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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 696742, 15 pages
http://dx.doi.org/10.1155/2012/696742
Research Article

Probability-Dependent Static Output Feedback Control for Discrete-Time Nonlinear Stochastic Systems with Missing Measurements

Department of Control Science and Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

Received 21 June 2012; Accepted 2 August 2012

Academic Editor: Zidong Wang

Copyright © 2012 Wangyan Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is devoted to the problems of gain-scheduled control for a class of discrete-time stochastic systems with infinite-distributed delays and missing measurements by utilizing probability-dependent Lyapunov functional. The missing-measurement phenomenon is assumed to occur in a random way, and the missing probability is time varying with securable upper and lower bounds that can be measured in real time. The purpose is to design a static output feedback controller with scheduled gains such that, for the admissible random missing measurements, time delays, and noises, the closed-loop system is exponentially mean-square stable. At last, a simulation example is exploited to illustrate the effectiveness of the proposed design procedures.

1. Introduction

Gain-scheduling is one of the most popular methods of controller design and has been extensively applied in engineering, such as rotation speed control of engine, aircraft control and process control. Over the past decades, the gain-scheduled control problem has been extensively studied both from theoretical and practical viewpoint, see, for example, [16]. For the controller design problems for parameter-varying systems, the gain-scheduling approach has been found to be one of the most effective ones, whose main idea is to design controller gains as functions of the scheduling parameters, which are supposed to be available in real time and, therefore, have much less conservatism than the conventional ones.

On the other hand, instead of using the information of system states, static output feedback (SOF) control directly makes use of system outputs to design controllers, which has also attracted attentions of many researchers over the past two decades, see, for example, [712]. It is obvious that the structure of SOF controllers is simple and easy to implement. However, to the best of the authors’ knowledge, there has been little research attention on the control problem for discrete-time nonlinear stochastic systems with a missing phenomenon based on the time-varying occurring probability by a gain-scheduling method.

The missing-measurement phenomenon, due to various reasons such as probabilistic network congestion and intermittent mechanical failures, usually occurs in many real-world systems, which has attracted considerable attention during the past few years, see, for example, [1315]. The Bernoulli distribution has been successfully applied to model this phenomenon, in which 0 is used to stand for an entire signal missing and 1 denotes the intactness (i.e., there is no signal missing at all), and all sensors have the same missing probability, which is simple and effective and has become very popular during the past years, see, for example, [5, 13, 14, 16]. However, in the practical systems, the occurring probability of the missing-measurement phenomenon might be time varying; consequently, a time-varying Bernoulli distribution model is more suitable for such parameter-varying systems.

In another aspect, considering the signal propagation often distributed during a certain time period, then, a new kind of delays, namely, distributed time-delays, has drawn many researchers’ attention, see, for example, [1722], but most of the existing works on distributed delays have focused on continuous-time systems which are described either in the form of finite or infinite integral. As we all know, when it comes to implementing the control laws in a digital way, the discrete-time system is much better than continuous-time one. Naturally, it turns out to be meaningful to investigate the issue of how distributed delays influence the dynamical behavior of a discrete-time system. However, as far as authors know, based on gain-scheduled control methods, the SOF control problem for nonlinear stochastic systems with infinite-distributed delays and missing measurements with time-varying occurring probability has not been addressed yet and is still a very interesting and challenging problem.

The main contributions of this paper are summarized as follows: (1) a new SOF control problem is addressed for a class of discrete-time nonlinear stochastic systems with missing measurements and infinite-distributed delays via a gain-scheduling approach; (2) a sequence of stochastic variables satisfying Bernoulli distributions is introduced to describe the time-varying features of the missing measurements in the sensor; (3) a time-varying Lyapunov functional dependent on the missing probability is proposed and then applied to improve the performance of the gain-scheduled controller; and (4) a gain-scheduled controller is designed, in which the controller parameters can be adjusted online according to the missing probabilities estimated through statistical tests.

