New finite-time stability conditions of linear discrete switched singular systems with finite-time unstable subsystems

New Finite-time Stability Conditions of Linear Discrete Switched Singular Systems With Finite-time Unstable Subsystems

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New Finite-time Stability Conditions of Linear Discrete Switched Singular Systems With Finite-time Unstable Subsystems Jumei Wei, Xiuxiu Zhang, Huimin Zhi, Xunlin Zhu PII: DOI: Reference:

S0016-0032(19)30688-X https://doi.org/10.1016/j.jfranklin.2019.03.045 FI 4170

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Journal of the Franklin Institute

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2 September 2018 1 January 2019 10 March 2019

Please cite this article as: Jumei Wei, Xiuxiu Zhang, Huimin Zhi, Xunlin Zhu, New Finite-time Stability Conditions of Linear Discrete Switched Singular Systems With Finite-time Unstable Subsystems, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.03.045

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New Finite-time Stability Conditions of Linear Discrete Switched Singular Systems With Finite-time Unstable Subsystems Jumei Wei a , Xiuxiu Zhang a , Huimin Zhi a , Xunlin Zhu a,∗ a

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan, 450001, China

Abstract The finite-time stability problem of linear discrete switched singular systems with finite-time unstable subsystems is studied in this paper. By using dynamic decomposition technique, the original switched singular systems can be transformed into equivalent one that is a reduced-order switched normal systems. For linear discrete switched singular systems, based on the mode-dependent average dwell time (MADAT)switching signal, new sufficient conditions are presented to guarantee the considered systems with finite-time unstable subsystems being finite-time stability, finite-time bounded. At last, two numerical examples is employed to verify the efficiency of the preceding method. Key words: Discrete switched singular systems; Mode-dependent average dwell time; Finite-time stability

1. Introduction Singular systems which are also called as strong coupling systems, incomplete state systems, generalized systems have been widely studied due to their extensive applications in modeling and control of electrical circuits, power systems, economics and other areas [1, 2, 3, 4, 5, 6]. On the other hand, many practical systems are always subjected to abrupt variation in their structures and parameters, such as failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment changes, which cannot be well described by using pure continuous or discrete models. However, switched systems, which consist of a finite number of continuous-time (or discrete-time) subsystems and a logical rule orchestrating the switching between them, play an important role in studying the special class of hybrid systems. And because of their broad applications, lots of valuable results have been obtained in many aspects in recent years. For example, multiple Lyapunov functions [7], Markovian jump neural networks [8], T-S fuzzy systems [9, 10, 11], variational principles [12], observer design [13], stochastic switching [14], and other problems of stability and stabilization for linear or nonlinear switched systems [15, 16, 17, 18, 19, 20]. In particular, owing to the significance both in theoretical development and actual operations, switched singular systems have been received increasing attention in [21, 22, 23, 24, 25]. There is no doubt that stability is a fundamental and vital issue for switched singular systems. At present, we mainly focus on the stability analysis of the time-dependent switching signals under constrained switching. Typical switching signals comprise dwell time (DT) switching signal [26], average dwell time (ADT) switching signal [27]and mode-dependent average dwell time (MDADT) switching signal [28]. It has been illustrated that MDADT approach is more applicable than DT and ADT approaches in [28]. Since each mode not only has its own ADT but also has its own control strategy under MDADT switching. The stability for discrete-time nonlinear singular systems with switching actuators has been discussed in [29] by using ADT approach. As unstable subsystems are usually encountered in many practical switched systems, the authors of [30] have investigated the problem of stability and stabilization for switched systems with unstable subsystems in both continuous-time and discrete-time contexts by using MDADT approach. Unfortunately, the stability of switched singular systems with unstable subsystems under MDADT switching signal has not been investigated, especially, in discrete-time case, which motivated us to carry out our research. ∗ Corresponding

author. Email address: [email protected] (Jumei Wei a , Xiuxiu Zhang a , Huimin Zhi a , Xunlin Zhu a,∗ )

