Observer-Based 𝐻_∞ Control Design for Nonlinear Networked Control Systems with Limited Information

Abstract

This paper is concerned with the problem of designing a robust observer-based 𝐻_∞ controller for discrete-time networked systems with limited information. An improved networked control system model is proposed and the effects of random packet dropout, time-varying delay, and quantization are considered simultaneously. Based on the obtained model, a stability criterion is developed by constructing an appropriate Lyapunov-Krasovskii functional and sufficient conditions for the existence of a dynamic quantized output feedback controller which are given in terms of linear matrix inequalities (LMIs) such that the augmented error system is stochastically stable with an 𝐻_∞ performance level. An example is presented to illustrate the effectiveness of the proposed method.

1. Introduction

Networked control systems (NCSs) are distributed systems in which communication between sensors, actuators and controllers is supported by a shared real-time network. Compared with conventional point-to-point system connection, this new network-based control scheme reduces system wiring and has low cost, high reliability, information sharing, and remote control [1, 2]. However, NCSs also introduce many new challenges in control system design such as packet dropout, networked-induced delay and signal quantization variable transmission intervals, network security and other communication constraints [3–9]. The authors in [10] proposed different levels of network-induced imperfections and presented the main survey of the methodologies in the analysis and design of NCSs.

The time-delay and packet dropouts occur in various physical, industrial, and engineering systems, which is a source of poor performance and instability of systems. Therefore, the issues of network-induced time-delays and packet dropouts have been considered by many researchers. Markov chain has been used to model the randomness of the network-induced delays (packet dropouts) [11–15]. The Markov chain takes values, which correspond to network-induced delays (packet dropouts), in a finite set based on known probabilities. In addition, the transition probabilities of Markovian jump systems (MJSs) are partly unknown due to the complexity of network [16, 17]. The authors in [16] introduced a nonlinear delayed MJSs model with partially unknown transition probabilities established by multiple channels data transmission framework. An iterative method was proposed in [17] to model NCSs with bounded packet dropout as Markovian jump linear systems with partly unknown transition probabilities. On the other hand, much attention has been devoted to the time-delayed jump linear systems with Markovian jumping parameters [18–20], which applied a less-conservative delay-range-dependent method. The stability analysis and stabilization problems for a class of discrete-time Markov jump linear systems with partially known transition probabilities, and time-varying delays are investigated in [18].

Quantization process cannot be neglected in NCSs. Real communication networks only send finite precision data due to limited bandwidth. Quantizers with coarser quantization densities help in reducing the network congestion. Consequently, network-induced delays can be reduced because less information is transmitted. In [21, 22], the quantizing effects were transformed into sector-bounded uncertainties and the quantization density was given in the control synthesis. For a logarithmic quantizer, the quantization error was modeled as a norm-bounded uncertainty [23]. However, the quantization density in most studies is just neglected or given as sector bounded uncertainties, and the relationship between the quantization density and the network load has not been fully investigated in the aforementioned papers.

Motivated by the above discussion, in this paper, we are aiming at investigating the study of observer-based controller design for nonlinear discrete-time networked systems subject to sensor-controller packet losses, time-varying delays, and output quantization. In this work, the quantization density is designed to be a function of the network load condition which is modeled by a Markov chain [24]. Furthermore, by applying the Lyapunov-Krasovskii functional method, a mode-dependent observer is constructed and the bounded real lemma (BRL) for the resulting error system is derived in terms of LMIs. Also, an improved version of the BRL is given by introducing additional slack matrix variables to eliminate the cross-coupling between system matrices and Lyapunov matrices among different operation modes, which can reduce the computational complexity as far as possible.

The paper is organized as follows. In Section 2, the NCS description, quantization error modeling, packet dropout and time-varying delay formulation are presented. Main results for stability criterion and observer-based controller synthesis are given in Section 3. A numerical example is provided in Section 4. Finally, Section 5 concludes this paper.

Notation. Throughout the paper, the superscripts “−1” and “𝑇” stand for the inverse and transpose of a matrix, respectively; 𝑅^𝑛 denotes an 𝑛-dimensional Euclidean space and the notation 𝑃>0 means that 𝑃 is a real symmetric positive definite matrix. 𝐸{𝑥} is the expectation of a stochastic variable 𝑥. 𝐼 and 0 represent an identity matrix and a zero matrix with appropriate dimensions in different place, respectively. In symmetric block matrices or complex matrix expressions, we use an asterisk * to represent a term that is induced by symmetry and diag{⋯} stands for a block diagonal matrix. ‖⋅‖ refers to the Euclidean norm for vectors and induced 2-norm for matrices. 𝑙₂[𝑘₀,∞) stands for the space of square summable infinite sequence on [𝑘₀,∞).

