Energy-to-peak filtering for T–S fuzzy systems with Markovian jumping: The finite-time case

Neurocomputing 168 (2015) 348–355

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Energy-to-peak filtering for T–S fuzzy systems with Markovian jumping: The finite-time case Shuping He n School of Electrical Engineering and Automation, Anhui University, Hefei 230601, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 23 January 2015 Received in revised form 1 April 2015 Accepted 25 May 2015 Available online 6 June 2015

This paper studies the robust finite-time energy-to-peak filtering problem for a class of uncertain nonlinear Markovian jumping systems (MJSs) with energy constraint external disturbances. The Takagi– Sugeno (T–S) fuzzy model is employed to represent the nonlinear MJSs. By selecting the appropriate Lyapunov–Krasovskii functional, a sufficient condition is derived such that the filtering error dynamic fuzzy MJSs are finite-time stable and have a prescribed level of L2–L1 disturbance attenuation in a finite time-interval. The sufficient condition on the existence of robust finite-time energy-to-peak fuzzy filtering criterion is formulated in the form of linear matrix inequalities and the designed finite-time energy-to-peak fuzzy filter is described as an optimization one. A numerical example is given to illustrate the effectiveness of the proposed design approach. & 2015 Elsevier B.V. All rights reserved.

Keywords: Markovian jumping systems Finite-time stable Energy-to-peak filtering Uncertainties Takagi–Sugeno fuzzy models

1. Introduction In general, the model of a practical dynamic system always contains some types of uncertainties and nonlinearities. It stems from the fact that, we can not obtain the exact mathematical model of a practical dynamics due to the complexity process, the environmental noises, time-varying parameters and the difficulties if measuring nonlinear and uncertain parameters, etc. Due to these nonlinearities and uncertainties, it is difficult to design the relevant controller and filter by using general nonlinear approaches. A successful approach to overcome this kind of difficulties is to model the considered nonlinear systems as Takagi–Sugeno (T–S) fuzzy systems, which are locally linear dynamical systems connected by IF–THEN rules. It can provide an effective solution to the controlled plants that are complex, uncertain, ill-defined, and have available qualitative knowledge from domain experts for the controller and filter design. Moreover, it has been proved that a linear T–S fuzzy model is a universal approximator of any smooth nonlinear system on a compact set [1]. Over the past few decades, T–S fuzzy systems have been intensively studied and many significant advances have been achieved; see the references [1–6]. In fact, many dynamical systems have variable parameters or structures subject to random abrupt changes due to, for instance, sudden environment changes, system noises and subsystem switching, etc. But some general dynamics systems can not represent these random abrupt cases. As a special kind of stochastic systems

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http://dx.doi.org/10.1016/j.neucom.2015.05.091 0925-2312/& 2015 Elsevier B.V. All rights reserved.

which involve both time-evolving and event-driven mechanisms, Markovian jump systems (MJSs) may be employed to model these phenomena. It forms a class of processes which describe random abrupt changes of parameters or structures according to an underlying Markov chain. In this kind of stochastic systems, the dynamics of jump modes and continuous states are respectively modeled by finite state Markov chains and differential equations. The existing results about MJSs cover a large variety of problems such as stochastic stability, stochastic stabilization, robust control, robust state estimation and fault detection; see, for example, [7–12] and the references therein. On another research front, the energy-to-peak filtering (L2–L1 for continuous-time system and l2–l1 for discrete-time system) is an important issue in the area of state estimation [13,14]. The energy-to-peak performance criterion was firstly discussed in [15], where it is shown that the energy-to-peak gain can be computed from the controllability Grammian and the state-space representation of the system. In the energy-to-peak filter design, the problem is to find an estimator such that the dynamic error system is stable and satisfies an energy-to-peak performance constraint gain from L2 to L1 (l2 to l1 for discrete-time system) is below a prespecified level. Since the pioneering work of energyto-peak filtering for linear-invariant systems [16], this filtering scheme has regained increasing interest, see [17–19] and the references therein. In the work of [20], the reduced-order energy-to-peak filter design was designed for discrete-time MJSs. The extensions to the discrete time-delayed MJSs were given by [21] and [22]. In [21], the authors studied the energy-to-peak filtering problem for MJSs with time-varying delays; while in [22], the l2–l1 filter was designed for a class of MJSs with

S. He / Neurocomputing 168 (2015) 348–355

mode-dependent time-delays. The authors in [23] investigated the energy-to-peak filtering problem for a class of continuous-time stochastic time-delay systems. Very recently, the exponential L2–L1 filtering problems for distributed delay MJSs were respectively solved in [24] and [25]. However, to the best of the authors’ knowledge, very little effort has been made to study the energyto-peak filtering for the nonlinear MJSs with respect to the finitetime interval. These motivate us to research this topic. As we all know, the robust controller and filter design schemes pay more attention to the steady-state behavior of controlled dynamics over an infinite time interval by Lyapunov stability theory, and they are inclined to deal with the asymptotic property of system trajectories. But in some practical process, the main attention may be related to the behavior of the dynamical systems over a fixed finite-time interval, for instance, biochemistry reaction system, communication network system and robot control system. The concept of finite-time stability (or short-time stability [26]) related to these transient performances of control dynamics dates back to the Sixties, and some earlier results on this transient performance can be found in [27,28]. Recently, with the aid of linear matrix inequalities (LMIs) techniques [29], more issues of finite-time concepts have been proposed for linear continuous-time or discrete-time control system, such as finite-time boundedness and finite-time control. For more results on this matter, we refer readers to [30–33] and the references therein. But in each case above, most results are related to linear dynamical systems, and very few literatures consider the filtering problems for nonlinear MJSs. By using the T–S fuzzy modeling technology, we have studied the problem of finite-time energy-to-peak filtering problem for uncertain nonlinear MJSs in this paper. Different with the general filter design [11,13,14,16,17,18,21–24], we add two different terms in the filter index. By selecting the appropriate Lyapunov–Krasovskii functional and using the relevant Newton–Leibniz calculus formula, a sufficient condition is established on the existence of the finite-time filter such that the filtering error dynamical fuzzy MJSs are finite-time bounded and satisfies a prescribed level of L2– L1 disturbance attenuation in a finite time-interval. The finitetime term can be guaranteed by the relevant inequality constraint. The finite-time filter design criterion is presented in the form of LMIs, and can be formulated as an LMIs optimization algorithm. At last, a numerical example is provided to illustrate the effectiveness of the developed techniques.

