Observer-based asynchronous H∞ control for a class of discrete time nonlinear switched singular systems

Observer-based asynchronous H control for a class of discrete time nonlinear switched singular systems

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Observer-based asynchronous H control for a class of discrete time nonlinear switched singular systems Yuchen Sun, Shuping Ma PII: DOI: Reference:

S0016-0032(20)30129-0 https://doi.org/10.1016/j.jfranklin.2020.02.045 FI 4457

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

9 September 2019 8 January 2020 19 February 2020

Please cite this article as: Yuchen Sun, Shuping Ma, Observer-based asynchronous H control for a class of discrete time nonlinear switched singular systems, Journal of the Franklin Institute (2020), doi: https://doi.org/10.1016/j.jfranklin.2020.02.045

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Observer-based asynchronous H∞ control for a class of discrete time nonlinear switched singular systems 1 Shuping Ma∗

Yuchen Sun

School of Mathematics, Shandong University, Jinan, Shandong, 250100, P. R. China Abstract This paper discusses the observer-based asynchronous H∞ control for switched singular systems with quadratically inner-bounded nonlinearities. Different from the Lyapunov functional is piecewise decreasing in synchronous control, a less conservative Lyapunov functional method is constructed, which allows to increase in the time interval when a subsystem is active. By utilizing average dwell time (ADT) approach and free-weighting matrix technique, sufficient conditions are obtained to guarantee the closed-loop systems be regular, causal, have a unique solution and be globally uniformly asymptotically stable (GUAS) with a weighted H∞ performance. Then less conservative linear matrix inequality (LMI) conditions are derived by using Finsler’s lemma and the observer design method is given. Finally, two examples are given to illustrate the effectiveness of the proposed methods.

Keywords: Switched singular system; Asynchronous H∞ control; Quadratically inner-bounded nonlinearities; The existence of unique solution.

1

Introduction

Singular systems are also called the differential (difference) algebraic systems, which can better describe physical systems such as aircraft modelings, circuit systems, economic systems, mechanical systems and many other practical models than ordinary systems, therefore many core control problems for singular systems have attracted widespread concern in recent years. The classical issues such as stability and stabilization problems for singular systems were discussed in the papers [1]-[3]. Inspired by the research of time-invariant systems, the strong observability for time-varying singular systems was analyzed in [4]. In [5], finite time stabilization and finite time H∞ control for a class of nonlinear Hamiltonian singular systems were discussed. As another type of system related to the research of this paper, switched systems represent a significant class of hybrid systems, which are formed by a finite number of subsystems and a switching rule orchestrates the subsystems at each moment [6]. Research on switched systems has received extensive attention due to the switched nature of many physical processes such as robot control systems, network systems, transportation control systems and so on. For example, Xiang and Xiao [7] discussed the GUAS issue for nonlinear switched systems under several types of switching including arbitrary switching, dynamical dwell time switching and asynchronous switching, respectively. The output reachable set estimation for discrete-time switched systems under persistent dwell time switching was investigated in [8]. The H∞ filter design for switched T-S fuzzy systems with persistent dwell time was studied in [9]. As for other problems, stability, asynchronous H∞ control [10], adaptive fuzzy tracking control [11], observer-based adaptive control [12]-[13] and asynchronous filtering design [14]-[16] for switched systems were discussed. When at least one subsystem in switched systems is a singular system, they become switched singular systems. Various techniques have been proposed to investigate the stability and stabilization problems for switched singular systems. Singular Markov jump systems as a special case of switched singular systems whose switching rules depend on a Markov 1 This work is supported by National Natural Science Foundation of China (61977042, 61473173), and the Foundation for Innovation Research Groups of National Natural Science Foundation of China (61821004). ∗ Corresponding author. E-mail address: [email protected] (Y. Sun), [email protected] (S. Ma).

1

chain have been widely studied in the past few decades [17]-[19]. To just mention a few, singular value decomposition method was proposed in the work [20]. In [21], switched singular systems were transformed to the dynamics decomposition form, and the stability is totally determined by the reduced-order switched dynamic subsystems and the switching law. In [22], under persistent dwell time switching, an asynchronous filter was constructed to make the error dynamic systems be exponentially admissible with a fixed H∞ performance. As for other issues, output feedback for switched singular systems was addressed in [23]. A two-phase method was investigated to study the robust H∞ control problem for continuous-time switched singular systems in [24], and in the work of Ma et al [25], stability, robust stabilization and H∞ control problems were considered for time delay switched singular systems by using novel state variables and restricted system equivalent transformation. Finite time H∞ control problem, filtering and fault estimation problem were addressed in [26] and [27] respectively. It is well known that nonlinearity is an inevitable phenomenon in production, which is undoubtedly a challenge to our research. T-S fuzzy technique is an effective method to deal with nonlinearities. The non-fragile H∞ filtering design for a class of T-S fuzzy systems was discussed in [28]-[29]. The adaptive fuzzy tracking control problem for nonlinear switched systems was investigated by Zhao et al in [30]. Recently, many works focus on several special types of nonlinearities such as Lipschitz nonlinearities [20], [31]-[34] and one-sided Lipschitz nonlinearities [35]-[39]. As described in [20], [31]-[34], the Lipschitz scalars are supposed to be positive, while the parameters in quadratically inner-bounded nonlinearity belong to the real number field, and quadratically innerbounded nonlinearity only needs to satisfy the second property of one-sided Lipschitz nonlinearity as discussed in [35]-[39], which implies that it is less conservative. Quadratically inner-bounded nonlinearities are widely applied in many well-known systems including the recurrent neural networks, the Chua’s circuit, the Lorenz systems and so on [35]. And the property such as the existence of the unique solution for switched singular systems is not parallel with previous results, which is one of the challenges in our study. Generally speaking, in real life, the state variables of a system are always partly unknown or unmeasurable, observer-based control provides a powerful tool to cope with this drawback [34]-[42]. To just list a few results, by using extended reciprocal convexity inequality, a less conservative nonlinear H∞ observer design method was given by Yang et al in [36]. In [40], two observer design methods were proposed based on exact and almost exact decoupling techniques respectively. Based on switched systems method, the observer-based dissipative control for wireless networked control systems was discussed in [41]. In many existing results about switched singular systems, such as the works [7], [20]-[27], [30]-[33], [40], [42], a common assumption is that the switching between subsystems and controllers is synchronous, which is quiet ideal in reality since network packet loss and channel blocking may occur during data transmission, the switching of the controllers has a certain delay in contrast with the system’s switching. As a result, asynchronous control accurately describes this phenomenon. Zhang et al in [10] considered asynchronous H∞ control for discrete time switched systems by utilizing a less conservative Lyapunov functional method, which allows to increase in the running time intervals, and the existence conditions of an asynchronous H∞ controller were derived via LMI formulation. In [14], the stabilization problem for switched systems with ADT was considered via asynchronous H∞ filtering. In a recent paper [38], an asynchronous observer design approach was given for discrete time switched systems with one-sided Lipschitz nonlinear terms. By using mode-dependent ADT approach, the stability and L1 -gain analysis for positive switched T-S fuzzy systems under asynchronous switching was considered in [43]. The problem dissipativity-based asynchronous control for discrete-time Markov jump systems with mixed time delays was studied in [44]. And asynchronous passive control for Markov jump systems was addressed in [45]. To the best of our knowledge, in the existing references, there are few corresponding results about observerbased asynchronous H∞ control for nonlinear switched singular systems, which motivates our study. The rest of the content is structured as follows. In Section 2, a mode-dependent asynchronous Luenberger-like observer is proposed, then several concepts and lemmas are recalled. In Section 3, sufficient conditions are derived, which en2

