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

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, [1–6]. 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, [7–12]. 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, [13–15]. 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, [17–22], 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, ℝ^𝑛, ℝ^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 ℝ^𝑛. 𝐼 denotes the identity matrix of compatible dimension. The notation 𝑋≥𝑌 (resp., 𝑋>𝑌), where 𝑋 and 𝑌 are symmetric matrices, means that 𝑋−𝑌 is positive semidefinite (resp., positive definite). For a matrix 𝑀, 𝑀^𝑇 and 𝑀⁻¹ represent its transpose and inverse, respectively. The shorthand diag{𝑀₁,𝑀₂,…,𝑀_𝑛} denotes a block diagonal matrix with diagonal blocks being the matrices 𝑀₁,𝑀₂,…,𝑀_𝑛. 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=1𝑥(𝑘)=𝜌(𝑘),𝑘𝜇_𝑑=𝑥−(𝑘𝑑,−−𝑑𝑑)++1𝑁,…𝑓(,𝑧0(,𝑘))+𝐸𝑥(𝑘)𝑤(𝑘),((22..12)) where 𝑥(𝑘)∈ℝ^𝑛 is the state, 𝑧(𝑘)∶=𝐺𝑥(𝑘)+𝐺_𝑑∑^+∞𝑑=1𝜇_𝑑𝑥(𝑘−𝑑). 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 𝐺_𝑑 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)^𝑇𝑓(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))=𝑓_𝑠(𝑧(𝑘))+𝐹₁𝑧(𝑘),(2.4) then, from (2.3), we have 𝑓^𝑇𝑠(𝑧(𝑘))𝑓_𝑠(𝑧(𝑘))−𝐹𝑧(𝑘)≤0,(2.5) where 𝐹=𝐹₂−𝐹₁>0.

On the other hand, 𝜇_𝑑≥0 is the convergence constant that satisfies the following condition: ^+∞_𝑑=1𝜇_𝑑≤^+∞_𝑑=1𝑑𝜇_𝑑<+∞.(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, [17–22]. The delay term ∑^+∞𝑑=1𝜇_𝑑𝑥(𝑘−𝑑) 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 𝜇_𝑑(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)∈ℝ is a random white sequence characterizing the probabilistic sensor-data missing, which obeys the following time-varying Bernoulli distribution: ProbP{ro𝜉b(𝑘{)𝜉=(𝑘0)}==1}1=−𝔼𝔼{{𝜉𝜉((𝑘𝑘))}}==1𝑝(−𝑘)𝑝,(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, [13–15]. 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)𝐾_𝑢,(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,𝐾_𝑢 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=1𝜇_𝑑𝑥(𝑘−𝑑)+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 𝔼‖𝜂(𝑘)‖²≤𝛼𝜏^𝑘sup_{−𝑑≤𝑖≤0}𝔼‖𝜂(𝑖)‖²,𝑘∈𝕀⁺.(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=Σ^𝑇1 and 0<Σ₂=Σ^𝑇2, then Σ₁+Σ^𝑇3Σ²⁻¹Σ₃≥0 if and only if ⎡⎢⎢⎣Σ₁Σ^𝑇3Σ₃−Σ₂⎤⎥⎥⎦≥0or⎡⎢⎢⎣−Σ₂Σ₃Σ^𝑇3Σ₁⎤⎥⎥⎦≥0.(3.1)

Lemma 3.2 (see [24]). Let M ∈ℝ^𝑛×𝑛 be a positive semidefinite matrix, 𝑥_𝑖∈ℝ^𝑛 and constant 𝑎_𝑖>0(𝑖=1,2,…). If the series concerned is convergent, then one has ^∞_𝑖=1𝑎_𝑖𝑥_𝑖^𝑇𝑀^∞_𝑖=1𝑎_𝑖𝑥_𝑖≤^∞_𝑖=1𝑎_𝑖^∞_𝑖=1𝑎_𝑖𝑥^𝑇𝑖𝑀𝑥_𝑖.(3.2)

