Distributed Wireless Networked 𝐻_∞ Control for a Class of Lurie-Type Nonlinear Systems

Abstract

A new approach to solving the distributed control problem for a class of discrete-time nonlinear systems via a wireless neural control network (WNCN) is presented in this paper. A unified Lurie-type model termed delayed standard neural network model (DSNNM) is used to describe these nonlinear systems. We assume that all neuron nodes in WNCN which have limited energy, storage space, and computing ability can be regarded as a subcontroller, then the whole WNCN is characterized by a mesh-like structure with partially connected neurons distributed over a wide geographical area, which can be considered as a fully distributed nonlinear output feedback dynamic controller. The unreliable wireless communication links within WNCN are modeled by fading channels. Based on the Lyapunov functional and the S-procedure, the WNCN is solved and configured for the DSNNM to absolutely stabilize the whole closed-loop system in the sense of mean square with a 𝐻_∞ disturbance attenuation index using LMI approach. A numerical example shows the effectiveness of the proposed design approaches.

1. Introduction

Artificial neural networks (ANNs) are one of the effective technologies in modeling and controlling complex nonlinear systems due to the universal nonlinear function approximation property of ANNs. Because all biological neural networks (BNNs) have the recursive properties and most industrial processes are nonlinear dynamic system, the recurrent neural networks (RNNs) which have internal feedback loops and are suitable for dynamic mapping have attracted increasing attention in the control field [1]. Many researchers have extensively investigated RNNs-based design methods for nonlinear control systems. For example, in [2], diagonal recurrent neural networks (DRNNs) are constructed to identify and control, respectively, for both BIBO and non-BIBO nonlinear plants. Lin et al. [3] studied an FPGA-based computed force control system based on the Elman neural network (ENN) considered as a particular class of RNN to achieve the high-accuracy position control of linear ultrasonic motor. A neural controller using recurrent learning (RTRL) network updating algorithm for nonlinear plants with unknown dynamics is presented in [4]. Guaranteed cost control for exponential synchronization of cellular neural networks (CNNs) with various activation functions and mixed time-varying delays is investigated in [5]. As a dynamic system, the stability analysis of RNNs and stabilization synthesis of RNNs-based control systems are a primary consideration. One of the main characteristics of RNNs is that the nonlinear activation functions in RNNs are of the sigmoidal type. Since the various sigmoidal functions in RNNs belong to a subset of nonlinear functions of Lurie-type system [6], during the past two decades, there have been a large number of research contributions concerning the absolute stability of RNNs such as [7–14]. It is worth noting that a new neural network model termed by the standard neural network model (SNNM) is proposed in [12]. Most nonlinear control systems based on delayed (or nondelayed) RNNs can be converted into the SNNMs, the absolute stability of which can be analyzed using a unified approach in the sense of Lurie [6, 13, 14]. However, the traditional static and dynamic control methods [12–14] for SNNMs are centralized and do not apply to distributed networked control systems.

Boosted by advances in computing, communications, and sensing technologies, cyber-physical systems (CPSs) in which computational and physical components are closely conjoined and coordinated are becoming increasingly ubiquitous [15, 16]. A large number of embedded devices (such as sensors, actuators, and controllers) distributed over a vast geographical area in CPSs will depend more and more on communications networks to achieve information interaction and manipulate physical entities; therefore, wireless networked control systems (WNCSs) represent a new research frontier of CPSs and have recently received a great deal of attention [17]. Employing wireless networks for CPSs will enhance the flexibility and expandability of system (e.g., network nodes are easy to move or be deployed in scenes which have difficulties in wiring) whilst reducing installation, maintenance, debugging, and labour costs. However, the unreliable communication channels, resource constraints, and limited bandwidth that characterize the wireless technology require special care and raise new challenges to communication, signal processing, closed-loop control, and so forth. Recently, many researchers have investigated these issues and some significant results were obtained and many are in progress. Shi and Zhang [18] investigate the remote state estimation and optimal schedule for two sensors under bandwidth constraint. Guo et al. [19] consider the control and actuators/sensors scheduling problem for linear system and then propose a novel stability criterion based on the modes of Markov chains and the transmission delays. The problem of joint design of an output feedback controller and the medium access scheduling policy are investigated for networked control systems in [20, 21]. The analysis and design of state feedback controllers for linear systems where there are limitations on the number of active actuators and transmission delays are studied in [22]. A decentralized event-triggered control method over wireless sensor and actuator network (WSAN) of centralized controllers is discussed in [23]. Furthermore, with network scale unceasingly expanding, any of the sensing/actuating nodes cannot access/act to the full state of the physical plant, so development and design of distributed control methods for a large-scale WNCSs are still hotspot issues in both engineering and academic fields [24].

At present, the wireless network (WN) is considered primarily as a communication medium in most of the research results [17, 23–26] for WNCSs. It means that the nodes in WN will only achieve the data communication and transmission tasks among sensing/actuating nodes and one or more dedicated controllers. However, these works have potential drawbacks such as that the WNCS is susceptible to the failures of those dedicated controllers and the packet losses and delays over unreliable wireless communication links among nodes. Pajic et al. [27] propose the basic concept of wireless control network (WCN), a new fully distributed control method for WNCSs, in which the control function is achieved over a multihop WN. For WCN, the entire multihop network fulfills itself as a distributed controller where every node can be regarded as a local (small) linear dynamical controller for linear physical plants [28].

In this paper, we focus on the distributed networked 𝐻_∞ control and absolute stability analysis of delayed standard neural network model (DSNNM) based on a wireless neuron control network (WNCN) introduced in [29], which is an improved nonlinear WCN. In summary, the aim for introducing WNCN stems from the need of a distributed control approach for WNCSs. There are many practical application requirements that also motivate this study. Typical examples include industrial humidity, ventilation, air conditioning (HVAC) control systems in [24], the networked process control for the distillation column in [30], the drip irrigation control for agriculture using wireless sensor and actuator network (WSAN) [31], and so forth.

Compared with normal RNN being the fully connected among neurons and having a layered architecture as shown in Figure 1(a), the WNCN, as a special kind of control-oriented RNN, is characterized by a mesh-like structure with partially connected neurons distributed over a wide geographical area. Consider a scenario where several neural nodes forming with limited computation and wireless communication capabilities are deployed around an industrial plant and can exchange information with immediate neighbor neuron nodes to form a wireless mesh network, some of which can also receive state values of the plant from neighbor sensors or send control signals to neighbor actuators, respectively, as shown in Figure 1(b). Compared with WCN behaving as a linear dynamical system, WNCN is essentially a nonlinear wireless mesh RNN system. To the best of our knowledge, the problem formulation is novel.

(a)

(b)

Figure 1 

(a) A three-layer fully connected RNN. (b) Example of WNCN with 10 neurons consisting of a wireless mesh network (WMN) where dashed lines represent radio connectivity.

The remainder of this paper is organized as follows. In Section 2, we first briefly cover the delayed standard neural network model (DSNNM) and then describe the nonlinear dynamic behaviors of WNCN. Section 3 investigates the absolute stability and the 𝐻_∞ performance of the closed-loop system. The criteria to synthesis of the optimal 𝐻_∞ controller based on WNCN without stochastic packet dropping are first presented in Section 4, and then the result is extended to study the robust case based on stochastic WNCN with fading communication channels in Section 5. In Section 6, a numerical example is given to demonstrate the effectiveness of the derived results. And finally, conclusions are drawn in Section 7.

Notation. ℝ^𝑛 is the 𝑛-dimensional Euclidean space. ℝ^𝑛×𝑚 is the set of real 𝑛×𝑚 matrices. 𝐴^𝑇 denotes the transpose of matrix 𝐴. 𝐓𝐫(𝐴) denotes the trace of a square matrix 𝐴. 𝕊^𝑛 denotes the set of symmetric 𝑛×𝑛 matrices. 𝕊^𝑛₊ denotes the set of positive semidefinite 𝑛×𝑛 matrices. 𝕊^𝑛₊₊ denotes the set of positive definite 𝑛×𝑛 matrices. The curled inequality symbol ⪰(≻,⪯,≺) is used to denote generalized inequality: 𝐴,𝐵∈𝕊^𝑛, the matrix inequality 𝐴⪰(≻,⪯,≺)0 means that 𝐴∈𝕊^𝑛₊(𝕊^𝑛₊₊,−𝕊^𝑛₊,−𝕊^𝑛₊₊), and 𝐴⪰(≻,⪯,≺)𝐵⇔𝐴−𝐵⪰(≻,⪯,≺)0. 𝑘(𝐽) denotes the cardinality of set 𝐽. 𝐼 denotes an identity matrix of appropriate order. 𝑒_𝑖 denotes the 𝑖th vector of the standard basis of ℝ^𝑛.  𝔼(⋅) denotes the estimation operator. diag(⋅) denotes a diagonal matrix. ∗ is used as an ellipsis for terms induced by symmetry. If 𝑋∈ℝ^𝑝 and 𝑌∈ℝ^𝑞, 𝐶(𝑋;𝑌) denotes the space of all continuous functions mapping ℝ^𝑝→ℝ^𝑞.  𝑙₂[0;∞) is the space of square integrable vectors. ‖⋅‖ denotes the Euclidean norm for vectors or the spectral norm of matrices.