Notation 1. In this paper, 𝑛, 𝑛×𝑚, and 𝕀+ denote, respectively, the 𝑛-dimensional Euclidean space, and the set of all 𝑛×𝑚 real matrices, the set of all positive integers. || refers to the Euclidean norm in 𝑛. 𝐼 denotes the identity matrix of compatible dimension. The notation 𝑋𝑌 (resp., 𝑋>𝑌), where 𝑋 and 𝑌 are symmetric matrices, means that 𝑋𝑌 is positive semidefinite (resp., positive definite). For a matrix 𝑀, 𝑀𝑇 and 𝑀1 represent its transpose and inverse, respectively. The shorthand diag{𝑀1,𝑀2,,𝑀𝑛} denotes a block diagonal matrix with diagonal blocks being the matrices 𝑀1,𝑀2,,𝑀𝑛. In symmetric block matrices, the symbol is used as an ellipsis for terms induced by symmetry. Matrices, if they are not explicitly stated, are assumed to have compatible dimensions. In addition, 𝔼{𝑥} and Prob{𝑦} will, respectively, mean expectation of 𝑥 and probability of 𝑦.

2. Problem Formulation

Consider the following discrete-time nonlinear stochastic systems with infinite-distributed delays: 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵𝑢(𝑘)+𝐷+𝑑=1𝜇𝑑𝑥(𝑘𝑑)+𝑁𝑓(𝑧(𝑘))+𝐸𝑥(𝑘)𝑤(𝑘),(2.1)𝑥(𝑘)=𝜌(𝑘),𝑘=𝑑,𝑑+1,,0,(2.2) where 𝑥(𝑘)𝑛 is the state, 𝑧(𝑘)=𝐺𝑥(𝑘)+𝐺𝑑+𝑑=1𝜇𝑑𝑥(𝑘𝑑). 𝜔(𝑘) is a one-dimensional Gaussian white noise sequence satisfying 𝔼{𝜔(𝑘)}=0 and 𝔼{𝜔2(𝑘)}=𝜎2, 𝜌(𝑘) is the initial state of the system. 𝐴, 𝐵, 𝐷, 𝑁, 𝐸, 𝐺, and 𝐺𝑑 are constant real matrices of appropriate dimensions and 𝐵 is of full-column rank.

The nonlinear function 𝑓() with (𝑓(0)=0) is assumed as nonlinear disturbances and satisfies the following sector-bounded condition:𝑓(𝑧(𝑘))𝐹1𝑧(𝑘)𝑇𝑓(𝑧(𝑘))𝐹2𝑧(𝑘)0,(2.3) where 𝑓() is called to belong to the sector [𝐹1,𝐹2] and 𝐹1 and 𝐹2 are given constant real matrices.

For the technique convenience, the nonlinear function 𝑓(𝑧(𝑘)) can be decomposed into a linear and a nonlinear part as 𝑓(𝑧(𝑘))=𝑓𝑠(𝑧(𝑘))+𝐹1𝑧(𝑘),(2.4) then, from (2.3), we have 𝑓𝑇𝑠𝑓(𝑧(𝑘))𝑠(𝑧(𝑘))𝐹𝑧(𝑘)0,(2.5) where 𝐹=𝐹2𝐹1>0.

On the other hand, 𝜇𝑑0 is the convergence constant that satisfies the following condition: +𝑑=1𝜇𝑑+𝑑=1𝑑𝜇𝑑<+.(2.6)