2. Problem Formulation

A simple-networked control system is shown in Figure 1. A class of discrete-time nonlinear systems under consideration is described by the following equations: 𝑥()(𝑘)(𝑘)((𝑘(𝑘),𝑦(𝑘)(𝑘),𝑧(𝑘)(𝑘),𝑘+1=𝐴𝑥+𝐵𝑢+𝑓𝑘,𝑥))+𝐻𝑤=𝐶𝑥=𝐺𝑥 (1) where 𝑥(𝑘)∈𝑅^𝑛, 𝑢(𝑘)∈𝑅^𝑚, 𝑧(𝑘)∈𝑅^𝑝, and 𝑦(𝑘)∈𝑅^𝑠 are the state vector, control input vector, controlled output vector, and measured output, respectively. 𝑤(𝑘)∈𝑅^𝑑 is the exogenous disturbance signal belonging to 𝑙₂[0,∞). 𝐴, 𝐵, 𝐻, 𝐶, and 𝐺 are known real matrices with appropriate dimensions. The nonlinear function𝑓(⋅) satisfies 𝑓(⋅,0)=0 and the following sector-bounded condition [25]: 𝑓(𝑘,𝑥(𝑘))−𝑆₁𝑥(𝑘)^𝑇𝑓(𝑘,𝑥(𝑘))−𝑆₂𝑥(𝑘)≤0, (2) where 𝑆₁, 𝑆₂∈𝑅^𝑛×𝑛 are known real constant matrices, and 𝑆=𝑆₁−𝑆₂ is a symmetric positive definite matrix.

Figure 1 

Configuration of the networked control systems with quantizer.

The network load condition is related to the network-induced delay. In [26], a Markov chain has been used to model the network load condition. Varying network load can be modeled by making the Markov chain do a transition and transfer every time over the communication network. Modes 1, 2, and 3 are, respectively, corresponded to light load, medium load, and heavy load.

Then, a discrete homogeneous Markov chain {𝜎(𝑘),𝑘} with a finite number of states ℜ={1,2,3} is used to model a network load. The transition probability from mode 𝑖 at 𝑘 to mode 𝑗 at time 𝑘+1 is governed by 𝜋_𝑖𝑗=prob𝜎(𝑘+1)=𝑗∣𝜎(𝑘)=𝑖,𝑖,𝑗∈ℜ, (3) where 𝜋_𝑖𝑗≥0(𝑖,𝑗∈ℜ) is the transition probability element and satisfies ∑³_𝑗=1𝜋_𝑖𝑗=1 (for all 𝑖∈ℜ).

In addition, the transition probability considered in this paper is partly accessed, and some elements are unknown. For notation clarity, we denote ℜ=ℜ^𝑖_𝐾+ℜ^𝑖_𝑈𝐾 (for all 𝑖∈ℜ) with ℜ^𝑖_𝐾≜{𝑗∶𝜋_𝑖𝑗isknown} and ℜ^𝑖_𝑈𝐾≜{𝑗∶𝜋_𝑖𝑗isunknown}.

Also, we can obtain _𝑗∈ℜ𝜋_𝑖𝑗𝑃_𝑗=𝑃^𝑖_𝐾+_{𝑗∈ℜ𝑖𝑈𝐾}𝜋_𝑖𝑗𝑃_𝑗, (4) where 𝑃^𝑖_𝐾=∑_{𝑗∈ℜ𝑖𝐾}𝜋_𝑖𝑗𝑃_𝑗.

In this paper, the quantization density is designed to be a function of the network load condition. Hence, the quantization density function 𝛿=𝛿(𝜎(𝑘)) is described as a finite state Markov chain. On the basis of this technique, a networked load dependent on the logarithmic quantizer [23, 24] is proposed as follows: 𝑞()=𝜌𝜐,𝑖_ℎ(𝑖)if1(𝑖)𝜌1+𝛿^ℎ(𝑖)1<𝜐≤(𝑖)𝜌1−𝛿^ℎ(𝑖),0𝜐>0,ℎ=0,±1,±2,…if()𝜐=0−𝑞−𝜐,𝑖if𝜐<0, (5) where 0<𝜌(𝑖)<1 is the quantization density of 𝑞(⋅,⋅), and 𝛿(𝑖) is related to 𝜌(𝑖) by 𝛿(𝑖)=1−𝜌(𝑖)(𝑖).1+𝜌 (6)

This means that the smaller 𝜌(𝑖) results in the smaller 𝛿(𝑖). For this reason, we will call 𝛿(𝑖) the quantization density in the rest of this paper.

The associated quantized set 𝑈 is given by 𝑈=±𝜌^ℎ(𝑖),ℎ=0,±1,±2,…{0}. (7)

Now define the quantization error as 𝑒_𝑦(𝑘,𝑖)=𝑞𝑦(𝑘),𝑖−𝑦(𝑘)=Δ_𝑞(𝑘,𝑖)𝑦(𝑘), (8) where 𝑦(𝑘) is the signal to be quantized and 𝑞(𝑦(𝑘),𝑖) is the quantized signal. The uncertainty matrix Δ_𝑞(𝑘,𝑖)≜diag(Δ_𝑞1(𝑘,𝑖),…,Δ_𝑞𝑠(𝑘,𝑖)) satisfies Δ_𝑞𝜄(𝑘,𝑖)∈[−𝛿(𝑖),𝛿(𝑖)], 𝜄=1,2,…,𝑠.