sure the regularity, causality, the existence and uniqueness of solution, and GUAS for nonlinear switched singular systems with an H∞ performance. However, the conditions obtained in Section 3 cannot be solved by Matlab toolbox, in Section 4, by means of equivalent transformation and Finsler’s lemma, strict LMI conditions are obtained for the existence of asynchronous H∞ observer. Finally, several examples are given to illustrate the effectiveness of the proposed approaches. The main contributions of this paper are summarized as the following three aspects. (i). Inspired by [10], by constructing a less conservative multiple Lyapunov functional which is allowed increasing in the certain initial period of each subsystem is active, the observer-based asynchronous H∞ stabilization conditions are obtained for the nonlinear switched singular systems. By regulating the upper bound of the increasing time of the Lyapunov functional, a smaller ADT condition is obtained compared with [15]. (ii). The stability conditions obtained in Theorem 1 not only guarantee the regularity and causality of the resulting closed-loop systems, but also ensure the existence of the unique solution for the nonlinear switched singular systems. (iii). Based on Finsler’s lemma, a less conservative observer design method is proposed compared with the Young’s inequality investigated in [15] and the method proposed in [42]. Furthermore, the approach proposed in this paper also can be applied to solve the GUAS issue for nonlinear switched systems with Lipschitz and one-sided Lipschitz nonlinear terms. Notation: In this paper, the following symbols are adopted if there is no special explanation. Rn denotes the n− dimensional Euclidean space. N stands for the natural number set. For a matrix A, AT and A−1 denote its transpose and inverse, respectively. For a square matrix A, A > 0(A ≥ 0) means that A is positive definite (semi-positive definite), He(A) = A + AT . I and 0 represent the identity matrix and zero matrix with appropriate dimensions respectively. Furthermore, in symmetric block matrices or long matrix expressions, ‘∗’ represents the symmetric element of a matrix. “ > ” indicates the element that we don’t need to know the specific form in the +∞ P T proof. diag{·} represents the diagonal matrix. A real vector v(k) ∈ l2 [0, +∞) means v (k)v(k) < +∞. k=0

hm, ni = mT n represents the inner product of two real vectors m and n. The symbol x(k − ) represents the value of the state at the left limit of time k. col(x1 , x2 , · · · , xn ) represents the column vector [xT1 , xT2 , · · · , xTn ]T , A⊥ represents a matrix with the property AA⊥ = 0.

2

System Description and Preliminaries In this section, consider the following discrete time nonlinear switched singular system    Ex(k + 1) = Aσ(k) x(k) + Bσ(k) u(k) + Dσ(k) w(k) + fσ(k) (x(k)), y(k) = Cσ(k) x(k),    z(k) = Fσ(k) x(k) + Gσ(k) w(k),

(1)

where x(k) ∈ Rn , u(k) ∈ Rm , y(k) ∈ Rp are respectively the state, controlled input and measurable output vectors, w(k) ∈ Rr represents the disturbance input that belongs to l2 [0, +∞), and z(k) ∈ Rnz is the signal to be estimated. E ∈ Rn×n is a singular matrix with rank(E) = re < n. The piecewise continuous function σ(k) : [0, +∞) → IM = {1, 2, · · · , M }, k 7→ σ(k) characterizes the switching signal at each moment k. In the following description, denote σ(k) = i which belongs to IM . For some given time k, the subsystem (Ai , Bi , Ci , Di , Fi , Gi ) is active, where Ai , Bi , Ci , Di , Fi , Gi are the prescribed given matrices with appropriate dimensions. The nonlinear term fi (x(k)) is assumed to satisfy the

3

quadratically inner-bounded condition, i.e. for ∀x, x ˆ ∈ Rn , there exist scalars ε1 , ε2 ∈ R such that T

2

(fi (x) − fi (ˆ x)) (fi (x) − fi (ˆ x)) ≤ ε1 kx − x ˆk + ε2 hx − x ˆ, fi (x) − fi (ˆ x)i .

(2)

Remark 1. It should be pointed out that condition (2) is a generalization of the Lipschitz and one-sided Lipschitz property. As stated in the work [36], the Lipschitz constant must be positive, however the scalars ε1 , ε2 in (2) also can be negative and zero. And quadratically inner-bounded condition contains the one-sided Lipschitz property as a special case. Therefore many properties such as the existence and uniqueness of solution for system (1) is not a parallel extension as before. As pointed in [38], the quadratically inner-bounded constants are smaller than Lipschitz constants, and quadratically inner-bounded nonlinearities are widely applied in many well-known systems including the recurrent neural networks, the Chua’s circuit, the Lorenz systems and so on [35]. Therefore considering systems with this type of nonlinearities is valuable in theory and practice. It is necessary to point out that it seems to have some limitations to choose mode-independent parameters ε1 and ε2 rather than modedependent ones, however it is easy to realize and has no essential impact on the issue we discuss. Furthermore, this selection will simplify the calculation. Definition 1. [20] System Ex(k + 1) = Ai x(k) (or the pair (E, Ai ) is said to be (i) regular if det(zE − Ai ) 6≡ 0 for ∀i ∈ IM ; (ii) causal if it is regular and degree (det(zE − Ai )) = rank(E) for ∀i ∈ IM .

Definition 2. [10, 24] For a switched system (1), let Nσ (k0 , k) be the number of the switching times over a time interval [k0 , k). For each switching signal σ and k ∈ [k0 , +∞), if there exist an integer N0 ≥ 0 and τa > 0 such that the following condition holds Nσ (k0 , k) ≤ N0 +

k − k0 , τa

then τa is called average dwell time (ADT) and N0 is the chatter bound. Definition 3. [7] System (1) with w(k) = 0 is said to be uniformly stable (US), if for ∀ε > 0, there exists a δ(ε) > 0, when k x(k0 ) k< δ, it has k x(k) k< ε. For ∀δ > 0, ∃k0 > 0, when ∀k > k0 , it has k x(k) k< ε, system (1) is said to be globally uniformly stable (GUS). Furthermore, system (1) is said to be globally uniformly asymptotically stable (GUAS) if it is GUS and satisfies lim x(k) = 0. k→+∞

Definition 4. [25] For a scalar γ > 0, system (1) is said to be GUAS with a weighted H∞ disturbance attenuation performance, if system (1) is GUAS when w(k) = 0 and satisfies the inequality ∞ X s=0

T

z (s)z(s) ≤

∞ X

γ 2 wT (s)w(s)

s=0

under zero initial condition. Remark 2. The intuitive meaning of ADT is that system switches no more than N0 times within an ADT interval. As stated in [6], [8]-[9], when N0 = 0 corresponds the case of no switching, N0 = 1 is exactly the switching signal with dwell time τa , and N0 > 1 is the ADT switching. The minimal γ in Definition 4 is called l2 induced gain, which can be viewed as a indicator of the influence of a system’s disturbance input on output, and it is a significant index to investigate the input-output performance of systems. Lemma 1. [19] For a scalar ρ, matrices M, N and symmetric matrix P with appropriate dimensions, if P + ρI > 0, then the following inequality holds −MT PM ≤ −MT N − N T M + ρMT M + N T (P + ρI)−1 N .

4

Lemma 2. (Finsler’s lemma)[46] For ζ(t) ∈ Rn , Ω = ΩT ∈ Rn×n and H ∈ Rm×n which satisfies rank(H) < n, the following items are equivalent. (i). ζ T (t)Ωζ(t) < 0 ∀Hζ(t) = 0, ζ(t) 6= 0; (ii). (H ⊥ )T ΩH ⊥ < 0;

(iii). ∃X ∈ Rn×m such that Ω + XH + (XH)T < 0. The objective of this paper is to design a controller based on mode-dependent asynchronous observer as the form

  ˆ(k + 1) = Aσ0 (k) x ˆ(k) + Bσ0 (k) u(k) + fσ0 (k) (ˆ x(k)) + Lσ0 (k) (ˆ y (k) − y(k)),  E x   

yˆ(k) = Cσ0 (k) x ˆ(k),

(3)

u(k) = Kσ0 (k) x ˆ(k),

subject to the augment system formed by (1) and (3) is GUAS with a prescribed weighted H∞ performance 0 γ, where x ˆ(k) ∈ Rn is the state of observer which is used to estimate x(k). σ (k) = σ(k − d(k)), where d(k) : [0, +∞) → [0, tmax ) represents the time-varying delays between the system and the observer. Supposing that tmax is the upper bound of d(k) which is given in advance. For the convenience of description, denote 0 σ (k) = l ∈ IM in the following discussion.