Remark 2.1. The distributed delay is one important type of time delays and has been widely recognized and intensively studied, see, for example, [1722]. The delay term +𝑑=1𝜇𝑑𝑥(𝑘𝑑) in the resulted stochastic system (2.1) called infinitely distributed delay. However, almost all existing references concerning distributed delays are concerned with the continuous-time systems, where the distributed delays are described in the form of a finite or infinite integral. In this paper, the constants 𝜇𝑑(𝑑=1,2,) are assumed to satisfy the convergence conditions (2.6), which can guarantee the convergence of the terms of infinite delays as well as the Lyapunov-Krasovskii functional defined later.
The measurement output with missing sensor data is described as 𝑦(𝑘)=𝜉(𝑘)𝐶𝑥(𝑘),(2.7) where 𝐶 is a constant real matrix of appropriate dimensions and 𝜉(𝑘) is a random white sequence characterizing the probabilistic sensor-data missing, which obeys the following time-varying Bernoulli distribution: Prob{𝜉(𝑘)=1}=𝔼{𝜉(𝑘)}=𝑝(𝑘),Prob{𝜉(𝑘)=0}=1𝔼{𝜉(𝑘)}=1𝑝(𝑘),(2.8) where 𝑝(𝑘) is a time-varying positive scalar sequence and belongs to [𝑝1𝑝2][01] with 𝑝1 and 𝑝2 being the lower and upper bounds of 𝑝(𝑘), respectively. In this paper, for simplicity, we assume that 𝜉(𝑘), 𝜔(𝑘) and 𝜌(𝑘) are uncorrelated.

Remark 2.2. In (2.7), a random white sequence satisfying the time-varying Bernoulli distribution is introduced to reflect the missing-measurement phenomenon that has attracted considerable attention in the past few years, see, for example, [1315]. However, the missing probability in most relevant literatures has always been assumed to be a constant. Such an assumption, unfortunately, tends to be conservative in handling time-varying missing measurements. In this paper, the missing probability is allowed to be time-varying with known lower and upper bounds, which will then be used to schedule controller gains, thereby reducing the possible conservatism.
In this paper, we are interested in designing the following gain-scheduled controller: 𝑢(𝑘)=𝐾(𝑝)𝑦(𝑘),(2.9) where 𝐾(𝑝) is the controller gain sequence to be designed and assumed as the following structure: 𝐾(𝑝)=𝐾0+𝑝(𝑘)𝐾𝑢,(2.10) for every time step 𝑘, 𝑝(𝑘) is the time-varying parameter of the controller gain, which takes value in [𝑝1,𝑝2] and 𝐾0,𝐾𝑢 are the constant parameters of the controller gain to be designed.
The closed-loop system of the static output feedback gain-scheduled controller is as follows: 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝜉(𝑘)𝐵𝐾(𝑝)𝐶𝑥(𝑘)+𝐷+𝑑=1𝜇𝑑𝑥(𝑘𝑑)+𝑁𝑓(𝑧(𝑘))+𝐸𝑥(𝑘)𝑤(𝑘).(2.11)
Before formulating the problem to be investigated, we first introduce the following stability concepts.

Definition 2.3. The closed-loop system (2.11) is said to be exponentially mean-square stable if, with 𝑤(𝑘)=0, there exist constants 𝛼>0 and 𝜏(0,1) such that 𝔼𝜂(𝑘)2𝛼𝜏𝑘sup𝑑𝑖0𝔼𝜂(𝑖)2,𝑘𝕀+.(2.12)
In this paper, our purpose is to design a probability-dependent gain-scheduled controller of the form (2.9) for the system (2.1) by exploiting a probability-dependent Lyapunov functional and LMI method such that, for all admissible infinite-distributed delays, missing measurements with time-varying probability, and exogenous stochastic noises, the closed-loop system (2.11) is exponentially mean-square stable.

3. Main Results

The following lemmas will be used in the proofs of our main results in this paper.