Lemma 3.3 (see [25]). Let the matrix 𝐵∈𝑅^𝑛×𝑚 be of full-column rank. There always exist two orthogonal matrices 𝑈∈𝑅^𝑛×𝑛 and 𝑉∈𝑅^𝑛×𝑛 such that Σ=di𝐵ag=𝑈𝜎₁⎡⎢⎢⎣Σ0⎤⎥⎥⎦𝑉^𝑇,,𝜎₂,…,𝜎_𝑚.(3.3) If matrix 𝑆 has the following structure: 𝑆=𝑈⎡⎢⎢⎣𝑆¹¹𝑆₁₂0𝑆₂₂⎤⎥⎥⎦𝑈_𝑇,(3.4) where 𝑆₁₁∈𝑅^𝑛×𝑚, 𝑆₁₂∈𝑅^{𝑛×(𝑛−𝑚)}, 𝑆₂₂∈𝑅^{(𝑛−𝑚)×(𝑛−𝑚)}, then there exists a nonsingular matrix 𝑅∈𝑅^𝑚×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 𝑄_𝜏, slack matrix 𝑆 and nonsingular matrices 𝑌(𝑝) and 𝑅, such that the following LMIs hold: ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜇𝑄_𝜏−𝑄0(𝑝(𝑘))−1∗∗∗∗∗𝜇𝑄^𝜏𝐹𝐺𝐹𝐺_𝑑∗∗∗∗𝑆−2𝐼∗∗∗_𝑇𝐴+𝜎𝑝(𝑘)𝐵𝑌(𝑝)𝐶𝑆^𝑇𝐷𝑆^𝑇𝑁−Λ∗∗₂Δ𝑆^𝑇𝐸000−𝜎²Λ∗_𝑝(𝑘)𝐵𝑌(𝑝)𝐶0000−Δ_𝑝(𝑘)Λ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.5) where Λ=−𝑄(𝑝(𝑘+1))+𝑆+𝑆^𝑇,𝜇=^+∞_𝑑=1𝜇_𝑑,Δ_𝑃(𝑘)=𝑃(𝑘)(1−𝑃(𝑘)),𝑆_𝑇𝐴=𝐴+𝑁𝐹₁𝐺,𝐷=𝐷+𝑁𝐹₁𝐺_𝑑,𝐵=𝐵𝑅,𝑅𝐾(𝑝)=𝑌(𝑝),𝐾(𝑝)=𝑅⁻¹𝑌(𝑝),(3.6) in this case, the constant gains of the desired controller can be obtained as follows: 𝐾₀=𝑅⁻¹𝑌₀,𝐾_𝑢=𝑅⁻¹𝑌_𝑢,(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)∶=𝑥^𝑇(𝑘)𝑄(𝑝(𝑘))𝑥(𝑘)+^+∞_𝑑=1𝜇_{𝑑𝑘−1𝑠=}_𝑘−𝑑𝑥^𝑇(𝑠)𝑄_𝜏𝑥(𝑠).(3.8) Then, noting 𝔼{𝜉(𝑘)−𝑝(𝑘)}=0, 𝔼{𝜔(𝑘)}=0 and 𝔼{[𝜉(𝑘)−𝑝(𝑘)]²}=𝑝(𝑘)(1−𝑝(𝑘)), we can get that 𝔼{Δ𝑉(𝑘)}=𝔼−𝑥^𝑇(𝑘+1)𝑄(𝑝(𝑘+1))𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑄(𝑝(𝑘))𝑥(𝑘)+𝜇𝑥^𝑇(𝑘)𝑄_𝜏𝑥(𝑘)_+∞_𝑑=1≤E⎧⎪⎨⎪⎩𝜇_𝑑𝑥^𝑇(𝑘−𝑑)𝑄_𝜏𝑥(𝑘−𝑑)𝐴+𝑝(𝑘)𝐵𝐾(𝑝)𝐶𝑥(𝑘)+𝐷^+∞_𝑚₌₁×𝜇_𝑑𝑥(𝑘−𝑑)+𝑁𝑓_𝑠(𝑧(𝑘))^𝑇𝑄(𝑝(𝑘+1))𝐴+𝑝(𝑘)𝐵𝐾(𝑝)𝐶𝑥(𝑘)+𝐷^+∞_𝑚₌₁+[𝑝(𝑘)(1−(𝑝(𝑘))𝐵𝐾(𝑝)𝐶)𝑥(𝑘)]_𝑇𝜇_𝑑𝑥(𝑘−𝑑)+𝑁𝑓_𝑠(𝑧(𝑘))𝑄(𝑝(𝑘+1))𝐵𝐾(𝑝)𝐶𝑥(𝑘)+𝜎²𝑥^𝑇(𝑘)𝐸^𝑇×𝑄(𝑝(𝑘+1))𝐸𝑥(𝑘)−𝑥^𝑇(𝑘)𝑄(𝑝(𝑘))𝑥(𝑘)−^+∞_𝑑=1+𝜇_𝑑𝑥^𝑇(𝑘−𝑑)𝑄_𝜏𝑥(𝑘−𝑑)𝜇𝑥^𝑇(𝑘)𝑄_𝜏𝑥(𝑘)+2𝑓^𝑇𝑠(𝑧(𝑘))𝐹𝐺𝑥(𝑘)+2𝑓^𝑇𝑠(𝑧(𝑘))𝐹𝐺_𝑑+∞𝑚₌₁−2𝑓_𝑇𝑠𝜇_𝑑𝑥(𝑘−𝑑)(𝑧(k))𝑓_𝑠(𝑧(𝑘)).(3.9) From Lemma 3.2, it is obvious that −^+∞_𝑑=1𝜇_𝑑𝑥^𝑇(𝑘−𝑑)𝑄_𝜏𝑥(𝑘−𝑑)≤−1𝜇^+∞_𝑑=1𝜇_𝑑𝑥^𝑇(𝑘−𝑑)𝑄_𝜏^+∞_𝑑=1𝜇_𝑑𝑥(𝑘−𝑑).(3.10) Denote the following matrix variables 𝜂(𝑘)=𝑥^𝑇(𝑘)^+∞_𝑑=1𝜇_𝑑𝑥^𝑇(𝑘−𝑑)𝑓^𝑇𝑠(𝑧(𝑘))^𝑇.