependent MCLFs, a sufficient stability condition has been deduced for linear context in terms of LMIs. Moreover, the obtained results have also been applied to SPSSs with stable modes. All obtained results are free of ill-conditioning and stiffness problems. Two examples have been presented to illustrate the effectiveness of the proposed results. Future research will be focussed on the stability of SPSSs with time delay and impulsive effects. References

Fig. 3. Simulation results of Example 2 with destabilizing switching law when ε = 0.0008.

Fig. 4. Simulation results of Example 2 with stabilizing switching law when ε = 0.0008.

parameters: A11 1 A11 2

[

0 = − 1

1 0.1 0 , A12 , A21 1 = 1 = 0.01 1 0

[

0 = − 118

]

[

]

[

]T

, A22 1,2 = −1,

] [ ] [ ]T −1 0 0.1 12 21 , A2 = , A2 = . 38.5 0.5 0

Obviously, both A0i and A22 are Hurwitz for any i ∈ {1, 2}. i However, the exponential stability of both the subsystems cannot guarantee the exponential stability of the whole system, this fact can be confirmed by Fig. 3, where the switching law is designed to satisfy: (1) Mode 1 is activated, if x2 ≥ 0 and x2 ≥ −3x1 , or x2 ≤ 0 and x2 ≤ −3x1 ; (2) Mode 2 is activated, if x2 ≥ 0 and x2 < −3x1 , or x2 ≤ 0 and x2 > −3x1 . Besides, as can be verified, the conditions of Theorem 1 in Malloci, Daafouz and Iung (2009) are infeasible, which means that the obtained result in Malloci, Daafouz and Iung (2009) is invalid here. Based on Theorem 2, select the length of the dwell time interval τ1 = τ2 = 2.87, β1 = −0.1, β2 = 1.99, and set ν = 0.33, γ3 = 0.72, d = 0.5, it can be found that all conditions of Theorem 2 are satisfied with ε ∗ = 0.000857, which means that the system in Example 2 is GUES for any ε ∈ (0, 0.000857). Fig. 4 depicts the state response of the system and its corresponding switching law. 5. Conclusions In this paper, the GUES problem of SPSSs with all modes being unstable has been studied. First, the so-called MCLFs method has

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W. Yang, Y.-W. Wang, C. Wen et al. / Automatica 113 (2020) 108800 Wang, Q., Zhou, L., Ma, X., & Yang, C. (2018). Disturbance rejection of singularly perturbed switched systems subject to actuator saturation. International Journal of Robust and Nonlinear Control, 28(6), 2231–2248. Xiang, W., & Xiao, J. (2014). Stabilization of switched continuous-time systems with all modes unstable via dwell time switching. Automatica, 50(3), 940–945. Yang, W., Wang, Y.-W., Shen, Y., & Pan, L. (2017). Cluster synchronization of coupled delayed competitive neural networks with two time scales. Nonlinear Dynamics, 90(4), 2767–2782. Yang, W., Wang, Y.-W., Xiao, J.-W., & Liu, Z.-W. (2017). Coordination of networked delayed singularly perturbed systems with antagonistic interactions and switching topologies. Nonlinear Dynamics, 89(1), 741–754. Wu Yang received the B.S. degree in Information and Computing Science and the M.S. degree in Applied Mathematics from Guangxi University, Nanning, China, in 2011 and 2014, respectively, and the Ph.D. degree in Control Science and Engineering from Huazhong University of Science and Technology (HUST), Wuhan, China, in 2017, where he is currently post-doctoral fellow at School of Artificial Intelligence and Automation, Huazhong University of Science and Technology. His current research interests include singularly perturbed systems, hybrid systems, complex systems and networks, and so on. Yan-Wu Wang (M’10–SM’13) received the B.S. degree in automatic control, the M.S. degree and the Ph.D. degree in control theory and control engineering from Huazhong University of Science and Technology (HUST), Wuhan, China, in 1997, 2000, and 2003, respectively. Currently, she is a Professor with the School of Automation, HUST, and with the Key Laboratory of Image Processing and Intelligent Control, Ministry of Education, China. Her research interests include hybrid systems, cooperative control, and multi-agent systems, with applications in smart grid. Dr. Wang was a recipient of several awards, including the first prize of Hubei Province Natural Science in 2014, the first prize of the Ministry of Education of China in 2005, and the Excellent PhD Dissertation of Hubei Province in 2004, China. In 2008, she was awarded the title of ‘‘New Century Excellent Talents" by the Chinese Ministry of Education.

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Changyun Wen received the B.Eng. degree from Xi’an Jiaotong University, Xi’an, China, in 1983 and the Ph.D. degree from the University of Newcastle, Newcastle, Australia in 1990. From August 1989 to August 1991, he was a Research Associate and then Postdoctoral Fellow at University of Adelaide, Adelaide, Australia. Since August 1991, he has been with School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently a Full Professor. His main research activities are in the areas of control systems and applications, autonomous robotic system, intelligent power management system, smart grids, cyber–physical systems, complex systems and networks, and so on. He is a Fellow of IEEE, was a member of IEEE Fellow Committee from January 2011 to December 2013 and a Distinguished Lecturer of IEEE Control Systems Society from February 2010 to February 2013. Prof. Wen is an Associate Editor of a number of journals including Automatica, IEEE Transactions on Industrial Electronics and IEEE Control Systems Magazine. He is the Executive Editor-inChief of Journal of Control and Decision. He served the IEEE Transactions on Automatic Control as an Associate Editor from January 2000 to December 2002. He has been actively involved in organizing international conferences playing the roles of General Chair, General Co-Chair, Technical Program Committee Chair, Program Committee Member, General Advisor, Publicity Chair and so on. He was the recipient of a number of awards, including the Prestigious Engineering Achievement Award from the Institution of Engineers, Singapore in 2005, and the Best Paper Award of IEEE Transactions on Industrial Electronics from the IEEE Industrial Electronics Society in 2017. Jamal Daafouz is a professor of automatic control at Université de Lorraine in Nancy, France. He is a researcher at the Research Centre of Automatic Control (CRAN UMR 7039 CNRS). He prepared his Ph.D at LAAS-CNRS and received the Ph.D. degree from INSA Toulouse, in 1997. He got the French Habilitation degree from the Institut National Polytechnique de Lorraine (INPL) in 2005. He is senior editor of the journal IEEE Control Systems Letters (L-CSS). He also served as an associate editor for the journals: IEEE Transactions on Automatic Control, Automatica and European Journal of Control. His research interests include hybrid and switched systems, networked control systems, robust control and applications in securecommunications and metallurgy.