2. Problem Formulation

2.1. Delayed Standard Neural Network Model

Consider the following discrete-time DSNNM with input-output: 𝒫𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐴_𝑑𝑥(𝑘−𝑑)+𝐵_𝜙𝜙(𝜀(𝑘))+𝐵_𝜔𝜔(𝑘)+𝐵_𝑢𝑢(𝑘),𝜀(𝑘)=𝐶_𝜀𝑥(𝑘)+𝐶_𝑑𝑥(𝑘−𝑑)+𝐷_𝜙𝜙(𝜀(𝑘))+𝐷_𝜔𝜔(𝑘)+𝐷_𝑢𝑢(𝑘),𝑦(𝑘)=𝐶_𝑦𝑥(𝑘), (1) with the initial condition function 𝑥(𝑘)=𝜛(𝑘), ∀𝑘∈[−𝑑,0], where 𝑥(𝑘)∈ℝ^𝑛 is the state vector, 𝑢(𝑘)∈ℝ^𝑚 is the control input vector, 𝑦(𝑘)∈ℝ^𝑙 is the measured output vector, 𝜔(𝑘)∈ℝ^𝑟 is the disturbance that belongs to 𝑙₂[0,∞), 𝜙(𝜀(𝑘))∈𝐶(ℝ^𝐿;ℝ^𝐿) is the activation function with the input vector 𝜀(𝑘)∈ℝ^𝐿, 𝐿∈ℝ is the number of nonlinear activation functions, 𝑑>0 is the time delay, 𝐴∈ℝ^𝑛×𝑛, 𝐴_𝑑∈ℝ^𝑛×𝑛, 𝐵_𝜙∈ℝ^𝑛×𝐿, 𝐵_𝜔∈ℝ^𝑛×𝑟, 𝐵_𝑢∈ℝ^𝑛×𝑚, 𝐶_𝜀∈ℝ^𝐿×𝑛, 𝐶_𝑑∈ℝ^𝐿×𝑛, 𝐷_𝜙∈ℝ^𝐿×𝐿, 𝐷_𝜔∈ℝ^𝐿×𝑟, 𝐷_𝑢∈ℝ^𝐿×𝑚, and 𝐶_𝑦∈ℝ^𝑙×𝑛. Assume that the activation function 𝜙 satisfies 𝜙(0)=0 and belongs to a type of set Ω(𝐾) as follows: Ω(𝐾)≜𝜙𝜙∣0≤^𝑖𝜀_𝑖(𝑘)𝜀_𝑖(𝑘)≤𝑘_𝑖,𝑖=1,…,𝐿, (2) which means that 𝜙 is sector restricted to the interval [0,𝐾], where 𝐾=diag(𝑘₁,…,𝑘_𝐿)≻0.

2.2. Wireless Neural Control Network

The traditional design approaches of dynamic controllers based on SNNMs are centralized and the dimension of controller and plant must remain consistent [12–14]. However, without losing system stability, the WNCN can be structured as a distributed recurrent neurocontroller (RNC) with arbitrary dimension which will be in favor of controlling the complex nonlinear systems with multiple geographically distributed sensors (multi-output) and actuators (multi-input). So, the motivation for introducing WNCN stems from the need for distributed control approaches for WNCSs.

Assume that we use a WNCN consisting of 𝑁 neuron nodes to control the aforementioned DSNNM 𝒫. The wireless network in the whole system can be described by a directed graph as follows: 𝒢≜𝒜∪𝒱∪𝒮_vertexset,ℰ^ℐ∪ℰ^𝒞∪ℰ^𝒪_edgeset, (3) where 𝒱={𝑣₁,…,𝑣_𝑁} is the set of 𝑁 neuron nodes, 𝒜={𝑎₁,…,𝑎_𝑚} is the set of 𝑚 actuators which can execute the input vector 𝑢(𝑘)=[𝑢₁(𝑘),…,𝑢_𝑚(𝑘)]^𝑇, 𝒮={𝑠₁,…,𝑠_𝑙} is the set of 𝑙 sensors used to measure the output vector 𝑦(𝑘)=[𝑦₁(𝑘),…,𝑦_𝑙(𝑘)]^𝑇, and edge sets ℰ^ℐ={(𝑣_𝑖,𝑎_𝑝)∣𝑣_𝑖∈𝒱,𝑎_𝑝∈𝒜}, ℰ^𝒞={(𝑣_𝑖,𝑣_𝑗)∣𝑣_𝑗,𝑣_𝑖∈𝒱}, and ℰ^𝒪={(𝑠_𝑞,𝑣_𝑖)∣𝑠_𝑞∈𝒮,𝑣_𝑖∈𝒱} correspond to the physical radio communication links in the wireless network. Define the following three sets: the neighbor sensors of 𝑣_𝑖, ∀𝑖∈{1,…,𝑁}𝒮^𝑣𝑖≜𝑠_𝑞∣𝑠_𝑞∈𝒮,∃𝑠_𝑞,𝑣_𝑖∈ℰ^𝒪=𝑠_𝑞∣𝑤^𝒪_𝑖𝑞≠0, (4) the neighbor neurons of 𝑎_𝑝, ∀𝑝∈{1,…,𝑚}𝒱^𝑎𝑝≜𝑣_𝑖∣𝑣_𝑖∈𝒱,∃𝑣_𝑖,𝑎_𝑝∈ℰ^ℐ=𝑣_𝑖∣𝑤^ℐ_𝑝𝑖≠0, (5) the neighbor neurons of 𝑣_𝑖, ∀𝑖∈{1,…,𝑁}𝒱^𝑣𝑖≜𝑣_𝑗∣𝑣_𝑗∈𝒱,∃𝑣_𝑗,𝑣_𝑖∈ℰ^𝒞=𝑣_𝑗∣𝑤^𝒞_𝑖𝑗≠0, (6)where 𝑤^ℐ_𝑝𝑖,  𝑤^𝒞_𝑖𝑗,  𝑤^𝒪_𝑖𝑞 are the weights of edge (𝑣_𝑖,𝑎_𝑝), (𝑣_𝑗,𝑣_𝑖), and (𝑠_𝑞,𝑣_𝑖), respectively. This implies that 𝑠_𝑞∈𝒮^𝑣𝑖 if 𝑣_𝑖 can receive data directly from 𝑠_𝑞, 𝑣_𝑖∈𝒱^𝑎𝑝 if 𝑎_𝑝 can receive data directly from 𝑣_𝑖, and 𝑣_𝑗∈𝒱^𝑣𝑖 if 𝑣_𝑖 can receive data directly from 𝑣_𝑗.