2. Preliminaries 2.1. Notations The adopted notation of this paper is as follows: The symbols ℜn and ℜnm stand for an n-dimensional Euclidean space and the set of all n  m real matrices, respectively,
AT andA  1 denote the matrix transpose and matrix inverse, diag A B represents the block-diagonal matrix of A and B, σ max ðC Þ denote the maximal eigenvalue of a positive-define matrix C, ‖n‖ denotes the Euclidean norm of vectors, Efng denotes the mathematics statistical   expectation of the stochastic process or vector, Ln2 0 1 is the space of  n dimensional square integrable function vector over 0 1 , P o 0 (or P 4 0) stands for a negative-definite (or positive-define) matrix, I is the unit matrix with appropriate dimensions, 0 is the zero matrix with appropriate dimensions, n means the symmetric terms in a symmetric matrix. 2.2. Problem formulation Given a probability space ðΩ; F; ρÞ where Ω is the sample space, F is the algebra of events and ρ is the probability measure defined

349

on F. Let the random form process fr t ; t Z 0g be the continuoustime discrete-state Markovian stochastic process taking values in a finite
set Λ ¼ f1; 2;  …; N g with transition probability matrix ρr ¼ ρrk ðt Þ; r; k A Λ given by ( π rk Δt þ oðΔt Þ; rak
 ρr ¼ ρrk ðt Þ ¼ ρr r t þ Δt ¼ kj r t ¼ r ¼ ð1Þ 1 þ π rr Δt þ oðΔt Þ; r ¼ k where Δt 4 0 and limoðΔt Þ=Δt-0. π rk Z0 is the transition probΔt↓0

ability rates from mode r at time t to mode kðr a kÞ at time t þ Δt, N P π rk ¼  π rr . The finite set Λ comprises the operation and k ¼ 1;k a r

modes of the controlled system. In this paper, we consider the following nonlinear system: 8 _ > < xðt Þ ¼ f ðxðt Þ; r t Þ þ g ðxðt Þ; r t Þdðt Þ yðt Þ ¼ f y ðxðt Þ; r t Þ þ g y ðxðt Þ; r t Þdðt Þ ð2Þ > : zðt Þ ¼ f ðxðt Þ; r Þ þ g ðxðt Þ; r Þdðt Þ t

z

z

t

n

where xðt Þ A ℜ is the state, yðt Þ A ℜl is the measurement,   þ 1 is the unknown input, zðt Þ A ℜq is the output dðt Þ A Lm 2 0 signal to be estimated, f ð U Þ A ℜn , f y ð U Þ A ℜl , f z ð U Þ A ℜq , g ð U Þ A ℜnm , g y ð U Þ A ℜlm , g z ð U Þ A ℜqm are nonlinear function matrices. In this paper, a class of T–S fuzzy dynamic model, proposed by Takagi and Sugeno [1], is used to represent local linear input– output relations of nonlinear system (2). The fuzzy linear model is described by fuzzy IF–THEN rules and will be employed in this paper to deal with the energy-to-peak filtering problem for the nonlinear system. The ith rule of the uncertain MJSs linear model is described by T–S fuzzy models as follows: Plant Rule i: IF μ1 ðt Þ is F i1 , μ2 ðt Þ is F i2 , and …, μg ðt Þ is F ig , THEN 8 x_ ðt Þ ¼ ½Ai ðr t Þ þ ΔAi ðr t Þxðt Þ þ ½Bi ðr t Þ þΔBi ðr t Þdðt Þ > > > > < yðt Þ ¼ ½C i ðr t Þ þ ΔC i ðr t Þxðt Þ þ ½Di ðr t Þ þ ΔDi ðr t Þdðt Þ ð3Þ > > zðt Þ ¼ Li ðr t Þxðt Þ > > : xðt Þ ¼ x0 ; r t ¼ r 0 ; t ¼ 0; i ¼ 1; 2; …; S where x0 is the initial state and r 0 is the initial mode. μ1 ðt Þ; μ2 ðt Þ; …; μg ðt Þ are the available premise variables. F ij , i ¼ 1; 2; …; S, j ¼ 1; 2; …; g are the fuzzy sets, S is the numbers of IF–THEN rules. Ai ðr t Þ, Bi ðr t Þ, C i ðr t Þ, Di ðr t Þ and Li ðr t Þ are known mode-dependent matrices with appropriate dimensions, r t represents a continuous-time discrete state Markovian stochastic process with values in the finite set Λ. For presentation convenience, we denote Ai ðr t Þ, ΔAi ðr t Þ, Bi ðr t Þ, ΔBi ðr t Þ, C i ðr t Þ, ΔC i ðr t Þ, Di ðr t Þ, ΔDi ðr t Þ, Li ðr t Þ as Ai ðr Þ, ΔAi ðr Þ, Bi ðr Þ, ΔBi ðr Þ, C i ðr Þ, ΔC i ðr Þ, Di ðr Þ, ΔDi ðr Þ, Li ðr Þ respectively. The matrices with the symbol ΔðnÞ are considered as the uncertain matrices satisfying the following condition: " # " # ΔAi ðr Þ ΔBi ðr Þ M 1i ðr Þ   ð4Þ ¼ Γ ðr; t Þ N 1i ðr Þ N 2i ðr Þ ΔC i ðr Þ ΔDi ðr Þ M 2i ðr Þ i where M 1i ðr Þ, M 2i ðr Þ, N1i ðr Þ, N 2i ðr Þ are known mode-dependent matrices with appropriate dimensions and Γ i ðr; t Þ is the timevarying unknown matrix function with Lebesgue norm measurable elements satisfying Γ Ti ðr; t ÞΓ i ðr; t Þ r I. Remark 1. In general, the model of a practical dynamic system always contains some types of uncertainties. It stems from the fact that, we can not obtain the exact mathematical model of a practical dynamics due to the complexity process, the environmental noises, time-varying parameters and the difficulties if measuring various and uncertain parameters, etc. In fact, the uncertainties described in (3) have been widely used in the schemes of stochastic robust control and filtering of uncertain MJSs [5–8,12,16]. And the uncertainties ΔðnÞ satisfying the