The measured output 𝑦(𝑘) which is used as feedback information for controller may not be available all the time due to packet dropouts. To describe these packet dropouts, a stochastic variable following Bernoulli sequence is used. In this paper, the measured output received in the controller side is involved with the effects of quantization, packet losses, and time-varying delay, and it is described by 𝑦(𝑘)=𝜃_𝑘𝑞𝑦𝑘−𝜏_𝑘, (9) where 𝜏=min𝜏_𝑘 and 𝜏=max𝜏_𝑘 are the lower and upper bounds of the delay, respectively. Stochastic variable 𝜃_𝑘∈𝑅 is a Bernoulli distributed white sequence with a probability distribution as follows: Prob𝜃_𝑘=1=𝐸𝜃_𝑘=𝜃,Prob𝜃_𝑘=0=1−𝐸𝜃_𝑘=1−𝜃, (10) where 0≤𝜃≤1 is a known positive constant which is used to denote the probability that the packet will be transmitted successfully from sensor to controller, and we have 𝐸𝜃_𝑘−𝜃=0,𝐸𝜃_𝑘−𝜃²=𝜃(1−𝜃)=𝜃². (11)

Remark 1. For NCSs, network-induced delay, packet dropout, and quantization error often occur simultaneously due to limited channel capacity and noise. Actually, pattern (9) describes a class of NCSs with data loss, time-varying delay and the quantization error simultaneously, so it is more comprehensive than separate consideration of delay pattern [27] or data loss pattern [28].

In this work, the following dynamic observer-based control scheme is employed for system (1): 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵𝑢(𝑘)+𝐿_𝜎(𝑘)𝑦(𝑘)−𝑦(𝑘)𝑦(𝑘)𝑥(𝑘)𝑢(𝑘)=𝜃𝐶=𝐾_𝜎(𝑘)𝑥(𝑘), (12) where 𝑥(𝑘)∈𝑅^𝑛 and 𝑦(𝑘)∈𝑅^𝑠 represent the state estimation and output estimation vectors of system (1), respectively. 𝐿_𝜎(𝑘)∈𝑅^𝑛×𝑠 and 𝐾_𝜎(𝑘)∈𝑅^𝑚×𝑛 are the observer gain and controller feedback gain to be designed, respectively.

We define the following error variable as 𝑒(𝑘)=𝑥(𝑘)−𝑥(𝑘). (13)

Combining (12) with (1) and considering (9) and (11), the error system is derived as follows: 𝑒(𝑘+1)=𝜃𝐿_𝜎(𝑘)𝐶𝑥(𝑘)+𝐴−𝜃𝐿_𝜎(𝑘)𝐶𝑒(𝑘)−𝜃𝐿_𝜎(𝑘)𝐼+Δ_𝑞(𝑘,𝜎(𝑘))𝐶𝑥𝑘−𝜏_𝑘−𝜃_𝑘−𝜃𝐿_𝜎(𝑘)𝐼+Δ_𝑞(𝑘,𝜎(𝑘))𝐶𝑥𝑘−𝜏_𝑘+𝑓(𝑘,𝑥(𝑘))+𝐻𝑤(𝑘). (14)

On the other hand, system (1) can be rewritten as 𝑥(𝑘+1)=𝐴+𝐵𝐾_𝜎(𝑘)𝑥(𝑘)−𝐵𝐾_𝜎(𝑘)𝑒(𝑘)+𝑓(𝑘,𝑥(𝑘))+𝐻𝑤(𝑘). (15)

By setting 𝜂(𝑘)=𝑥^𝑇(𝑘)𝑒^𝑇(𝑘)^𝑇, the augmented system can be obtained as follows: 𝜂(𝑘+1)=Φ_1𝜎(𝑘)𝜂(𝑘)+𝜃Φ_2𝜎(𝑘)𝜂𝑘−𝜏_𝑘+𝜃_𝑘−𝜃Φ_2𝜎(𝑘)𝜂𝑘−𝜏_𝑘+𝐷𝐹𝑘,𝜂(𝑘)+(𝑘),𝑧(𝑘)=𝐻𝑤(𝑘)𝜂(𝑘)(𝑘)𝐺𝜂=𝜙;𝑘=−𝜏,−𝜏+1,…,0, (16) where Φ_1𝜎(𝑘)=𝐴+𝜃𝐼₁𝐿_𝜎(𝑘)𝐶₁+𝐵𝐾_𝜎(𝑘)𝐼₂,Φ_2𝜎(𝑘)=−𝐼₁𝐿_𝜎(𝑘)𝐼+Δ_𝑞(𝑘,𝜎(𝑘))𝐶₂,𝐺=𝐺0,𝐻𝐻𝐼𝐼,𝐹(𝑘)(𝑘),𝐻=,𝐷=𝑘,𝜂=𝑓𝑘,𝐸𝜂 (17) with 𝐴=𝐴00𝐴,𝐼₁=0𝐼,𝐶₁=𝐶−𝐶,𝐵0𝐵=,𝐼₂=𝐼−𝐼,𝐶₂=𝐶0,𝐸=𝐼0. (18)

Lemma 2 (see [29]). Let 𝐷, 𝐸, and 𝐹 be matrices with appropriate dimensions. Suppose 𝐹^𝑇𝐹≤𝐼, and then for any scalar 𝜀>0, we have 𝐷𝐹𝐸+𝐸^𝑇𝐹^𝑇𝐷^𝑇≤𝜀𝐷𝐷^𝑇+𝜀⁻¹𝐸^𝑇𝐸. (19)

Definition 3 (see [30]). System (16) with 𝑤(𝑘)≡0 is said to be stochastically stable if, for every finite 𝜂₀=𝜂(0), the following inequality holds: 𝐸^∞_𝑘=0𝜂(𝑘)²∣𝜂₀<∞. (20)

Definition 4 ([31]). For a given scalar 𝛾>0, system (16) is stochastically stable and has an 𝐻_∞ noise attenuation performance index 𝛾 if the following conditions are satisfied. (a)System (16) with 𝜔(𝑘)≡0 is stochastically stable.(b)Under the zero-initial condition, it holds that ^∞_𝑘=0𝐸𝑧^𝑇(𝑘)𝑧(𝑘)<𝛾^2∞_𝑘=0𝐸𝑤^𝑇(𝑘)𝑤(𝑘,(𝑘))∀𝑤∈𝑙₂[0,∞). (21)

In this paper, the main objective is to design controller (12) for system (1) in the presence of output quantization, sensor-controller packet losses, and time-varying delays such that system (16) is stochastically stable with a prescribed 𝐻_∞ performance index 𝛾.