The dynamic behavior of the neuron node 𝑣_𝑖 may be represented by the following pair of nonlinear equations: 𝑧_𝑖(𝑘+1)=𝜓_𝑖𝜉_𝑖(𝑘),𝜉_𝑖(𝑘)=𝑤^𝒞_𝑖𝑖𝑧_𝑖(𝑘)+_{𝑣𝑗∈𝒱𝑣𝑖}𝑤^𝒞_𝑖𝑗𝑧_𝑗(𝑘)+_{𝑠𝑞∈𝒮𝑣𝑖}𝑤^𝒪_𝑖𝑞𝑦_𝑞(𝑘), (7) where 𝑧_𝑖(𝑘) is the state of neuron node 𝑣_𝑖, 𝑧_𝑗(𝑘) is the state of neuron node 𝑣_𝑗,  𝑣_𝑗∈𝒱^𝑣𝑖, 𝑦_𝑞(𝑘) is the measurement value of sensor 𝑠_𝑞, 𝑠_𝑞∈𝒮^𝑣𝑖, 𝜉_𝑖(𝑘) is the weighted linear combination of 𝑣_𝑖’s present state and exogenous input signals (from neuron nodes in 𝒱^𝑣𝑖 or sensors in 𝒮^𝑣𝑖), and  𝜓_𝑖(⋅)∈Ω(𝐾) is the activation function of neuron node 𝑣_𝑖, where 𝐾=diag(𝑘_𝐿+1,…,𝑘_𝐿+𝑁)≻0. Each plant input 𝑢_𝑝(𝑘), 𝑝∈{1,…,𝑚} is a weighted linear combiner output due to neighbor neuron nodes of the actuator 𝑎_𝑝 as follows: 𝑢_𝑝(𝑘)=_{𝑣𝑖∈𝒱𝑎𝑝}𝑤^ℐ_𝑝𝑖𝑧_𝑖(𝑘). (8) If each neuron node is regarded as a nonlinear dynamical subcontroller, the whole WNCN consisting of 𝑁 neuron nodes may act as a fully distributed RNC whose dynamic behavior may be described as 𝒦𝑧()(𝜉(𝑘,𝜉(𝑘)𝑘+1=𝜓))=𝑊^𝒞𝑧(𝑘)+𝑊^𝒪𝑦(𝑘),𝑢(𝑘)=𝑊^ℐ𝑧(𝑘), (9) where 𝑧∈ℝ^𝑁 is the state vector of WNCN, 𝜓∈𝐶(ℝ^𝑁;ℝ^𝑁), 𝜉∈ℝ^𝑁, 𝑊^𝒞∈ℝ^𝑁×𝑁, 𝑊^𝒪∈ℝ^𝑁×𝑙, and 𝑊^ℐ∈ℝ^𝑚×𝑁. In the above-mentioned equations, ∀𝑖∈{1,…,𝑁}, 𝑤^𝒞_𝑖𝑗=0 if 𝑣_𝑗∉𝒱^𝑣𝑖∪{𝑣_𝑖}, 𝑤^𝒪_𝑖𝑞=0 if 𝑠_𝑞∉𝒮^𝑣𝑖, and 𝑤^ℐ_𝑝𝑖=0 if 𝑣_𝑖∉𝒱^𝑎𝑝. Therefore, the weight matrices 𝑊^𝒞, 𝑊^𝒪, and 𝑊^ℐ have the sparsity constraints. This means that the WNCN has considerably fewer weights (accounting for little computational overhead) than the fully connected neural network, which is conducive to industrial real-time control.

In this paper, a MAC synchronized network protocol based on time division multiple access (TDMA) architecture is used to schedule neuron nodes in WNCN to accomplish the cooperative control for the system (1). Under the scheme, every neuron node 𝑣_𝑖, 𝑖∈{1,…,𝑁} transmits its state information once per time frame. In the beginning, 𝑣_𝑖 has an arbitrary initial state value and then successively receives information from its neighbors in 𝒱^𝑣𝑖 and 𝒮^𝑣𝑖 in each time slot of frame. After 𝑣_𝑖 has received all the information from its neighbors, 𝑣_𝑖 will update its state by (7). Furthermore, in a similar way, every actuator 𝑎_𝑝, 𝑝∈{1,…,𝑚} can receive the combination of control signals from neighbor neuron nodes in 𝒱^𝑎𝑝 and then act to system (1) by (8).

Define vectors 𝑥=[𝑥^𝑇,𝑧^𝑇]^𝑇∈ℝ^𝑛+𝑁, 𝜀̃=[𝜀^𝑇,𝜉^𝑇]^𝑇∈ℝ^𝐿+𝑁, and 𝜙=[𝜙^𝑇,𝜓^𝑇]^𝑇∈Ω(𝐾),  𝐾=diag(𝑘₁,…,𝑘_𝐿+𝑁)≻0. Then the overall closed-loop system 𝒢 of the DSNNM 𝒫 and the WNCN 𝒦 is described as 𝒢𝑥(𝑘+1)=𝐴𝐵_𝑢𝑊^ℐ00_𝐴𝑥(𝑘)𝑧(𝑘)_𝑥(𝑘)+𝐴_𝑑000_𝐴𝑑𝑥()𝑧()𝑘−𝑑𝑘−𝑑_{𝑥(𝑘−𝑑)}+𝐵_𝜙00𝐼_𝐵𝜙𝜙(𝜀(𝑘𝜓(𝜉(𝑘))))_{̃𝜙(𝜀(𝑘))}+𝐵_𝜔𝐵0_𝜔𝜔(𝑘),𝜀̃(𝑘)=𝐶_𝜀𝐷_𝑢𝑊^ℐ𝑊^𝒪𝐶_𝑦𝑊^𝒞_𝐶𝜀𝑥(𝑘)𝑧(𝑘)_𝑥(𝑘)+𝐶_𝑑000_𝐶𝑑𝑥()𝑧()𝑘−𝑑𝑘−𝑑_{𝑥(𝑘−𝑑)}+𝐷_𝜙000_𝐷𝜙𝜙(𝜀(𝑘𝜓(𝜉(𝑘))))_{𝜙(̃𝜀(𝑘))}+𝐷_𝜔𝐷0_𝜔𝜔(𝑘). (10)

Consider that the performance output of the closed-loop system 𝒢 is described as 𝑦(𝑘)=𝐶_𝑦𝑥(𝑘), where 𝐶_𝑦=[𝐼0], then the following definition is introduced.

Definition 1 (see [6, 32, 33]). Given a scalar 𝛾>0, the closed-loop system 𝒢 is said to be absolutely stable with a 𝐻_∞-norm bound 𝛾 if there exists a distributed dynamic neural controller WNCN 𝒦 such that the following conditions are satisfied (Algorithm 1).(1)With zero disturbance, that is, 𝜔(𝑘)=0, the zero solution of the closed-loop system 𝒢 is globally asymptotically stable, ∀𝑥(0), ∀𝜙∈Ω(𝐾).(2)Under the zero-initial condition, the performance output 𝑦(𝑘) satisfies ^∞_𝑘=0𝑦(𝑘)²≤𝛾^2∞_𝑘=0‖𝜔(𝑘)‖²,∀nonzero𝜔(𝑘). (11)Then the WNCN 𝒦 is said to be a 𝐻_∞ controller for the DSNNM 𝒫. Furthermore, if we can find a minimal 𝛾^∗ to satisfy the above conditions, the WNCN 𝒦 is an optimal 𝐻_∞ controller.
Our aim is to design the WNCN 𝒦 for DSNNM 𝒫 such that the closed-loop system 𝒢 satisfies the requirements (1) and (2) in Definition 1.

Step  1. Set 𝑘=0. If there exists an initial feasible solution set Υ0=𝑃,𝑄,𝑅,Σ,𝑊𝒪,𝑊𝒞,𝑊ℐ satisfying the constraints    (25)-(26), let 𝒳0=𝑃, 𝒴0=𝑄. Otherwise, exit. Step  2. If 𝑘≤𝜅, go to Step  3 where 𝜅 is the assumed maximum number of iteration. Otherwise, exit. Step  3. At 𝑘≥0, obtain the feasible solution set Υ𝑘+1=𝑃,𝑄,𝑅,Σ,𝑊𝒪,𝑊𝒞,𝑊ℐ by solving the following LMI problem:     min𝐓𝐫𝒳𝑘𝑄+𝒴𝑘𝑃 s.t. (25)-(26). Step  4. Substitute Υ𝑘+1 into (15). If (15) holds, stop the algorithm. Otherwise, set 𝑘=𝑘+1, 𝒳𝑘=𝑃, 𝒴𝑘=𝑄 and go to Step  2.

Algorithm 1 

Given a scalar 𝛾>0, solving the 𝐻_∞ WNCN for closed-loop system 𝒢.

3. 𝐻_∞ Performance Analysis of the Closed-Loop System

In this section, we will investigate the absolute stability and 𝐻_∞ performance of the closed-loop system 𝒢. Before deducing the main results, we need to make use of the following two lemmas.

Lemma 2 (S-procedure [34]). Let 𝑇₀,𝑇₁,…,𝑇_𝑝∈𝕊^𝑛. If there exists 𝜏_𝑖≥0,  𝑖=1,…,𝑝 such that 𝑇₀−^𝑝_𝑖=1𝜏_𝑖𝑇_𝑖≺0 (12) then 𝜁^𝑇𝑇₀𝜁<0 for all 𝜁≠0 such that 𝜁^𝑇𝑇_𝑖𝜁≤0,  𝑖=1,…,𝑝.