350

S. He / Neurocomputing 168 (2015) 348–355

restraining conditions above are said to be admissible if both condition (3) and Γ Ti ðr; t ÞΓ i ðr; t Þ r I hold. Note that the unknown mode-dependent matrix Γ i ðr; t Þ in (3) can also be allowed to be state-dependent, i.e., Γ i ðr; t Þ ¼ Γ i ðr; t; xðt ÞÞ, as long as ‖Γ i ðr; t; xðt ÞÞ‖ r 1 is satisfied. Assumption 1. The energy constraint external disturbance dðt Þ RT T satisfies 0 d ðt Þdðt Þdt r d with respect to the finite-time interval   0 T . By using a standard fuzzy singleton inference method, i.e., a singleton fuzzifier to produce a fuzzy inference and weighted center-average defuzzifer [1–6,9,32], we can get the following fuzzy dynamical MJSs: 8 S X
 > > > hi ðμðt ÞÞ ½Ai ðr Þ þ ΔAi ðr Þxðt Þ þ ½Bi ðr Þ þ ΔBi ðr Þdðt Þ > x_ ðt Þ ¼ > > > i¼1 > > > > S X >
 > < hi ½μðt Þ ½C i ðr Þ þ ΔC i ðr Þxðt Þ þ ½Di ðr Þ þ ΔDi ðr Þdðt Þ yðt Þ ¼ ð5Þ i ¼ 1 > > > S > X > > > z ðt Þ ¼ hi ½μðt ÞLi ðr Þxðt Þ > > > > i¼1 > > : xðt Þ ¼ λðt Þ; r ¼ r 0 ; t ¼ 0: h i where μðt Þ ¼ μ1 ðt Þ μ2 ðt Þ ⋯ μS ðt Þ . And for 8 i ¼ 1; 2; …; S, it follows 8 S X > > > hi ½μðt Þ ¼ wi ½μðt Þ= wi ½μðt Þ > < i¼1 ð6Þ g >   > > wi ½μðt Þ ¼ Π F i μ ðt Þ > j j : j¼1   F ij μj ðt Þ is the grade of membership of μj ðt Þ in the fuzzy set F ij , and for t, it is assumed as, 8 w ½μðt Þ Z 0 > > < i S X : ð7Þ wi ½μðt Þ 4 0 > > :i¼1 Therefore, we have 8 S X > > < hi ½μðt Þ ¼ 1; i¼1 > > : 0 r h ½μðt Þ r1; i ¼ 1; 2; …; S i

:

ð8Þ

Then, we consider the following fuzzy filter: Filter Rule i: IF μ1 ðt Þ is F i1 , μ2 ðt Þ is F i2 , and …, μg ðt Þ is F ig , THEN

8_ > < x^ ðt Þ ¼ Af i ðr Þx^ ðt Þ þ Bf i ðr Þyðt Þ z^ ðt Þ ¼ C f i ðr Þx^ ðt Þ > :^ xðt Þ ¼ x^ 0 ; r t ¼ r 0 ; t ¼ 0; i ¼ 1; 2; …; S

and the dynamic global filter model can be constructed as 8 S X >   > > > x_^ ðt Þ ¼ hi ½μðt Þ Af i ðr Þx^ ðt Þ þ Bf i ðr Þyðt Þ > > > > i¼1 < S X > hi ½μðt ÞC f i ðr Þx^ ðt Þ z^ ðt Þ ¼ > > > > i¼1 > > > : x^ ðt Þ ¼ x^ ; r ¼ r ; t ¼ 0: 0 0

ð9Þ

ð10Þ

where x^ ðt Þ A ℜn is the filter state, z^ ðt Þ A ℜq is the filer output, x^ 0 is the initial estimated state and the mode-dependent matrices Af i ðr Þ, Bf i ðr Þ, C f i ðr Þ are unknown filter parameters to be designed for each value r A Λ. The objective of this paper consists of designing the finite-time energy-to-peak filter of uncertain MJSs in (1) and obtaining an estimate z^ ðt Þ of the signal zðt Þ such that the defined guaranteed

performance criteria are minimized. Define eðt Þ ¼ xðt Þ  x^ ðt Þ and vðt Þ ¼ zðt Þ  z^ ðt Þ, such that the filtering error dynamic MJSs is given by: 8 > x_~ ðt Þ ¼ A^ ðr Þx~ ðt Þ þ B^ ðr Þdðt Þ > > < vðt Þ ¼ C^ ðr Þx~ ðt Þ ð11Þ h iT > > > x~ ¼ x0 x0  x^ 0 ; r ¼ r ; t ¼ 0 : 0 0 " # x ðt Þ where x~ ðt Þ ¼ , eðt Þ S S X X A^ ðr Þ ¼ hi ½μðt Þ hj ½μðt Þ i¼1

" 

j¼1

Ai ðr Þ þ ΔAi ðr Þ

0

Ai ðr Þ þ ΔAi ðr Þ  Af j ðr Þ  Bf j ðr Þ½C i ðr Þ þ ΔC i ðr Þ

Af j ðr Þ

B^ ðr Þ ¼

S X

hi ½μðt Þ

i¼1

C^ ðr Þ ¼

S X

S X

" hj ½μðt Þ

j¼1

hi ½μðt Þ

i¼1

S X

# ;

# Bi ðr Þ þ ΔBi ðr Þ ; Bi ðr Þ þ ΔBi ðr Þ  Bf j ðr Þ½Di ðr Þ þ ΔDi ðr Þ

h hj ½μðt Þ Li ðr Þ  C f j ðr Þ

i C f j ðr Þ :

j¼1

Definition 1. Given a time-constant T 40, the filtering error dynamic MJSs (11) is stochastically finite-time bounded (FTB) with respect to  c1 c2 T R~ ðr Þ d , if there exist a positive matrix   R~ ðr Þ ¼ diag Rðr Þ Rðr Þ 40 and scalars c1 40 and c2 4 0, such that n o n o   ð12Þ E x~ T0 R~ ðr Þx~ 0 r c1 ) E x~ T ðt ÞR~ ðr Þx~ ðt Þ o c2 ; t A 0 T : Definition 2. (Mao [4]) Let V ðxðtÞ; r t ; t 4 0Þ ¼ V ðxðtÞ; r Þ be a stochastic positive functional, and define its weak infinitesimal operator as ℑV ðxðt Þ; r Þ ¼ lim