Lemma 3.1 ([Schur complement] see[23]). Given constant matrices Σ1,Σ2,Σ3 where Σ1=Σ𝑇1 and 0<Σ2=Σ𝑇2, then Σ1+Σ𝑇3Σ21Σ30 if and only if Σ1Σ𝑇3Σ3Σ20orΣ2Σ3Σ𝑇3Σ10.(3.1)

Lemma 3.2 (see [24]). Let M 𝑛×𝑛 be a positive semidefinite matrix, 𝑥𝑖𝑛 and constant 𝑎𝑖>0(𝑖=1,2,). If the series concerned is convergent, then one has 𝑖=1𝑎𝑖𝑥𝑖𝑇𝑀𝑖=1𝑎𝑖𝑥𝑖𝑖=1𝑎𝑖𝑖=1𝑎𝑖𝑥𝑇𝑖𝑀𝑥𝑖.(3.2)

Lemma 3.3 (see [25]). Let the matrix 𝐵𝑅𝑛×𝑚 be of full-column rank. There always exist two orthogonal matrices 𝑈𝑅𝑛×𝑛 and 𝑉𝑅𝑛×𝑛 such that Σ0𝑉𝐵=𝑈𝑇,𝜎Σ=diag1,𝜎2,,𝜎𝑚.(3.3) If matrix 𝑆 has the following structure: 𝑆𝑆=𝑈11𝑆120𝑆22𝑈𝑇,(3.4) where 𝑆11𝑅𝑛×𝑚, 𝑆12𝑅𝑛×(𝑛𝑚), 𝑆22𝑅(𝑛𝑚)×(𝑛𝑚), then there exists a nonsingular matrix 𝑅𝑅𝑚×𝑚 such that 𝑆𝐵=𝐵𝑅.

In the following theorem, a probability-dependent gain-scheduled static output feedback control problem is dealt with for a class of discrete-time nonlinear stochastic systems (2.1) by exploiting Lyapunov theory and LMI method. A sufficient condition is derived to guarantee the solvability of the desired gain-scheduled control problem and, simultaneously, the parameters of the gain-scheduled controller can be obtained by solving the LMIs and the measured time-varying probability.

Theorem 3.4. Consider the discrete-time nonlinear stochastic systems (2.11). If there exist positive-definite matrices 𝑄(𝑝(𝑘)) and 𝑄𝜏, slack matrix 𝑆 and nonsingular matrices 𝑌(𝑝) and 𝑅, such that the following LMIs hold: 𝜇𝑄𝜏1𝑄(𝑝(𝑘))0𝜇𝑄𝜏𝐹𝐺𝐹𝐺𝑑𝑆2𝐼𝑇𝐴+𝑝(𝑘)𝐵𝑌(𝑝)𝐶𝑆𝑇𝐷𝑆𝑇𝑁𝜎Λ2𝑆𝑇𝐸000𝜎2ΔΛ𝑝(𝑘)𝐵𝑌(𝑝)𝐶0000Δ𝑝(𝑘)Λ<0,(3.5) where Λ=𝑄(𝑝(𝑘+1))+𝑆+𝑆𝑇,𝜇=+𝑑=1𝜇𝑑,Δ𝑃(𝑘)=𝑃(𝑘)(1𝑃(𝑘)),𝐴=𝐴+𝑁𝐹1𝐺,𝐷=𝐷+𝑁𝐹1𝐺𝑑,𝑆𝑇𝐵=𝐵𝑅,𝑅𝐾(𝑝)=𝑌(𝑝),𝐾(𝑝)=𝑅1𝑌(𝑝),(3.6) in this case, the constant gains of the desired controller can be obtained as follows: 𝐾0=𝑅1𝑌0,𝐾𝑢=𝑅1𝑌𝑢,(3.7) and the closed-system (2.11) is then exponentially mean-square stable for all 𝑝(𝑘)[𝑝1𝑝2].