Lemma 3 (Schur complement [35]). Consider a matrix 𝑋∈𝕊^𝑛 partitioned as 𝐵𝑋=𝐴𝐵_𝑇𝐶, (13) where 𝐴∈𝕊^𝑘. If 𝐴 is nonsingular, the matrix 𝑆=𝐶−𝐵^𝑇𝐴⁻¹𝐵 is called the Schur complement of A in 𝑋. Then, the following characterizations of positive definiteness or semidefiniteness of the block matrix 𝑋 hold: (1)(2)𝑋≻0,𝑖𝑓𝑓𝐴≻0,𝑆≻0,𝐼𝑓𝐴≻0,𝑡ℎ𝑒𝑛𝑋⪰0𝑖𝑓𝑓𝑆⪰0. (14)

Theorem 4. Given 𝛾>0 and WNCN 𝒦 with parameter set 𝒦={𝑊^𝒪,𝑊^𝒞,𝑊^ℐ}, if there exist appropriate dimension matrices 𝑃≻0, 𝑅≻0, and 𝑇⪰0, such that the following matrix inequality holds: Ξ₁=𝐴^𝑇𝑃𝐶𝐴−𝑃+𝑅+^𝑇_𝑦𝐶_𝑦𝐴^𝑇𝑃𝐴_𝑑𝐴^𝑇𝑃𝐵_𝜙+𝐶^𝑇_𝜀𝑇𝐾𝐴^𝑇𝑃𝐵_𝜔∗𝐴^𝑇_𝑑𝑃𝐴_𝑑−𝑅𝐴^𝑇_𝑑𝑃𝐵_𝜙+𝐶^𝑇_𝑑𝑇𝐾𝐴^𝑇_𝑑𝑃𝐵_𝜔𝐵∗∗^𝑇𝜙𝑃𝐵_𝜙+𝐷^𝑇_𝜙𝑇𝐾+𝑇𝐾𝐷_𝜙𝐵−2𝑇^𝑇𝜙𝑃𝐵_𝜙+𝑇𝐾𝐷_𝜔∗∗∗𝐵^𝑇_𝜔𝑃𝐵_𝜔−𝛾²𝐼≺0, (15)where 𝐾=diag(𝑘₁,…,𝑘_𝐿+𝑁), then the zero solution of closed-loop system 𝒢 is absolutely stable and the 𝐻_∞-norm constraint (11) is achieved for all nonzero 𝜔(𝑘).

Proof. From system 𝒢 with 𝜔(𝑘)=0, one can obtain 𝒢𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐴_𝑑𝑥(𝑘−𝑑)+𝐵_𝜙𝜙(𝜀̃(𝑘,𝜀̃(𝑘)=𝐶))_𝜀𝑥(𝑘)+𝐶_𝑑𝑥(𝑘−𝑑)+𝐷_𝜙𝜙(𝜀̃(𝑘)). (16) Assume that the 𝑥(𝑘)=0 is the only equilibrium of 𝒢. Consider the following Lyapunov-Krasovskii functional for systems 𝒢 as 𝑉(𝑥(𝑘))=𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+^𝑑_𝑖=1𝑥^𝑇(𝑘−𝑖)𝑅𝑥(𝑘−𝑖). (17) According to the sector bound set Ω(𝐾) of 𝜙, we have 𝜙_𝑖𝜀_𝑖̃(𝑘)⋅𝜏_𝑖⋅𝜙_𝑖𝜀_𝑖̃(𝑘)−𝑘_𝑖𝜀_𝑖̃(𝑘)=𝜏_𝑖𝜙²_𝑖𝜀_𝑖̃(𝑘)−𝜏_𝑖𝑘_𝑖𝜀_𝑖̃(𝑘)𝜙_𝑖𝜀_𝑖̃(𝑘)≤0, (18) where 𝜀_𝑖≥0, 𝑖=1,…,𝐿+𝑁.
Now, by defining the difference of 𝑉(𝑥(𝑘)) along 𝒢 as Δ𝑉(𝑥(𝑘))≜𝑉(𝑥(𝑘+1))−𝑉(𝑥(𝑘)) and using Lemma 2 (S-procedure), we can obtain (𝑥(𝑘=𝑥Δ𝑉))_𝑇(𝑘+1)𝑃𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+𝑥(𝑘)^𝑇𝑅𝑥(𝑘)−𝑥^𝑇()𝑅𝑥()≤𝐴𝑥(𝑘)+𝐴𝑘−𝑑𝑘−𝑑_𝑑𝑥(𝑘−𝑑)+𝐵_𝜙𝜙(𝜀̃(𝑘))^𝑇×𝑃𝐴𝑥(𝑘)+𝐴_𝑑𝑥(𝑘−𝑑)+𝐵_𝜙𝜙(𝜀̃(𝑘−𝑥))_𝑇(𝑘)𝑃𝑥(𝑘)+𝑥(𝑘)^𝑇𝑅𝑥(𝑘)−𝑥^𝑇(𝑘−𝑑)𝑅𝑥(𝑘−𝑑)−2^𝐿+𝑁_𝑖=1𝜏_𝑖𝜙²_𝑖𝜀_𝑖̃(𝑘)+2^𝐿+𝑁_𝑖=1𝜏_𝑖𝑘_𝑖𝜀_𝑖̃(𝑘)𝜙_𝑖𝜀_𝑖̃(𝑘=𝑥(𝑘)𝑥()𝜙(𝜀̃(𝑘)𝑘−𝑑))^𝑇×𝐴^𝑇𝑃𝐴−𝑃+𝑅𝐴^𝑇𝑃𝐴_𝑑𝐴^𝑇𝑃𝐵_𝜙+𝐶^𝑇_𝜀𝑇𝐾∗𝐴^𝑇_𝑑𝑃𝐴_𝑑−𝑅𝐴^𝑇_𝑑𝑃𝐵_𝜙+𝐶^𝑇_𝑑𝑇𝐾𝐵∗∗^𝑇𝜙𝑃𝐵_𝜙+𝐷^𝑇_𝜙𝑇𝐾+𝑇𝐾𝐷_𝜙−2𝑇_Ξ0×𝑥(𝑘)𝑥()𝜙(𝜀̃(𝑘𝑘−𝑑,)) (19) where 𝑇=diag(𝜏₁,𝜏₂,…,𝜏_𝐿+𝑁)⪰0. By Lemma 3 (Schur complement), if Ξ₁≺0 (15) holds, Ξ₀≺0 also holds. So, if Ξ₁≺0, system 𝒢 with 𝜔(𝑘)=0, that is, system 𝒢, is globally asymptotically stable, ∀𝜙∈Ω(𝐾).
Next, for ∀ℓ>0, define 𝐽_ℓ=^ℓ_𝑘=0𝑦(𝑘)²−𝛾^2ℓ_𝑘=0‖𝜔(𝑘)‖²=^ℓ_𝑘=0𝑦^𝑇(𝑘)𝑦(𝑘)−𝛾²𝜔^𝑇(𝑘)𝜔(𝑘). (20) Consider the zero initial condition 𝑉(𝑥(0))=0, that is 𝑉(𝑥(ℓ+1))=^ℓ_𝑘=0[𝑉(𝑥((𝑥(𝑘=𝑘+1))−𝑉))]_ℓ𝑘=0Δ𝑉(𝑥(𝑘))>0. (21) Therefore, for system 𝒢, defining vector 𝜁(𝑘)=[𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−𝑑)𝜙^𝑇(𝜀(̃𝑘))𝜔^𝑇(𝑘)]^𝑇 and according to (19) and (20), we have 𝐽_ℓ=^ℓ_𝑘=0𝑦^𝑇(𝑘)𝑦(𝑘)−𝛾²𝜔^𝑇(𝑘)𝜔(𝑘)(𝑥(𝑘(𝑥(≤+Δ𝑉))−𝑉ℓ+1))_ℓ𝑘=0𝑦^𝑇(𝑘)𝑦(𝑘)−𝛾²𝜔^𝑇(𝑘)𝜔(𝑘)(𝑥(𝑘=+Δ𝑉))_ℓ𝑘=0𝜁^𝑇(𝑘)Ξ₁𝜁(𝑘). (22) If Ξ₁≺0 (15) holds, lim_ℓ→∞𝐽_ℓ=lim_ℓ→∞∑^ℓ_𝑘=0𝜁^𝑇(𝑘)Ξ₁𝜁(𝑘)<0. Thus, ∀nonzero𝜔(𝑘)∈𝑙₂[0,∞) and the 𝐻_∞-norm constraint (11) is achieved. This completes the proof.