Preprint submitted to Elsevier

November 6, 2019

It is worth noting that most of the aforementioned results in the literature are focused on the classical Lyapunov stability and exponential stability, which describe the dynamic behavior in an infinite-time interval. However, in practical engineering, there are systems which work in a short time (such as missile system, communication network system, robot operating, etc.). Hence, it is important to investigate their transient behaviors in a finite-time interval. The concept of finite-time stability (FTS) is originated from [31], which requires that the state can not exceed a certain threshold during a fixed finite-time interval. It has considered the transient behavior over a finite-time interval rather than the asymptotic behavior of a system response [32]. It not only has dealt with systems whose operation is limited to a fixed finite interval of time, but also requires prescribed bounds on system variables [33, 34]. However, among these studies, the results are concentrated on the FTS problem of switched systems with normal subsystems. Few efforts are devoted to dealing with the FTS of switched systems with singular subsystems [35, 36, 37, 38]. Existing attempts for FTS analysis of linear switched singular systems are mere extensions of the Lyapunov methods for individual linear switched singular systems. However, the robust stability problems have been discussed based on the restricted systems equivalent transformation for discrete switched singular systems in [39]. In addition, the dynamic decomposition technique considered in [40] is a novel method to study the FTS problems of linear switched singular systems. It can reveal the relationship among switching, algebraic equation and the parameters of the systems. By using the dynamic decomposition technique, a switched singular system can be converted to an equivalent reduced-order switched normal system, and based on MDADT switching, the FTS problems of the linear switched singular systems with finite-time unstable subsystems could be solved, which is the second motivation of the current research. Inspired by the above results, we investigate the FTS of the linear discrete switched singular systems with finite-time unstable subsystems. The main contributions of this paper are stated as follows. Firstly, based on the dynamic decomposition technique, an equivalent condition of FTS for linear discrete switched singular systems is given, that is, the original finite-time stabilization problem is converted into the one for the reduced-order switched normal systems. Secondly, the MDADT switching signals where fast switching and slow switching are respectively applied to finite-time unstable subsystems and stable subsystems are designed to discuss the FTS of linear discrete switched singular systems. Compared to ADT, the modedependent average dwell time (MDADT) is more flexible. The main reason is that under MDADT switching, each mode not only has its own ADT but also has its own control strategy. Moreover, new sufficient conditions to determine the FTS of the linear discrete switched singular systems with finite-time unstable subsystems are obtained. Finally, we illustrate that the advantages of our proposed methods over the existing results [38] by two numerical examples, and tighter bounds are obtained. The structure of this paper is as follows: section 2 is preliminaries and problem formulation. The main results to the FTS of linear discrete switched singular systems with finite-time unstable subsystems in section 3. Section 4 gives two examples to illustrate the stability problem for linear discrete switched singular systems with finite-time unstable subsystems. The conclusions are summarized in section 5. Notations. Some notations may be used throughout the paper: For a matrix M, the kernel (null space) of M is kerM, the image(range, column space) of M is imM, and the transpose of M is M T . For a matrix M ∈ Rn×n and a set S ⊂ Rn , the image of S under M is MS := {Mx ∈ Rn |x ∈ S} and the pre-image of S under M is M −1 S := {x ∈ Rn |∃y ∈ S : Mx = y}. 2. Preliminaries and problem formulation Consider a class of linear discrete switched singular systems as follows: ( Eσ (k) x(k + 1) = Aσ (k) x(k) x(0) = x0

(1)

where x(k) ∈ Rn is the system state, x0 ∈ Rn is a vector-valued initial state. The index function σ (k) : [0, ∞) → M = {1, 2, ..., m}(m ∈ N + ) is the switching signal, which is a piecewise constant function depending on time k or state x(k), m ∈ N + . Throughout this paper, we denote σ (k) as ik (ik ∈ M). Ei , Ai are constant matrices for any i ∈ {1, 2, ..., m}, and it is assumed that rank(Ei ) = r < n. For simplicity, we use (Ei , Ai ) to denote the ith subsystem. The positive integer m shows the number of the subsystems. Consider the linear discrete switched singular systems with disturbance:

2

  Eσ (k) x(k + 1) = Aσ (k) x(k) + Gσ (k) ω(k) ω(k + 1) = Fσ (k) ω(k)   x(0) = x0

(2)

where ω(k) ∈ Rr is the exogenous disturbance. Gi and Fi are constant matrix for any i ∈ {1, 2, ..., m}. Based on the switching signal σ (k), the switching sequence can be described as: {Eσ (0) x0 ; (i0 , k0 ), ..., (i j , k j ), ...|i j ∈ M, j = 0, 1, ...}