Remark 3. As is well known, the mode-dependent observer is a more effective tool to achieve the stabilization objective than mode-independent one since the information of each subsystem can be fully utilized. In many existing literatures, such as [7], [20]-[27], [30]-[33], [40], [42], a common assumption is that the modes between controllers and systems are consistent at each moment. In practice, affected by network transmission delay, the switching of controllers inevitably has a delay compared to the system, therefore it is meaningful to study asynchronous control. In particular, when IM = {1}, observer (3) degenerates into a mode-independent one, and if 0 σ(k) = σ (k) at all the time, in other words, there is no information transmission delay between systems and the controllers, asynchronous control turns into a synchronous one, in this sense, asynchronous switching control considered in this paper is more general. Remark 4. The observer (3) employed in this paper is based on the assumption that no uncertainty or disturbance affects the controller. A more general situation concerned in [28]-[29], the non-fragile filtering was designed for the T-S fuzzy systems with a prescribed H∞ performance. For further study, the non-fragile observer-based stabilization for system (1) can be viewed as a future topic. Remark 5. In this paper, what we consider is, in the process of asynchronous control, it allows the Lyapunov functional to increase in the running time interval when a subsystem is motivated, and its terminal value is less than the initial one. In other words, for a switching sequence 0 ≤ k0 < k1 < · · · < ks < ks+1 · · · < ∞, each time interval [ks , ks+1 ) is divided into two sets, one is the Lyapunov functional (or the energy functional) increasing marked as tup (ks , ks+1 ), correspondingly, the energy functional decreasing is marked as tdown (ks , ks+1 ). And they have S the following relationship: tup (ks , ks+1 ) tdown (ks , ks+1 ) = [ks , ks+1 ). The upper bound of tup (ks , ks+1 ) is denoted as tmax , which is given in advance. Define a new state vector ν(k) = col(x(k), x ˆ(k)), the closed-loop system formed by (1) and (3) can be expressed in the following form  ¯ ¯ i w(k) + f¯il (ν(k)), Eν(k + 1) = A¯il ν(k) + D    ∀k ∈ [ks , ks + tmax )  ¯ z(k) = Fi ν(k) + Gi w(k), ¯ ¯ i w(k) + f¯i (ν(k)),  Eν(k + 1) = A¯i ν(k) + D   ∀k ∈ [ks + tmax , ks+1 )  ¯ z(k) = Fi ν(k) + Gi w(k), 5

(4)

where

    A B K  i i l E ¯ = diag{E, E}, A¯il =  ,     −L C A + B K + L C  l i l l l l l     h i     h i  fi ( I 0 ν(k)) Di ¯ i =   , F¯i = Fi 0 , f¯il (ν(k)) =  h i , D  0 f ( ν(k))  0 I l      h i      fi ( I 0 ν(k)) A B K i i i    , f¯i (ν(k)) =  h i , A¯i =    −Li Ci Ai + Bi Ci + Li Ci fi ( 0 I ν(k))

and for ∀ν1 (k) = col(ς1 (k), ςˆ1 (k)), ν2 (k) = col(ς2 (k), ςˆ2 (k)), the nonlinearity f¯il (·) satisfies
T
f¯il (ν1 (k)) − f¯il (ν2 (k)) f¯il (ν1 (k)) − f¯il (ν2 (k)) h ih iT = fiT (ς1 (k)) − fiT (ς2 (k)) flT (ˆ ς1 (k)) − flT (ˆ ς2 (k)) fiT (ς1 (k)) − fiT (ς2 (k)) flT (ˆ ς1 (k)) − flT (ˆ ς2 (k)) T

≤ ε1 (ν1 (k) − ν2 (k)) (ν1 (k) − ν2 (k)) + ε2 hν1 (k) − ν2 (k), f¯il (ν1 (k)) − f¯il (ν2 (k))i,

(5a)

similarly,
T
f¯i (ν1 (k)) − f¯i (ν2 (k)) f¯i (ν1 (k)) − f¯i (ν2 (k))

T ≤ ε1 (ν1 (k) − ν2 (k)) (ν1 (k) − ν2 (k)) + ε2 hν1 (k) − ν2 (k), f¯i (ν1 (k)) − f¯i (ν2 (k))i.

3

(5b)

GUAS and H∞ Performance Analysis

In this section, sufficient conditions are obtained to guarantee that system (4) be regular, causal, have a unique solution and be GUAS with a weighted H∞ performance γ.

Theorem 1. For given scalars α ∈ (0, 1), β > 0, ρ > 0, µ > 1, γ¯ > 0, for ∀ i, l ∈ IM , if there exist ¯ i1 , M ¯ i2 , N ¯il1 , N ¯il2 with appropriate dimensions such that symmetric matrix P¯i , matrices M



¯ T P¯i E ¯ ≥ 0, E ¯ T P¯i E ¯ ≤ µE ¯ T P¯l E, ¯ ∀ i 6= l, i, l ∈ IM , E

(6) 

¯ i1 D ¯i ¯ i1 ¯ i1 A¯i − M ¯T M M 0 M i2 ¯ + He(M ¯ i2 A¯i ) + ρI M ¯ i2 D ¯i M ¯ i2 + ( ε2 − 1 )I F¯ T  ∗ −(1 − α)E P¯i E i  2  2 (7) ∗ ∗ −¯ γ I 0 GTi   < 0,  ∗ ∗ ∗ [(−ε1 + ρ)−1 − 1]I 0  ∗ ∗ ∗ ∗ −I   ¯il1 ) ¯il1 A¯il − N ¯T ¯il1 D ¯i ¯il1 P¯i − He(N N N N 0 il2   ε − 1 T 2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯  ∗ −(1 + β)E Pi E + He(Nil2 Ail ) + ρI Nil2 Di Nil2 + ( 2 )I FiT     0 GTi  ∗ ∗ −¯ γ2I   < 0, (i 6= l), (8)   −1 ∗ ∗ ∗ [(−ε1 + ρ) − 1]I 0   ∗ ∗ ∗ ∗ −I       

¯ i1 ) P¯i − He(M

¯T

then system (4) is regular, causal, has a unique solution and is GUAS with an H∞ performance no greater than p γ = (θtmax µ)N0 θ(tmax −1) γ¯ with ADT satisfying τa > τa∗ = −

lnµ + tmax lnθ , ln(1 − α) 6

(9)

where θ=

1+β , tmax , max tup (kl , kl+1 ). l∈N 1−α

Proof. The proof contains four parts, in the first two parts, the regularity, causality and the existence of the unique solution will be proved, and in the last two sections, the GUAS of system (4) with w(k) = 0 and the weighted H∞ performance γ are verified. Firstly, we prove that system (4) is regular and causal under conditions (6)-(8). We use (8) as an example, similar conclusion can be obtained from (7). From (8), it follows that "

¯il1 ) ¯il1 A¯il − N ¯T P¯i − He(N N il2 T ¯ ¯ ¯ ¯il2 A¯il ) + ρI ∗ −(1 + β)E Pi E + He(N

#

< 0,

(10)

"

# I A¯il let Ail = , pre- and post- multiplying ATil and Ail on (10) respectively, it yields 0 I "

> > T ¯ ¯ ¯ + ρI ¯ ¯ T P¯i E > Ail Pi Ail − (1 + β)E

#

< 0,

which implies ¯ T P¯i E ¯ < 0. A¯Til P¯i A¯il − (1 + β)E

(11)

Since rank(E) = re , there exist two nonsingular matrices U and V such that "

# I 0 2r e ¯ = U EV , 0 0 accordingly, define

"

# " # ¯il1 A¯il2 ¯i1 P¯i2 A P −T −1 U A¯il V = , U P¯i U = , T ¯ A¯il3 A¯il4 P¯i2 Pi4

multiplying V T and V on the both sides of (11) respectively, it yields T ¯ Ail2 + A¯Til2 P¯i2 A¯il4 + A¯Til4 P¯i4 A¯il4 < 0, A¯Til2 P¯i1 A¯il2 + A¯Til4 P¯i2

(6) implies that P¯i1 ≥ 0, thus A¯il4 is nonsingular, therefore system (4) is regular and causal.