3. Main Results

Theorem 5. For a given scalar 𝛾>0, quantization densities 𝜌(𝑖) and a packet loss rate 0<𝜃<1, system (16) with partly unknown transition probabilities is stochastically stable with a prescribed 𝐻_∞ performance index 𝛾, if there exist matrices 𝑃_𝑖>0(𝑖∈ℜ) and 𝑄>0 such that 𝜋^𝑖_𝐾Θ^𝑖₁Γ^𝑖₁𝑃^𝑖_𝐾Γ^𝑖₂𝑃^𝑖_𝐾∗−𝑃^𝑖_𝐾0∗∗−𝑃^𝑖_𝐾<0,Θ^𝑖₁Γ^𝑖₁𝑃_𝑗Γ^𝑖₂𝑃_𝑗∗−𝑃_𝑗0∗∗−𝑃_𝑗<0,∀𝑗∈ℜ^𝑖_𝑈𝐾, (22) where Θ^𝑖₁=𝜏−𝜏+1𝑄−𝑃_𝑖−𝑆₁+𝐺^𝑇𝐺0−𝑆₂0∗−𝑄00∗∗−𝐼0∗∗∗−𝛾²𝐼,Γ^𝑖₁=Φ_1𝑖𝜃Φ_2𝑖𝐷𝐻^𝑇,Γ^𝑖₂=0𝜃Φ_2𝑖00^𝑇,𝜋^𝑖_𝐾=_{𝑗∈ℜ𝑖𝐾}𝜋_𝑖𝑗,𝑃^𝑖_𝐾=_{𝑗∈ℜ𝑖𝐾}𝜋_𝑖𝑗𝑃_𝑗. (23)