(3)

Where k j denotes the jth switching time, and i j is the appropriate subsystem which is activated at the time k j. In this paper, we obey the following assumptions: Assumption 1. ([41]) The states of switched systems do not jump at the switching instants, i.e., the trajectory x(k) is everywhere continuous. In every finite interval, the number of switches is limited. Assumption 2. ([42])The initial value ω(0) satisfies the constraint: ω T (0)ω(0) ≤ d, d > 0 We denote S = {1, 2, ..., s} as the set of finite-time stable subsystems. U = {s + 1, ..., m} denotes the set of finite-time unstable subsystems, and S ∪ U = M. Let L denote the set of switching signal which has a finite number of switching on any finite interval time. In this paper, we only consider the switching signal which belongs to L. Next, some necessary definitions are presented for the main results. Definition 1. ([3, 5]) For every i ∈ M, the singular system (Ei , Ai ) is said to be (i) regular if det(sEi − Ai ) is not identically zero. (ii) causal if deg(det(sEi − Ai )) = rank(Ei ). Definition 2. Given positive constants c1 , c2 , with c1 < c2 , K f ∈ N + , a positive definite matrix R, and a given switching signal σ (k) ∈ L, the linear discrete switched singular system (1) is said to be finite-time stable with respect to (c1 , c2 , K f , R, σ (k)), if x0T E0T RE0 x0 ≤ c1 =⇒ xT (k)EσT (k) REσ (k) x(k) ≤ c2 ,

k = 1, 2, · · · , K f

(4)

Definition 2 is extended from the definition of finite-time stable for discrete switched singular positive systems in [38]. Definition 3. Given positive constants d, c1 , c2 , with c1 < c2 , K f ∈ N + , a positive definite matrix R, and a given switching signal σ (k) ∈ L, the linear discrete switched singular system (2) is said to be finite-time bounded with respect to (c1 , c2 , K f , R, σ (k), d), if x0T E0T RE0 x0 ≤ c1 =⇒ xT (k)EσT (k) REσ (k) x(k) ≤ c2 ,

k = 1, 2, · · · , K f

(5)

This definition 3 is extended from [42]. Definition 4. ([38]) For any k ≥ k0 ≥ 0 and a switching signal σ (k), let Nσ p (k, 0) denote the number that the pth subsystem is activated over the interval [0, k), and Tp (k, 0) be the total running time of the pth subsystem over the interval [0, k), ∀p ∈ S. If there exist constants τap > 0, N0p ≥ 0 such that the inequalities Nσ p (k, 0) ≤ N0p +

Tp (k, 0) , ∀p ∈ S τap

(6)

hold, then the positive constant τap is called the slow mode-dependent average dwell time (SMDADT) of the switching signal σ (k). N0p is called the mode-dependent chattering bound.

3

Definition 5. ([19], [30]) For any k ≥ k0 ≥ 0 and a switching signal σ (k), let Nσ q (k, 0) denote the number that the qth subsystem is activated over the interval [0, k), and Tq (k, 0) be the total running time of the qth subsystem over the interval [0, k), ∀q ∈ U. If there exist constants τaq > 0, N0q ≥ 0 such that the inequalities Nσ q (k, 0) ≥ N0q +

Tq (k, 0) , ∀q ∈ U τaq

(7)

hold, then the positive constant τaq is called the fast mode-dependent average dwell time (FMDADT) of the switching signal σ (k). N0q is called the mode-dependent chattering bound. that

Due to the fact rank(Ei ) = r, we can always find nonsingular matrices Mi and Hi (i = 1, 2, ..., m) such   Ir 0 Mi Ei Hi = =: E¯i (8) 0 0   A (i) A12 (i) Mi Ai Hi = 11 =: A¯i (9) A21 (i) A22 (i)

Under Definition 1, we know that A22 (σ (k)) is nonsingular. Thus, we can obtain a reduced-order switched normal system via dynamic decomposition technique: x1 (k + 1) = (A11 (σ (k)) − A12 (σ (k))A−1 22 (σ (k))A21 (σ (k)))x1 (k) = W (σ (k))x1 (k)

(10)

A11 (σ (k)) − A12 (σ (k))A−1 22 (σ (k))A21 (σ (k)),

Where W (σ (k)) = and we can derive equivalent finite-time stability condition based on the reduced-order switched normal system (10). Lemma 1. Consider the linear discrete switched singular system (1). The system (1) is finite-time stable with respect to (c1 , c2 , K f , R, σ (k)) if and only if there exists matrices R11 > 0, R21 > 0 such that the state of the reduced-order switched normal systems (10) satisfy Where

x1T (0)R11 x1 (0) ≤ c1 =⇒ x1T (k)R21 x1 (k) ≤ c2 , k = 1, 2, · · · , K f R1 = M0−T RM0−1 =

and R2 = Mk−T RMk−1 = According to the [40], the Lemma 1 can be derived.