The following part is to prove the existence and uniqueness of the solution for system (4). Let Qi = ∂fi (x(k)) ∂f (ˆ x(k)) ,Q = l , thus ∂x(k) x(k)=x1 (k) l ∂x ˆ(k) xˆ(k)=ˆx1 (k) Qil0 =

∂ f¯il (ν(k)) ν(k)=ν1 (k) = diag{Qi , Ql }. ∂ν(k)

According to Taylor’s formula, in a neighborhood of ν1 (k), f¯il (ν(k)) can be expanded into f¯il (ν(k)) − f¯il (ν1 (k)) = Qil0 (ν(k) − ν1 (k)) + o (k ν(k) − ν1 (k) k) ,

7

(12)

therefore,
f¯il (ν(k)) − f¯il (ν1 (k)) = (ν(k) − ν1 (k))T QTil0 Qil0 (ν(k) − ν1 (k))
+ 2(ν(k) − ν1 (k))T QTil0 o (k (ν(k) − ν1 (k)) k) + o k (ν(k) − ν1 (k)) k2 ,

f¯il (ν(k)) − f¯il (ν1 (k))


T

(13)

according to (5) and Lemma 1, the nonlinear term f¯il (ν(k)) satisfies


f¯il (ν(k)) − f¯il (ν1 (k))

 ≤ ε1 (ν(k) − ν1 (k))T (ν(k) − ν1 (k)) + ε2 ν(k) − ν1 (k), f¯il (ν(k)) − f¯il (ν1 (k)) ≤ −(ν(k) − ν1 (k))T (f¯il (ν(k)) − f¯il (ν1 (k))) − (f¯il (ν(k)) − f¯il (ν1 (k))T (ν(k) − ν1 (k)) + ρ(ν(k) − ν1 (k))T (ν(k) − ν1 (k)) + (f¯il (ν(k)) − f¯il (ν1 (k))T (−ε1 I + ρI)−1 (f¯il (ν(k)) − f¯il (ν1 (k))

 + ε2 ν(k) − ν1 (k), f¯il (ν(k)) − f¯il (ν1 (k)) . (14) f¯il (ν(k)) − f¯il (ν1 (k))


T

(13) together with (12) and (14), it is obtained that QTil0 Qil0 ≤ (

ε2 ε2 − 1)Qil0 + ( − 1)QTil0 + ρI + (−ε1 + ρ)−1 QTil0 Qil0 . 2 2

(15)



 0 0 0 I   Define Qil =  0 QTil0 0 , pre- and post- multiplying Qil and QTil on (8) respectively, it has 0 I 

 ¯il1 − N ¯T N ¯il1 (A¯il + Qil0 − N ¯T ) P¯i − N il1 il2 >  ∗ Ωil22   < 0, > >

where

  ¯il2 (A¯il + Qil0 ) + ( ε2 − 1)Qil0 + ρI + [(−ε1 + ρ)−1 − 1]QTil0 Qil0 − (1 + β)E ¯ T P¯i E, ¯ Ωil22 = He N 2

together with (15), it yields that

¯il2 (A¯il + Qil0 )) − (1 + β)E ¯ T P¯i E, ¯ Ωil22 ≥ He(N therefore

"

¯il1 − N ¯T ¯il1 (A¯il + Qil0 ) − N ¯T P¯i − N N il1 il2 ¯il2 (A¯il + Qil0 )) − (1 + β)E ¯ T P¯i E ¯ ∗ He(N

#

< 0.

¯ A¯il ) being regular and causal, it can be concluded that (E, ¯ A¯il + Qil0 ) is regular and causal. From the proof of (E, Similar to the proof in Theorem 1 [42], let V

−1

"

# " # " # x11 (k) x ˆ11 (k) Q1il0 Q2il0 −1 ν(k) = , V ν1 (k) = , U Qil0 V = , x21 (k) x ˆ21 (k) Q3il0 Q4il0 "

# " # ¯il1 (x11 (k), x21 (k)) ¯ i1 D f ¯i = UD , U f¯il (ν(k)) = ¯ , ¯ i2 D fil2 (x11 (k), x21 (k))

8

the first system of (4) is restricted system equivalent to

Since

 x (k + 1) = A¯ x (k) + A¯ x (k) + D ¯ i1 w(k) + f¯il1 (x11 (k), x21 (k)), 11 il1 11 il2 21 0 = A¯il3 x11 (k) + A¯il4 x21 (k) + D ¯ i2 w(k) + f¯il2 (x11 (k), x21 (k)).

(16)

∂(A¯il4 x21 (k) + f¯il2 (x11 (k), x21 (k))) 4 ¯ (x11 (k),x21 (k))=(ˆ x11 (k),ˆ x21 (k)) = Ail4 + Qil0 ∂x21 (k)

is nonsingular, from the implicit function theorem, it follows that in the neighborhood of each point x11 (k) = x ˆ11 (k), x21 (k) = x ˆ21 (k), there exists a unique function f˜il (x11 (k), w(k)) such that x21 (k) = f˜il (x11 (k), w(k)),

(17)

and ¯ i2 w(k) + f¯il (x11 (k), f˜il (x11 (k), w(k))), 0 = A¯il3 x11 (k) + A¯il4 f˜i (x11 (k), w(k)) + D x ˆ21 (k) = f˜il (ˆ x11 (k), w(k))), substituting (17) into the first equation of (16), it is obtained that the existence of the unique solution of the closedloop system (4) at each point col(ˆ x11 , x ˆ21 ). Next, the focus is to prove the GUAS for system (4) when w(k) = 0 and ADT satisfying (9). Based on Schur complement lemma, (8) is equivalent to  ¯ ¯il1 − N ¯T N ¯il1 A¯il − N ¯T ¯il1 D ¯i ¯il1 Pi − N N N il1 il2  ¯il2 D ¯ i + F¯ T Gi N ¯il2 + ( ε2 − 1 )I  ∗ Πi22 N  i ¯ il =  2 Φ  < 0,    ∗ ∗ −¯ γ 2 I + GTi Gi 0 −1 ∗ ∗ ∗ [(−ε1 + ρ) − 1]I

(18)

¯ + He(N ¯il2 A¯il ) + ρI + F¯ T F¯i . ¯ T P¯i E where Πi22 = −(1 + β)E i ¯ ¯ i w(k) + f¯il (ν(k)) = 0, hence for arbitrary matrices From system (4), it has −Eν(k + 1) + A¯il ν(k) + D ¯il1 , N ¯il2 with appropriate dimensions, one obtains N 



  ¯il1 N  N h  i   ¯il2  ¯ i I η(k) = η T (k)Nil η(k) He η T (k)   −I A¯il D  = 0,   0  

(19)

0

¯ with η(k) = col(Eν(k + 1), ν(k), w(k), f¯il (ν(k)) and 

 ¯il1 ) N ¯il1 A¯il − N ¯T N ¯il1 D ¯i N ¯il1 −He(N il2  ¯il2 A¯il ) N ¯il2 D ¯i N ¯il2  ∗ He(N   Nil =  .  ∗ ∗ 0 0  ∗ ∗ ∗ 0 By (14), (18) and (19), it can be obtained that ¯ il η(k) = η T (k)(Φil + Nil )η(k) = η T (k)Φil η(k) < 0, η T (k)Φ

9

(20)

where

¯  Pi 0 0 0   ¯ T P¯i E ¯ + F¯ T F¯i + ρI F¯iT Gi ( ε2 2− 1 )I  ∗ −(1 + β)E  i Φil =  . ∗  ∗ −¯ γ I + GTi Gi 0 −1 ∗ ∗ ∗ [(−ε1 + ρ) − 1]I

¯ T P¯i Eν(k), ¯ Choose the multiple Lyapunov functional as Vi (k) = ν T (k)E (20) is equivalent to

Vi (k + 1) − Vi (k) ≤ βVi (k) − z T (k)z(k) + γ¯ 2 wT (k)w(k), ∀k ∈ tup (ks , ks+1 ). Similarly, it can be obtained from (7) that Vi (k + 1) − Vi (k) ≤ −αVi (k) − z T (k)z(k) + γ¯ 2 wT (k)w(k), ∀k ∈ tdown (ks , ks+1 ). Therefore,

 −αV (k) − (z T (k)z(k) − γ¯ 2 wT (k)w(k)), ∀k ∈ t i down (ks , ks+1 ), ∆V (k) ≤ βVi (k) − (z T (k)z(k) − γ¯ 2 wT (k)w(k)), ∀k ∈ tup (ks , ks+1 ),

(21)

notice that w(k) = 0 and z T (k)z(k) ≥ 0, (21) becomes

 −αV (k), ∀k ∈ t i down (ks , ks+1 ), ∆V (k) ≤ βVi (k), ∀k ∈ tup (ks , ks+1 ).