 1
E V ðxðt þ Δt Þ; r t þ Δt ; t þ Δt Þ xðt Þ; r t ¼ r  V ðxðt Þ; r; t Þ:

Δt-0Δt

ð13Þ Remark 2. Letting d  0, the concepts of FTB leads to finite-time stable (FTS). Notice that the concepts of Lyapunov stability and FTS are different. The former is largely known to the control characteristic in infinite time-interval, but the latter concerns the boundedness analysis of the controlled states within a finite time-interval. Obviously, a stochastic MJSs may not be Lyapunov stochastically stable and vice versa. And the stochastic finite-time filtering problem in this paper concerns fuzzy filter design via a finite-time interval which guarantees the stochastic finite-time boundedness of the filtering error dynamic MJSs (11). Our task now is to formulate the finite-time L2-L1 filter design problem as follows: give a disturbance attenuation level γ 4 0, and design the finite-time energy-to-peak fuzzy filter parameters Af i ðr Þ, Bf i ðr Þ, and C f i ðr Þ such that the filtering error dynamic MJSs (11) is stochastically FTB and  under the zero-initial condition for any nonzero dðt Þ A Lm 2 0 T , it satisfies
 E ‖vðt Þ‖21 r γ 2 ‖dðt Þ‖22 ð14Þ   T 
 RT T 2 2 where E ‖vðt Þ‖1 ¼ E sup v ðt Þvðt Þ , ‖dðt Þ‖2 ¼ 0 d ðt Þdðt Þdt. t40

Remark 3. In stochastic finite-time energy-to-peak filtering process, the unknown noises dðt Þ are assumed to be arbitrary deterministic signals of bounded energy and the problem of this paper is to design a filter that guarantees a prescribed bounded for the finite-time interval induced L2–L1 norm of the operator from the unknown noise inputs dðt Þ to the output error vðt Þ, i.e. the designed stochastic finite-time energy-to-peak filter is supposed to satisfy inequality (13) with attenuation γ.

S. He / Neurocomputing 168 (2015) 348–355

Before proceeding with the study, the following lemma is required. Lemma 1. [34] Let T, M and N be real matrices with appropriate dimensions. Then for all time-varying unknown matrix function F ðt Þ satisfying F T ðt ÞF ðt Þ r I, the following relation: T þMF ðt ÞN þ NT F T ðt ÞM T o0

Rðr Þ oP ðr Þ o σ 1 Rðr Þ "

T þβMM T þ β  1 N T N o 0:

ð16Þ

  e  αT c2 þ α eαT  1 pffiffiffiffiffi c1

where

2

6 6 Ψ ij ðr Þ ¼ 6 6 4 2

3. Main results In this section, we will study the robust stochastic finite-time energy-to-peak filtering problem for the filtering error fuzzy dynamic MJSs (11). Theorem 1. For given T 4 0, α 4 0, c1 4 0, d 4 0, R~ ðr Þ 4 0, the filtering error (11) is stochastically FTB with  fuzzy dynamic MJSs respect to c1 c2 T R~ ðr Þ d , if there exist scalars γ 4 0, c2 4 0, and mode-dependent symmetric positive-definite matrices P~ ðr Þ, r A Λ, such that: " # Ξ ðr Þ P~ ðr ÞB^ ðr Þ o0 ð17Þ n γ 2 e  αT I T C^ ðr ÞC^ ðr Þ o γ 2 P~ ðr Þ

c1 eαT σ P^ þ

ð18Þ

σ 1

o0

Ψ 11 ðr Þ

Ψ 12 ðr Þ

Ψ 13 ðr Þ

n

Ψ 22 ðr Þ

Ψ 23 ðr Þ

n

n

Ψ 33 ðr Þ

n

n

n

P ðr Þ

6 Ωij ðr Þ ¼ 6 4 0 n

ð25Þ

Ψ 14 ðr Þ

3

7 Ψ 24 ðr Þ 7 7; 7 0 5

 βi ðr ÞI

0

LTi ðr Þ  C Tfj ðr Þ

P ðr Þ

C Tfj ðr Þ

n

γ2I

3 7 7; 5

N X

Ψ 11 ðr Þ ¼ ATi ðr ÞP ðr Þ þ P ðr ÞAi ðr Þ þ

π rk P ðkÞ  αP ðr Þ þ βi ðr ÞN T1i ðr ÞN 1i ðr Þ;

k¼1

Ψ 12 ðr Þ ¼ ATi ðr ÞP ðr Þ  X Tj ðr Þ  C Ti ðr ÞY Tj ðr Þ; Ψ 13 ðr Þ ¼ P ðr ÞBi ðr Þ þβi ðr ÞN T1i ðr ÞN 2i ðr Þ; Ψ 14 ðr Þ ¼ P ðr ÞM 1i ðr Þ; N X

Ψ 22 ðr Þ ¼ X Tj ðr Þ þ X j ðr Þ þ

π rk P ðkÞ  αP ðr Þ;

k¼1

 d 1  e  αT oc2 σ P^ α

ð19Þ

P T  1=2 π rk P~ ðkÞ  αP~ ðr Þ, P^ ðr Þ ¼ R~ where Ξ ðr Þ ¼ A^ ðr ÞP~ ðr Þ þ P~ ðr ÞA^ ðr Þ þ k¼1    1=2 ðr ÞP~ ðr ÞR~ ðr Þ, σ P^ ¼ maxσ max P^ ðr Þ , σ P^ ¼ minσ min P^ ðr Þ . N

iAΛ

iAΛ

Proof. Let the mode at time t be r; that is r t ¼ r A Λ. Take the stochastic Lyapunov–Krasovskii functional V ðx~ ðt Þ; r t ; t 4 0Þ :ℜn  Λ  ℜ þ -ℜ þ as: V ðx~ ðt Þ; r Þ ¼ x~ T ðt ÞP~ ðr Þx~ ðt Þ. The time derivative of V ðx~ ðt Þ; r Þ along the trajectories of the filtering error fuzzy dynamic MJSs (11) is given by: ℑV ðx~ ðt Þ; r Þ ¼ 2x~ T ðt ÞP~ ðr Þx_~ ðt Þ þ x~ T ðt Þ

N X

π rk P~ ðkÞx~ ðt Þ

Ψ 23 ðr Þ ¼ P ðr ÞBi ðr Þ Y j ðr ÞDi ðr Þ; Ψ 24 ðr Þ ¼ P ðr ÞM 1i ðr Þ  Y j ðr ÞM 2i ðr Þ: Ψ 33 ðr Þ ¼  γ 2 e  αT I þβi ðr ÞN T2i ðr ÞN 2i ðr Þ: Moreover, the mode-dependent fuzzy filter parameters are given by Af i ðr Þ ¼ P  1 ðr ÞX i ðr Þ; Bf i ðr Þ ¼ P  1 ðr ÞY i ðr Þ; C f i ðr Þ ¼ C f i ðr Þ:   Proof. Considering Theorem 1 and letting P~ ðr Þ ¼ diag P ðr Þ P ðr Þ , we can get the following relations according to matrix inequality (16), S X

k¼1

¼ x~ ðt ÞΞ ðr Þx~ ðt Þ þ 2x~ ðt ÞP~ ðr ÞB^ ðr Þdðt Þ þ x~ T ðt ÞαP~ ðr Þx~ ðt Þ: T