 R11 R13

 R21 R23

R12 R14 R22 R24

(11)

 

Similarly, we can decompose these matrices Gi , Fi and ω(k) as follows:   G1 (i) 0 Mi Gi Hi = =: G¯ i 0 G2 (i)   F1 (i) 0 −1 Hi Fi Hi = =: F¯i 0 F2 (i)   ω (k) Hi−1 ω(k) = 1 =: ω¯ i (k) ω2 (k) Then, we can gain a reduced-order switched system with disturbance: x1 (k + 1) = W (σ (k))x1 (k) + G1 (σ (k))ω1 (k) − T (σ (k))ω2 (k)

(12) (13) (14)

(15)

where T (σ (k)) = A12 (σ (k))A−1 22 (σ (k))G2 (σ (k)), and we can derive equivalent finite-time bounded condition based on the system (15). Lemma 2. Consider the linear discrete switched singular system (2). It is finite-time bounded with respect to (c1 , c2 , K f , R, σ (k), d) if and only if there exists matrices R11 > 0, R21 > 0 such that the state of the reduced-order switched normal systems (15) satisfy x1T (0)R11 x1 (0) ≤ c1 =⇒ x1T (k)R21 x1 (k) ≤ c2 ,

k = 1, 2, · · · , K f

(16)

Attention now turns to these questions of finite-time stability of switched reduced-order normal system (10) and (15). In other words, if the sub-state of x1 (k) is finite-time stable or finite-time bounded, then the original linear discrete switched singular systems is also finite-time stable or finite-time bounded. 4

3. Main results 3.1. Finite-time stability Now, we applied the definition of FMDADT to analyse the finite-time properties of linear discrete switched singular systems. First of all, we introduce a class of quasi-alternative switching signals satisfying the following conditions: (a) If σ (ki ) ∈ S, then σ (ki+1 ) ∈ M; (b) If σ (ki ) ∈ U, then σ (ki+1 ) ∈ S. This class of switching signals implies that a switched subsystem cannot switch from a finite-time unstable subsystem to another finite-time unstable subsystem. Next, finite-time stability conditions for linear discrete switched singular system (1) with finite-time unstable subsystems are given by designing quasi-alternative switching signals with MDADT property. Theorem 1. Consider linear discrete switched singular system (1), and let ρ p > 1, µ p > 1, p ∈ S, ρq > 1, 0 < µq < 1, q ∈ U, be given constants. Suppose there exists a set of matrices Pp > 0, p ∈ S, Pq > 0, q ∈ U, such that WpT PpWp − ρ p Pp ≤ 0, ρ p > 1, p ∈ S (17) WqT PqWq − ρq Pq ≤ 0, ρq > 1, q ∈ U

(18)

Pq ≤ µq Pp , ∀q ∈ U, ∀p ∈ S

(20)

Pp ≤ µ p Pr , ∀p ∈ S, ∀r ∈ M, p 6= r

(19)

λmax (Pσ (k) )λmax (R21 ) c2 −K f < ρp λmin (Pσ (k) )λmin (R11 ) c1

(21)

Then, the linear discrete switched singular system (1) is finite-time stable with respect to (c1 , c2 , K f , R, σ (k)) for any switching signal with MDADT ∗ τap ≥ τap =

ln

K f ln µ p c2 λmin (Pσ (k) )λmin (R11 ) c1 λmax (Pσ (k) )λmax (R21 )

∗ τaq ≤ τaq =−

− K f ln ρ p

, ∀p ∈ S

ln µq , ∀q ∈ U ln ρq

(22)

(23)