Therefore,

(22)

Vσ(k) (k) ≤ (1 − α)tdown (ks ,k) (1 + β)tup (ks ,k) Vσ(ks ) (ks )

= (1 − α)(k−ks −tup (ks ,k)) (1 + β)tup (ks ,k) Vσ(ks ) (ks )

≤ (1 − α)k−ks −tmax (1 + β)tmax Vσ(ks ) (ks ) ≤ µ(1 − α)k−ks θtmax Vσ(ks−1 ) (ks− ) ≤ ···

≤ µNσ (k0 ,k) (1 − α)k−k0 (θtmax )Nσ (k0 ,k) Vσ(k0 ) (k0 ) 1

≤ µN0 θtmax [µ τa (1 − α)θ 1

when τa satisfies (9), µ τa (1 − α)θ GUAS.

tmax τa

tmax τa

](k−k0 ) Vσ(k0 ) (k0 ),

< 1, therefore Vσ(k) (k) converges to zero as k → ∞, hence system (4) is

Finally, we need to verify that the H∞ performance is no greater than γ under zero initial condition. Let Υ(k) = z T (k)z(k) − γ¯ 2 wT (k)w(k), it can be obtained by (21), Vσ(k) (k) ≤ (1 − α)k−kl θtmax Vσ(kl ) (kl ) −

kl +tX max −1

(1 − α)k−(kl +tmax ) (1 + β)kl +tmax −¯s−1 Υ(¯ s) −

s¯=kl

≤ µ(1 − α)k−kl θtmax Vσ(kl−1 ) (kl ) −

kl +tX max −1 s¯=kl

(1 − α)k−(kl +tmax ) (1 + β)kl +tmax −¯s−1 Υ(¯ s) −

10

k−1 X

s¯=kl +tmax

k−1 X

s¯=kl +tmax

(1 − α)k−¯s−1 Υ(¯ s)

(1 − α)k−¯s−1 Υ(¯ s)

≤ ···

≤ (1 − α)k−k0 (θtmax µ)Nσ (k0 ,k) Vσ(k0 ) (k0 ) − −

k0 +t max −1 X s¯=k0 k−1 X

s¯=k0 +tmax

(θtmax µ)Nσ (¯s,k) (1 − α)k−(k0 +tmax ) (1 + β)k0 +tmax −¯s−1 Υ(¯ s) (θtmax µ)Nσ (¯s,k) (1 − α)k−¯s−1 Υ(¯ s)

= (1 − α)k−k0 (θtmax µ)Nσ (k0 ,k) Vσ(k0 ) (k0 ) − −

k0 +t max −1 X s¯=k0 k−1 X

s¯=k0 +tmax

(θtmax µ)Nσ (¯s,k) (1 − α)k−¯s−1 θk0 +tmax −¯s−1 Υ(¯ s) (θtmax µ)Nσ (¯s,k) (1 − α)k−¯s−1 Υ(¯ s).

Under zero initial condition, it has k0 +t max −1 X s¯=k0

(θtmax µ)Nσ (¯s,k) (1 − α)k−¯s−1 θk0 +tmax −¯s−1 Υ(¯ s) +

k−1 X

s¯=k0 +tmax

(θtmax µ)Nσ (¯s,k) (1 − α)k−¯s−1 Υ(¯ s) ≤ 0,

multiplying (θtmax µ)−Nσ (k0 ,k) on the both sides of the above inequality, from the fact of µ−Nσ (k0 ,¯s) < 1, it is obtained that k0 +t max −1 X s¯=k0

≤

ˆ T (¯ s)z(¯ s) + (θtmax µ)−Nσ (k0 ,¯s) (1 − α)k−¯s−1 θz

k0 +t max −1 X s¯=k0

k−1 X

ˆγ 2 wT (¯ (1 − α)k−¯s−1 θ¯ s)w(¯ s) +

s¯=k0 +tmax

k−1 X

s¯=k0 +tmax

(θtmax µ)−Nσ (k0 ,¯s) (1 − α)k−¯s−1 z T (¯ s)z(¯ s)

(1 − α)k−¯s−1 γ¯ 2 wT (¯ s)w(¯ s),

where θˆ = θk0 +tmax −¯s−1 , since 1 < θˆ < θtmax −1 , together with (9), it is obtained that k−1 X

s¯=k0

(1 − α)k−¯s−1 z T (¯ s)z(¯ s) ≤

k−1 X

s¯=k0

(θtmax µ)N0 θtmax −1 (1 − α)k−¯s−1 γ¯ 2 wT (¯ s)w(¯ s),

therefore, +∞ k−1 X X

k=k0 s¯=k0

⇔ ⇔ ⇔

(1 − α)k−¯s−1 z T (¯ s)z(¯ s) ≤

+∞ X +∞ X

s¯=k0

k=k0 s¯=k0

(1 − α)k−¯s−1 z T (¯ s)z(¯ s) ≤

s¯=k0 k=¯ s ∞ X s¯=k0 ∞ X

+∞ k−1 X X

1 z T (¯ s)z(¯ s) ≤ α(1 − α)

z T (¯ s)z(¯ s) ≤

∞ X

∞ X

s¯=k0

(θtmax µ)N0 θtmax −1 (1 − α)k−¯s−1 γ¯ 2 wT (¯ s)w(¯ s),

+∞ X +∞ X

s¯=k0 k=¯ s

(θtmax µ)N0 θtmax −1 (1 − α)k−¯s−1 γ¯ 2 wT (¯ s)w(¯ s),

1 (θtmax µ)N0 θtmax −1 γ¯ 2 wT (¯ s)w(¯ s), α(1 − α)

γ 2 wT (¯ s)w(¯ s).

s¯=k0

It can be seen that the H∞ performance is no greater than γ =

p

(θtmax µ)N0 θ(tmax −1) γ¯ .

Remark 6. For discrete time linear singular systems or switched singular systems, the regularity guarantees

11

the existence and uniqueness of the solution for the systems, and the causality guarantees that the systems have the impulse-free unique solution. While for nonlinear singular systems, even if the system is regular and causal, the solution of system may not exist for compatible initial condition [32]. In [20], the existence and uniqueness of solution for switched singular systems with Lipschitz nonlinearities was proved by using implicit function theorem. As presented in Remark 1, quadratically inner-bounded nonlinearity is more general than the Lipschitz one since the parameters can be negative and zero. The existence and uniqueness of solutions for a wider class of nonlinear switched singular systems is ensured by the criteria proposed in Theorem 1, which is not a parallel extension of [20]. Remark 7. Conditions proposed in Theorem 1 guarantee system (1) be GUAS based on an asynchronous ¯ i1 , M ¯ i2 , N ¯il1 , N ¯il2 in Theorem 1 avoids the coupling of unobserver. The introduction of slack matrices M ¯ known variables Pi and system coefficient matrices. Since the nonlinearity condition (2) is less conservative than Lipschitz nonlinearity as mentioned in Remark 1, the approach in Theorem 1 can also be applied to solve the H∞ asynchronous control problem for switched singular systems with Lipschitz nonlinearities, the corresponding result is given in Corollary 1. In this case, fi (·) in system (1) satisfies Lipschitz property, for ∀x1 , x2 ∈ Rn , kfi (x1 ) − fi (x2 )k ≤ ci kx1 − x2 k, ci > 0. Corollary 1. For given scalars α ∈ (0, 1), β > 0, µ > 1, γ¯ > 0, if there exist symmetric matrices P¯i , matrices ¯ ¯ i2 , N ¯il1 , N ¯il2 with appropriate dimensions satisfy (6) and Mi1 , M 

 ¯ i1 ) ¯ i1 A¯i − M ¯T ¯ i1 D ¯i M ¯ i1 0 P¯i − He(M M M i2  ¯ + He(M ¯ i2 A¯i ) + ci I M ¯ i2 D ¯i M ¯ i2 F¯i  ¯ T P¯i E   ∗ −(1 − α)E   2 T  < 0,  ∗ ∗ −¯ γ I 0 Gi     ∗ ∗ ∗ −I 0   ∗ ∗ ∗ ∗ −I

 ¯il1 ) ¯il1 A¯il − N ¯T ¯il1 D ¯i N ¯il1 0 P¯i − He(N N N il2  ¯ + He(N ¯il2 A¯il ) + ci I N ¯il2 D ¯i N ¯il2 F¯ T  ¯ T P¯i E  ∗ −(1 + β)E i    2 T  < 0,  ∗ ∗ −¯ γ I 0 G i     ∗ ∗ ∗ −I 0   ∗ ∗ ∗ ∗ −I 

(23)

(24)

then system (4) with Lipschitz nonlinearities is GUAS with an H∞ performance no greater than γ when ADT satisfying (9). Proof. It can be obtained directly from Theorem 1. Remark 8. This study considers the asynchronous switching phenomenon between the observer and the system, however in [42] the switching between them is assumed to be synchronous. As a result, a less conservative Lyapunov functional method is provided in this paper. In the period of one subsystem is active, ∆V (k) satisfies (21), in other words, it allows increasing for certain time before decreasing as long as the final value is less than the initial value, which is less conservative than the Lyapunov functional is piecewise decreasing in synchronous switching of [42]. Furthermore, the observer-based finite time stabilization problem for a class of nonlinear switched singular system was considered in [42]. It should be pointed out that finite time stability and asymptotic stability discussed in Theorem 1 are two independent concepts, a system is finite time stable, however it may not be asymptotic stable and vice versa. Therefore, the results of this paper are not a parallel extension of those in [42]. Remark 9. For system (1), if E = I and fi (·) = 0, it turns into a switched linear system, asynchronous H∞ filtering for this system was designed in [15]. And in [34], the non-fragile asynchronous observer design for switched systems with Lipschitz nonlinearities was discussed, which implies that our result is more general.