4. 𝐻_∞ Controller Design Based on WNCN

In the previous stage, the matrix inequality condition Ξ₁≺0 (15) is not an LMI, which cannot be solved by LMI tools. In what follows, we first convert the matrix inequality condition Ξ₁≺0 (15) into a cone complementarity problem (CCP) and then use the cone complementarity linearization (CCL) algorithm introduced in [36] to formulate a convex optimization problem with LMI constraints to obtain the appropriate parameters of WNCN (i.e., interconnection weight matrices set 𝒦={𝑊^𝒪,𝑊^𝒞,𝑊^ℐ}).

Lemma 5 (see [27]). There exist matrices 𝑃, 𝑄∈𝕊^𝑛₊₊ satisfying the constraint 𝑄=𝑃⁻¹ if and only if they are optimal points for the problem min𝐓𝐫{𝑄𝑃}𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑡𝑜𝑄𝐼𝐼𝑃⪰0,𝑄,𝑃∈𝕊^𝑛₊₊, (23) and the optimal cost of the problem is 𝑛.

Theorem 6. Given a scalar 𝛾>0, the closed-loop system 𝒢 is said to be absolutely stabilizable by using a WNCN 𝒦 and the 𝐻_∞-norm constraint (11) is achieved for all nonzero 𝜔(𝑘) if there exist appropriate dimension matrices 𝑃≻0, 𝑄≻0, 𝑅≻0, Σ⪰0, and 𝑊^𝒪,𝑊^𝒞,𝑊^ℐ∈𝒦 such that the following optimization problem: min𝐓𝐫{𝑄𝑃}s.t.Ξ₂=−𝑃+𝑅0𝐶^𝑇_𝜀0𝐴^𝑇𝐶^𝑇_𝑦∗−𝑅𝐶^𝑇_𝑑0𝐴^𝑇_𝑑0∗∗𝐷_𝜙Σ𝐾+Σ𝐾𝐷^𝑇_𝜙−2Σ𝐷_𝜔Σ𝐾𝐵^𝑇_𝜙0∗∗∗−𝛾²𝐼𝐵^𝑇_𝜔0∗∗∗∗−𝑄0∗∗∗∗∗−𝐼≺0,𝑄𝐼𝐼𝑃⪰0,𝑄,𝑃∈𝕊^𝑛+𝑁₊₊, (24)(25)(26)where 𝐴=𝐴𝐵_𝑢𝑊^ℐ,𝐶00^𝜀=𝐶_𝜀𝐷_𝑢𝑊^ℐ𝑊^𝒪𝐶_𝑦𝑊^𝒞, (27) is feasible with optimal cost 𝑛+𝑁.

Proof. Inequality Ξ₁≺0 (15) in Theorem 4 can be rewritten as −𝑃+𝑅0𝐶^𝑇_𝜀𝑇𝐶𝐾0∗−𝑅^𝑇_𝑑𝑇𝐷𝐾0∗∗^𝑇_𝜙𝑇𝐾+𝑇𝐾𝐷_𝜙−2𝑇𝑇𝐾𝐷_𝜔∗∗∗−𝛾²𝐼+𝐴^𝑇𝐴^𝑇_𝑑𝐵^𝑇_𝜙𝐵^𝑇_𝜔𝑃𝐴𝐴_𝑑𝐵_𝜙𝐵^𝑇_𝜔+𝐶^𝑇_𝑦000𝐶_𝑦000≺0. (28) Then, using Lemma 3 (Schur complement), the inequality (28) is equivalent to −𝑃+𝑅0𝐶^𝑇_𝜀𝑇𝐾0𝐴^𝑇𝐶^𝑇_𝑦∗−𝑅𝐶^𝑇_𝑑𝑇𝐾0𝐴^𝑇_𝑑0∗∗𝐷^𝑇_𝜙𝑇𝐾+𝑇𝐾𝐷_𝜙−2𝑇𝑇𝐾𝐷_𝜔𝐵^𝑇_𝜙0∗∗∗−𝛾²𝐼𝐵^𝑇_𝜔0,∗∗∗∗−𝑄0∗∗∗∗∗−𝐼≺0, (29) where 𝑄=𝑃⁻¹. By defining 𝑆=𝑇𝐾⁻¹,Σ=𝑇𝐾⁻¹𝑇𝑇𝐾⁻¹ (30) and pre- and postmultiplying the left-hand side matrix of (29) by diag(𝐼,𝐼,𝑆,𝐼,𝐼,𝐼), respectively, the inequality (29) is equivalent to −𝑃+𝑅0𝐶^𝑇_𝜀0𝐴^𝑇𝐶^𝑇_𝑦∗−𝑅𝐶^𝑇_𝑑0𝐴^𝑇_𝑑0∗∗𝑆𝐷^𝑇_𝜙+𝐷_𝜙𝑆−2Σ𝐷_𝜔𝑆𝐵^𝑇_𝜙0∗∗∗−𝛾²𝐼𝐵^𝑇_𝜔0∗∗∗∗−𝑄0∗∗∗∗∗−𝐼≺0. (31) According to (30), the following equations hold: 𝑆⁻¹=𝑇𝐾,𝑇=𝑆⁻¹Σ𝑆⁻¹. (32) Form (32), we know 𝑆=Σ𝐾. (33) Substituting (33) into (31), one can obtain Ξ₂≺0 (25). By using Lemma 5, the nonconvex constraint 𝑄=𝑃⁻¹ is approximated with an optimization problem. This completes the proof.

So far the WNCN has been designed to guarantee the absolute stability with a given 𝐻_∞-norm bound 𝛾 of the closed-loop system. In what follows, we give the Algorithm 2 based on the bisection method to design WNCN for the optimal 𝐻_∞ control problem: min 𝛾 s.t. (15), 𝑃≻0, 𝑅≻0, 𝑇⪰0.

Step  1. Set 𝑘=0. Let 𝛾0− and 𝛾0+ be the initial lower and upper bounds of 𝛾, that is, 𝛾0∈[𝛾0−,𝛾0+] where 𝛾0−=0, 𝛾0+ can be    assigned an arbitrarily sufficiently large value to make inequalities (25)-(26) have initial feasible solution set     Υ0=𝑃,𝑄,𝑅,Σ,𝑊𝒪,𝑊𝒞,𝑊ℐ. Step  2. At 𝑘≥0, compute 𝛾𝑘=(𝛾𝑘−+𝛾𝑘+)/2. Step  3. Use Algorithm 1 to check whether the feasible solution set Υ𝑘+1 satisfying inequality (15). If Υ𝑘+1 exists, set 𝛾𝑘+=𝛾𝑘.    Otherwise, set 𝛾𝑘−=𝛾𝑘. Step  4. If 𝛾𝑘+−𝛾𝑘−≤𝜖, where 𝜖 is the assumed calculation accuracy, set optimal 𝐻∞ performance index 𝛾∗=𝛾𝑘+ and exit.    Otherwise, set 𝑘=𝑘+1, go to Step  2.

Algorithm 2 

Minimizing 𝛾 to solve the optimal 𝐻_∞ WNCN for closed-loop system 𝒢.

5. Robust 𝐻_∞ Controller Design Based on Fading WNCN

Due to the large geographical nature of the closed-loop system 𝒢 over a WN, a realistic distributed control design approach for WNCN should take the communication packet losses into account.