Where Wi = A11 (i) − A12 (i)A−1 22 (i)A21 (i). Proof. Choose Lyapunov functions as Vσ (k) (x(k)) ¯ = x¯T (k)E¯σ (k) P¯σ (k) E¯σ (k) x(k) ¯  Pσ (k) > be converted to

Where P¯σ (k) =

(24)

 0 , where > represents a matrix term that does not require attention, then (24) can > Vσ (k) (x(k)) ¯ = x1T (k)Pσ (k) x1 (k)

(25)

We get from (17), (18) that Vσ (k) (x(k ¯ + 1)) − ρσ (k)Vσ (k) (x(k)) ¯

= x1T (k + 1)Pσ (k) x1 (k + 1) − ρσ (k) x1T (k)Pσ (k) x1 (k)

= x1T (k)WσT(k) Pσ (k)Wσ (k) x1 (k) − ρσ (k) x1T (k)Pσ (k) x1 (k) " # = x1T (k) WσT(k) Pσ (k)Wσ (k) − ρσ (k) Pσ (k) x1 (k) ≤0

(26)

For any k ∈ [ki , ki+1 ], we gain from (26) that Vσ (k) (x(k ¯ + 1)) ≤ ρσ (k)Vσ (k) (x(k)) ¯ 5

(27)

By (19) and (20), we have

For k ∈ [0, K f )

Vp (x(k ¯ i )) ≤ µ pVr (x(k ¯ i− )), σ (ki ) = p, σ (ki− ) = r

(28)

Vq (x(k ¯ i )) ≤ µqVp (x(k ¯ i− )), σ (ki ) = q, σ (ki− ) = p

(29)

V (x(k)) ¯ ≤ ρσ (k)Vσ (k) (x(k ¯ − 1))

≤ ρσ (k) µσ (k)Vσ (k−1) (x(k ¯ − 1)− )

≤ ρσ (k) µσ (k) ρσ (k−1)Vσ (k−1) (x(k ¯ − 1)) ≤ · · · ≤ ∏ µp σ p N

p∈S

(0,k)

(

∏ µq σ q N

q∈U

(0,k)

∏ ρp p

K (0,k)

p∈S

∏ ρq q

K (0,k)

q∈U

Vσ (0) (x(0)) ¯

) ln µq ln µ p ≤ exp ∑ [ + ln ρ p ]K p (0, k) + ∑ [ + ln ρq ]Kq (0, k) Vσ (0) (x(0)) ¯ q∈U τaq p∈S τap ) ( ln µ p + ln ρ p ]K p (0, k) Vσ (0) (x(0)) ¯ ≤ exp ∑ [ p∈S τap Moreover,

V (x(k)) ¯ = x1T (k)Pσ (k) x1 (k) ≥ λmin (Pσ (k) )x1T (k)x1 (k)

(31)

V (x(0)) ¯ = x1T (0)Pσ (0) x1 (0) ≤ λmax (Pσ (k) )x1T (0)x1 (0) So, we can obtain that V (x(0)) ¯ ≤ λmax (Pσ (k) )x1T (0)x1 (0) ≤ This implies x1T (k)R21 x1 (k) ≤ λmax (R21 )x1T (k)x1 (k)

λmax (Pσ (k) )λmax (R21 ) ≤ c1 exp λmin (Pσ (k) )λmin (R11 )

(

c1 λmax (Pσ (k) ) λmin (R11 )

(30)

(32)

from the x1T (0)R11 x1 (0) ≤ c1 .

) ln µ p + ln ρ p ]K p (0, k) ∑[ p∈S τap

(33)

From (22) and (33), we know λmax (Pσ (k) )λmax (R21 ) x1T (k)R21 x1 (k) ≤ c1 λmin (Pσ (k) )λmin (R11 )

c2 λmin (Pσ (k) )λmin (R11 ) exp ln c1 λmax (Pσ (k) )λmax (R21 )

!

= c2 Therefore, we can conclude that linear discrete switched singular system (1) is finite-time stable by Lemma 1. Remark 1. In theorem 1, we design fast and slow switching for unstable subsystems and stable subsystems respectively in our switching scheme. It not only gives the lower bounds that stable subsystems should dwell on but also provide the upper bounds for unstable subsystems. Moreover, one can note that if the stable subsystem is activated, any subsystem can be activated at the next switching instance. However, if unstable subsystem is activated, the next activated system must be the stable subsystem. In addition, dynamic decomposition technique converts the original system into a reduced-order switched normal system, which simplifies the complexity of the problems studied in this paper. Remark 2. In theorem 1, if system (1) is positive system and only composed of finite-time stable subsystems, then the conclusion in [38] can be directly obtained by using the average dwell time method(reduced from the MDADT).

