Nonfragile Gain-Scheduled Control for Discrete-Time Stochastic Systems with Randomly Occurring Sensor Saturations

Abstract

This paper is devoted to tackling the control problem for a class of discrete-time stochastic systems with randomly occurring sensor saturations. The considered sensor saturation phenomenon is assumed to occur in a random way based on the time-varying Bernoulli distribution with measurable probability in real time. The aim of the paper is to design a nonfragile gain-scheduled controller with probability-dependent gains which can be achieved by solving a convex optimization problem via semidefinite programming method. Subsequently, a new kind of probability-dependent Lyapunov functional is proposed in order to derive the controller with less conservatism. Finally, an illustrative example will demonstrate the effectiveness of our designed procedures.

1. Introduction

In reality, virtually almost all dynamic systems are subject to stochastic perturbation, and stochastic model has been successfully established to describe many practical systems, such as economic systems, process control systems, networked control systems (NCSs), and sensor network. For several decades, the study of stabilization, control, and filtering problem has drawn many researchers’ attention; some results can be found in [1–16]. On the other hand, time delays also serve as one of the main sources for poor performance and instability. Consequently, the stochastic control issue for time-delay systems has also been intensively investigated; see, for example, [2, 4, 7, 8, 10, 11, 13, 15].

The randomly occurring phenomenon is a newly emerged research topic which has drawn many researchers' attention; see, for example, [1–3, 5, 6, 8, 9, 12–16]. It refers to these phenomena appearing in a random way based on a certain kind of probabilistic law including randomly occurring nonlinearities (RONs), missing measurements, randomly occurring actuator faults, randomly varying sensor delays (RVSDs), and randomly occurring sensor saturations (ROSSs), and so on. For more details about randomly occurring phenomena, the reader is referred to [9]. If not handled appropriately, these phenomena could cause a reduction of performance and/or launch a threat to the safety and reliability of the plant. Therefore, it is not surprising that various filtering and control techniques have been developed to deal with such randomly occurring phenomena, in addition to 𝐻_∞ control [16]/filtering [12] and 𝐻_∞ state estimation [1] methods. In [2], a robust sliding mode control has been designed for system with mixed time-delays, randomly occurring uncertainties, and RONs; while gain-constrained recursive filter approach has been used in [5] for system with probabilistic sensor delays, the extended Kalman filtering and quantized recursive filtering problem for system with missing measurements have been studied in [3, 6], respectively. Therefore, in this paper, the ROSS (one of the important randomly occurring phenomena) is studied by exploiting gain-scheduling method, which is another motivation of this paper.

Sensor saturation phenomenon is very common in practical engineering. It means that sensors cannot provide signals of unlimited amplitude due mainly to the physical or technological constraints. In another aspect, because of random occurrences of networked induced phenomena in networked control systems (NCSs), such as random sensor failures leading to intermittent saturation and sensor aging resulting in changeable saturation level, sensor saturation may occur in a random way. We consider this phenomenon as randomly occurring sensor saturation, which has received increasing attention, for instance, [1, 12]. Reference [1] discussed the 𝐻_∞ state estimation problem for discrete-time complex networks with ROSSs and RVSDs, while [12] turned to design an 𝐻_∞ filter for system with ROSSs and missing measurements. However, to the best of authors' knowledge, rare published literature has dealt with ROSSs; therefore, this paper tries to flourish the research on this phenomenon by designing a nonfragile gain-scheduled controller.

Over the past decades, gain-scheduling method is one of the most popular methods of controller designing and has been extensively studied from theoretical and practical viewpoints; see, for example, [8, 14, 15, 17–19]. The gain-scheduling method is to design controller gains as functions of the scheduling parameters, which can update the controller with a set of tuning parameters in order to optimize the closed-loop performance when outside environment changes (e.g., the occurrences of a variety of randomly occurring phenomena). It should be noted that the designed gain-scheduling controller has not only the constant part but also time-varying part which can be scheduled online according to the corresponding time-varying parameters; see [8, 14, 15]. Therefore, it will naturally lead to less conservatism than the conventional ones with fixed gains only.

On the other hand, it is well known that in order to get better performance of the system, an accuracy controller is needed to resist the impact by the uncertainties occurring in the course of the implementation of a designed controller. Such uncertainties can be due to the existence of parameter drift, round-off errors in numerical computation during controller implementation, and the safe-tuning margins provided for engineering application. In these cases, the nonfragile controller is a good choice, as it can tolerate some level of controller parameter variations; see [7, 20–22]. However, the controller with uncertainties and outside environment changes often occur simultaneously; unfortunately, few papers have tackled this phenomenon, and therefore, we proposed a nonfragile gain-scheduled controller in this paper to fill the gap by making a few first attempts to deal with this problem.

The main contributions of this paper are summarized as follows: (1) a new nonfragile gain-scheduled control problem is addressed for a class of discrete-time nonlinear stochastic systems with randomly occurring phenomenon; (2) a sequence of stochastic variables satisfying Bernoulli distribution is introduced to describe the time-varying features of the ROSSs; (3) a time-varying Lyapunov functional dependent on the saturation probability is proposed and applied to improve the performance of system; (4) the parameters of the nonfragile gain-scheduled controller can be adjusted online according to the saturating probability estimated through statistical tests.