According to [37], we adopt the fading channel models to simulate the unreliable wireless communication links in WNCN as shown in Figure 2(a). First, define a bijective mapping Ω∶{(𝑎,𝑏)}→{𝜏}, (𝑎,𝑏)∈ℰ^𝒪∪ℰ^𝒞∪ℰ^𝐼, where 𝜏={1,…,𝜌} and 𝜌=𝑘(ℰ^𝒪)+𝑘(ℰ^𝒞)+𝑘(ℰ^𝐼) is the total number of wireless links in WNCN, to concisely enumerate all links in the network. Therefore, the weights of links {(𝑎,𝑏)} can be mapped to 𝑤^𝒪_𝜏,𝑤^𝒞_𝜏,𝑤^ℐ_𝜏, ∀𝜏=Ω(𝑎,𝑏) and then compacted into the following weight vector as 𝑤=𝑤^𝒪𝑇,𝑤^𝒞𝑇,𝑤^ℐ𝑇𝑇=𝑤^𝒪_{𝜏𝜏=Ω(𝑠𝑞,𝑣𝑖)},𝑤^𝒞_{𝜏𝜏=Ω(𝑣𝑗,𝑣𝑖)},𝑤^ℐ_{𝜏𝜏=Ω(𝑣𝑖,𝑎𝑝)}^𝑇, (34) where 𝑤^𝒪∈ℝ^𝜌𝒪, 𝑤^𝒞∈ℝ^𝜌𝒞, 𝑤^ℐ∈ℝ^𝜌ℐ, and 𝑤∈ℝ^𝜌. Let 𝑡_𝑘,𝜏 denote the data packet transmitted over the 𝜏th communication link at time 𝑘. Then, aggregating all of 𝑡_𝑘,𝜏 in a vector 𝑡_𝑘∈ℝ^𝜌, we can obtain 𝑡(𝑘)=𝑊^𝒪_𝑡00𝑊^𝒞_𝑡^ℐ_𝑡𝑊_𝑊or𝐶_𝑦00𝐼_𝑁𝑥(𝑘)𝑧(𝑘)=𝑊^or𝑥(𝑘), (35) where 𝑊^𝒪_𝑡=diag(𝑤^𝒪)[𝑒^𝑇_𝑞]_{𝑞∣𝜏=Ω(𝑠𝑞,𝑣𝑖)}, 𝑒_𝑞∈ℝ^𝑙, 𝑊^𝒞_𝑡=diag(𝑤^𝒞)[𝑒^𝑇_𝑗]_{𝑗∣𝜏=Ω(𝑣𝑗,𝑣𝑖)}, 𝑒_𝑗∈ℝ^𝑁, and 𝑊^ℐ_𝑡=diag(𝑤^ℐ)[𝑒^𝑇_𝑖]_{𝑖∣𝜏=Ω(𝑣𝑖,𝑎𝑝)}, 𝑒_𝑖∈ℝ^𝑁.

(a)

(b)

Figure 2 

(a) The wireless communication links simulated as fading channels and (b) WNCN is transformed into a robust sense.

Remark 7. 𝑊^or∈ℝ^{𝜌×(𝑁+𝑙)} is a row selection matrix whose each row contains a single nonzero element which equals to a corresponding weight 𝑤^𝒪_𝜏, 𝑤^𝒞_𝜏 or 𝑤^ℐ_𝜏.
Next, let 𝑟(𝑘) denote the received date from 𝑡(𝑘) via the unreliable wireless communication links. 𝛾_𝜏(𝑘), 𝜏∈{1,…,𝜌} is independent and identically distributed (I.I.D) Bernoulli random variable with mean 𝜇_𝜏=𝔼[𝛾_𝜏(𝑘)] and variance 𝜎²_𝜏=𝔼[(𝛾_𝜏(𝑘)−𝜇_𝜏)²]. 𝛾_𝜏(𝑘) indicates whether packet 𝑡_𝜏(𝑘) is successfully received by 𝑟_𝜏(𝑘); that is, 𝛾_𝜏(𝑘)=1 if packet arrives and 𝛾_𝜏(𝑘)=0 otherwise. If Δ_𝜏(𝑘), 𝜏∈{1,…,𝜌} denotes an I.I.D Bernoulli random variable with zero-mean and variance 𝜎²_𝜏, 𝛾_𝜏(𝑘) can be transformed into a robust form such that 𝛾_𝜏(𝑘)=𝜇_𝜏+Δ_𝜏(𝑘), where 𝜇_𝜏 is nominal value and Δ_𝜏(𝑘) is random perturbation value. Thus, the fading channel model is described by the following bijective mapping: Γ∶𝑡(𝑘)⟶𝑟(𝑘)=Γ(𝑘)𝑡(𝑘)=(M+Δ(𝑘))𝑡(𝑘), (36) where Γ(𝑘)=diag(𝛾₁(𝑘),…,𝛾_𝜌(𝑘)), M=diag(𝜇₁,…,𝜇_𝜌), Δ(𝑘)=diag(Δ₁(𝑘),…,Δ_𝜌(𝑘)), 𝔼[Δ(𝑘)Δ(𝑘)^𝑇]=diag(𝜎²₁,…,𝜎²_𝜌), and 𝔼[Δ(𝑘)]=0.
Thus, the dynamic behavior of the fading WNCN with stochastic packet losses can be described as follows: 𝒦𝑧()(𝜉(𝑘,𝜉(𝑘)𝑘+1=𝜓))=𝑊^𝒞_𝜇𝑧(𝑘)+𝑊^𝒪_𝜇𝑦(𝑘)+𝑊_Δ^𝒞𝒪Δ(𝑘)𝑡(𝑘),𝑢(𝑘)=𝑊^ℐ_𝜇𝑧(𝑘)+𝑊^ℐ_ΔΔ(𝑘)𝑡(𝑘), (37) where 𝑊_Δ^𝒞𝒪=𝑊^𝒞_Δ+𝑊^𝒪_Δ and 𝑊^𝒪_𝜇=𝑤^𝒪_{𝜇𝑖𝑞𝑁×𝑙},where𝑤^𝒪_𝜇𝑖𝑞=𝜇_𝜏𝑤^𝒪_𝜏,∃𝑣_𝑖∈𝒱,𝑠_𝑞Ω𝑠∈𝒮,_𝑞,𝑣_𝑖=𝜏0,else,𝑊^𝒞_𝜇=𝑤^𝒞_{𝜇𝑖𝑗𝑁×𝑁},where𝑤^𝒞_𝜇𝑖𝑗=𝜇_𝜏𝑤^𝒞_𝜏,if𝑖≠𝑗,∃𝑣_𝑖,𝑣_𝑗Ω𝑣∈𝒱,_𝑗,𝑣_𝑖𝑤=𝜏_𝒞𝑖𝑖,if𝑖=𝑗,∃𝑣_𝑖∈𝒱0,else,𝑊^ℐ_𝜇=𝑤^ℐ_{𝜇𝑝𝑖𝑚×𝑁},where𝑤^ℐ_𝜇𝑝𝑖=𝜇_𝜏𝑤^ℐ_𝜏,∃𝑣_𝑖∈𝒱,𝑎_𝑝Ω𝑣∈𝒜,_𝑖,𝑎_𝑝=𝜏0,else,𝑊^𝒪_Δ=𝑤^𝒪_{Δ𝑖𝜏𝑁×𝜌},where𝑤^𝒪_Δ𝑖𝜏=1,∃𝑣_𝑖∈𝒱,𝑠_𝑞Ω𝑠∈𝒮,_𝑞,𝑣_𝑖=𝜏0,else,𝑊^𝒞_Δ=𝑤^𝒞_{Δ𝑖𝜏𝑁×𝜌},where𝑤^𝒞_Δ𝑖𝜏=1,∃𝑣_𝑖,𝑣_𝑗Ω𝑣∈𝒱,_𝑗,𝑣_𝑖=𝜏0,else,𝑊^ℐ_Δ=𝑤^ℐ_{Δ𝑝𝜏𝑚×𝜌},where𝑤^ℐ_Δ𝑝𝜏=1,∃𝑣_𝑖∈𝒱,𝑎_𝑝Ω𝑣∈𝒜,_𝑖,𝑎_𝑝=𝜏0,else. (38)

Remark 8. Similar to 𝑊^𝑜𝑟, matrices 𝑊_Δ^𝒞𝒪 and 𝑊^ℐ_Δ are used to select which elements of Δ(𝑘)𝑡(𝑘) are added to 𝜉(𝑘) and 𝑢(𝑘), respectively.
As shown in Figure 2(b), DSNNM (1) is controlled by a fading network composed by the mean WNCN (MWNCN) and the stochastic perturbation Δ. Consider the following stochastic closed-loop system: 𝒢𝑥(𝑘+1)=𝐴_𝜇𝑥(𝑘)+𝐴_𝑑𝑥(𝑘−1)+𝐵_𝜙𝜙(𝜀̃(𝑘+𝐵))_𝜔𝜔(𝑘)+𝑊^dst₁Δ(𝑘)𝑡(𝑘),𝜀̃(𝑘)=𝐶^𝜇_𝜀𝑥(𝑘)+𝐶_𝑑𝑥(𝑘−1)+𝐷_𝜙𝜙(𝜀̃(𝑘+𝐷))_𝜔𝜔(𝑘)+𝑊^dst₂Δ(𝑘)𝑡(𝑘), (39) where 𝐴_𝜇=^{𝐴𝐵𝑢𝑊ℐ𝜇}₀₀, 𝐶^𝜇_𝜀=^{𝐶𝜀𝐷𝑢𝑊ℐ𝜇}_{𝑊𝒪𝜇𝐶𝑦𝑊𝒞𝜇}, and 𝑊^dst₁=^{𝐵𝑢𝑊ℐΔ}₀, 𝑊^dst₂=^{𝐷𝑢𝑊ℐΔ}_{𝑊Δ𝒞𝒪}.