6

3.2. Finite-time bounded Theorem 2. Consider linear discrete switched singular system (2), and let γ p > 1, µ p > 1, p ∈ S, γq > 1, 0 < µq < 1, q ∈ U, be given constants. Suppose there exists a set of matrices Pp > 0, Q1 (p) > 0, Q2 (p) > 0, p ∈ S, Pq > 0, Q1 (q) > 0, Q2 (q) > 0, q ∈ U, such that  T Wp PpWp − γ p Pp  ∗ ∗

 WpT Pp G1 (p) −WpT Pp Tp Ω1p −GT1 (p)Pp Tp  ≤ 0 ∗ Ω2p

 T  Wq PqWq − γq Pq WqT Pq G1 (q) −WqT Pq Tq  ∗ Ω1q −GT1 (q)Pq Tq  ≤ 0 ∗ ∗ Ω2q  Pp ≤ µ p Pr , ∀p ∈ S, ∀r ∈ M, p 6= r   Q1 (p) ≤ µ p Q1 (r), ∀p ∈ S, ∀r ∈ M, p 6= r   Q2 (p) ≤ µ p Q2 (r), ∀p ∈ S, ∀r ∈ M, p 6= r  Pq ≤ µq Pp , ∀p ∈ S, ∀q ∈ U   Q1 (q) ≤ µq Q1 (p), ∀p ∈ S, ∀q ∈ U   Q2 (q) ≤ µq Q2 (p), ∀p ∈ S, ∀q ∈ U

(34)

(35)

(36)

(37)

c2 λ1 −K f c1 λ2 + (λ5 + λ6 )d ≤ γp λ4 λ3

(38)

λ1 = min(λmin Pr ), λ2 = max(λmax Pr ), ∀r ∈ M

(39)

λ3 = λmax R21 , λ4 = λmin R11

(40)

λ5 = max(λmax Q1 (i)), λ6 = max(λmin Q2 (i)), ∀i ∈ M

(41)

r

r

i

i

Then, the linear discrete switched singular system (2) is finite-time bounded with respect to (c1 , c2 , K f , R, σ (k), d) for any switching signal with MDADT ∗ τap ≥ τap =

K f ln µ p ln c2λλ1 − ln[ c1λλ2 + (λ5 + λ6 )d] − K f ln γ p 3

4

∗ =− τaq ≤ τaq

Where

, (∀p ∈ S)

ln µq (∀q ∈ U) γq

(42)

(43)

−1 Wi = A11 (i) − A12 (i)A−1 22 (i)A21 (i), Ti = A12 (i)A22 (i)G2 (i)

Ω1i = GT1 (i)Pi G1 (i) + F1T (i)Q1 (i)F1 (i) − γi Q1 (i), Ω2i = TiT Pi Ti + F2T (i)Q2 (i)F2 (i) − γi Q2 (i). Proof. Construct Lyapunov function as ¯ ¯ Vσ (k) (x(k), ¯ ω(k)) = x¯T (k)E¯σT (k) P¯σ (k) E¯σ (k) x(k) ¯ + ω¯ T (k)Q¯ 1 (σ (k))ω(k)

(44)

After matrix transformation, (44) is also equivalent to the following (45) ¯ Vσ (k) (x(k), ¯ ω(k)) = x1T (k)Pσ (k) x1 (k) + ω1T (k)Q1 (σ (k))ω1 (k) + ω2T (k)Q2 (σ (k))ω2 (k)

(45)

¯ Therefore, the difference equation for Vσ (k) (x(k), ¯ ω(k)) is as:   x1 (k)  T  ¯ + 1)) − γkVσ (k) (x(k), ¯ Vσ (k) (x(k ¯ + 1), ω(k ¯ ω(k)) = x1 (k) ω1T (k) ω2T (k) Ωσ (k) ω1 (k) ω2 (k) 7

(46)