Notation. In this paper, ℝ^𝑛, ℝ^𝑛×𝑚, and 𝕀⁺ denote, respectively, the 𝑛-dimensional Euclidean space, the set of all 𝑛×𝑚 real matrices, and 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: 𝑥()(𝑘)()(𝑘)(𝑧(𝑘(𝑘)𝜔(𝑘),𝑥(𝑘)(𝑘)𝑘+1=𝐴𝑥+𝐷𝑥𝑘−𝑑+𝐵𝑢+𝑁𝑓))+𝐸𝑥=𝜌,𝑘=−𝑑,−𝑑+1,…,0, (1)(2) where 𝑥(𝑘)∈ℝ^𝑛 is the state, 𝑑 is a constant delay and 𝑧(𝑘)∶=𝐺𝑥(𝑘)+𝐺_𝑑𝑥(𝑘−𝑑), 𝜔(𝑘) is a one-dimensional Gaussian white noise sequence satisfying 𝔼{𝜔(𝑘)}=0 and 𝔼{𝜔²(𝑘)}=𝜎², and 𝜌(𝑘) is the initial state of the system. 𝐴, 𝐵, 𝐷, 𝐸, 𝑁, 𝐺, and 𝐺_𝑑 are constant real matrices of appropriate dimensions and 𝐵 is of full column. The nonlinear function 𝑓(⋅) with (𝑓(0)=0) is assumed as nonlinear disturbance and satisfies the following sector-bounded condition: 𝑓(𝑧(𝑘))−𝐹₁𝑧(𝑘)^𝑇𝑓(𝑧(𝑘))−𝐹₂𝑧(𝑘)≤0, (3) where 𝑓(⋅) belongs to the sector [𝐹₁,𝐹₂], 𝐹₁ and 𝐹₂ are given constant real matrices.

For the technique convenience, the nonlinear function 𝑓(𝑧(𝑘)) can be decomposed into a linear part and a nonlinear part as 𝑓(𝑧(𝑘))=𝑓_𝑠(𝑧(𝑘))+𝐹₁𝑧(𝑘); (4) then, from (3), we have 𝑓^𝑇_𝑠(𝑧(𝑘))𝑓_𝑠(𝑧(𝑘))−𝐹𝑧(𝑘)≤0, (5) where 𝐹=𝐹₂−𝐹₁>0.

The measurement output with sensor saturation is described as 𝑦(𝑘)=𝜉(𝑘)𝜚(𝐶𝑥(𝑘))+(1−𝜉(𝑘))𝐶𝑥(𝑘), (6) where 𝐶 is a constant real matrix of appropriate dimensions and 𝜚(𝑥)=sign(𝑥)min{1,|𝑥|}. Here, the notation of “sign” means the signum function, and we use the notation 𝜚 to denote saturation functions. Note that, without loss of generality, the saturation level is taken as unity.

According to the definition of the saturation function, we can get that the nonlinear function 𝜚 satisfies [𝜚(𝑥)−𝑎𝑥][𝜚(𝑥)−𝑥]≤0 and |𝑥|≤𝑎⁻¹, where 𝑎 is a positive scalar satisfying 0<𝑎<1, so the nonlinear function 𝜚(𝐶𝑥(𝑘)) satisfies [𝜚(𝐶𝑥(𝑘))−𝑎𝐶𝑥(𝑘)]^𝑇[𝜚(𝐶𝑥(𝑘))−𝐶𝑥(𝑘)]≤0, while |𝑎𝐶𝑥(𝑘)|≤1 and 𝑎 satisfies 0<𝑎<1.

The variable 𝜉(𝑘)∈ℝ is a random white sequence characterizing the probabilistic sensor saturation, which obeys the following time-varying Bernoulli distribution: {𝜉(𝑘)}{𝜉(𝑘(𝑘),{𝜉(𝑘)}{𝜉(𝑘(𝑘),Prob=1=𝔼)}=𝑝Prob=0=1−𝔼)}=1−𝑝 (7) 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. Throughout the paper, for simplicity, we assume that 𝜉(𝑘), 𝜔(𝑘) and 𝜌(𝑘) are uncorrelated.

Remark 1. In many practical systems, especially in NCSs, the measurement output is often subject to ROSSs, and the Bernoulli distribution model has been proven to be a very flexible and effective way to model randomly occurring phenomenon; see, for example, [1–3, 5, 6, 8, 13–15]. Furthermore, in practical engineering, the occurring probability of sensor saturation phenomenon usually changes with time. Therefore, in this paper, the occurrence of sensor saturation is described by a random variable sequence 𝜉(𝑘) satisfying a time-varying instead of time-invariant Bernoulli distribution model, which will reduce the conservatism when used to deal with the systems with time-varying ROSSs.

