# Distributed Wireless NetworkedH∞ Control for a Class of Lurie-Type Nonlinear Systems

Authors: Wen Ren; Bugong Xu
Journal: Mathematical Problems in Engineering
 (2014)

---

## 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 aH∞ disturbance attenuation index using LMI approach. A numerical example shows the effectiveness of the proposed design approaches.


---

## Body

## 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 aunified 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 networkedH∞ 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 Figure1(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) 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.
(a)
(b)The remainder of this paper is organized as follows. In Section2, 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. ℝn is the n-dimensional Euclidean space. ℝn×m is the set of real n×m matrices. AT denotes the transpose of matrix A. Tr(A) denotes the trace of a square matrix A. 𝕊n 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∈𝕊n, 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. ei denotes the ith vector of the standard basis of ℝn.  𝔼(·) denotes the estimation operator. diag⁡(·) denotes a diagonal matrix. * is used as an ellipsis for terms induced by symmetry. If X∈ℝp and Y∈ℝq, C(X;Y) denotes the space of all continuous functions mapping ℝp→ℝq.  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:(1)𝒫{x(k+1)=Ax(k)+Adx(k-d)+Bϕϕ(ε(k))x(k+1)=+Bωω(k)+Buu(k),ε(k)=Cεx(k)+Cdx(k-d)+Dϕϕ(ε(k))ε(k)=+Dωω(k)+Duu(k),y(k)=Cyx(k),
with the initial condition function x(k)=ϖ(k), ∀k∈[-d,0], where x(k)∈ℝn is the state vector, u(k)∈ℝm is the control input vector, y(k)∈ℝl is the measured output vector, ω(k)∈ℝr is the disturbance that belongs to l2[0,∞), ϕ(ε(k))∈C(ℝL;ℝL) is the activation function with the input vector ε(k)∈ℝL, L∈ℝ is the number of nonlinear activation functions, d>0 is the time delay, A∈ℝn×n, Ad∈ℝn×n, Bϕ∈ℝn×L, Bω∈ℝn×r, Bu∈ℝn×m, Cε∈ℝL×n, Cd∈ℝL×n, Dϕ∈ℝL×L, Dω∈ℝL×r, Du∈ℝL×m, and Cy∈ℝl×n. Assume that the activation function ϕ satisfies ϕ(0)=0 and belongs to a type of set Ω(K) as follows:
(2)Ω(K)≜{ϕ∣0≤ϕi(εi(k))εi(k)≤ki,i=1,…,L},
which means that ϕ is sector restricted to the interval [0,K], where K=diag⁡(k1,…,kL)≻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 ofN neuron nodes to control the aforementioned DSNNM 𝒫. The wireless network in the whole system can be described by a directed graph as follows:
(3)𝒢≜{𝒜∪𝒱∪𝒮︸vertexset,ℰℐ∪ℰ𝒞∪ℰ𝒪︸edgeset},
where 𝒱={v1,…,vN} is the set of N neuron nodes, 𝒜={a1,…,am} is the set of m actuators which can execute the input vector u(k)=[u1(k),…,um(k)]T, 𝒮={s1,…,sl} is the set of l sensors used to measure the output vector y(k)=[y1(k),…,yl(k)]T, and edge sets ℰℐ={(vi,ap)∣vi∈𝒱,ap∈𝒜}, ℰ𝒞={(vi,vj)∣vj,vi∈𝒱}, and ℰ𝒪={(sq,vi)∣sq∈𝒮,vi∈𝒱} correspond to the physical radio communication links in the wireless network. Define the following three sets:theneighbor sensors of