Proof. Choose a Lyapunov-Krasovskii functional of the form: 𝑉𝜂(𝑘),𝑘=𝑉₁𝜂(𝑘),𝑘+𝑉₂𝜂(𝑘),𝑘+𝑉₃𝜂(𝑘),𝑘, (24) with 𝑉₁𝜂(𝑘),𝑘=𝜂^𝑇(𝑘)𝑃_𝜎(𝑘)𝜂(𝑘)𝑉₂𝜂(𝑘),𝑘=^𝑘−1_{𝑙=𝑘−𝜏𝑘}𝜂^𝑇(𝑙)(𝑙)𝑉𝑄𝜂₃𝜂(𝑘),𝑘=^−𝜏_{ℎ=−𝜏+1}^𝑘−1_{𝑙=𝑘+ℎ}𝜂^𝑇(𝑙)𝑄𝜂(𝑙), (25) where 𝑃_𝜎(𝑘)>0(𝜎(𝑘)∈ℜ) and 𝑄>0 are symmetric positive definite matrices.
Let the mode at times 𝑘 and 𝑘+1 be 𝑖 and 𝑗, respectively. That is, 𝜎(𝑘)=𝑖 and 𝜎(𝑘+1)=𝑗 for any 𝑖,𝑗∈ℜ. Taking the difference of above functional along the solution of system (16), we have 𝐸𝜂(𝑘)𝑉𝜂()(𝑘𝜂(𝑘)Δ𝑉,𝑘=𝐸𝑘+1,𝑘+1∣𝜂)−𝑉,𝑘=Δ𝑉₁𝜂(𝑘),𝑘+Δ𝑉₂𝜂(𝑘),𝑘+Δ𝑉₃𝜂(𝑘),𝑘, (26) with Δ𝑉₁𝜂(𝑘),𝑘=𝜂^𝑇(𝑘+1)_𝑗∈ℜ𝜋_𝑖𝑗𝑃_𝑗𝜂(𝑘+1)−𝜂^𝑇(𝑘)𝑃_𝑖𝜂(𝑘)Δ𝑉₂𝜂(𝑘)=,𝑘_{𝑘𝑙=𝑘+1−𝜏𝑘+1}𝜂^𝑇(𝑙)𝑄𝜂(𝑙)−^𝑘−1_{𝑙=𝑘−𝜏𝑘}𝜂^𝑇(𝑙)𝑄𝜂(𝑙)=𝜂^𝑇(𝑘)𝑄𝜂(𝑘)−𝜂^𝑇𝑘−𝜏_𝑘𝑄𝜂𝑘−𝜏_𝑘+^𝑘−1_{𝑙=𝑘+1−𝜏𝑘+1}𝑥^𝑇(𝑙)𝑄𝑥(𝑙)−^𝑘−1_{𝑙=𝑘+1−𝜏𝑘}𝑥^𝑇(𝑙)𝑄𝑥(𝑙)≤𝜂^𝑇(𝑘)𝑄𝜂(𝑘)−𝜂^𝑇𝑘−𝜏_𝑘𝑄𝜂𝑘−𝜏_𝑘+^𝑘−𝜏_{𝑙=𝑘+1−𝜏}𝜂^𝑇(𝑙)𝑄𝜂(𝑙),Δ𝑉₃𝜂(𝑘),𝑘=^−𝜏_ℎ=−^𝑘_{𝜏+1𝑙=𝑘+1+ℎ}𝜂^𝑇(𝑙)(𝑙)−𝑄𝜂_{−𝜏ℎ=−𝜏+1𝑘−1𝑙=𝑘+ℎ}𝜂^𝑇(𝑙)(𝑙)=𝑄𝜂𝜏−𝜏𝜂^𝑇(𝑘)𝑄𝜂(𝑘)−^𝑘−𝜏_{𝑙=𝑘+1−𝜏}𝜂^𝑇(𝑙)𝑄𝜂(𝑙). (27)
Therefore, 𝐸Δ𝑉𝜂(𝑘),𝑘≤𝜂^𝑇(𝑘+1)_𝑗∈ℜ𝜋_𝑖𝑗𝑃_𝑗𝜂(𝑘+1)−𝜂^𝑇(𝑘)𝑃_𝑖𝜂(𝑘)+𝜏−𝜏+1𝜂^𝑇(𝑘)𝑄𝜂(𝑘)−𝜂^𝑇𝑘−𝜏_𝑘𝑄𝜂𝑘−𝜏_𝑘. (28) From inequality (2), we know 𝜂(𝑘)𝐹𝑘,𝜂(𝑘)^𝑇𝑆₁𝑆₂𝜂(𝑘)∗𝐼𝐹𝑘,𝜂(𝑘)≤0, (29) where 𝑆₁=𝐸^𝑇𝑆^𝑇₁𝑆₂+𝑆^𝑇₂𝑆₁2𝐸,𝑆₂=−𝐸^𝑇𝑆^𝑇₁+𝑆^𝑇₂2. (30)
Denoting 𝜁(𝑘)=[𝜂^𝑇(𝑘),𝜂^𝑇(𝑘−𝜏_𝑘),𝐹^𝑇(𝑘,𝜂(𝑘))]^𝑇 and combining (28) and (29) lead to 𝐸Δ𝑉𝜂(𝑘),𝑘≤𝜂^𝑇(𝑘+1)_𝑗∈ℜ𝜋_𝑖𝑗𝑃_𝑗𝜂(𝑘+1)+𝜁^𝑇(𝑘)_{𝑗∈ℜ𝑖𝐾}𝜋_𝑖𝑗+_{𝑗∈ℜ𝑖𝑈𝐾}𝜋_𝑖𝑗Θ^𝑖₁𝜁(𝑘)=𝜂^𝑇(𝑘+1)𝑃^𝑖_𝐾+_{𝑗∈ℜ𝑖𝑈𝐾}𝜋_𝑖𝑗𝑃_𝑗𝜂(𝑘+1)+𝜁^𝑇(𝑘)𝜋^𝑖_𝐾+_{𝑗∈ℜ𝑖𝑈𝐾}𝜋_𝑖𝑗Θ^𝑖₁𝜁(𝑘)=𝜁^𝑇(𝑘)𝜋^𝑖_𝐾Θ^𝑖₁+Θ^𝑖₂+_{𝑗∈ℜ𝑖𝑈𝐾}𝜋_𝑖𝑗Θ^𝑖₁+Θ^𝑖₃𝜁(𝑘), (31) where Θ^𝑖₁=𝜏−𝜏+1𝑄−𝑃_𝑖−𝑆₁0−𝑆₂,Θ∗−𝑄0∗∗−𝐼_𝑖2=Φ^𝑇_1𝑖𝑃^𝑖_𝐾Φ_1𝑖𝜃Φ^𝑇_1𝑖𝑃^𝑖_𝐾Φ_2𝑖Φ^𝑇_1𝑖𝑃^𝑖_𝐾𝐷∗𝜃²+𝜃²Φ^𝑇_2𝑖𝑃^𝑖_𝐾Φ_2𝑖𝜃Φ^𝑇_2𝑖𝑃^𝑖_𝐾𝐷∗∗𝐷^𝑇𝑃^𝑖_𝐾𝐷,Θ^𝑖₃=Φ^𝑇_1𝑖𝑃_𝑗Φ_1𝑖𝜃Φ^𝑇_1𝑖𝑃_𝑗Φ_2𝑖Φ^𝑇_1𝑖𝑃_𝑗𝐷∗𝜃²+𝜃²Φ^𝑇_2𝑖𝑃_𝑗Φ_2𝑖𝜃Φ^𝑇_2𝑖𝑃_𝑗𝐷∗∗𝐷^𝑇𝑃_𝑗𝐷. (32)