In this paper, we are interested in designing the following nonfragile gain-scheduled static output feedback controller: 𝑢(𝑘)=𝐾𝑝(𝑘)+Δ𝐾𝑦(𝑘), (8) where 𝐾(𝑝(𝑘)) is the controller gain sequence to be designed and assumed as the following structure: 𝐾𝑝(𝑘)=𝐾₀+𝑝(𝑘)𝐾_𝑢; (9) for every time step 𝑘, 𝑝(𝑘) is the time-varying parameter of the controller gain, and 𝐾₀ and 𝐾_𝑢 are the constant parameters of the controller gain to be designed, while Δ𝐾 is an unknown matrix of appropriate dimensions and represents the uncertainty in the controller, which is assumed to be of the form Δ𝐾=𝐿𝐻(𝑘)𝑀, (10) where 𝐿 and 𝑀 are known constant matrices with the structured information of the uncertainty, and 𝐻(𝑘) is an unknown, real, and time-varying matrix with Lebesgue-measurable elements satisfying 𝐻^𝑇(𝑘)𝐻(𝑘)≤𝐼,∀𝑘. (11)

Remark 2. Instead of using the information of system states, static output feedback control directly makes use of system outputs to design controllers, which has proven to be much simpler and easier to implement and has been extensively used in various kinds of engineering fields; for more details, we recommend some papers such as [23–27].

Remark 3. Owing to the pervasive existence of the uncertainties during controller implementation, an accuracy controller is needed to resist such an impact by the uncertainties, and the nonfragile controller has been proven to be an effective one; see, for example, [7, 20–22]. In another aspect, ROSSs are ubiquitous during the process of measurement, especially in NCSs, and gain-scheduling method has been successfully utilized to tackle with randomly occurring phenomenon in [8, 14, 15]. Therefore, in this paper, we design a nonfragile gain-scheduled static output feedback controller for nonlinear stochastic systems to deal with uncertainties and ROSSs simultaneously.

From the aforementioned, the closed-loop system with the nonfragile gain-scheduled controller is 𝑥()(𝑘)()𝑝(𝑘)×𝜉(𝑘)𝜚((𝑘+((𝑘(𝑘(𝑧(𝑘(𝑘)𝜔(𝑘).𝑘+1=𝐴𝑥+𝐷𝑥𝑘−𝑑+𝐵𝐾𝐶𝑥))1−𝜉))𝐶𝑥)+𝑁𝑓))+𝐸𝑥 (12)

Before formulating the problem to be investigated, we first introduce the following stability concepts.

Definition 4. The closed-loop system (12) is said to be exponentially mean-square stable if, with 𝜔(𝑘)=0, there exist constant 𝛼>0 and 𝜏∈(0,1) such that 𝔼𝑥_𝑘²≤𝛼𝜏^𝑘sup_{−𝑑≤𝑖≤0}𝔼𝑥_𝑖²,𝑘∈𝕀⁺. (13)

In this paper, our purpose is to design a probability-dependent nonfragile gain-scheduled controller of the form (8) for the system (1) by exploiting a probability-dependent Lyapunov functional and LMI method such that, for all admissible sensor saturations and exogenous stochastic noise, the closed-loop system (12) 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 5 ((Schur complement) [28]). 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. (14)

Lemma 6 (see [13]). 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 𝐵=𝑈^Σ₀𝑉^𝑇 and Σ=diag{𝜎₁,𝜎₂,…,𝜎_𝑚}. If matrix 𝑆 has the following structure: 𝑆=𝑈^𝑆11𝑆12_0𝑆22𝑈^𝑇, where 𝑆₁₁,𝑆₁₂∈𝑅^{𝑛×(𝑛−𝑚)} and 𝑆₂₂∈𝑅^{(𝑛−𝑚)×(𝑛−𝑚)}, then there exists a nonsingular matrix 𝑅∈𝑅^𝑚×𝑚 such that 𝑆𝐵=𝐵𝑅.

Lemma 7 ((𝑆-procedure) [28]). For given matrices 𝑄=𝑄^𝑇, 𝐻, and 𝐸 with appropriate dimensions, 𝑄+𝐻𝐹(𝑘)𝐸+𝐸^𝑇𝐹^𝑇(𝑘)𝐻^𝑇<0 (15) holds for all 𝐹(𝑘) satisfying 𝐹^𝑇(𝑘)𝐹(𝑘)≤𝐼 if and only if there exists 𝜀>0 such that 𝑄+𝜀⁻¹𝐻𝐻^𝑇+𝜀𝐸^𝑇𝐸<0. (16)

For convenience of presentation, we first consider the desired controller without uncertainty (i.e., Δ𝐾=0), and the result will be shown in Theorem 8. Then, we design the nonfragile gain-scheduled controller in Theorem 10 based on the conclusion in Theorem 8.