12

4

Observer-based Controller Design

In this section, the design method of observer-based controller will be given. " First, we # make an equivalent transI 0 formation for (6)-(8) in Theorem 1, to this end, a transformation matrix X = = X −1 is introduced, and I −I denote   ˆ = X −1 EX ¯ = diag{E, E} = E, ¯  E         A + B K −B K i i l i l  Aˆil = X −1 A¯il X =  , (25) Ai − Al + Bi Kl − Bl Kl + Ll Ci − Ll Cl Al − Bi Kl + Bl Kl + Ll Cl          D A + Bi Ki −Bi Ki  ˆ i = X −1 D ¯ i =  i  , Aˆi = X −1 A¯i X =  i D  , Fˆi = F¯i X .    Di 0 Ai + Li Ci

Based on the above transformation, the following Theorem is given.

Theorem 2. For given parameters α ∈ (0, 1), β > 0, ρ > 0, µ > 1, γ¯ > 0, for ∀i, l ∈ IM , if there exist symmetric matrices Pi , matrices Mi1 , Mi2 , Nil1 , Nil2 with appropriate dimensions satisfy ¯ T Pi E ¯ ≥ 0, E ¯ T Pi E ¯ ≤ µE ¯ T Pl E, ¯ ∀i 6= l, i, l ∈ IM , E

 T ˆi Pi − He(Mi1 ) Mi1 Aˆi − Mi2 Mi1 D Mi1 0  ˆ i Mi2 + ( ε2 − 1 )X T X Fˆ T   ∗ Ξi22 Mi2 D i  2   2 T  < 0,  ∗ ∗ −¯ γ I 0 G  i    ∗ ∗ ∗ [(−ε1 + ρ)−1 − 1]X T X 0   ∗ ∗ ∗ ∗ −I   T ˆi Pi − He(Nil1 ) Nil1 Aˆil − Nil2 Nil1 D Nil1 0  ˆ i Nil2 + ( ε2 − 1 )X T X Fˆ T   ∗ Λil22 Nil2 D i  2   2  ∗ ∗ −¯ γ I 0 GTi    < 0,   −1 T ∗ ∗ ∗ [(−ε + ρ) − 1]X X 0   1 ∗ ∗ ∗ ∗ −I 

(26)

(27)

(28)

then system (4) is GUAS with an H∞ performance γ with ADT satisfying (9), where ¯ T Pi E ¯ + He(Mi2 Aˆi ) + ρX T X , Λil22 = −(1 + β)E ¯ T Pi E ¯ + He(Nil2 Aˆil ) + ρX T X . Ξi22 = −(1 − α)E

Proof. Multiplying diag{X T , X T , I, X T , I} and diag{X , X , I, X , I} on the both sides of (6)-(8) respec¯ i1 X = Mi1 , X T M ¯ i2 X = Mi2 , X T N ¯il1 X = Nil1 , X T N ¯il2 X = Nil2 , tively. Denote X T P¯i X = Pi , X T M together with (25), conditions (26)-(28) can be obtained directly by Theorem 1. Remark 10. After the above transformation, although Aˆil is still in a relatively complex form, Aˆi becomes an upper triangular matrix, which is simpler for controllers’ design compared with the original forms A¯i and A¯il . Remark 11. The bilinear matrix inequalities (BMIs) conditions proposed in Theorem 2 cannot efficiently be solved by using the standard numerical software, next we have to consider how to transform the BMIs into −1 −1 LMIs. In [15], in order to eliminate the nonlinear terms such as Pj,i , Pi,h+1 , the well-known Young’s inequality −1 T T −Mjτ Pj,i Mjτ ≤ Pj,i −Mjτ −Mjτ was utilized to perform the inequality amplification. In the following section, LMI conditions will be derived based on Lemma 2 without increasing the conservation of the conditions (26)-(28) obtained in Theorem 2. Theorem 3. For given scalars α ∈ (0, 1), β > 1, µ > 1, a family of parameters {pki }4k=1 , {qki }4k=1 , {rki }2k=1

13

"

# H1 H2 , positive definite matrix Pˆi = H2T H3

(∀i ∈ IM ), ρ > 0, γ¯ > 0, if there exist symmetric matrix H = " # Pˆi1 Pˆi2 > 0, invertible matrix Mi12 , matrices Mijk , Niljk , (j, k ∈ {1, 3}), Ti , Si , W1i with appropriate diPˆ T Pˆi3 i2

mensions such that (29)-(30) hold, system (4) is GUAS with an H∞ performance no greater than γ under ADT p ¯ = diag{R1 , R1 }, R1 ∈ R(n−re )×n satisfies R1 E = 0 and condition (9), where γ = (θtmax µ)N0 θ(tmax −1) γ¯ , R −1 −1 rank(R1 ) = n − re , the feedback gain matrices Ki = W1i Ti and the observer gains Li = Mi12 Si . "

¯ T Pˆi E ¯ ≤ µE ¯ T Pˆl E, ¯ (∀i 6= l, i, l ∈ IM ), E "

# Ψi1 Ψi2 < 0, ∗ He(−r1i W1i )

# Θil1 Θil2 < 0, ∗ He(−r2i W1l )

where  Ψi11 Ψi12 Ψi13 Ψi14 Ψi15 Mi11 Mi12 0      ∗ Ψi22 Ψi23 Ψi24 Ψi25 Mi13 Mi12 0    M B − p B W i11 i 1i i 1i  ∗ ∗ Ψi33 Ψi34 Ψi35 Ψi36 Ψi37 FiT        Mi13 Bi − p2i Bi W1i  ∗   ∗ ∗ Ψi44 Ψi45 Ψi46 Ψi47 0    T  =  , Ψi2 =  Mi21 Bi − p3i Bi W1i + r1i Ti  , 2 T  ∗ ∗ ∗ ∗ −¯ γ I 0 0 Gi   T   M B − p B W − r T  i23 i 4i i 1i 1i i   ∗ ∗ ∗ ∗ ∗ Ψ66 Ψ67 0    0   ∗ ∗ ∗ ∗ ∗ Ψ77 0   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I 

Ψi1

T Ψi11 = Pˆi1 − R1T H1 R1 − He(Mi11 ), Ψi12 = Pˆi2 − R1T H2 R1 − Mi12 − Mi13 , T T Ψi13 = Mi11 Ai + p1i Bi Ti − Mi21 , Ψi14 = Mi12 Ai + Si Ci − Mi23 − p1i Bi Ti , Ψi15 = (Mi11 + Mi12 )Di , Ψi22 = Pˆi3 − R1T H3 R1 − He(Mi12 ), T T Ψi23 = Mi13 Ai + p2i Bi Ti − Mi12 , Ψi24 = Mi12 Ai + Si Ci − Mi12 − p2i Bi Ti , T ˆ Ψi25 = (Mi13 + Mi12 )Di , Ψi33 = −(1 − α)E Pi1 E + He(Mi21 Ai + p3i Bi Ti ) + 2ρI, T − p4i Bi Ti + (p3i Bi Ti )T − ρI, Ψi34 = −(1 − α)E T Pˆi2 E + Mi12 Ai + Si Ci + ATi Mi12 Ψi35 = (Mi21 + Mi12 )Di , Ψi36 = Mi21 + (ε2 − 1)I, Ψi37 = Mi12 − ε2 2− 1 I, Ψi44 = −(1 − α)E T Pˆi3 E + He(Mi12 Ai + Si Ci − p4i Bi Ti ) + ρI, Ψi45 = (Mi23 + Mi12 )Di , Ψi46 = Mi23 − ε2 2− 1 I, Ψi47 = Mi12 + ε2 2− 1 I, Ψ66 = 2((−ε1 + ρ)−1 − 1)I, Ψ67 = −((−ε1 + ρ)−1 − 1)I, Ψ77 = ((−ε1 + ρ)−1 − 1)I,