Proof. Define the Lyapunov functional: 𝑉(𝑘)=𝑥𝑇(𝑘)𝑄(𝑝(𝑘))𝑥(𝑘)++𝑑=1𝜇𝑑𝑘1𝑠=𝑘𝑑𝑥𝑇(𝑠)𝑄𝜏𝑥(𝑠).(3.8) Then, noting 𝔼{𝜉(𝑘)𝑝(𝑘)}=0, 𝔼{𝜔(𝑘)}=0 and 𝔼{[𝜉(𝑘)𝑝(𝑘)]2}=𝑝(𝑘)(1𝑝(𝑘)), we can get that 𝑥𝔼{Δ𝑉(𝑘)}=𝔼𝑇(𝑘+1)𝑄(𝑝(𝑘+1))𝑥(𝑘+1)𝑥𝑇(𝑘)𝑄(𝑝(𝑘))𝑥(𝑘)+𝜇𝑥𝑇(𝑘)𝑄𝜏𝑥(𝑘)+𝑑=1𝜇𝑑𝑥𝑇(𝑘𝑑)𝑄𝜏𝑥(𝑘𝑑)E𝐴+𝑝(𝑘)𝐵𝐾(𝑝)𝐶𝑥(𝑘)+𝐷+𝑚=1𝜇𝑑𝑥(𝑘𝑑)+𝑁𝑓𝑠(𝑧(𝑘))𝑇×𝑄(𝑝(𝑘+1))𝐴+𝑝(𝑘)𝐵𝐾(𝑝)𝐶𝑥(𝑘)+𝐷+𝑚=1𝜇𝑑𝑥(𝑘𝑑)+𝑁𝑓𝑠+[](𝑧(𝑘))𝑝(𝑘)(1(𝑝(𝑘))𝐵𝐾(𝑝)𝐶)𝑥(𝑘)𝑇𝑄(𝑝(𝑘+1))𝐵𝐾(𝑝)𝐶𝑥(𝑘)+𝜎2𝑥𝑇(𝑘)𝐸𝑇×𝑄(𝑝(𝑘+1))𝐸𝑥(𝑘)𝑥𝑇(𝑘)𝑄(𝑝(𝑘))𝑥(𝑘)+𝑑=1𝜇𝑑𝑥𝑇(𝑘𝑑)𝑄𝜏+𝑥(𝑘𝑑)𝜇𝑥𝑇(𝑘)𝑄𝜏𝑥(𝑘)+2𝑓𝑇𝑠(𝑧(𝑘))𝐹𝐺𝑥(𝑘)+2𝑓𝑇𝑠(𝑧(𝑘))𝐹𝐺𝑑+𝑚=1𝜇𝑑𝑥(𝑘𝑑)2𝑓𝑇𝑠(𝑧(k))𝑓𝑠.(𝑧(𝑘))(3.9) From Lemma 3.2, it is obvious that +𝑑=1𝜇𝑑𝑥𝑇(𝑘𝑑)𝑄𝜏1𝑥(𝑘𝑑)𝜇+𝑑=1𝜇𝑑𝑥𝑇𝑄(𝑘𝑑)𝜏+𝑑=1𝜇𝑑.𝑥(𝑘𝑑)(3.10) Denote the following matrix variables 𝑥𝜂(𝑘)=𝑇(𝑘)+𝑑=1𝜇𝑑𝑥𝑇(𝑘𝑑)𝑓𝑇𝑠(𝑧(𝑘))𝑇.(3.11) Combining (3.9), (3.10), and (3.11), we can get 𝔼𝜂{Δ𝑉(𝑘)}𝔼𝑇,Ω(𝑘)Ω𝜂(𝑘)Ω=1Ω2Ω3Ω4Ω5Ω6,Ω1=𝐴+𝑝(𝑘)𝐵𝐾(𝑝)𝐶𝑇𝑄(𝑝(𝑘+1))𝐴+𝑝(𝑘)𝐵𝐾(𝑝)𝐶+𝜎2𝐸𝑇𝑄(𝑝(𝑘+1))𝐸+𝑝(𝑘)(1𝑝(𝑘))(𝐵𝐾(𝑝)𝐶)𝑇𝑄(𝑝(𝑘+1))𝐵𝐾(𝑝)𝐶+𝜇𝑄𝜏Ω𝑄(𝑝(𝑘)),2=𝐷𝑇𝑄(𝑝(𝑘+1)),Ω𝐴+𝑝(𝑘)𝐵𝐾(𝑝)𝐶3=𝐷𝑇𝑄(𝑝(𝑘+1))1𝐷𝜇𝑄𝜏,Ω4=𝑁𝑇𝑄(𝑝(𝑘+1))Ω𝐴+𝑝(𝑘)𝐵𝐾(𝑝)𝐶+𝐹𝐺,5=𝑁𝑇𝑄(𝑝(𝑘+1))𝐷+𝐹𝐺𝑑,Ω6=𝑁𝑇𝑄(𝑝(𝑘+1))𝑁2𝐼.