Theorem 8. Consider the discrete-time nonlinear stochastic systems with ROSSs (12). If there exist positive-definite matrices 𝑄(𝑝(𝑘)) and 𝑄_𝜏, slack matrix 𝑆, and nonsingular matrices 𝑌(𝑝(𝑘)) and 𝑅, such that the following LMIs hold: 𝑄Υ∶=^𝜏−𝑄𝑝(𝑘)−2𝑎𝐶^𝑇𝐶∗∗∗∗∗∗0−𝑄_𝜏()∗∗∗∗∗𝑎+1𝐶0−2𝐼∗∗∗∗𝐹𝐺𝐹𝐺_𝑑𝑆0−2𝐼∗∗∗_𝑇𝐴+1−𝑝(𝑘)𝐵𝑌𝑝(𝑘)𝐶𝑆^𝑇𝐷𝑝(𝑘)𝐵𝑌𝑝(𝑘)𝑆^𝑇𝑁−𝜎Λ∗∗₂𝑆^𝑇𝐸0000−𝜎²ΔΛ∗_𝑝(𝑘)𝐵𝑌𝑝(𝑘)𝐶0Δ_𝑝(𝑘)𝐵𝑌𝑝(𝑘)000−Δ_𝑝(𝑘)Λ<0, (17)where Λ=−𝑄𝑝(𝑘+1)+𝑆+𝑆^𝑇,Δ_𝑝(𝑘)=𝑝(𝑘)1−𝑝(𝑘),𝐴=𝐴+𝑁𝐹₁𝐺,𝐷=𝐷+𝑁𝐹₁𝐺_𝑑,𝑆^𝑇𝑝(𝑘)𝑝(𝑘)𝑝(𝑘)𝐵=𝐵𝑅,𝑅𝐾=𝑌,𝐾=𝑅⁻¹𝑌𝑝(𝑘),𝑌𝑝(𝑘)=𝑌₀+𝑝(𝑘)𝑌_𝑢, (18) in this case, the constant gains of the desired controller can be obtained as follows: 𝐾₀=𝑅⁻¹𝑌₀,𝐾_𝑢=𝑅⁻¹𝑌_𝑢, (19) and the closed-system (12) is then exponentially mean-square stable for all 𝑝(𝑘)∈𝑝₁𝑝₂.