By Schur complement, (22) implies 𝜋^𝑖_𝐾Θ^𝑖₁+Θ^𝑖₂<0 and Θ^𝑖₁+Θ^𝑖₃<0, and then we obtain from (31) that 𝐸Δ𝑉(𝜁(𝑘),𝑘)≤−𝜆_min−𝜋^𝑖_𝐾Θ^𝑖₁−Θ^𝑖₂𝜁^𝑇(𝑘)𝜁(𝑘)−_{𝑗∈ℜ𝑖𝑈𝐾}𝜋_𝑖𝑗min_𝑗𝜆_min−Θ^𝑖₁+Θ^𝑖₃𝜁^𝑇(𝑘)𝜁(𝑘)≤−𝛽₁+𝛽₂𝜁^𝑇(𝑘)𝜁(𝑘), (33) where 𝛽₁=inf_𝑖∈ℜ{−𝜆_min[−𝜋^𝑖_𝐾Θ^𝑖₁−Θ^𝑖₂]} and 𝛽₂=inf_𝑖∈ℜ{−(1−𝜋^𝑖_𝐾)min_𝑗{𝜆_min[−(Θ^𝑖₁+Θ^𝑖₃)]}}. From (33), if we do mathematical expectations in both sides, we obtain that for any 𝑁≥0𝐸𝑉(𝜁())(𝜁(0)𝛽𝑁+1,𝑁+1−𝑉,0)≤−₁+𝛽₂^𝑁_𝑘=0𝐸𝜁^𝑇(𝑘)𝜁(𝑘), (34) which yields ^∞_𝑘=0𝐸𝜁^𝑇(𝑘)𝜁(𝑘≤1)𝛽₁+𝛽₂𝐸𝑉(𝜁(0),0)<∞. (35)
Thus, from Definition 3, the closed-loop system (16) with 𝑤(𝑘)=0 is stochastically stable.
Now, to establish an 𝐻_∞ performance, consider the following performance index: 𝐽_𝑁≜𝐸^𝑁_𝑘=0𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘). (36) Under zero-initial condition, we have 𝐽_𝑁=𝐸^𝑁_𝑘=0𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘)𝜂(𝑘)[𝑉(+Δ𝑉,𝑘−𝐸𝑁+1)]≤𝐸^∞_𝑘=0𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘)𝜂(𝑘)=+Δ𝑉,𝑘_∞𝑘=0𝜉^𝑇(𝑘)𝜋^𝑖_𝐾Θ^𝑖₁+Θ^𝑖₂+_{𝑗∈ℜ𝑖𝑈𝐾}𝜋_𝑖𝑗Θ^𝑖₁+Θ^𝑖₃𝜉(𝑘), (37) where 𝜉(𝑘)=𝜁^𝑇(𝑘),𝑤^𝑇(𝑘)^𝑇,Θ^𝑖₂=Φ^𝑇_1𝑖𝑃^𝑖_𝐾Φ_1𝑖𝜃Φ^𝑇_1𝑖𝑃^𝑖_𝐾Φ_2𝑖Φ^𝑇_1𝑖𝑃^𝑖_𝐾𝐷Φ^𝑇_1𝑖𝑃^𝑖_𝐾𝐻∗𝜃²+𝜃²Φ^𝑇_2𝑖𝑃^𝑖_𝐾Φ_2𝑖𝜃Φ^𝑇_2𝑖𝑃^𝑖_𝐾𝐷𝜃Φ^𝑇_2𝑖𝑃^𝑖_𝐾𝐻∗∗𝐷^𝑇𝑃^𝑖_𝐾𝐷𝐷^𝑇𝑃^𝑖_𝐾𝐻∗∗∗𝐻^𝑇𝑃^𝑖_𝐾𝐻,Θ^𝑖₃=Φ^𝑇_1𝑖𝑃_𝑗Φ_1𝑖𝜃Φ^𝑇_1𝑖𝑃_𝑗Φ_2𝑖Φ^𝑇_1𝑖𝑃_𝑗𝐷Φ^𝑇_1𝑖𝑃_𝑗𝐻∗𝜃²+𝜃²Φ^𝑇_2𝑖𝑃_𝑗Φ_2𝑖𝜃Φ^𝑇_2𝑖𝑃_𝑗𝐷𝜃Φ^𝑇_2𝑖𝑃_𝑗𝐻∗∗𝐷^𝑇𝑃_𝑗𝐷𝐷^𝑇𝑃_𝑗𝐻∗∗∗𝐻^𝑇𝑃_𝑗𝐻. (38) Similarly, by Schur complement, it can be obtained from (22) that [𝜋^𝑖_𝐾Θ^𝑖₁+Θ^𝑖₂]+∑_{𝑗∈ℜ𝑖𝑈𝐾}𝜋_𝑖𝑗(Θ^𝑖₁+Θ^𝑖₃)<0, which means that 𝐽_𝑁<0.
Then, ^∞_𝑘=0𝐸‖𝑧(𝑘)‖²≤𝛾^2∞_𝑘=0‖𝑤(𝑘)‖². (39) This completes the proof.