(3.12) If Ω0, we can conclude the following matrix inequalities by Schur complement: 𝜇𝑄𝜏1𝑄(𝑝(𝑘))0𝜇𝑄𝜏𝐹𝐺𝐹𝐺𝑑2𝐼𝐴+𝑝(𝑘)𝐵𝐾(𝑝)𝐶𝐷𝑁Λ𝐸000𝜎2Λ𝐵𝐾(𝑝)𝐶0000Δ𝑝1(𝑘)Λ<0,(3.13) with Λ=𝑄1(𝑝(𝑘+1)).
At this time, preforming the congruence transformation diag{𝐼,𝐼,𝐼,𝑆,𝜎2𝑆,Δ𝑝(𝑘)𝑆} to (3.13), we can have 𝜇𝑄𝜏1𝑄(𝑝(𝑘))0𝜇𝑄𝜏𝐹𝐺𝐹𝐺𝑑𝑆2𝐼𝑇𝐴+𝑝(𝑘)𝑆𝑇𝐵𝐾(𝑝)𝐶𝑆𝑇𝐷𝑆𝑇𝜎𝑁Λ2𝑆𝑇𝐸000𝜎2ΔΛ𝑝(𝑘)𝑆𝑇𝐵𝐾(𝑝)𝐶0000Δ𝑝(Λ𝑘)<0,Λ=𝑆𝑇𝑄1(𝑝(𝑘+1))𝑆,(3.14) then from inequality 𝑆𝑇𝑄1(𝑝(𝑘+1))𝑆𝑆𝑇+𝑆Q(𝑝(𝑘+1))=Λ,(3.15) we can get 𝜇𝑄𝜏1𝑄(𝑝(𝑘))0𝜇𝑄𝜏𝐹𝐺𝐹𝐺𝑑𝑆2𝐼𝑇𝐴+𝑝(𝑘)𝑆𝑇𝐵𝐾(𝑝)𝐶𝑆𝑇𝐷𝑆𝑇𝑁𝜎Λ2𝑆𝑇𝐸000𝜎2ΔΛ𝑝(𝑘)𝑆𝑇𝐵𝐾(𝑝)𝐶0000Δ𝑝(𝑘)Λ<0,(3.16) and from lemma 3, we have 𝑆𝑇𝐵=𝐵𝑅 denoting 𝑅𝐾(𝑝)=𝑌(𝑝), and 𝐾(𝑃)=𝑅1𝑌(𝑝). Then (3.16) can be written as 𝜇𝑄𝜏1𝑄(𝑝(𝑘))0𝜇𝑄𝜏𝐹𝐺𝐹𝐺𝑑𝑆2𝐼𝑇𝐴+𝑝(𝑘)𝐵𝑌(𝑝)𝐶𝑆𝑇𝐷𝑆𝑇𝑁𝜎Λ2𝑆𝑇𝐸000𝜎2ΔΛ𝑝(𝑘)𝐵𝑌(𝑝)𝐶0000Δ𝑝(𝑘)Λ<0,(3.17) Furthermore, by Lemma 3.1, we can know from that Ω<0 and, subsequently, 𝔼{Δ𝑉(𝑘)}<𝜆min||||(Ω)𝔼𝜂(𝑘)2,(3.18) where 𝜆min(Ω) is the minimum eigenvalue of (Ω). Finally, we can confirm from Lemma 1 of [13] that the closed-loop system is exponentially mean-square stable, then the proof of this theorem is complete.