Where

 T Wi PiWi − γi Pi Ωσ (k) =  GT1 (i)PiWi −TiT PiWi

 ∗ ∗ Ω1i ∗  −TiT Pi G1 (i) Ω2i

(∗ is the symmetric term, i = σ (k) ∈ M). We gain from (34), (35) that

¯ + 1)) ≤ γkVσ (k) (x(k), ¯ Vσ (k) (x(k ¯ + 1), ω(k ¯ ω(k))

(47)

Combining (36) and (37), one can lead to ¯ ¯ − )), σ (k) = p, σ (k− ) = r Vp (x(k), ¯ ω(k)) ≤ µ pVr (x(k ¯ − ), ω(k

(48)

¯ ¯ − )), σ (k) = q, σ (k− ) = p Vq (x(k), ¯ ω(k)) ≤ µqVp (x(k ¯ − ), ω(k

(49)

Therefore, for k ∈ [0, K f ]

¯ ¯ − 1)) ≤ γσ (k) µσ (k)Vσ (k−1) (x(k ¯ − 1)− ) ≤ · · · Vσ (k) (x(k), ¯ ω(k)) ≤ γσ (k)Vσ (k) (x(k ¯ − 1), ω(k ¯ − 1)− , ω(k ≤ ∏ µp σ p N

p

(

(0,k)

∏ µq σ q N

q

(0,k)

∏ γpp

T (0,k)

p

∏ γq q

T (0,k)

¯ Vσ (0) (x(0), ¯ ω(0))

q

)     = exp ∑ Nσ p (0, k) ln µ p + Tp (0, k) ln γ p + ∑ Nσ q (0, k) ln µq + Tq (0, k) ln γq p

q

¯ ×Vσ (0) (x(0), ¯ ω(0)) ) (

ln µq ln µ p ¯ + ln γ p Tp (0, k) + ∑ + ln γq Tq (0, k) Vσ (0) (x(0), ¯ ω(0)) ≤ exp ∑ τap τaq q p

By switching signal (43), the above inequalities can be rewritten as ( )
ln µ p ¯ ≤ exp ∑ + ln γ p Tp (0, k) Vσ (0) (x(0), ¯ ω(0)) τap p

(50)

Moreover,

¯ Vσ (k) (x(k), ¯ ω(k)) ≥ x1T (k)Pσ (k) x1 (k) ≥ λ1 x1T (k)x1 (k) ¯ Vσ (0) (x(0), ¯ ω(0)) ≤ λ2 x1T (0)x1 (0) + (λ5 + λ6 )d ≤

c1 λ2 + (λ5 + λ6 )d λ4

(51)

(52)

Which implies λ3 ¯ x1T (k)R21 x1 (k) ≤ λ3 x1T (k)x1 (k) ≤ Vσ (k) (x(k), ¯ ω(k)) λ1 ( )( )
ln µ p λ3 c1 λ2 ≤ exp ∑ + ln γ p Tp (0, k) + (λ5 + λ6 )d λ1 τap λ4 p

(53)

Since (38), it follows that

ln

c2 λ1 c1 λ2 − ln[ + (λ5 + λ6 )d] − ln γ p K f > 0 λ3 λ4

By switching signal (42), we can get

∑ p


 c1 λ2  ln µ p c2 λ1 + ln γ p Tp (0, k) ≤ ln − ln + (λ5 + λ6 )d τap λ3 λ4

(54)

Clearly, x1T (k)R21 x1 (k) ≤ c2 . This means the linear discrete switched singular system (2) is finite-time bounded with respect to (c1 , c2 , K f , R, σ (k), d). 8

system mode

2.5 2 1.5 1 0.5

0

50

100

150

Time (s)

State response

10 x1 x2

0 −10 −20 −30

0

50

100

150

Time (s)

Fig. 1. System Mode and Response of the System under the Designed MDADT Switching (Based on Theorem 1) ∗ Between Theorem 2 of [38] and Theorem 1 Table 1. Comparisons of τa1

µ1 1.4 1.5 1.6 1.6 1.7 1.8

ρ1 1.01 1.02 1.01 1.02 1.01 1.02

Theorem 2 of [38] 2.4826 3.2262 3.4678 3.7397 3.9151 4.6768

Theorem 1 in this paper 2.3149 2.9888 3.2331 3.4649 3.6516 4.3385

4. Numerical example Example 1. Consider the finite-time stability problem for linear discrete switched singular system (1) with two subsystems. The corresponding subsystem matrices are given below:    0 1 1 E1 = , A1 = 0 0 1