Remark 6. Note that 𝛽₁+𝛽₂ will be reduced to only 𝛽₁ (resp., 𝛽₂) if all the transition probabilities are known (resp., unknown).

Remark 7. The considered systems are more general than these systems with completely known or completely unknown transition probabilities, which can be viewed as two special cases of the ones tackled here; that is, when all the transition probabilities are known or unknown, the underlying systems are the traditional MJSs (ℜ^𝑖_𝐾=ℜ, ℜ^𝑖_𝑈𝐾=Ø) or the switched systems under arbitrary switching (ℜ^𝑖_𝐾=Ø, ℜ^𝑖_𝑈𝐾=ℜ), respectively. Therefore, observer-based controller with partly unknown transition probabilities of this paper is a more natural assumption to the Markovian jump systems and hence covers the existing ones.

Corollary 8. For a given scalar 𝛾>0, quantization densities 𝜌(𝑖) and a packet losses rate 0<𝜃<1, the closed-loop system (16) with completely known transition probabilities is stochastically stable with an 𝐻_∞ attenuation level if there exist matrices 𝑃_𝑖>0(𝑖∈ℜ) and 𝑄>0 such that 𝜋^𝑖_𝐾Θ^𝑖₁Γ^𝑖₁𝑃^𝑖_𝐾Γ^𝑖₂𝑃^𝑖_𝐾∗−𝑃^𝑖_𝐾0∗∗−𝑃^𝑖_𝐾<0, (40) where 𝑃^𝑖_𝐾=∑_{𝑗∈ℜ𝑖𝐾}𝜋_𝑖𝑗𝑃_𝑗 with ℜ^𝑖_𝐾=ℜ and ℜ^𝑖_𝑈𝐾=Ø.

Proof. Similar to the proof line of Theorem 5, the result can be obtained. The detailed proof is omitted here.

Theorem 5 provides Markov-chain-element-dependent sufficient conditions which guarantee the 𝐻_∞ performance as well as the stochastic stability of the augmented error system (16). The following theorem gives sufficient conditions on the existence of a mode-dependent observer-based controller.

Theorem 9. For a given scalar 𝛾>0, quantization densities 𝜌(𝑖) and a packet loss rate 0<𝜃<1, if there exist matrices 𝐵_𝑖, 𝐿_𝑖, 𝑅_𝑖, 𝑃_𝑖>0(𝑖∈ℜ), 𝑄>0 and scalar 𝜀_𝑖>0 such that Θ^𝑖₁Σ^𝑖₁Σ^𝑖₂0∗Υ_𝑗−𝑅_𝑖−𝑅^𝑇_𝑖0−𝜃𝐿_𝑖∗∗Υ_𝑗−𝑅_𝑖−𝑅^𝑇_𝑖𝜃𝐿_𝑖∗∗∗−𝜀_𝑖𝐼<0, (41) where Θ^𝑖₁=𝜏−𝜏+1𝑄−𝑃_𝑖−𝑆₁+𝐺^𝑇𝐺0−𝑆₂0∗−𝑄+𝜀_𝑖𝛿²(𝑖)𝐶^𝑇₂𝐶₂00∗∗−𝐼0∗∗∗−𝛾²𝐼,Σ^𝑖₁=𝑅_𝑖𝐴+𝐵_𝑖𝐼₂+𝜃𝐿_𝑖𝐶₁−𝜃𝐿_𝑖𝐶₂𝑅_𝑖𝐷𝑅_𝑖𝐻^𝑇,Σ^𝑖₂=0𝜃𝐿_𝑖𝐶₂00^𝑇, (42) then, there exists a mode-dependent observer-based controller as (12) such that the augmented system (16) is stochastically stable with a prescribed 𝐻_∞ performance index 𝛾. Moreover, if LMIs (41) have a feasible solution, the desired controller gains and observer gains are given by 𝐵_𝑖=𝑅_𝑖𝐵𝐾_𝑖,𝐿_𝑖=𝑅_𝑖𝐼₁𝐿_𝑖,𝑖∈ℜ. (43)

Proof. From Theorem 9, inequalities (22) can be rewritten as Θ^𝑖₁Γ^𝑖₁Υ_𝑗Γ^𝑖₂Υ_𝑗∗−Υ_𝑗0∗∗−Υ_𝑗<0, (44) where Υ_𝑗≜1𝜋^𝑖_𝐾𝑃^𝑖_𝐾,∀𝑗∈ℜ^𝑖_𝐾𝑃_𝑗,∀𝑗∈ℜ^𝑖_𝑈𝐾. (45) Performing a congruence transformation to (44) by diag{𝐼,…,𝐼₄,Υ_𝑗⁻¹𝑅^𝑇_𝑖,Υ_𝑗⁻¹𝑅^𝑇_𝑖}, one has Θ^𝑖₁Γ^𝑖₁𝑅^𝑇_𝑖Γ^𝑖₂𝑅^𝑇_𝑖∗−𝑅_𝑖Υ_𝑗⁻¹𝑅^𝑇_𝑖0∗∗−𝑅_𝑖Υ_𝑗⁻¹𝑅^𝑇_𝑖<0. (46) For an arbitrary matrix 𝑅_𝑖, for all 𝑖∈ℜ, we have the following fact: 1𝜋^𝑖_𝐾𝑃^𝑖_𝐾−𝑅_𝑖1𝜋^𝑖_𝐾𝑃^𝑖_𝐾⁻¹1𝜋^𝑖_𝐾𝑃^𝑖_𝐾−𝑅_𝑖⁻¹𝑃≥0,_𝑗−𝑅_𝑖𝑃_𝑗⁻¹𝑃_𝑗−𝑅_𝑖⁻¹≥0. (47) Then, we obtain Υ_𝑗−𝑅^𝑇_𝑖−𝑅_𝑖≥−𝑅_𝑖Υ_𝑗⁻¹𝑅^𝑇_𝑖, so (46) holds if the following condition is satisfied: Θ^𝑖₁Γ^𝑖₁𝑅^𝑇_𝑖Γ^𝑖₂𝑅^𝑇_𝑖∗Υ_𝑗−𝑅_𝑖−𝑅^𝑇_𝑖0∗∗Υ_𝑗−𝑅_𝑖−𝑅^𝑇_𝑖<0. (48) Then, using the condition (8), we have Δ_𝑞(𝑘,𝑖)=𝛿(𝑖)Δ_𝑖, (49) where Δ_𝑖=diag{Δ_𝑞1(𝑘,𝑖)/𝛿(𝑖),…,Δ_𝑞𝑠(𝑘,𝑖)/𝛿(𝑖)} and satisfies Δ^𝑇_𝑖Δ_𝑖≤𝐼.
Substituting (49) into (48) and using Lemma 2, we get that (48) is equivalent to Λ+Π_1𝑖Δ^𝑇_𝑖Π_2𝑖+Π^𝑇_2𝑖Δ_𝑖Π^𝑇_1𝑖≤Λ+𝜀_𝑖Π_1𝑖Π^𝑇_1𝑖+𝜀_𝑖⁻¹Π^𝑇_2𝑖Π_2𝑖<0, (50) where Λ=𝜏−𝜏+1𝑄−𝑃_𝑖−𝑆₁0−𝑆₂0𝑅_𝑖Φ_1𝑖^𝑇0∗−𝑄00−𝜃𝑅_𝑖𝐼₁𝐿_𝑖𝐶₂^𝑇𝜃𝑅_𝑖𝐼₁𝐿_𝑖𝐶₂^𝑇∗∗−𝐼0𝑅_𝑖𝐷^𝑇0∗∗∗−𝛾²𝐼𝑅_𝑖𝐻^𝑇0∗∗∗∗Υ_𝑗−𝑅_𝑖−𝑅^𝑇_𝑖0∗∗∗∗∗Υ_𝑗−𝑅_𝑖−𝑅^𝑇_𝑖,Π_1𝑖=0𝛿(𝑖)𝐶₂0000^𝑇,Π_2𝑖=0000−𝜃𝑅_𝑖𝐼₁𝐿_𝑖^𝑇𝜃𝑅_𝑖𝐼₁𝐿_𝑖^𝑇. (51)
Applying Schur complement and substituting 𝐵_𝑖=𝑅_𝑖𝐵𝐾_𝑖 and 𝐿_𝑖=𝑅_𝑖𝐼₁𝐿_𝑖 into (50), we have that (50) is equivalent to (41). Therefore, we obtain that system (16) is stochastically stable with a prescribed 𝐻_∞ performance index 𝛾. The proof is completed.