Remark 3.5. In the above theorem, a static output feedback controller has been designed based on a set of LMIs. However, the LMIs are actually infinite owing to the time-varying parameter 𝑝(𝑘)[𝑝1𝑝2]. In this case, the desired controller cannot be obtained directly from Theorem 3.4 due to the infinite number of LMIs. To handle such a problem, in the next theorem, we have to convert this problem to a computationally accessible one by assigning a specific form to 𝑝(𝑘). Let us set 𝑄(𝑝(𝑘))=𝑄0+𝑝(𝑘)𝑄𝑢.

Theorem 3.6. Consider the discrete-time nonlinear stochastic system with infinite-distributed delays and missing measurements (2.11). If there exist positive-difinite matrices 𝑄0, 𝑄𝑢 and 𝑄𝜏, slack matrix 𝑆 and nonsingular matrices 𝑌(𝑝) and 𝑅, such that the following LMIs hold: 𝑀𝑖𝑗𝑙𝑚=𝜇𝑄𝜏𝑄𝑖1(𝑝(𝑘))0𝜇𝑄𝜏𝐹𝐺𝐹𝐺𝑑𝑆2𝐼𝑇𝐴+𝑝𝑖𝐵𝑌𝑚𝐶𝑆𝑇𝐷𝑆𝑇𝑁Λ𝑙𝜎2𝑆𝑇𝐸000𝜎2Λ𝑙Δ𝑖𝑗𝐵𝑌𝑚𝐶0000Δ𝑖𝑗Λ𝑙<0,(3.19) where Λ𝑙=𝑄0𝑝𝑙𝑄𝑢+𝑆+𝑆𝑇,Δ𝑖𝑗=𝑝𝑖1𝑝𝑗,𝑄𝑖(𝑝(𝑘))=𝑄0+𝑝𝑖𝑄𝑢,𝑌𝑚=𝑌0+𝑝𝑚𝑌𝑢,𝑆𝑇𝐵=𝐵𝑅,𝑅𝐾(𝑝)=𝑌(𝑝),𝐾(𝑝)=𝑅1𝑌(𝑝),(3.20) the constant gains of the desired controller can be obtained as follows: 𝐾0=𝑅1𝑌0,𝐾𝑢=𝑅1𝑌𝑢,(3.21) and the closed-system (2.11) is then exponentially mean-square stable for all 𝑝(𝑘)[𝑝1𝑝2].

Proof. Firstly, set 𝛼1𝑝(𝑘)=2𝑝(𝑘)𝑝2p1,𝛼2(𝑘)=𝑝(𝑘)𝑝1𝑝2𝑝1,(3.22) then, we have 𝑝(𝑘)=𝛼1(𝑘)𝑝1+𝛼2(𝑘)𝑝2,(3.23) with 𝛼𝑖(𝑘)0(𝑖=1,2) and 𝛼1(𝑘)+𝛼2(𝑘)=1. Similarly, let 𝛽1𝑝(𝑘)=2𝑝(𝑘+1)𝑝2𝑝1,𝛽2(𝑘)=𝑝(𝑘+1)𝑝1𝑝2𝑝1,(3.24) then we have 𝑝(𝑘+1)=𝛽1(𝑘)𝑝1+𝛽2(𝑘)𝑝2,(3.25) with 𝛽𝑖(𝑘)0(𝑖=1,2), 𝛽1(𝑘)+𝛽2(𝑘)=1. From the above transformation, we can easily get 𝑄(𝑝(𝑘))=2𝑖=1𝛼𝑖(𝑘)𝑄𝑖,Λ=2𝑙=1𝛽𝑙(𝑘)Λ𝑙,𝑌(𝑝(𝑘))=2𝑚=1𝛼𝑚(𝑘)𝑌𝑚(𝑝).(3.26) On the other hand, it is easy to find that 2𝑖,𝑗,𝑙,𝑚=1𝛼𝑖(𝑘)𝛼𝑗(𝑘)𝛼𝑚(𝑘)𝛽𝑙(𝑘)𝕄𝑖𝑗𝑙𝑚<0.(3.27) From (3.22)–(3.27), we can have that (3.5) in Theorem 3.4 is true, then the proof is now complete.