(3.11) Combining (3.9), (3.10), and (3.11), we can get 𝔼{Δ𝑉Ω(𝑘)=}⎡⎢⎢⎢⎢⎣≤Ω𝔼₁𝜂^𝑇(𝑘)Ω𝜂(𝑘),Ω∗∗₂Ω₄Ω₃∗Ω₅Ω₆Ω₁⎤⎥⎥⎥⎥⎦,=+𝑝𝐴(𝑘+)(𝑝1(𝑘−)𝑝𝐵(𝐾𝑘)()𝑝(𝐵)𝐶𝐾(^𝑇𝑝𝑄)𝐶()𝑝_𝑇(𝑘+1))𝐴+𝑝(𝑘)𝐵𝐾(𝑝)𝐶+𝜎²𝐸^𝑇𝑄(𝑝(𝑘+1))𝐸Ω𝑄(𝑝(𝑘+1))𝐵𝐾(𝑝)𝐶+𝜇𝑄_𝜏−𝑄(𝑝(𝑘)),₂Ω=𝐷^𝑇𝑄(𝑝(𝑘+1))𝐴+𝑝(𝑘)𝐵𝐾(𝑝)𝐶₃,=𝐷^𝑇𝑄(𝑝(𝑘+1))𝐷−1𝜇𝑄^𝜏Ω₄,Ω=𝑁^𝑇𝑄(𝑝(𝑘+1))𝐴+𝑝(𝑘)𝐵𝐾(𝑝)𝐶+𝐹𝐺,₅Ω₆=𝑁^𝑇𝑄(𝑝(𝑘+1))𝐷+𝐹𝐺_𝑑,=𝑁^𝑇𝑄(𝑝(𝑘+1))𝑁−2𝐼.(3.12) If Ω≤0, we can conclude the following matrix inequalities by Schur complement: ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜇𝑄_𝜏−𝑄0(𝑝(𝑘))−1∗∗∗∗∗𝜇𝑄^𝜏𝐹𝐺𝐹𝐺_𝑑∗∗∗∗−2𝐼∗∗∗𝐴+𝑝(𝑘)𝐸𝐵𝐾(𝑝)𝐶𝐷0𝑁0−0Λ−𝜎∗₋₂∗𝐵𝐾(𝑝)𝐶0000Λ−Δ_𝑝−1∗(𝑘)Λ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.13) with Λ=𝑄⁻¹(𝑝(𝑘+1)).
At this time, preforming the congruence transformation diag{𝐼,𝐼,𝐼,𝑆,𝜎²𝑆,Δ_𝑝(𝑘)𝑆} to (3.13), we can have ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜇𝑄_𝜏−𝑄0(𝑝(𝑘))−1∗∗∗∗∗𝜇𝑄^𝜏𝐹𝐺𝐹𝐺_𝑑∗∗∗∗𝑆−2𝐼∗∗∗_𝑇𝐴+𝑝𝜎(𝑘)𝑆^𝑇𝐵𝐾(𝑝)𝐶𝑆^𝑇𝐷𝑆^𝑇𝑁−Λ∗∗₂Δ𝑆^𝑇𝐸000−𝜎²Λ∗_𝑝(𝑘)𝑆^𝑇𝐵𝐾(𝑝)𝐶Λ=𝑆_𝑇0000−Δ_𝑝(𝑘)Λ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,𝑄⁻¹(𝑝(𝑘+1))𝑆,(3.14) then from inequality 𝑆^𝑇𝑄⁻¹(𝑝(𝑘+1))𝑆≥𝑆^𝑇+𝑆−Q(𝑝(𝑘+1))=Λ,(3.15) we can get ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜇𝑄_𝜏−𝑄0(𝑝(𝑘))−1∗∗∗∗∗𝜇𝑄^𝜏𝐹𝐺𝐹𝐺_𝑑∗∗∗∗𝑆−2𝐼∗∗∗_𝑇𝐴+𝑝𝜎(𝑘)𝑆^𝑇𝐵𝐾(𝑝)𝐶𝑆^𝑇𝐷𝑆^𝑇𝑁−Λ∗∗₂Δ𝑆^𝑇𝐸000−𝜎²Λ∗_𝑝(𝑘)𝑆^𝑇𝐵𝐾(𝑝)𝐶0000−Δ_𝑝(𝑘)Λ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.16) and from lemma 3, we have 𝑆^𝑇𝐵=𝐵𝑅 denoting 𝑅𝐾(𝑝)=𝑌(𝑝), and 𝐾(𝑃)=𝑅⁻¹𝑌(𝑝). Then (3.16) can be written as ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜇𝑄_𝜏−𝑄0(𝑝(𝑘))−1∗∗∗∗∗𝜇𝑄^𝜏𝐹𝐺𝐹𝐺_𝑑∗∗∗∗𝑆−2𝐼∗∗∗_𝑇𝐴+𝜎𝑝(𝑘)𝐵𝑌(𝑝)𝐶𝑆^𝑇𝐷𝑆^𝑇𝑁−Λ∗∗₂Δ𝑆^𝑇𝐸000−𝜎²Λ∗_𝑝(𝑘)𝐵𝑌(𝑝)𝐶0000−Δ_𝑝(𝑘)Λ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.17) Furthermore, by Lemma 3.1, we can know from that Ω<0 and, subsequently, 𝔼{Δ𝑉(𝑘)}<−𝜆_min(−Ω)𝔼||𝜂(𝑘)||²,(3.18) where 𝜆_min(−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)𝑄_𝑢.