  −2 1 , E2 = −1 0

   1 5 2 , A2 = , c = 1, c2 = 35, K f = 20, R = I2 0 3 1 1

It is easy to check that subsystem 1 is finite-time stable, while the subsystem 2 is finite-time unstable. Using Theorem 1, if we choose ρ1 = 1.01, µ1 = 1.5, ρ2 = 2.04, µ2 = 0.6, the feasible solutions obtained are as follows:     70.5295 0 34.2455 0 ¯ ¯ P1 = , P2 = 0 70.5295 0 34.2455 Furthermore, the MDADT of two subsystems are obtained:

∗ ∗ τa1 = 3.0789, τa2 = 0.7165

Let τa1 = 8 > 3.0789, τa2 = 0.50 < 0.7165, we generate the switching sequence alternating between the two subsystems, the corresponding state responses of the linear switched singular systems are show in Fig.1. From Fig.1, we can see that the system is finite-time stable by utilizing the designed MDADT switching signals. In order to illustrate the proposed results, the comparison between Theorem 1 and Theorem 2 of [38] is shown in Table 1. [38] investigated the finite-time stability problem of discrete switched singular positive systems, based on the MDADT, by constructing the quasi-linear Lyapunov function. Obviously, as long as we choose appropriate parameters, we can get the tighter bounds on average dwell time. Example 2. Consider the finite-time bounded problem for linear discrete switched singular system (2) with two subsystems. The corresponding subsystem matrices are given below:             1 −3 1 1 5 2 18 −12 12 0 1 0 , A1 = , E2 = , A2 = , G1 = , G2 = E1 = 0 0 1 −1 0 0 3 1 4 −4 12 4     0.003 0 0.285 0.075 F1 = , F2 = c = 1, c2 = 30, K f = 15, R = I2 −0.007 0.01 0.21 0.07 1 9

system mode

2.5 2 1.5 1 0.5

0

2

4

6

8 10 Time (s)

12

14

16

18

State response

40 x1 x2

20 0 −20

0

2

4

6

8 10 Time (s)

12

14

16

18

Fig. 2. System Mode and Response of the System under the Designed MDADT Switching (Based on Theorem 2)

The subsystem 1 is finite-time stable, while the subsystem 2 is finite-time unstable. Using Theorem 2, if we choose γ1 = 3, µ1 = 2, γ2 = 15, µ2 = 0.5, the feasible solutions obtained are as follows:     0.0382 0 0.1060 0 ¯ ¯ P1 = , P2 = 0 0.0382 0 0.1060     0.7510 0 0.2019 0 Q¯ 1 = , Q¯ 2 = 0 0.6070 0 0.1177

Furthermore, the MDADT of two subsystems are obtained:

∗ ∗ τa1 = 0.4338, τa2 = 0.0462

Let τa1 = 8 > 0.4338, τa2 = 0.02 < 0.0462, we generate the switching sequence alternating between the two subsystems, the corresponding state responses of the linear discrete switched singular systems are show in Fig.2. From Fig.2, we can see that the system is finite-time bounded by utilizing the designed MDADT switching signals. 5. Conclusion In this paper, the finite-time stability problems for linear switched singular systems are investigated under MDADT switching in discrete case. The proposed MDADT switching signal is different from traditional MDADT switching and less restricted than the corresponding ADT switching. The switched strategy where fast switching and slow switching are respectively applied to unstable subsystems and stable subsystems is developed. Sufficient conditions are given for finite-time stability, finite-time bounded of the linear discrete switched singular systems with finite-time stable and finite-time unstable subsystems based on the proposed switching signals and dynamic decomposition technique. Finally, two numerical examples is given to illustrate the effectiveness of the theoretical results and tighter bounds are obtained than the existing results. 6. Acknowledgement This work is supported by the National Natural Science Foundation of China (grants No. 11401540, 61873245, 11602224 and 61772476), and the Foundation of Henan Educational Committee, China (grant No. 17A110031). The authors would like to thank the Editor, the Associate Editor, and anonymous reviewers for their constructive comments. References [1] J. L. Mills, A. A. Golddenberg, Force and position control of manipulators during constrained motion tasks, IEEE Trans. Robot. Autom. 15(1) (1989) 30-46. 10

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