4. Numerical Example

Consider the following state-space representation of a nonlinear networked system in (1) with the following parameters: 10,,𝑓((𝑘=𝑓𝐴=−0.35−0.10.10.2,𝐵=,𝐻=−0.40.2𝐶=0.10,𝐺=0.2−0.3𝑘,𝑥))^𝑇₁(𝑘,𝑥(𝑘))𝑓^𝑇₂(𝑘,𝑥(𝑘))^𝑇,with𝑓₁(𝑘,𝑥(𝑘))=−tanh𝑥₁+0.5𝑥₁+0.1𝑥₂,𝑓₂(𝑘,𝑥(𝑘))=0.1𝑥₁−tanh𝑥₂+0.5𝑥₂. (52) It is easy to verify that 𝑆₁=−0.50.10.1−0.1,𝑆₂=.0.50.10.10.5 (53)

For the given system with data loss and time-varying delay, delay bounds are given as 𝜏=3 and 𝜏=1, and the packet losses parameters are selected as 𝜃=0.8. The network load is modeled by a Markov chain which takes values from a finite set {1,2,3} and the transition probability matrix is Ω=,0.10.60.3?0.4?0.5?? (54) where ? stands for the unknown elements.

The quantization density for each mode is assumed as 𝛿(1)=0.2, 𝛿(2)=0.3, and 𝛿(3)=0.4. Then the corresponding augmented system (16) can be obtained.

Taking 𝛾=1.2, then solving (41) in Theorem 9, we obtain the observer gain as 𝐿₁=0.42780.1599,𝐿₂=0.58820.8301,𝐿₃=,−0.7726−0.5458 (55) and controller gains can be computed as 𝐾₁=0.7088−0.2844,𝐾₂=,𝐾0.4107−0.2327₃=0.7023−0.2856. (56) Thus according to Theorem 9, we can conclude that, under the designed controller, the corresponding augmented system (16) is stochastically stable with a given 𝐻_∞ performance level 𝛾=1.2.

The initial condition is selected as 𝑥(0)=1−1, 𝑒(0)=00, and the noise signal 𝑤(𝑘)=0.01exp(−0.01𝑘)sin(0.02𝜋𝑘). The switching sequence and the trajectories of plant state 𝑥(𝑘) and error 𝑒(𝑘) are shown in Figures 2, 3, 4, and 5.

Figure 2 

Markovian switching signal.

Figure 3 

Trajectory of system state 𝑥(𝑘).

Figure 4 

Trajectory of error state 𝑒1(𝑘).

Figure 5 

Trajectory of error state 𝑒2(𝑘).

It is clearly observed from the simulation results that the corresponding augmented system (16) is stochastically stable. This demonstrates the effectiveness of the proposed results.

5. Conclusions

In this work, the problem of robust observer-based 𝐻_∞ control for discrete-time nonlinear networked systems with limited information is proposed. The quantization density is designed to be a function of the network load condition which is modeled by a Markov chain with partly unknown transition probability. Bernoulli random sequences are used to model packet dropouts in the network. An estimation technique is then developed to obtain the estimation of system state and the observer-based controller is proposed to stabilize the networked controlled systems by using the Lyapunov-Krasovskii functional approach. An example is presented to illustrate the effectiveness of the proposed method.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 61273120 and the Alexander von Humboldt Foundation in Germany.