Θil1

 Θil11 Θil12 Θil13 Θil14 Θil15 Nil11 Ml12 0    ∗ Θil22 Θil23 Θil24 Θil25 Nil13 Ml12 0     ∗ ∗ Θil33 Θil34 Θil35 Θil36 Ψl37 FiT     ∗ ∗ ∗ Θil44 Θil45 Θil46 Ψl47 0    = , 2 T  ∗  I 0 0 G ∗ ∗ ∗ −¯ γ i    ∗ ∗ ∗ ∗ ∗ Ψ66 Ψ67 0      ∗ ∗ ∗ ∗ ∗ Ψ77 0   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I

14

(29) (30)

 Nil11 Bi + Ml12 (Bi − Bl ) − q1l Bl W1l     Nil13 Bi + Ml12 (Bi − Bl ) − q2l Bl W1l   T , = N B + M (B − B ) − q B W + r T l12 i l 3l l 1l 2l l   il21 i    Nil23 Bi + Ml12 (Bi − Bl ) − q4l Bl W1l − r2l TlT  0 

Θil2

Θil11 Θil13 Θil15 Θil23 Θil33

T = Pˆi1 − R1T H1 R1 − He(Nil11 ), Θil12 = Pˆi2 − R1T H2 R1 − Ml12 − Nil13 , T ˆ il − N , Θil14 = −q1l Bl Tl + Ml12 Al + Sl Cl − N T , = q1l Bl Tl + Nil11 Ai + M il21 il23 = (Nil11 + Ml12 )Di , Θil22 = Pˆi3 − R1T H3 R1 − He(Ml12 ), Θil25 = (Nil13 + Ml12 )Di , ˆ il − M T , Θil24 = −q2l Bl Tl + Ml12 Al + Sl Cl − M T , = q2l Bl Tl + Nil13 Ai + M l12 l12   ˆ il + 2ρI, = −(1 + β)E T Pˆi1 E + He q3l Bl Tl + Nil21 Ai + M ˆ T − ρI, = −(1 + β)E T Pˆi2 E − q4l Bl Tl + q3l T T B T + Ml12 Al + Sl Cl + AT N T + M

Θil34 i l l il23 il Θil35 = (Nil21 + Ml12 )Di , Θil36 = Nil21 + (ε2 − 1)I, Θil44 = −(1 + β)E T Pˆi3 E + He (−q4l Bl Tl + Ml12 Al + Sl Cl ) + ρI, Θil45 = (Nil23 + Ml12 )Di , ˆ il = Ml12 (Ai − Al ) + Sl (Ci − Cl ). Θil46 = Nil23 − ε2 2− 1 I, M ¯T H R ¯ into (26), it yields Proof. Substitute Pi = Pˆi − R ¯ T Pi E ¯=E ¯ T (Pˆi − R ¯ T H R) ¯ E ¯=E ¯ T Pˆi E ¯ ≥ 0, E ¯ T Pi E ¯=E ¯ T (Pˆi − R ¯ T H R) ¯ E ¯=E ¯ T Pˆi E ¯ ≤ µE ¯ T Pˆl E ¯ = µE ¯ T (Pˆl + R ¯ T H R) ¯ E ¯ = µE ¯ T Pl E. ¯ E denote

"

# " Mi11 Mi12 Mi21 Mi1 = , Mi2 = Mi13 Mi12 Mi23 " # " Nil11 Ml12 Nil21 Nil1 = , Nil2 = Nil13 Ml12 Nil23

# Mi12 , Mi12 # Ml12 , Ml12

(31)

(32)

substituting (31) into (27), it is obtained that
¯ iK ¯ i < 0, ˆ i + He M Ψ

where 

 ˆ i13 Ψ ˆ i14 Ψi15 Mi11 Mi12 0 Ψi11 Ψi12 Ψ  ˆ i23 Ψ ˆ i24 Ψi25 Mi13 Mi12 0       ∗ Ψi22 Ψ    M B 0 i11 i T ˆ i33 Ψ ˆ i34 Ψi35 Ψi36 Ψi37 F   ∗ ∗ Ψ     i    Mi13 Bi   0   ∗ ˆ i44 Ψi45 Ψi46 Ψi47 0      ∗ ∗ Ψ   ˆi =  ¯ i =  Mi21 Bi  , K ¯ iT =  K T  , Ψ ,M i     2 T  ∗ ∗ ∗ ∗ −¯ γ I 0 0 Gi         Mi23 Bi   −KiT   ∗ ∗ ∗ ∗ ∗ Ψ66 Ψ67 0    0 0   ∗ ∗ ∗ ∗ ∗ Ψ77 0   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I ˆ i13 Ψ ˆ i33 Ψ ˆ i34 Ψ ˆ Ψi44

T ˆ i14 = Mi12 Ai − M T , Ψ ˆ i23 = Mi13 Ai − M T , Ψ ˆ i24 = Mi14 Ai − M T , = Mi11 Ai − Mi21 ,Ψ i23 i12 i12 = −(1 − α)E T Pˆi1 E + He(Mi21 Ai ) + 2ρI, T = −(1 − α)E T Pˆi2 E + Mi12 Ai + ATi Mi12 − ρI − Bi Ti , T ˆ = −(1 − α)E Pi3 E + He(Mi12 Ai ) + ρI,

15

(33)

(33) can be rewritten as

h

" #" # i Ψ ˆi M ¯i I ¯T < 0. I K i ¯i ∗ 0 K

(34)

Applying Lemma 2, (34) is equivalent to " let

# " # ! h i ˆi M ¯i Ψ Wi ¯ + He < 0, Ki −I ¯T 0 M r1i W1i i

h i T T T T T T T T WiT = p1i W1i Bi p2i W1i Bi p3i W1i Bi p4i W1i Bi 0 ,

then the first inequality of (30) can be obtained.

Similar to the above process, substitute (32) into (28), it yields ¯ il + He(N ¯il K ¯ l ) < 0, Θ

(35)

where 

 ˆ il13 Θ ˆ il14 Θil15 Nil11 Ml12 0 Θil11 Θil12 Θ  ˆ il23 Θ ˆ il24 Θil25 Nil13 Ml12 0     ∗ Θil22 Θ    N B + M (B − B ) il11 i l12 i l T ˆ il33 Θ ˆ il34 Θil35 Ψl36 Θil37 F   ∗ ∗ Θ   i    Nil13 Bi + Ml12 (Bi − Bl )   ∗  ˆ   ∗ ∗ Θil44 Θil45 Ψl46 Θil47 0  ¯ ¯ il =  Θ Nil21 Bi + Ml12 (Bi − Bl )    , Nil =   , 2 T  ∗ ∗ ∗ ∗ −¯ γ I 0 0 Gi       Nil23 Bi + Ml12 (Bi − Bl )   ∗ ∗ ∗ ∗ ∗ Ψ66 Ψ67 0    0   ∗ ∗ ∗ ∗ ∗ Ψ77 0   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I (2,1) T ˆ il13 = Nil11 Ai + Ml12 A(2,1) − N T , Θ ˆ Θ + Ml12 Ail − Ml12 , il  il21 il23 = Nil13 Ai  (2,1) T ˆ ˆ + 2ρI, Θil33 = −(1 + β)E Pi1 E + He Nil21 Ai + Ml12 Ail T ˆ il24 = Ml12 Al + Ml12 Ll Cl − M T , ˆ il14 = Ml12 Al + Ml12 Ll Cl − N , Θ Θ l12 il23 ˆ il34 = −(1 + β)E T Pˆi2 E + Ml12 Al + Ml12 Ll Cl + AT N T + (A(2,1) )T M T − ρI, Θ i il23 ll2 il ˆ il44 = −(1 + β)E T Pˆi3 E + He (Ml12 Al + Ml12 Ll Cl ) + ρI, A(2,1) = Ai − Al + Ll (Ci − Cl ). Θ il

(35) can be rewritten as

which equals to

Let

h "

"

i Θ ¯ il N ¯il I KlT ∗ 0

#"