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, 𝑄_𝑢 and 𝑄_𝜏, slack matrix 𝑆 and nonsingular matrices 𝑌(𝑝) and 𝑅, such that the following LMIs hold: 𝑀^{𝑖𝑗𝑙𝑚}=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜇𝑄_𝜏−𝑄0^𝑖(𝑝(𝑘))−1∗∗∗∗∗𝜇𝑄^𝜏𝐹𝐺𝐹𝐺_𝑑∗∗∗∗𝑆−2𝐼∗∗∗_𝑇𝐴𝜎+𝑝_𝑖𝐵𝑌^𝑚𝐶𝑆^𝑇𝐷𝑆^𝑇𝑁−Λ^𝑙∗∗₂Δ_𝑖𝑗𝑆^𝑇𝐸000−𝜎²Λ^𝑙∗𝐵𝑌^𝑚𝐶0000−Δ^𝑖𝑗Λ^𝑙⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.19) where Λ^𝑙=𝑄_𝑖−𝑄₀−𝑝_𝑙𝑄_𝑢+𝑆+𝑆^𝑇,Δ^𝑖𝑗=𝑝_𝑖1−𝑝_𝑗,𝑆_𝑇(𝑝(𝑘))=𝑄₀+𝑝_𝑖𝑄_𝑢,𝑌^𝑚=𝑌₀+𝑝_𝑚𝑌_𝑢,𝐵=𝐵𝑅,𝑅𝐾(𝑝)=𝑌(𝑝),𝐾(𝑝)=𝑅⁻¹𝑌(𝑝),(3.20) the constant gains of the desired controller can be obtained as follows: 𝐾₀=𝑅⁻¹𝑌₀,𝐾_𝑢=𝑅⁻¹𝑌_𝑢,(3.21) and the closed-system (2.11) is then exponentially mean-square stable for all 𝑝(𝑘)∈[𝑝₁𝑝₂].