ð24Þ pffiffiffiffiffi # c1

 d

ð15Þ

holds if and only if there exists a positive scalar β 40, such that

351

T

hi ðμðt ÞÞ

S X

i¼1

j¼1

S X

S X

  hj ðμðt ÞÞ Πij ðr Þ þΔΠij ðr Þ o 0 2

Referring inequality (14), we introduce,

 T J ðT Þ ¼ E ℑV ðx~ ðt Þ; r Þ  αE V ðx~ ðt Þ; r Þ þ e  αT d ðt Þdðt Þ: Then following the similar proof of [32], we can get the main results of Theorem 1. This completes the proof. Theorem 2. For given T 4 0, α 4 0, c1 4 0, d 4 0, R~ ðr Þ 4 0, the filtering error fuzzy dynamic MJSs (11) is stochastically FTB with  respect to c1 c2 T R~ ðr Þ d , if there exist scalars γ 4 0, c2 4 0, σ 1 4 0, and mode-dependent symmetric positive-definite matrices P ðr Þ, mode-dependent matrices X i ðr Þ, Y i ðr Þ, C f i ðr Þ, and a sequence
 βi ðr Þ 4 0 , such that the following LMIs hold for all r A Λ and i; j ¼ 1; 2; …; S, Ψ ii ðr Þ o 0; i ¼ 1; 2; …; S

ð20Þ

Ψ ij ðr Þ þ Ψ ji ðr Þ o0; i o j; i; j ¼ 1; 2; …; S

ð21Þ

Ωii ðr Þ 4 0; i ¼ 1; 2; …; S

ð22Þ

Ωij ðr Þ þ Ωji ðr Þ 4 0; i o j; i; j ¼ 1; 2; …; S

ð23Þ

hi ðμðt ÞÞ

i¼1

where

j¼1

2

6 Πij ðr Þ ¼ 4

0

LTi ðr Þ  C Tfj ðr Þ

P ðr Þ

C Tfj ðr Þ

n

γ2 I

P ðr Þ

6 hj ðμðt ÞÞ6 4 0 n

Π11 ðr Þ

Π12 ðr Þ

Π13 ðr Þ

n

Π22 ðr Þ

Π23 ðr Þ

n

2

ΔΠ11 ðr Þ 6 Πij ðr Þ ¼ 4 ΔΠ21 ðr Þ n

n n

0 n

ΔΠ13 ðr Þ

3 7 740 5

3

2  αT

γ e

ð26Þ

7 5; I

3

7 ΔΠ23 ðr Þ 5; 0

with Π11 ðr Þ ¼ ATi ðr ÞP ðr Þ þ P ðr ÞAi ðr Þ þ

N X

π rk P ðkÞ  αP ðr Þ;

k¼1

Π12 ðr Þ ¼ ATi ðr ÞP ðr Þ  ATfj ðr ÞP ðr Þ  C Ti ðr ÞBTfj ðr ÞP ðr Þ;

ð27Þ

352

S. He / Neurocomputing 168 (2015) 348–355

Π13 ðr Þ ¼ P ðr ÞBi ðr Þ; N X

Π22 ðr Þ ¼ ATfj ðr ÞP ðr Þ þ P ðr ÞAf j ðr Þ þ

following the similar proof of [32]. Without considering the condition of c1 eαT σ P^ þ dα 1  e  αT o c2 σ P^ and the boundedness of unknown disturbances, the main results of Theorems 1 and 2 will reduce to the general filter design, see [18,22]. Thank you. In this part, we obtain the sufficient condition such that the filtering error dynamic MJSs are finite-time stable and satisfy the given L2–L1 index. It is noted that the conditions in Theorem 2 are LMIs which can be easily solved. When the LMIs in (20–25) are feasible, the filter coefficients can be obtained. By applying the approach developed in this paper together with the methods on robust control, the results obtained in this paper can be extension to the case of MJSs with both uncertainties and time delays.

π rk P ðkÞ  αP ðr Þ;

k¼1

Π23 ðr Þ ¼ P ðr ÞBi ðr Þ  P ðr ÞBf j ðr ÞDi ðr Þ; ΔΠ11 ðr Þ ¼ ΔATi ðr ÞP ðr Þ þ P ðr ÞΔAi ðr Þ; ΔΠ13 ðr Þ ¼ P ðr ÞΔBi ðr Þ; ΔΠ21 ðr Þ ¼ P ðr ÞΔAi ðr Þ  P ðr ÞBf j ðr ÞΔC i ðr Þ; ΔΠ23 ðr Þ ¼ P ðr ÞΔBi ðr Þ  P ðr ÞBf j ðr ÞΔDi ðr Þ: According to Assumption 1, ΔΠij ðr Þ can be presented as the following form ΔΠ ij ðr Þ ¼ L11 ðr ÞΓ i ðr; t ÞL12 ðr Þ þ LT12 ðr ÞΓ Ti ðr; t ÞLT11 ðr Þ o βi ðr ÞL11 ðr Þ LT11 ðr Þ þ βi 1 ðr ÞLT12 ðr ÞL12 ðr Þ h where L11 ðr Þ ¼ col P ðr ÞM 1iðr Þ P ðr ÞM 1i ðr Þ  P ðr ÞBf j ðr ÞM 2i ðr Þ L12 ðr Þ ¼ N 1i ðr Þ 0 N 2i ðr Þ . Referring to Lemma 1, we can get S X

hi ðμðt ÞÞ

i¼1

S X

hj ðμðt ÞÞSij ðr Þ o0

i 0 ,

ð28Þ

j¼1

where

2

6 6 Sij ðr Þ ¼ 6 6 4

S11 ðr Þ

Π12 ðr Þ

S13 ðr Þ

n

Π22 ðr Þ

Π23 ðr Þ

n

n

S33 ðr Þ

n

n

n

S14 ðr Þ

3

7 S24 ðr Þ 7 7; 7 0 5

 βi ðr ÞI

S11 ðr Þ ¼ Π11 ðr Þ þ βi ðr ÞN T1i ðr ÞN 1i ðr Þ; S13 ðr Þ ¼ Π13 ðr Þ þ βi ðr ÞN T1i ðr ÞN 2i ðr Þ; S33 ðr Þ ¼ γ 2 e  αT I þ βi ðr ÞN T2i ðr ÞN 2i ðr Þ; S14 ðr Þ ¼ P ðr ÞM 1i ðr Þ; S24 ðr Þ ¼ P ðr ÞM 1i ðr Þ  P ðr ÞBf j ðr ÞM 2i ðr Þ: Let X i ðr Þ ¼ P ðr ÞAf i ðr Þ, Y i ðr Þ ¼ P ðr ÞBf i ðr Þ, and it can be easily shown that the above inequalities (26) and (27) are equivalent to the following conditions S X