Proof. Define the Lyapunov functional 𝑉(𝑘)∶=𝑥^𝑇(𝑘)𝑄𝑝(𝑘)𝑥(𝑘)+^𝑘−1_{𝑠=𝑘−𝑑}𝑥^𝑇(𝑠)𝑄_𝜏𝑥(𝑠); (20) noting that 𝔼{𝜉(𝑘)−𝑝(𝑘)}=0, 𝔼{𝜔(𝑘)}=0, and 𝔼{[𝜉(𝑘)−𝑝(𝑘)]²}=𝑝(𝑘)(1−𝑝(𝑘))≜Δ_𝑝(𝑘), we can get 𝔼{Δ𝑉(𝑘)}=𝔼𝑥^𝑇(𝑘+1)𝑄𝑝(𝑘+1)𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑄𝑝(𝑘)−𝑄_𝜏𝑥(𝑘)−𝑥^𝑇(𝑘−𝑑)𝑄_𝜏𝑥((𝑘)(𝑘)𝑝(𝑘)×𝜚((𝑘(𝑘+𝜉(𝑘)(𝑘𝑝(𝑘×𝜚((𝑘(𝑘𝑝(𝑘)(𝑘)(𝑧(𝑘(𝑘)𝜔(𝑘𝑘−𝑑)=𝔼𝐴𝑥+𝑝𝐵𝐾𝐶𝑥))−𝐶𝑥)−𝑝)𝐵𝐾)𝐶𝑥))−𝐶𝑥)+𝐵𝐾𝐶𝑥+𝑁𝑓))+𝐷𝑥(𝑘−𝑑)+𝐸𝑥)^𝑇𝑄𝑝(×(𝑘)(𝑘)𝑝(𝑘)×𝜚((𝑘(𝑘+𝜉(𝑘)(𝑘𝑝(𝑘×𝜚((𝑘(𝑘𝑝(𝑘)(𝑘)()(𝑧(𝑘(𝑘)𝜔(𝑘𝑘+1)𝐴𝑥+𝑝𝐵𝐾𝐶𝑥))−𝐶𝑥)−𝑝)𝐵𝐾)𝐶𝑥))−𝐶𝑥)+𝐵𝐾𝐶𝑥+𝐷𝑥𝑘−𝑑+𝑁𝑓))+𝐸𝑥)−𝑥^𝑇(𝑘)𝑄𝑝(𝑘)𝑥(𝑘)+𝑥^𝑇(𝑘)𝑄_𝜏𝑥(𝑘)−𝑥^𝑇(𝑘−𝑑)𝑄_𝜏𝑥(𝑘−𝑑)≤𝔼(𝑘)𝑝(𝑘)𝐶𝑥(𝑘)(𝑘)𝑝(𝑘)𝜚((𝑘+𝐴+1−𝑝𝐵𝐾+𝑝𝐵𝐾𝐶𝑥))𝐷𝑥(𝑘−𝑑)+𝑁𝑓_𝑠(𝑧(𝑘))^𝑇𝑄𝑝()×𝑘+1(𝑘)𝑝(𝑘)𝐶𝑥(𝑘)(𝑘)𝑝(𝑘)𝜚((𝑘+𝐴+1−𝑝𝐵𝐾+𝑝𝐵𝐾𝐶𝑥))𝐷𝑥(𝑘−𝑑)+𝑁𝑓_𝑠(𝑧(𝑘(𝑘)(𝑘)×𝑝(𝑘)𝜚((𝑘(𝑘)))+𝑝1−𝑝𝐵𝐾𝐶𝑥))−𝐶𝑥^𝑇𝑝(𝑝(𝑘×𝜚((𝑘(𝑘×𝑄𝑘+1)𝐵𝐾)𝐶𝑥))−𝐶𝑥)+𝜎²𝑥^𝑇(𝑘)𝐸^𝑇𝑄𝑝(𝑘+1)𝐸𝑥(𝑘)−𝑥^𝑇(𝑘)𝑄𝑝(𝑘)𝑥(𝑘)−𝑥^𝑇(𝑘−𝑑)𝑄_𝜏𝑥(𝑘−𝑑)+𝑥^𝑇(𝑘)𝑄_𝜏𝑥(𝑘)+2𝑓^𝑇_𝑠(𝑧(𝑘))𝐹𝐺𝑥(𝑘)+2𝑓^𝑇_𝑠(𝑧(𝑘))𝐹𝐺_𝑑𝑥(𝑘−𝑑)−2𝑓^𝑇_𝑠(𝑧(𝑘))𝑓_𝑠(𝑧(𝑘))−2𝜚^𝑇((𝑘𝜚((𝑘+()𝜚𝐶𝑥))𝐶𝑥))2+2𝑎^𝑇((𝑘(𝑘)((𝑘𝐶𝑥))𝐶𝑥−2𝑎𝐶𝑥))^𝑇𝐶𝑥(𝑘). (21)
Denote the following matrix variables: 𝜂(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−𝑑)𝜚^𝑇(𝐶𝑥(𝑘))𝑓^𝑇_𝑠(𝑧(𝑘))^𝑇; (22) then, it is obvious that 𝔼{Δ𝑉(𝑘)}≤𝔼𝜂^𝑇(𝑘)Ω𝜂(𝑘), (23) where ΩΩ=¹Ω∗∗∗₂Ω₃Ω∗∗₄Ω₅Ω₆∗Ω₇Ω₈Ω₉Ω₁₀,Ω₁=𝐴+1−𝑝(𝑘)𝐵𝐾𝑝(𝑘)𝐶^𝑇×𝑄𝑝(𝑘+1)𝐴+1−𝑝(𝑘)𝐵𝐾𝑝(𝑘)𝐶+𝜎²𝐸^𝑇𝑄𝑝(𝑘+1)𝐸+Δ_𝑝(𝑘)𝐵𝐾𝑝(𝑘)𝐶^𝑇𝑄𝑝()𝑝(𝑘𝑘+1×𝐵𝐾)𝐶+𝑄_𝜏−𝑄𝑝(𝑘)−2𝑎𝐶^𝑇Ω𝐶,₂=𝐷^𝑇𝑄𝑝(𝑘+1)(𝑘)𝑝(𝑘)𝐶,Ω𝐴+1−𝑝𝐵𝐾₃=𝐷^𝑇𝑄𝑝(𝑘+1)𝐷−𝑄_𝜏,Ω₄=𝑝(𝑘)𝐵𝐾𝑝(𝑘)^𝑇𝑄𝑝(×𝑘+1)𝐴+1−𝑝(𝑘)𝐵𝐾𝑝(𝑘)𝐶+Δ_𝑝(𝑘)𝐵𝐾𝑝(𝑘)^