Proof. Firstly, set 𝛼₁(𝑘)=𝑝²−𝑝(𝑘)𝑝₂−p₁,𝛼₂(𝑘)=𝑝(𝑘)−𝑝¹𝑝₂−𝑝₁,(3.22) then, we have 𝑝(𝑘)=𝛼₁(𝑘)𝑝₁+𝛼₂(𝑘)𝑝₂,(3.23) with 𝛼_𝑖(𝑘)≥0(𝑖=1,2) and 𝛼₁(𝑘)+𝛼₂(𝑘)=1. Similarly, let 𝛽₁(𝑘)=𝑝²−𝑝(𝑘+1)𝑝₂−𝑝₁,𝛽₂(𝑘)=𝑝(𝑘+1)−𝑝¹𝑝₂−𝑝₁,(3.24) then we have 𝑝(𝑘+1)=𝛽₁(𝑘)𝑝₁+𝛽₂(𝑘)𝑝₂,(3.25) with 𝛽_𝑖(𝑘)≥0(𝑖=1,2), 𝛽₁(𝑘)+𝛽₂(𝑘)=1. From the above transformation, we can easily get 𝑄(𝑝(𝑘))=²_𝑖=1𝛼_𝑖(𝑘)𝑄^𝑖,Λ=²_𝑙=1𝑌(𝑝(𝑘))=₂𝛽_𝑙(𝑘)Λ^𝑙,_𝑚₌₁𝛼_𝑚(𝑘)𝑌^𝑚(𝑝).(3.26) On the other hand, it is easy to find that ²_𝑖,_{𝑗,𝑙,𝑚=1}𝛼_𝑖(𝑘)𝛼_𝑗(𝑘)𝛼_𝑚(𝑘)𝛽_𝑙(𝑘)𝕄^{𝑖𝑗𝑙𝑚}<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: 𝐶𝐴==⎡⎢⎢⎣⎡⎢⎢⎣000..1.01957000.2..02231⎤⎥⎥⎦⎤⎥⎥⎦,,𝐷𝑁==⎡⎢⎢⎣⎡⎢⎢⎣0000..21..123538000.01..23813⎤⎥⎥⎦⎤⎥⎥⎦,,𝐹𝐵₁=⎡⎢⎢⎣0.0060.016⎤⎥⎥⎦,𝐹=⎡⎢⎢⎣0.0060.007₂⎤⎥⎥⎦,=⎡⎢⎢⎣0.0610.025⎤⎥⎥⎦,𝑝𝐺=⎡⎢⎢⎣00..111800..1122⎤⎥⎥⎦,𝐺_𝑑=⎡⎢⎢⎣00..111800..2099⎤⎥⎥⎦,𝐸=⎡⎢⎢⎣00..023100..1393₁⎤⎥⎥⎦,=0.19,𝑝₂=0.51,𝜎²=1,𝜇=2⁻³.(4.1)

Set the time-varying Bernoulli distribution sequences as 𝑝(𝑘)=𝑝₁+(𝑝₂−𝑝₁)|sin(𝑘)| and the sector nonlinear function 𝑓(𝑢) is taken as 𝑓(𝑢)=𝐹¹+𝐹₂2𝑢+𝐹²−𝐹₁2sin(𝑢),(4.2) which satisfies (2.3). Also, select the initial state as follows: 𝜌=[2−2]^𝑇.

According to Theorem 3.6, the constant controller parameters 𝐾₀, 𝐾_𝑢 can be obtained as follows: 𝐾₀=⎡⎢⎢⎣51712..47898194−−51060.4.9987028⎤⎥⎥⎦,𝐾_𝑢=⎡⎢⎢⎣−721..45491895−−602..35964192⎤⎥⎥⎦.(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.

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.

Figure 1 

State evolution 𝑥(𝑘) of uncontrolled systems.

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.