Remark 3.7. The above conclusions can be extended to multiple sensor case of measurement output. In this paper, to make the main idea and the proof more clear and concise, we choose the single sensor.

4. An Illustrative Example

In this section, the gain-scheduled static output feedback controller is designed for the discrete-time nonlinear stochastic systems with infinite-distributed delays and missing measurements.

The system parameters are given as follows: ,𝐴=0.97000.21,𝑁=0.130.210.280.33,𝐵=0.06000.16𝐶=0.10.20.150.23,𝐷=0.2300.150.18,𝐹1=,𝐹0.06000.072=0.61000.25,𝐺=0.110.120.180.12,𝐺𝑑=,𝑝0.110.290.180.09,𝐸=0.030.190.210.331=0.19,𝑝2=0.51,𝜎2=1,𝜇=23.(4.1)

Set the time-varying Bernoulli distribution sequences as 𝑝(𝑘)=𝑝1+(𝑝2𝑝1)|sin(𝑘)| and the sector nonlinear function 𝑓(𝑢) is taken as 𝐹𝑓(𝑢)=1+𝐹22𝐹𝑢+2𝐹12sin(𝑢),(4.2) which satisfies (2.3). Also, select the initial state as follows: 𝜌=[22]𝑇.

According to Theorem 3.6, the constant controller parameters 𝐾0, 𝐾𝑢 can be obtained as follows: 𝐾0=572.7914500.980811.488916.4972,𝐾𝑢=72.441962.56121.59850.3949.(4.3)

Then, according to the measured time-varying probability parameters 𝑝(𝑘), the gain-scheduled controller gain 𝐾(𝑝) and parameter-dependent Lyapunov matrix can be calculated at every time step 𝑘 as in Table 1.

tab1
Table 1: Computing results.

Figure 1 gives the response curves of state 𝑥(𝑘) of uncontrolled systems. Figure 2 depicts the simulation results of state 𝑥(𝑘) of the controlled systems. The simulation results have illustrated our theoretical analysis.

696742.fig.001
Figure 1: State evolution 𝑥(𝑘) of uncontrolled systems.
696742.fig.002
Figure 2: State evolution 𝑥(𝑘) of controlled systems.

5. Conclusions

In this paper, the problem of gain-scheduled control for a class of discrete stochastic systems with infinite-distributed delays and missing measurements has been studied, the missing-measurement phenomenon is assumed to occur in a random way, the missing probability is governed by an individual random variable satisfying a certain probabilistic distribution in the interval [01], and distributed delays are described in a discrete way. By employing probability-dependent Lyapunov functional, we have designed a gain-scheduled controller with the gain including both constant parameters and time-varying parameters such that, for the admissible missing measurements with time-varying probability, infinite-distributed delays, and noise disturbances, the closed-loop system is exponentially mean-square stable. Moreover, we can extend the main results to more complex and realistic systems, for instance, system with norm-bounded or polytopic uncertainties. Meanwhile, we can also consider dynamic output feedback control problem for discrete stochastic systems with missing measurements by gain-scheduling approach as well as the relevant applications in networked control system or robotic manipulator.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61074016, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, the Program for New Century Excellent Talents in University under Grant NCET-11-1051, the Shanghai Natural Science Foundation of China under Grant 10ZR1421200, the Leverhulme Trust of the UK, the Alexander von Humboldt Foundation of Germany, and the Innovation Fund Project for Graduate Student of Shanghai under Grant JWCXSL1202.

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