I ¯l K

#

< 0,

# " # ! h i ¯ il N ¯il ¯ il Θ W ¯ l −I + He < 0. K ∗ 0 r2i W1l

h i ¯ ilT = q1l W T B T q2l W T B T q4l W T B T q1l W T B T 0 , W 1l l 1l l 1l l 1l l

by using Lemma 2 and denoting W1l Kl = Tl and Mi12 Li = Si , the second inequality of (30) is obtained. The −1 −1 feedback controller and observer gains are Ki = W1i Ti , Li = Mi12 Si . Remark 12. Finsler’s lemma is introduced to " # transform the BMIs into LMIs. Generally speaking, Mi1 can Mi1 Mi2 be selected in a free form like Mi1 = . Compared with the diagonal form of the free matrices when Mi3 Mi4 using Young’s inequality in Theorem 3.1 of [31], this selection increases the feasibility of conditions and reduces

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the conservatism. It should be pointed out that using Finsler’s lemma to reduce the conservativeness is at the cost of increasing computational complexity, which contains the increasing of dimensions of matrices and the number of variables. To overcome this problem to some extent, in Theorem 3, the slack matrices are selected in the special forms (31)-(32). Although the proposed stability conditions require more computational burdens than [31] due to the introduction of slack variables, such concerns have recently been alleviated by the development of computer technologies. In addition, if p1i = p2i = p3i = p4i = 1, q1i = q2i = q3i = q4i = 1, the result obtained in Theorem 3 degenerates into the matrix decoupling method proposed in [42]. In other words, the introduction of the adjustable scalars makes our result less conservative than [42]. The scalars pmi , qmi , rmi (m = 1, 2, 3, 4) should be given in advance, if not, the conditions in Theorem 3 are not LMIs, the selection of the optimal values of these scalars can referred to Remark 5 in [47].

5

Illustrative Examples

In this section, an electronic circuit example and a numerical example are given to verify the correctness and effectiveness of the methods proposed in this paper. Example 1. Consider a nonlinear electrical circuit which is depicted in Figure 1, it consists of two groups of nonlinear electrical resistances and inductances. The switch controls the entire circuit system to switch from one location to another in a random manner. In this example, it is assumed that the switching rule satisfies ADT switching law (9). Let iR (t) represents the current passing through the nonlinear electrical resistance R1 (or R2 if the switch is on location 2), iL (t) is the inductance of L1 (or L2 ). Denote x1 (t) = iL (t), x2 (t) = iR (t), and y1 (t) = 3x1 (t) − x2 (t), y2 (t) = −0.5x1 (t) + x2 (t) represent two output signals. Based on Kirchhoff Laws, by selecting a sampling interval, such as Ts = 1s, as described in [17], this electrical circuit can be modeled as    Ex(k + 1) = Ai x(k) + Bi u(k) + Di w(k) + fi (x(k)), y(k) = Ci x(k),    z(k) = Fi x(k) + Gi w(k).

Let

"

# −1 2 E= , 0 0

subsystem 1: "

# " # " # h i h i −2 3 2 0.5 A1 = , B1 = , C1 = 3 −1 , D1 = , F1 = 3 −1 , G1 = −1, 3 1 0.3 −0.4 subsystem 2: "

# " # " # h i h i 1 0 3 1 A2 = , B2 = , C2 = −0.5 1 , D2 = , F2 = −2 3 , G2 = 1. 3 −2 0.5 0.2 Supposing that the nonlinearities f1 (x(k)) = f2 (x(k)) = 3 ε2 . quadratically inner-bounded condition if ε1 ≥ − 16 2

"

# −x1 (x21 + x22 ) , as stated in [42], fi (·) satisfies −x2 (x21 + x22 )

Figures 2-3 give the open-loop trajectories of each subsystem. It can be seen that the open-loop of the two

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subsystems are both unstable. We choose the parameters α = 0.5, β = 0.4, ε1 = 2, ε2 = −5, λ1 = 0.5, ρ1 = ρ2 = 0.5, γ¯ = 3, µ = 1.5, p11 = 1.5, p21 = 0.5, p31 = −1, p41 = 1.5, p12 = 1.5, p22 = 1, p32 = −1.5, p42 = 1, q11 = 0.3, q21 = 1.5, q31 = 2, q41 = 1, q12 = 1.5, q22 = 1, q32 = 2, q42 = 1, r11 = r12 = r21 = r22 = 2. Solving conditions (29)-(30), it is feasible, and the feedback gain matrices and observer gains are obtained as # " # −1.3935 −0.0981 K1 = −0.1358 − 0.1757 , K2 = 0.1368 − 0.1164 , L1 = , L2 = . 0.7026 0.2191 h

i

h

i

"

According to (9), it can be calculated that τa = 3.5558. If we choose the maximum controller delay tmax = 2s. Figure 4 gives the switching sequences satisfying ADT constraint (9), the solid curve indicates the switching of the system, and the dotted curve indicates the observer switching law. In order to intuitively show the relationship between ADT and attenuation rates α, β, Table 1 gives the tendency of ADT when α, β take different values with µ = 1.5, it can be obtained that the ADT is negatively correlated with the change of α and positively correlated with the change of β. And τa becomes quiet large when α → 0. h iT h iT Set the initial states x(0) = 0.1 − 0.5 , x ˆ(0) = 0.3 1.5 , Figure 5 shows the trajectories of the closed-loop system (4), where x3 and x4 in this figure represent the state of observer x ˆ1 and x ˆ2 , respectively. It can be found that the closed-loop system is GUAS. Finally, by optimizing conditions (29)-(30), the H∞ performance can be obtained as γ = 5.1809. Figure 6 gives the rate of the output energy relative to the disturbance w(k) = e−k which belongs to l2 [0, ∞).

Example 2. If E = I, and the nonlinearity f (·) = 0, system (1) degenerates to the system which was discussed in [15], in this example, the system parameters were borrowed from Example 2 in [15], "

# " # " # " # 1.05 0.3 −0.6 0 0.4 0.23 E = I2 , A1 = , A2 = , D1 = , D2 = , G1 = G2 = 0, −0.2 0.4 −0.1 0.3 −0.2 −0.1 h i h i h i h i C1 = 0.56 0.2 , C2 = −0.16 0.27 , F1 = 0.5 − 0.3 , F2 = −0.2 0.25

it is feasible by solving Corollary 1 in this paper with ci = 0, α = 0.8, µ = 1.5, which implies that the method proposed in this paper can also solve the problem in [15]. Furthermore, in [15], the ADT constraint was τa > (lnα − lnβ)tmax − ln(µ1 µ2 ) . Table 2 gives the comparison of ADT (9) and that in [15] under different β when lnα α = 0.7, it can be found that ADT obtained in this paper is smaller than [15], which implies that the results of this paper is less conservative. Table 1: The relationship between τa and α, β with µ = 1.5 τa β α 0.01 0.1 0.5 0.9

0.4

0.9

1.5

109.3009 12.2354 3.5558 2.4683

170.0713 18.0323 4.4370 2.7336

224.6838 23.2418 5.2288 2.9720

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Table 2: Comparison of the ADT when α = 0.7 0.5 1.5 2 3.0103 3.8589 4.1618 infeasible 5.4104 7.0235

β ADT (9) ADT in [15]

1020

3.5

x1 x2

3 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5

0

5

10

15

20

25

30

Time

Figure 1: A circuit diagram

Figure 2: The open-loop trajectory of subsystem 1

106

7

switching sequence of system switching sequence of controller

x1 x2

6

2

5

4

3

2

1

0

1 0

5

10

15

20

25

0

30

10

20

30

40

50

Time

Figure 3: The open-loop trajectory of subsystem 2

Figure 4: The switching sequence

2

6 x1 x2 x3 x4

1.5

γ=5.1809 5

1

4

0.5

3

0

2

-0.5

1

-1

0

5

10

15

20

25

0

30

0

5

10

15

20

25

Time

Figure 5: The trajectory of the closed-loop system

Figure 6: The H∞ performance

19

30

6

Conclusions

As a result, the observer-based asynchronous H∞ control for a class of nonlinear discrete-time switched singular systems with quadratically inner-bounded nonlinearities was investigated. By using ADT method and freeweighting matrices, less conservative conditions are obtained to guarantee the resulting systems be regular, causal, have a unique solution and be GUAS with a weighted H∞ performance. By means of Finsler’s lemma, strict LMI conditions for the existence of observer-based asynchronous H∞ control for discrete-time singular switched systems are obtained. For further consideration, how to design a reduced-order observer to realize the stability of the resulting closed-loop systems can be viewed as a future topic.

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