2

hi ðμðt ÞÞΨii ðr Þ þ

S X

hi ðμðt ÞÞ

S X

i¼1

i¼1

j¼1

S X

S X

S X

2

hi ðμðt ÞÞΩii ðr Þ þ

i¼1

i¼1

hi ðμðt ÞÞ

  hj ðμðt ÞÞ Ψij ðr Þ þΨji ðr Þ o 0;   hj ðμðt ÞÞ Ωij ðr Þ þ Ωji ðr Þ 4 0:

ð29Þ

ð30Þ

j¼1

And then they lead to LMIs (20–23).   1=2  1=2 ðr ÞP~ ðr ÞR~ ðr Þ, σ P^ ¼ maxσ max P^ ðr Þ , σ P^ ¼ Defining P^ ðr Þ ¼ R~ iAΛ  minσ min P^ ðr Þ , one has the following inequalities from LMI (24), iAΛ   1 r σ P^ ¼ minσ min P^ ðr Þ ; σ P^ ¼ maxσ max P^ ðr Þ o σ 1 ð31Þ iAΛ

iAΛ

Then recalling condition (19), we can get the linear matrix inequality in (25). This completes the proof. Remark 4. By Theorems 1 and 2, we can easily get the finite-time filter by solving the LMIs in (20–25). Comparing with the general
 T filter design, the expressions E V ðx~ ðt Þ; r Þ and e  αT d ðt Þdðt Þ are added in J ðT Þ; Taking the Lyapunov–Krasovskii functional as V ðx~ ðt Þ; r Þ ¼ x~ T ðt ÞP~ ðr Þx~ ðt Þ, and using the relevant Newton–Leibniz calculus formula, we can get the main results of Theorem 1 by

Remark 5. The direct Lyapunov method using a quadratic Lyapunov function is investigated in the proof of Theorems 1 and 2. Many studies related the T–S fuzzy models are usually considered in this method. In general, to find a common matrix satisfying all Lyapunov inequalities, for a T–S fuzzy system with a large number of rules, may be conservative and sometimes it is not possible to ensure its stability. Some approaches, such as fuzzy Lyapunov functions (Lyapunov function dependent on the fuzzy rule) and polynomial Lyapunov functions, have been developed to overcome this conservativeness. In the fuzzy Lyapunov function approach, it consists of finding positive definite functions for each linear model and constructing a Lyapunov function by fuzzy blending them with the same membership functions of the T–S fuzzy system. Moreover, the diagonal mode-dependent matrix P~ ðr Þ in the proof will also bring some conservativeS performances. To overcome P these, we can choose V ðt Þ ¼ x~ T ðt Þ hl ðμðt ÞÞP~ l ðr Þx~ ðt Þ, where P~ l ðr Þ l ¼ 1 and the solution of fuzzy can be selected as full order matrices, filter will be quite difficult and can be considered in the further research. On the other hand, to solve the finite-time energy-topeak fuzzy filter, the mode-dependent positive-definite matrices Rðr Þ in inequality (24) should be pre-set. For convenience, we choose the initial value for Rðr Þ ¼ I nn . For the uncertain T–S fuzzy MJSs (3) without jump parameters, we can get the following dynamical system: Plant Rule i: IF μ1 ðt Þ is F i1 , μ2 ðt Þ is F i2 , and …, μg ðt Þ is F ig , THEN 8 x_ ðt Þ ¼ ðAi þ ΔAi Þxðt Þ þ ðBi þ ΔBi Þdðt Þ > > > > < yðt Þ ¼ ðC i þ ΔC i Þxðt Þ þ ðDi þ ΔDi Þdðt Þ : ð32Þ zðt Þ ¼ Li xðt Þ > > > > : xðt Þ ¼ x0 ; t ¼ 0; i ¼ 1; 2; …; S Then, Theorem 2 reduces to the following Corollary 1. Corollary 1. For given T 4 0, α 4 0, c1 40, d 4 0, R~ 40, the filtering error dynamic MJSs (32) is stochastically FTB with respect to    c1 c2 T R~ d , where R~ ¼ diag R R , if there exists scalars γ 4 0, c2 4 0, σ 1 4 0, and symmetric positive-definite matrices  
 P~ ¼ diag P P , matrices X i , Y i , C f i , and a sequence βi 4 0 , such that the following LMIs hold for all i; j ¼ 1; 2; …; S: Ξ ii o 0; i ¼ 1; 2; …; S

ð33Þ

Ξ ij þ Ξ ji o 0; i oj; i; j ¼ 1; 2; …; S

ð34Þ

Σ ii 4 0; i ¼ 1; 2; …; S

ð35Þ

Σ ij þ Σ ji 4 0; io j; i; j ¼ 1; 2; …; S

ð36Þ

R oP o σ 1 R

ð37Þ

"

   e  αT c2 þ dα eαT  1 pffiffiffiffiffi c1

pffiffiffiffiffi # c1 σ 1

o0

ð38Þ

S. He / Neurocomputing 168 (2015) 348–355

3

Example 2. Consider a tunnel diode circuit [9], which can be represented as 8  αðiÞ þ ΔαðiÞ 2 > x1 ðt Þ x1 ðt Þ þ 10x2 ðt Þ > > x_ 1 ðt Þ ¼  0:1x1 ðt Þ  C > > < Lx_ 2 ðt Þ ¼  x1 ðt Þ Rx2 ðt Þ þ 0:1dðt Þ ð40Þ > > yðt Þ ¼ Jxðt Þ þ 0:1dðt Þ > > > : zðt Þ ¼ 0:1x ðt Þ