Definition 9 (see [6, 32, 33]). Given scalar 𝛾>0 and 𝑀(𝑘), the state correlation matrix, as 𝑀(𝑘)≜𝔼{𝑥(𝑘)𝑥(𝑘)^𝑇}, the stochastic closed-loop system 𝒢 is said to be absolutely stable in mean-square with a 𝐻_∞-norm bound 𝛾, if there exists a distributed dynamic neural controller WNCN 𝒦 such that the following conditions are satisfied.(1)With zero disturbance, that is, 𝜔(𝑘)=0, lim_𝑘→∞𝑀(𝑘)=0, ∀𝑥(0), and ∀𝜙∈Ω(𝐾).(2)Under the zero-initial condition, the performance output 𝑦(𝑘) satisfies ^∞_𝑘=0𝔼𝑦(𝑘)²≤𝛾^2∞_𝑘=0𝔼‖𝜔(𝑘)‖²,∀nonzero𝜔(𝑘) (40)Then the WNCN 𝒦 is said to be a robust 𝐻_∞ controller for the DSNNM 𝒫. Furthermore, if we can find a minimal 𝛾^∗ to satisfy the above conditions, the WNCN 𝒦 is an optimal robust 𝐻_∞ controller.

Theorem 10. Given a scalar 𝛾>0, the stochastic closed-loop system 𝒢 is said to be absolutely stabilizable in mean-square by using a fading WNCN 𝒦 and the 𝐻_∞-norm constraint (40) is achieved for all nonzero 𝜔(𝑘) if there exist appropriate dimension matrices 𝑃≻0, 𝑄≻0, 𝑅≻0, Σ⪰0, 𝑊^𝒪_𝜇, 𝑊^𝒞_𝜇, 𝑊^ℐ_𝜇, and scalar 𝜃_𝑖,  𝑖=1,…,𝜌, such that the following optimization problem: min𝐓𝐫{𝑄𝑃}s.t.Ξ₃=−𝑃+𝑅+Π0𝐶^𝜇_𝜀^𝑇0𝐴^𝑇_𝜇𝐶^𝑇_𝑦∗−𝑅𝐶^𝑇_𝑑0𝐴^𝑇_𝑑0∗∗𝐷_𝜙Σ𝐾+Σ𝐾𝐷^𝑇_𝜙−2Σ𝐷_𝜔Σ𝐾𝐵^𝑇_𝜙0∗∗∗−𝛾²𝐼𝐵^𝑇_𝜔0∗∗∗∗−𝑄0∗∗∗∗∗−𝐼≺0,𝜃_𝑖≥𝜎²_𝑖𝑊₁^{𝑑𝑠𝑡𝑇}_𝑖𝑃𝑊₁^𝑑𝑠𝑡_𝑖,∀𝑖=1,…,𝜌,𝑄𝐼𝐼𝑃⪰0,𝑄,𝑃∈𝕊^𝑛+𝑁₊₊, (41)(42)(43)(44)where Π=(𝑊^𝑜𝑟)^𝑇Θ𝑊^𝑜𝑟, Θ=diag(𝜃₁,…,𝜃_𝜌), and 𝜃_𝑖=𝜎²_𝑖(𝑊₁^𝑑𝑠𝑡)^𝑇_𝑖𝑃(𝑊₁^𝑑𝑠𝑡)_𝑖, (𝑊₁^𝑑𝑠𝑡)_𝑖 denote the 𝑖th column of the matrix 𝑊₁^𝑑𝑠𝑡, is feasible with optimal cost 𝑛+𝑁.

Proof. Consider a Lyapunov candidate as follows: 𝑉[𝑀(𝑘)]=𝐓𝐫[𝑀(𝑘)𝑃]. (45) The difference of 𝑉[𝑀(𝑘)] along the trajectory of stochastic closed-loop system 𝒢 with 𝜔(𝑘)=0 is given by [𝑀(𝑘[𝑀([𝑀(𝑘[𝑀()𝑃][𝑀(𝑘)𝑃]𝔼𝑥Δ𝑉)]=𝑉𝑘+1)]−𝑉)]=𝐓𝐫𝑘+1−𝐓𝐫=𝐓𝐫^𝑇(𝑘+1)𝑃𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘𝑥)=𝔼^𝑇(𝑘+1)𝑃𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘). (46) By considering Δ(𝑘) with 𝔼[Δ(𝑘)]=0 and 𝔼[Δ(𝑘)Δ^𝑇(𝑘)]=diag(𝜎²₁,…,𝜎²_𝜌) is independent from 𝑥(𝑘) and 𝜙(𝜀(̃𝑘)) and using Lemma 2 (S-procedure) one can obtain [𝑀(𝑘=𝑥(𝑘)𝑥()𝜙(𝜀̃(𝑘Δ𝑉)]𝑘−𝑑))^𝑇×𝐴^𝑇_𝜇𝑃𝐴_𝜇𝐴−𝑃+𝑅+Π^𝑇𝜇𝑃𝐴_𝑑𝐴^𝑇_𝜇𝑃𝐵_𝜙+𝐶^𝜇_𝜀^𝑇𝑇𝐾∗𝐴^𝑇_𝑑𝑃𝐴_𝑑−𝑅𝐴^𝑇_𝑑𝑃𝐵_𝜙+𝐶^𝑇_𝑑𝑇𝐾𝐵∗∗^𝑇𝜙𝑃𝐵_𝜙+𝐷^𝑇_𝜙𝑇𝐾+𝑇𝐾𝐷_𝜙−2𝑇_Ξ0×𝑥(𝑘)𝑥()𝜙(𝜀̃(𝑘𝑘−𝑑,)) (47) where 𝑇=diag(𝜏₁,𝜏₂,…,𝜏_𝐿+𝑁), 𝐾=diag(𝑘₁,…,𝑘_𝐿+𝑁), Π=(𝑊^or)^𝑇Θ𝑊^or, Θ=diag(𝜃₁,…,𝜃_𝜌), 𝜃_𝑖=𝜎²_𝑖(𝑊^dst₁)^𝑇_𝑖𝑃(𝑊^dst₁)_𝑖, (𝑊^dst₁)_𝑖 denote the 𝑖th column of the matrix 𝑊^dst₁.
According to (20)–(22), ∀ℓ>0, we have 𝔼𝐽_ℓ=^ℓ_𝑘=0𝔼𝑦(𝑘)²−𝛾^2ℓ_𝑘=0𝔼‖𝜔(𝑘)‖²=^ℓ_𝑘=0𝔼𝑦^𝑇(𝑘)𝑦(𝑘)−𝛾²𝜔^𝑇(𝑘)𝜔(𝑘≤_ℓ𝑘=0)𝜁^𝑇(𝑘)Ξ₁𝜁(𝑘), (48) whereΞ₁=𝐴^𝑇_𝜇𝑃𝐴_𝜇+𝐶−𝑃+𝑅_𝑇𝑦𝐶_𝑦𝐴+Π^𝑇𝜇𝑃𝐴_𝑑𝐴^𝑇_𝜇𝑃𝐵_𝜙+𝐶^𝜇_𝜀^𝑇𝑇𝐾𝐴^𝑇_𝜇𝑃𝐵̃_𝜔∗𝐴^𝑇_𝑑𝑃𝐴_𝑑−𝑅𝐴^𝑇_𝑑𝑃𝐵_𝜙+𝐶^𝑇_𝑑𝑇𝐾𝐴^𝑇_𝑑𝑃𝐵_𝜔𝐵∗∗^𝑇𝜙𝑃𝐵_𝜙+𝐷^𝑇_𝜙𝑇𝐾+𝑇𝐾𝐷_𝜙𝐵−2𝑇^𝑇𝜙𝑃𝐵_𝜙+𝑇𝐾𝐷_𝜔∗∗∗𝐵^𝑇_𝜔𝑃𝐵_𝜔−𝛾².𝐼. (49)Since Ξ₀ is the principal minor of Ξ₁, if Ξ₁≺0, then Ξ₀≺0 such that 𝑀(𝑘) converges to zeros as 𝑘→∞ for system 𝒢 with 𝜔(𝑘)=0, ∀𝑥(0), and ∀𝜙∈Ω(𝐾). Further, if Ξ₁≺0 holds, lim_ℓ→∞𝔼{𝐽_ℓ}≤lim_ℓ→∞∑^ℓ_𝑘=0𝜁^𝑇(𝑘)Ξ₁𝜁(𝑘)<0. Thus, ∀nonzero𝜔(𝑘)∈𝑙₂[0,∞) and the 𝐻_∞-norm constraint (40) is achieved. Similar to the proof process of Theorem 6, by using Lemma 3 (Schur complement), the inequality Ξ₁≺0 is equivalent to optimization problems (41)–(44) described in Theorem 10. This completes the proof.