𝑇𝑄𝑝()𝑝(𝑘)()Ω𝑘+1×𝐵𝐾𝐶+𝑎+1𝐶,₅=𝑝(𝑘)𝐵𝐾𝑝(𝑘)^𝑇𝑄𝑝(𝑘+1)Ω𝐷,₆=𝑝²(𝑘)𝐵𝐾𝑝(𝑘)^𝑇𝑄𝑝()𝑝(𝑘)𝑘+1×𝐵𝐾+Δ_𝑝(𝑘)𝐵𝐾𝑝(𝑘)^𝑇𝑝(𝑝(𝑘Ω×𝑄𝑘+1)𝐵𝐾)−2𝐼,₇=𝑁^𝑇𝑄𝑝()×𝑘+1(𝑘)𝑝(𝑘)𝐶Ω𝐴+1−𝑝𝐵𝐾+𝐹𝐺,₈=𝑁^𝑇𝑄𝑝(𝑘+1)𝐷+𝐹𝐺_𝑑,Ω₉=𝑝(𝑘)𝑁^𝑇𝑄𝑝()𝑝(𝑘),Ω𝑘+1𝐵𝐾₁₀=𝑁^𝑇𝑄𝑝(𝑘+1)𝑁−2𝐼. (24)
If Ω≤0, we can conclude the following matrix by Schur complement: 𝑄_𝜏−𝑄𝑝(𝑘)−2𝑎𝐶^𝑇𝐶∗∗∗∗∗∗0−𝑄_𝜏()∗∗∗∗∗𝑎+1𝐶0−2𝐼∗∗∗∗𝐹𝐺𝐹𝐺_𝑑0−2𝐼∗∗∗𝐴+1−𝑝(𝑘)𝐵𝐾𝑝(𝑘)𝐶𝐷𝑝(𝑘)𝐵𝐾𝑝(𝑘)𝑁−Λ∗∗𝐸0000−𝜎⁻²𝑝(𝑘)𝑝(𝑘)Λ∗𝐵𝐾𝐶0𝐵𝐾000−Δ_𝑝⁻¹(𝑘)Λ<0, (25)where Λ=𝑄⁻¹𝑝(𝑘+1); (26) by preforming the congruence transformation diag{𝐼,𝐼,𝐼,𝐼,𝑆,𝜎²𝑆,Δ_𝑝(𝑘)𝑆} to (25), we have𝑄_𝜏−𝑄𝑝(𝑘)−2𝑎𝐶^𝑇𝐶∗∗∗∗∗∗0−𝑄_𝜏()∗∗∗∗∗𝑎+1𝐶0−2𝐼∗∗∗∗𝐹𝐺𝐹𝐺_𝑑𝑆0−2𝐼∗∗∗_𝑇𝐴+1−𝑝(𝑘)𝑆^𝑇𝐵𝐾𝑝(𝑘)𝐶𝑆^𝑇𝐷𝑝(𝑘)𝑆^𝑇𝐵𝐾𝑝(𝑘)𝑆^𝑇𝜎𝑁−Λ∗∗₂𝑆^𝑇𝐸0000−𝜎²ΔΛ∗_𝑝(𝑘)𝑆^𝑇𝐵𝐾𝑝(𝑘)𝐶0Δ_𝑝(𝑘)𝑆^𝑇𝐵𝐾𝑝(𝑘)000−Δ_𝑝(𝑘)Λ<0, (27)where Λ=𝑆^𝑇𝑄⁻¹𝑝(𝑘+1)𝑆. (28) From inequality 𝑆^𝑇𝑄⁻¹𝑝(𝑘+1)𝑆≥𝑆^𝑇+𝑆−𝑄𝑝(𝑘+1)≜Λ, (29) we can get 𝑄_𝜏−𝑄𝑝(𝑘)−2𝑎𝐶^𝑇𝐶∗∗∗∗∗∗0−𝑄_𝜏()∗∗∗∗∗𝑎+1𝐶0−2𝐼∗∗∗∗𝐹𝐺𝐹𝐺_𝑑𝑆0−2𝐼∗∗∗_𝑇𝐴+1−𝑝(𝑘)𝑆^𝑇𝐵𝐾𝑝(𝑘)𝐶𝑆^𝑇𝐷𝑝(𝑘)𝑆^𝑇𝐵𝐾𝑝(𝑘)𝑆^𝑇𝑁−𝜎Λ∗∗₂𝑆^𝑇𝐸0000−𝜎²ΔΛ∗_𝑝(𝑘)𝑆^𝑇𝐵𝐾𝑝(𝑘)𝐶0Δ_𝑝(𝑘)𝑆^𝑇𝐵𝐾𝑝(𝑘)000−Δ_𝑝(𝑘)Λ<0. (30)By using Lemma 6, we have 𝑆^𝑇𝐵=𝐵𝑅, and denoting that 𝑅𝐾(𝑝(𝑘))=𝑌(𝑝(𝑘)), then (30) can be written as (17); furthermore, we can know from Lemma 5 that Ω<0 and, subsequently, 𝔼{Δ𝑉(𝑘)}<−𝜆_min(−Ω)𝔼𝜂(𝑘)², (31) where 𝜆_min(−Ω) is the minimum eigenvalue of (−Ω). Finally, we can confirm from Lemma 1 in [13] that the closed-loop system is exponentially mean-square stable; then the proof of this theorem is complete.