7 7: 5

where xðt Þ ¼

where 2 6 6 6 Ξ ij ¼ 6 6 4

ATi P þ PAi  αP þ βi N T1i N 1i

ATi P  X Tj  C Ti Y Tj

n

X Tj þ X j  αP

2

n

n

n

n

6 Σ ij ¼ 6 40

0

LTi  C Tfj

P

C Tfj

n

n

γ2I

P

PBi þ βi N T1i N 2i PBi  Y j Di 2  αT

γ e

I þ βi N T2i N 2i n

3

PM 1i

7 PM 1i  Y j M 2i 7 7 7; 7 0 5  βi I

1

"

Moreover, the energy-to-peak fuzzy filter parameters are given by Af i ¼ P  1 X i ; Bf i ¼ P  1 Y i ; C f i ¼ C f i : Remark 6. To obtain the optimal finite-time fuzzy filter, we can take γ 2 as optimized variable such that LMIs (33–38) are satisfied. Meanwhile, we can also get the optimal finite-time energy-topeak fuzzy filter for uncertain MJSs (20–25), i.e., to obtain an optimal finite-time fuzzy filter, the attenuation lever γ 2 can be reduced to the minimum possible value such that LMIs (20–25) are satisfied. The optimization problem can be described as follows: ρ

min

P ðr Þ; X i ðr Þ; Y i ðr Þ; C f i ðr Þ; βi ðrÞ; ρ

s:t: LMIsð20  25Þ with ρ ¼ γ 2

:

ð39Þ

Remark 7. Notice that if α ¼ 0 in Theorem 2, then matrix inequalities (20) and (21) are equivalent to the relation ½Ai ðr Þ þ N P ΔAi ðr ÞT P ðr Þ þ P ðr Þ½Ai ðr Þ þΔAi ðr Þ þ π rk P ðkÞ o 0, which can guark¼1

antee the Lyapunov stochastic stability [7–12,25,31,32] (or almost asymptotically stable) of uncertain T–S fuzzy dynamic MJSs (3). If α o 0, then it is globally exponentially stochastically stable [24].

Example 1. Considering a class of nonlinear system described by the T–S fuzzy model (32) with uncertain parameters given by:       2 4 0:2 2 1 A1 ¼ , A2 ¼ , B1 ¼ ,  4:5 5 0:4 1 1       0:3 B2 ¼ , C 1 ¼ 0:2 0:3 ,C 2 ¼ 0:2 0:1 ,  0:2     L1 ¼  0:1 0:2 ,L2 ¼ 0:2  0:1 , D1 ¼ 0:1, D2 ¼ 0:2,   0 M 11 ¼ ,  0:2    0:1 M 12 ¼ , M 21 ¼  0:2, M 22 ¼ 0:2, N 11 ¼ N12 ¼ 0:2 0:2

 0:1 ,

    0:8385 ; Bf 2 ¼ ; 0:0188 0:5751 0:3578   C f 2 ¼ 0:100  0:0500 ;  8:7043

# is the deviation variables, dðt Þ is the unknown

x 2 ðt Þ

where hi presents the normalized time-varying fuzzy weighting functions for each rule, i ¼ 1; 2,      0:1 10  4:6 10 A1 ð1Þ ¼ A1 ð2Þ ¼ ; A 2 ð1Þ ¼ ; 1  10 1  10   9:1 10 A2 ð2Þ ¼ 1  10 

0



0:1

2:8601

with the optimization attenuation parameter given by γ ¼ 2:0226.

 ; C 1 ðr Þ ¼ C 2 ðr Þ ¼ 1

 0 ; D1 ðr Þ ¼ D2 ðr Þ ¼ 0:1

   0 0 ; M 11 ð1Þ ¼ M 11 ð2Þ ¼ ; 0      0:5 ; N11 ð1Þ ¼ N 11 ð2Þ ¼ 0 0 ; M 12 ð1Þ ¼ 0

 L1 ðr Þ ¼ L2 ðr Þ ¼ 0:1

 M 12 ð2Þ ¼

0:9

  ; N 12 ð1Þ ¼ 0:9

0 M 21 ðr Þ ¼ M 22 ðr Þ ¼ 0

 N11 ð1Þ ¼ N 11 ð2Þ ¼ 0

N 21 ¼ 0:1, N 22 ¼  0:1. In this paper, we choose the initial values for d ¼ 2, T ¼ 6, α ¼ 0:5, c1 ¼ 1. From Corollary 1 and the relevant optimal algorithm, we solve LMIs (34–38) and get the following optimal robust finite-time energy-to-peak filters as:     2:0196 4:4081 3:5482 Af 1 ¼ ; Bf 1 ¼ ;  3:5751 4:3150 2:9214   C f 1 ¼  0:0500 0:1000 ; 

x 1 ðt Þ

input, yðt Þ is the measured output, zðt Þ is the controlled output and J is the sensor matrix. The parameters in the circuit are C ¼ 20mF,   L ¼ 1H and R ¼ 10 Ω, J ¼ 1 0 . Assume that the uncertain modedependent parameters αðiÞ þΔαðiÞ are aggregated into 2 modes shown as αð1Þ þ Δαð1Þ ¼ 0:01 7 10% and αð2Þ þ Δαð2Þ ¼ 0:02 7 10%. The transition rate matrix that relates the three operation modes is    0:5 0:5 given as Π ¼ . 0:3  0:3 Supposing that ‖x1 ðt Þ‖ r 3, the nonlinear circuit network can be translated into the following T–S fuzzy model: 8 S X
 > > > _ hi ½Ai ðr Þ þ ΔAi ðr Þxðt Þ þ Bi ðr Þdðt Þ > < xðt Þ ¼ i¼1 ð41Þ > yðt Þ ¼ C i ðr Þxðt Þ þ Di ðr Þdðt Þ > > > : zðt Þ ¼ Li ðr Þx1 ðt Þ

B1 ðr Þ ¼ B2 ðr Þ ¼

4. Numeral examples

Af 2 ¼

353

  0 ; N 12 ð2Þ ¼  1

 0 ;

 0 ; N 21 ðr Þ ¼ N 22 ðr Þ ¼ 0:

The membership functions are selected as h1 ðx1 ðt ÞÞ ¼ h2 ðx1 ðt ÞÞ ¼ x1 ðt6Þ þ 3.