As in the previous section, we present Algorithms 3 and 4.

Step  1. Set 𝑘=0. If there exists an initial feasible solution set Υ0=𝑃,𝑄,𝑅,Σ,𝑊𝒪𝜇,𝑊𝒞𝜇,𝑊ℐ𝜇 satisfying the constraints    (42)–(44), let 𝒳0=𝑃, 𝒴0=𝑄. Otherwise, exit. Step  2. If 𝑘≤𝜅, go to Step  3 where 𝜅 is the assumed maximum number of iteration. Otherwise, exit. Step  3. At 𝑘≥0, obtain the feasible solution set Υ𝑘+1=𝑃,𝑄,𝑅,Σ,𝑊𝒪𝜇,𝑊𝒞𝜇,𝑊ℐ𝜇 by solving the following LMI problem:     min𝐓𝐫𝒳𝑘𝑄+𝒴𝑘𝑃 s.t. (42)–(44). Step  4. Substitute Υ𝑘+1 into matrix Ξ1. If inequality Ξ1≺0 holds, stop the algorithm. Otherwise, set 𝑘=𝑘+1, 𝒳𝑘=𝑃,     𝒴𝑘=𝑄 and go to Step  2.

Algorithm 3 

Given a scalar 𝛾>0, solving the 𝐻_∞  fading WNCN with unreliable communication links for closed-loop system 𝒢.

Step  1. Set 𝑘=0. Let 𝛾0− and 𝛾0+ be the initial lower and upper bounds of 𝛾, that is, 𝛾0∈[𝛾0−,𝛾0+] where 𝛾0−=0, 𝛾0+ can be    assigned an arbitrarily sufficiently large value to make inequalities (42)–(44) have initial feasible solution set     Υ0=𝑃,𝑄,𝑅,Σ,𝑊𝒪𝜇,𝑊𝒞𝜇,𝑊ℐ𝜇. Step  2. At 𝑘≥0, computer 𝛾𝑘=(𝛾𝑘−+𝛾𝑘+)/2. Step  3. Use Algorithm 3 to check whether the feasible solution set Υ𝑘+1 satisfying Ξ1≺0. If Υ𝑘+1 exists, set 𝛾𝑘+=𝛾𝑘.    Otherwise, set 𝛾𝑘−=𝛾𝑘. Step  4. If 𝛾𝑘+−𝛾𝑘−≤𝜖, where 𝜖 is the assumed calculation accuracy, set optimal 𝐻∞ performance index 𝛾∗=𝛾𝑘+ and exit.    Otherwise, set 𝑘=𝑘+1, go to Step  2.

Algorithm 4 

Minimizing 𝛾 to solve the optimal 𝐻_∞  fading WNCN with unreliable communication links for closed-loop system 𝒢.

6. Numerical Simulation

Consider the following nonlinear system [38]: 𝑥₁(𝑘+1)=−𝑥²₁(𝑘)+0.3𝑥₂(𝑘)+0.1𝑥²₁(𝑘−2)−0.2𝑥₁(𝑘−2)𝑥₂(𝑘−2)+𝑢₁(𝑘),𝑥₂(𝑘+1)=0.1𝑥₁(𝑘)+𝑥₂(𝑘)+0.5𝑢₂(𝑘),𝑦(𝑘)=0.6𝑥₁(𝑘). (50)

According to [39, 40], when we consider the disturbance 𝑤(𝑘), the nonlinear system (50) can be transformed into the discrete-time DSNNM (1), where 𝐴=^−0.50.3_0.11, 𝐴_𝑑=^0.05−0.1₀₀, 𝐵_𝜙=¹₀, 𝐵_𝑢=¹⁰_00.5, 𝐶_𝜀=10, 𝐶_𝑑=−0.10.2, 𝐷_𝜙=0, 𝐷_𝑢=0_1×2, 𝐶_𝑦=0.60, 𝐵_𝜔=^0.5_0.1, 𝐷_𝜔=0.1, and 𝐾=𝐼.

Consider that the double-input-single-output (DISO) discrete-time DSNNM described above is synthesized by a WNCN which consists of 6 wireless neuron nodes shown in Figure 3. In WNCN, each wireless communication link is modeled as a fading channel with same packet arrival rate (mean) 𝛿 and variance 𝜎²=𝛿(1−𝛿). For 𝛿=0.95%, Algorithm 4 can be solved by CVX, a package for specifying and solving convex programs [41]. Then, we obtain the minimum optimal 𝐻_∞ performance index 𝛾^∗=0.7921, the solutions of (41)–(44), and the interconnection weight matrix parameters of WNCN as follows:,,,(),𝑊𝑃=3.6031−1.0546−0.0825−0.08250.00470.0047−0.0617−0.0617−1.054611.33981.25461.2546−1.1665−1.1665−1.1256−1.1256−0.08251.25461.16100.0840−0.0821−0.0821−0.0785−0.0785−0.08251.25460.08401.1610−0.0821−0.0821−0.0785−0.07850.0047−1.1665−0.0821−0.08211.14950.07250.07080.07080.0047−1.1665−0.0821−0.08210.07251.14950.07080.0708−0.0617−1.1256−0.0785−0.07850.07080.07081.14540.0684−0.0617−1.1256−0.0785−0.07850.07080.07080.06841.1454𝑄=0.29310.0501−0.0192−0.01920.03780.03780.05450.05450.05010.1789−0.1421−0.14210.13590.13590.13420.1342−0.0192−0.14210.99290.0644−0.0579−0.0579−0.0576−0.0576−0.0192−0.14210.06440.9929−0.0579−0.0579−0.0576−0.05760.03780.1359−0.0579−0.05790.98840.05990.05940.05940.03780.1359−0.0579−0.05790.05990.98840.05940.05940.05450.1342−0.0576−0.05760.05940.05940.98900.06050.05450.1342−0.0576−0.05760.05940.05940.06050.9890𝑅=1.1713−0.11820.01220.0122−0.0342−0.0342−0.1895−0.189500.5427−0.0197−0.0197−0.0580−0.05800.04050.0405000.7390−0.71830.00780.0078−0.0614−0.06140000.17380.06570.0657−0.5153−0.515300000.7494−0.68790.07130.0713000000.29730.34470.34470000000.7422−0.708900000000.2198Σ=diag0.5313,1.2461,0.1357,0.4233,1.4452,0.7652,1.2891_𝒞𝜇=0.04840.04840.00030.0003000.04840.04840.00030.000300−0.4925−0.49250.01100.0110−0.1193−0.1193−0.4925−0.49250.01100.0110−0.1193−0.1193000.48700.4870−0.0879−0.0879000.48700.4870−0.0879−0.0879,𝑊^𝒪_𝜇=0000,𝑊−1.3588−1.3588_ℐ𝜇=.0000−0.0646−0.06460000−0.2989−0.2989 (51)

Figure 3 

A discrete-time DSNNM synthesized by a WNCN with 6 wireless neuron nodes.

Figure 4 shows the simulation results of the state trajectories of controlled discrete-time DSNNM, where state 𝑥(𝑘) is initialized arbitrarily in interval [−0.5,0.5] at 𝑘=0 and 𝑘=40, respectively, and the disturbance input is nonlinear load 1/𝑘². It is easily seen that the WNSN with a distributed architecture solved by Algorithm 4 using CVX toolbox can ensure the absolute stability of the closed-loop system in the mean-square sense with optimal 𝐻_∞ performance.

Figure 4 

State response of the closed-loop system under 𝐻∞ controller WNCN with 𝛾∗=0.7921.

7. Conclusions

A novel wireless networked 𝐻_∞ control approach based on WNCN has been considered for a class of Lurie-type nonlinear systems named DSNNM. The WNCN which can absolutely stabilize the closed-loop system in mean-square with a desired 𝐻_∞ disturbance rejection level can be obtained by solving LMIs using a CVX toolbox (release 2.0 (beta)). Simulation results have illustrated the feasibility of the distributed control methods presented in this paper.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61174070) and the Specialized Research Fund for the Doctoral Program (20110172110033).