Remark 9. The ROSSs have been studied in [1, 12] by constructing a concise and effective time-invariant Bernoulli distribution model; however, in many practical systems, ROSSs sometimes appear with time-varying probability. Therefore, in this case, we considered ROSSs satisfying time-varying Bernoulli distribution which is more reasonable in reality. On the other hand, unlike other time-varying parameters discussed in gain-scheduling technique or parameter-dependent Lyapunov functional; see, for example, [17–19], the parameter 𝑝(𝑘) considered in this paper is the time-varying occurrence probability of ROSSs, based on which a new kind of controller is designed and a novel probability-dependent Lyapunov functional is proposed to reduce the potential conservatism.

Next, we are in a position to consider the nonfragile gain-scheduled controller design for system (12) based on what we got in Theorem 8.

Theorem 10. Consider the discrete-time nonlinear stochastic systems with ROSSs (12) and the nonfragile gain-scheduled controller (8). If there exist positive-definite matrices 𝑄(𝑝(𝑘)) and 𝑄_𝜏, slack matrix 𝑆, and nonsingular matrices 𝑌(𝑝(𝑘)) and 𝑅, scalars 𝜀₁>0, 𝜀₂>0, LMIs (17), equations (18), and the following LMIs hold: 𝜀Υ∗∗∗∗₁Π^𝑇₂−𝜀₁Π𝐼∗∗∗₃0−𝜀₁𝜀𝐼∗∗₂Π^𝑇₄00−𝜀₂Π𝐼∗₅000−𝜀₂𝐼<0, (32) where Π₂=00001−𝑝(𝑘)𝑆^𝑇𝐵𝐿^𝑇0Δ_𝑝(𝑘)𝑆^𝑇𝐵𝐿^𝑇𝑇,Π₃=,Π𝑀𝐶0𝑀0000₄=00002𝑝(𝑘)−1𝑆^𝑇𝐵𝐿^𝑇00^𝑇,Π₅=00𝑀0000; (33) in this case, the constant gains of the desired controller can be obtained as follows: 𝐾₀=𝑅⁻¹𝑌₀,𝐾_𝑢=𝑅⁻¹𝑌_𝑢, (34) and the closed-system (12) is then exponentially mean-square stable for all 𝑝(𝑘)∈𝑝₁𝑝₂.

Proof. In order to get the nonfragile gain-scheduled controller, we replace the 𝐾(𝑝(𝑘)) with 𝐾(𝑝(𝑘))+Δ𝐾; then, 𝑅𝐾(𝑝(𝑘))=𝑌(𝑝(𝑘)) can be written as 𝑅[𝐾(𝑝(𝑘))+Δ𝐾]=𝑌(𝑝(𝑘))+Δ𝑌, Δ𝑌=𝑅Δ𝐾=𝑅𝐿𝐻(𝑘)𝑀. Noting that 𝑆^𝑇𝐵=𝐵𝑅, we can rewrite (17) as Υ+Π₂𝐻(𝑘)Π₃+Π^𝑇₃𝐻^𝑇(𝑘)Π^𝑇₂+Π₄𝐻(𝑘)Π₅+Π^𝑇₅𝐻^𝑇(𝑘)Π^𝑇₄<0. (35) From Lemma 7, we know that a necessary and sufficient condition guaranteeing (35) is that there exist scalars 𝜀₁>0, 𝜀₂>0 such that Υ+𝜀₁Π₂Π^𝑇₂+𝜀₁⁻¹Π^𝑇₃Π₃+𝜀₂Π₄Π^𝑇₄+𝜀₂⁻¹Π^𝑇₅Π₅<0; (36) by using the knowledge of Schur complement, we can find that (36) is equivalent to (32). Now, the proof is complete.

Remark 11. In Theorem 10, a nonfragile gain-scheduled 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 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). First of all, let us set 𝑄(𝑝(𝑘))=𝑄₀+𝑝(𝑘)𝑄_𝑢.