 x1 ðt Þ þ 3 , 6

In this paper, we choose the initial values for d ¼ 2, T ¼ 6, α ¼ 0:5. From Theorem 2 and optimal algorithm (39), we can look for the optimal admissible c2 of different c1 guaranteeing the stochastically finite-time boundedness of desired filtering error dynamic properties. For c1 ¼ 1, we solve LMIs (20–25) by Theorem 2 and optimization algorithm (39) and get the following optimal robust finite-time energy-to-peak filters as:      1:0220 19:2282 0:3579 ; Bf 1 ð1Þ ¼ ; Af 1 ð1Þ ¼ 1:6675  30:7040 2:0013   C f 1 ð1Þ ¼ 0:0500 0:1000 ; 

    0:3541 ; Bf 2 ð1Þ ¼ ; 1:6606  30:5875 1:9921   C f 2 ð1Þ ¼ 0:0500 0:1000 ; Af 2 ð1Þ ¼

 10:6984

19:5373

354

S. He / Neurocomputing 168 (2015) 348–355

0.6

System state x1(t)

0.4

Real state x1(t) Estimated state x1(t)

0.2

0

−0.2

−0.4

−0.6

−0.8

0

1

2

3

4

5

6

t Fig. 1. The response of system state x1(t).

1.2

System state x2(t)

1 Real state x2(t) Estimated state x2(t)

0.8

0.6

0.4

0.2

0

−0.2

0

1

2

3

4

5

6

5

6

t Fig. 2. The response of system state x2(t).

0.02

The response of output error v(t)

0.01 0 −0.01 −0.02 −0.03 −0.04 −0.05 −0.06 −0.07 −0.08

0

1

2

3

t Fig. 3. The filtering output error v(t).

4

S. He / Neurocomputing 168 (2015) 348–355



 1:0229

    0:3553 ; Bf 1 ð2Þ ¼ ; 1:6679  30:6302 1:9956   C f 1 ð2Þ ¼ 0:0500 0:1000 ; 

 20:3834

Af 1 ð2Þ ¼

 19:2045

    0:3550 ; Bf 2 ð2Þ ¼ ; 1:6497 30:6209 1:9955   C f 2 ð2Þ ¼ 0:0500 0:1000 : Af 2 ð2Þ ¼

19:9046

And then, we can also get the optimal attenuation lever as γ ¼ 0:2672. To show the effectiveness of the designed filter, we select the initial conditions as x1 ð0Þ ¼ xf 1 ð0Þ ¼  0:8, x2 ð0Þ ¼ xf 2 ð0Þ ¼ 1 and r 0 ¼ 2. And we assume the unknown inputs are unknown  white  noise with noise power 0.1 over a finite-time interval t A 0 6 . Thus, the simulation results of the response of system states (real states and estimated states) and the filtering output error are shown in Figs. 1–3. It is clear from the simulation figures that the estimated states can track the real states smoothly. Meanwhile, the presented fuzzy energyto-peak filter guarantees a prescribed bounded for the induced finitetime L2–L1 norm of the operator from the unknown disturbance to the filtering output error with the given optimal attenuation level. Remark 8. We present a tunnel diode circuit here to illustrate the efficiency of the proposed methods. By the simulation results, it can be seen that the system states can be tracked by the estimated states smoothly in the finite-time interval. The results in this paper also show the practical advantages in nonlinear systems in the area of state estimation and finite-time analysis. 5. Conclusion The fuzzy finite-time energy-to-peak filtering problems for a class of nonlinear MJSs with uncertain parameters have been studied. By using the fuzzy Lyapunov–Krasovskii functional approach and LMIs technique, a sufficient condition is derived such that the filtering error dynamic fuzzy MJSs are finite-time bounded and satisfies a prescribed level of L2–L1 disturbance attenuation in a finite timeinterval. And the results are extension to the nonlinear systems without Markovian jumping parameters. Simulation example illustrates the effectiveness of the proposed design approach. Acknowledgment This work was supported in part by the Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20123 401120010), the National Natural Science Foundation of China (Grant no. 61203051) and the Key Program of Natural Science Foundation of Education Department of Anhui Province (Grant no. KJ2012A014). References [1] K. Tanaka, H.O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, Wiley, New York, 2001. [2] J. Qiu, G. Feng, H. Gao, Asynchronous output feedback control of networked nonlinear systems with multiple packet dropouts: T–S fuzzy affine model based approach, IEEE Trans. Fuzzy Syst. 19 (6) (2011) 1014–1030. [3] H. Shen, S. Xu, J. Zhou, J. Lu, Fuzzy H1 filtering for nonlinear Markovian jump neutral delayed systems, Int. J. Syst. Sci. 42 (5) (2011) 767–780. [4] H.D. Tuan, P. Apkarian, T. Narikiyo, Y. Yamamoto, Parameterized linear matrix inequality techniques in fuzzy control system design, IEEE Trans. Fuzzy Syst. 9 (2) (2001) 324–332. [5] J. Dong, G. Yang, H. Zhang, Stability analysis of T–S fuzzy control systems by using set theory (in press), IEEE Trans. Fuzzy Syst. (2015), http://dx.doi.org/ 10.1109/TFUZZ.2014.2328016. [6] A. Sala, C. Ariño, Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: applications of Polya's theorem, Fuzzy Sets Syst. 158 (24) (2007) 2671–2686.

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Yue, Stochastic finite-time stability of nonlinear Markovian switching systems with impulsive effects, J. Dyn. Syst., Meas. Control 134 (1) (2011) 011011–011015. [32] S. He, F. Liu, Finite-time H1 control of nonlinear jump systems with timedelays via dynamic observer-based state feedback, IEEE Trans. Fuzzy Syst. 20 (4) (2012) 605–614. [33] T. Shi, Finite-time control of linear systems under time-varying sampling, Neurocomputing 151 (2015) 1327–1331. [34] Y. Wang, L. Xie, C.E. de Souza, Robust control of a class of uncertain nonlinear systems, Syst. Control Lett. 19 (2) (1992) 139–149. Shuping He was born in 1983. He received his B.Eng. and Ph.D. from Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, Jiangnan University, Wuxi, China. From 2011 to 2013, he was successively a senior lecturer in Anhui University. Since 2013, he has been a professor with School of Electrical Engineering and Automation, Anhui University, Hefei, China. From 2010 to 2011, he was a Visiting Scholar with the Control Systems Centre, School of Electrical and Electronic Engineering, The University of Manchester, UK. His current research focuses on control theory and applications, includes robust control of stochastic systems, nonlinear and optimal control of nonlinear systems and complex system modeling, control, filtering, fault detection and their applications.