Theorem 12. Consider the discrete-time nonlinear stochastic system with ROSSs (12). If there exist positive-definite matrices 𝑄₀, 𝑄_𝑢 and 𝑄_𝜏, slack matrix 𝑆 and nonsingular matrices 𝑌(𝑝(𝑘)) and 𝑅, such that the following LMIs hold: 𝕄^{𝑖𝑗𝑙𝑚}Υ∶=^{𝑖𝑗𝑙𝑚}𝜀∗∗∗∗₁Π₂^𝑇𝑖𝑗−𝜀₁Π𝐼∗∗∗₃0−𝜀₁𝜀𝐼∗∗₂Π^𝑖₄^𝑇00−𝜀₂Π𝐼∗₅000−𝜀₂𝐼Υ<0,_{𝑖𝑗𝑙𝑚}𝑄∶=^𝜏−𝑄^𝑖𝑝(𝑘)−2𝑎𝐶^𝑇𝐶∗∗∗∗∗∗0−𝑄_𝜏()∗∗∗∗∗𝑎+1𝐶0−2𝐼∗∗∗∗𝐹𝐺𝐹𝐺_𝑑𝑆0−2𝐼∗∗∗_𝑇𝐴+1−𝑝_𝑖𝐵𝑌^𝑚𝑝(𝑘)𝐶𝑆^𝑇𝐷𝑝_𝑖𝐵𝑌^𝑚𝑝(𝑘)𝑆^𝑇𝑁−Λ^𝑙𝜎∗∗₂𝑆^𝑇𝐸0000−𝜎²Λ^𝑙∗Δ^𝑖𝑗𝐵𝑌^𝑚𝑝(𝑘)𝐶0Δ^𝑖𝑗𝐵𝑌^𝑚𝑝(𝑘)000−Δ^𝑖𝑗Λ^𝑙, (37)where Λ=−𝑄₀−𝑝_𝑙𝑄_𝑢+𝑆+𝑆^𝑇,Δ^𝑖𝑗=𝑝_𝑖1−𝑝_𝑗,𝐴=𝐴+𝑁𝐹₁𝐺,𝐷=𝐷+𝑁𝐹₁𝐺_𝑑,𝑆^𝑇𝑝(𝑘)𝑝(𝑘),𝐾𝑝(𝑘𝐵=𝐵𝑅,𝑅𝐾=𝑌)=𝑅⁻¹𝑌𝑝(𝑘,𝑌_𝑚)𝑝(𝑘)=𝑌₀+𝑝_𝑚𝑌_𝑢,𝑄^𝑖𝑝(𝑘)=𝑄₀+𝑝_𝑖𝑄_𝑢,Π₂^𝑖𝑗=00001−𝑝_𝑖𝑆^𝑇𝐵𝐿^𝑇0Δ^𝑖𝑗𝑆^𝑇𝐵𝐿^𝑇𝑇,Π₃=,Π𝑀𝐶0𝑀0000_𝑖4=00002𝑝_𝑖−1𝑆^𝑇𝐵𝐿^𝑇00^𝑇,Π₅=00𝑀0000, (38) in this case, the constant gains of the desired controller can be obtained as follows: 𝐾₀=𝑅⁻¹𝑌₀,𝐾_𝑢=𝑅⁻¹𝑌_𝑢, (39) and the closed-system (12) 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; (40) therefore, we have 𝑝(𝑘)=𝛼₁(𝑘)𝑝₁+𝛼₂(𝑘)𝑝₂, (41) 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, (42) and we have 𝑝(𝑘+1)=𝛽₁(𝑘)𝑝₁+𝛽₂(𝑘)𝑝₂, (43) with 𝛽_𝑖(𝑘)≥0(𝑖=1,2), 𝛽₁(𝑘)+𝛽₂(𝑘)=1. From the pervious transformation, we can easily get 𝑄𝑝(𝑘)=²_𝑖=1𝛼_𝑖(𝑘)𝑄^𝑖𝑝(𝑘),Λ=²_𝑙=1𝛽_𝑙(𝑘)Λ^𝑙,𝑌𝑝(𝑘)=²_𝑚=1𝛼_𝑚(𝑘)𝑌^𝑚𝑝(𝑘). (44) 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. (45) From (40)–(45), we can have that (32) in Theorem 10 is true; then the proof is now complete.

Remark 13. By using the methods proposed in the proof of Theorem 10, we choose 4 variables; then, it is easy to calculate the number of LMIs as 2⁴ depending on the upper and lower bound of 𝑝(𝑘).

4. An Illustrative Example

In this section, the nonfragile gain-scheduled controller is designed for the discrete-time nonlinear stochastic systems with ROSSs.

The system parameters are given as follows: ,,𝐴=0.44000.81,𝑁=0.130.20.280.33𝐵=0.0109.22.8,𝐶=00.190.62.20𝐷=0.020.140.150.18,𝐹₁=,𝐹0.06000.01₂=,𝐺0.1000.01,𝐺=0.080.120.080.02_𝑑=,,𝐻(𝑘)0.010.090.180.09,𝐸=0.30.190.10.02𝐿=0.01000.02,𝑀=0.1000.02=𝐼,𝑝₁=0.49,𝑝₂𝜎=0.51,₂=1,𝑎=0.411,𝜀₁=0.21,𝜀₂=0.2. (46)

Set the time-varying Bernoulli distribution sequences as 𝑝(𝑘)=𝑝₁+(𝑝₂−𝑝₁)|sin(𝑘)|, and the sector nonlinear function 𝑓(𝑢) is taken by 𝑓(𝑢)=𝐹₁+𝐹₂2𝐹𝑢+²−𝐹₁2sin(𝑢), (47) which satisfies (3). Also, select the initial state 𝜌=2−2^𝑇.

According to Theorem 12, the constant controller parameters 𝐾₀, 𝐾_𝑢 can be obtained as follows: 𝐾₀=10.2565−12.4544−33.730640.8091,𝐾_𝑢=0.1041−0.0090−0.34210.0296. (48)

Then, according to the measured time-varying probability parameters 𝑝(𝑘), the gain-scheduled controller gain 𝐾(𝑝(𝑘)) and parameter-dependent Lyapunov matrix 𝑄(𝑝(𝑘)) 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 nonfragile gain-scheduled control problem for a class of discrete stochastic systems with ROSSs is tackled, and the sensor saturation phenomenon is assumed to occur in a random way based on the time-varying Bernoulli distribution with measurable probability in real time. By employing probability-dependent Lyapunov functional, we design a nonfragile gain-scheduled controller with the gain including both constant and time-varying parameters such that, for all admissible sensor saturations, time-delays and noise disturbances, the closed-loop system is still exponentially mean-square stable. Furthermore, we can extend the main results to more complex and realistic systems, for instance, complex networks and systems with several kinds of randomly occurring phenomena simultaneously. Meanwhile, we can also consider corresponding control/filtering problems for time-varying systems with time-varying ROSSs, such as robust sliding mode control, quantized recursive filtering, or extended Kalman filtering. The related references can be found; see, for example, [2, 3, 6].

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 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.