vi, ∀i∈{1,…,N}(4)𝒮vi≜{sq∣sq∈𝒮,∃(sq,vi)∈ℰ𝒪}={sq∣wiq𝒪≠0},theneighbor neurons of ap, ∀p∈{1,…,m}(5)𝒱ap≜{vi∣vi∈𝒱,∃(vi,ap)∈ℰℐ}={vi∣wpiℐ≠0},theneighbor neurons of vi, ∀i∈{1,…,N}(6)𝒱vi≜{vj∣vj∈𝒱,∃(vj,vi)∈ℰ𝒞}={vj∣wij𝒞≠0},where wpiℐ,  wij𝒞,  wiq𝒪 are the weights of edge (vi,ap), (vj,vi), and (sq,vi), respectively. This implies that sq∈𝒮vi if vi can receive data directly from sq, vi∈𝒱ap if ap can receive data directly from vi, and vj∈𝒱vi if vi can receive data directly from vj.The dynamic behavior of the neuron nodevi may be represented by the following pair of nonlinear equations:
(7)zi(k+1)=ψi(ξi(k)),ξi(k)=wii𝒞zi(k)+∑vj∈𝒱vi‍wij𝒞zj(k)+∑sq∈𝒮vi‍wiq𝒪yq(k),
where zi(k) is the state of neuron node vi, zj(k) is the state of neuron node vj,  vj∈𝒱vi, yq(k) is the measurement value of sensor sq, sq∈𝒮vi, ξi(k) is the weighted linear combination of vi’s present state and exogenous input signals (from neuron nodes in 𝒱vi or sensors in 𝒮vi), and  ψi(·)∈Ω(K-) is the activation function of neuron node vi, where K-=diag⁡(kL+1,…,kL+N)≻0. Each plant input up(k), p∈{1,…,m} is a weighted linear combiner output due to neighbor neuron nodes of the actuator ap as follows:
(8)up(k)=∑vi∈𝒱ap‍wpiℐzi(k).
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
(9)𝒦{z(k+1)=ψ(ξ(k)),ξ(k)=W𝒞z(k)+W𝒪y(k),u(k)=Wℐz(k),
where z∈ℝN is the state vector of WNCN, ψ∈C(ℝN;ℝN), ξ∈ℝN, W𝒞∈ℝN×N, W𝒪∈ℝN×l, and Wℐ∈ℝm×N. In the above-mentioned equations, ∀i∈{1,…,N}, wij𝒞=0 if vj∉𝒱vi∪{vi}, wiq𝒪=0 if sq∉𝒮vi, and wpiℐ=0 if vi∉𝒱ap. Therefore, the weight matrices W𝒞, W𝒪, and Wℐ 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 vi, i∈{1,…,N} transmits its state information once per time frame. In the beginning, vi has an arbitrary initial state value and then successively receives information from its neighbors in 𝒱vi and 𝒮vi in each time slot of frame. After vi has received all the information from its neighbors, vi will update its state by (7). Furthermore, in a similar way, every actuator ap, p∈{1,…,m} can receive the combination of control signals from neighbor neuron nodes in 𝒱ap and then act to system (1) by (8).Define vectorsx~=[xT,zT]T∈ℝn+N, ε~=[εT,ξT]T∈ℝL+N, and ϕ~=[ϕT,ψT]T∈Ω(K~),  K~=diag⁡(k1,…,kL+N)≻0. Then the overall closed-loop system 𝒢 of the DSNNM 𝒫 and the WNCN 𝒦 is described as
(10)𝒢{x~(k+1)=[ABuWℐ00]︸A~[x(k)z(k)]︸x~(k)x~(k+1)=+[Ad000]︸A~d[x(k-d)z(k-d)]︸x~(k-d)x~(k+1)=+[Bϕ00I]︸B~ϕ[ϕ(ε(k))ψ(ξ(k))]︸ϕ~(ε~(k))+[Bω0]︸B~ωω(k),ε~(k)=[CεDuWℐW𝒪CyW𝒞]︸C~ε[x(k)z(k)]︸x~(k)ε~(k)=+[Cd000]︸C~d[x(k-d)z(k-d)]︸x~(k-d)ε~(k)=+[Dϕ000]︸D~ϕ[ϕ(ε(k))ψ(ξ(k))]︸ϕ~(ε~(k))+[Dω0]︸D~ωω(k).Consider that the performance output of the closed-loop system𝒢 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 scalarγ>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 outputy~(k) satisfies
(11)∑k=0∞‍∥y~(k)∥2≤γ2∑k=0∞‍∥ω(k)∥2,∀nonzeroω(k).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 WNCN𝒦 for DSNNM 𝒫 such that the closed-loop system 𝒢 satisfies the requirements (1) and (2) in Definition 1.Algorithm 1:Given a scalar γ>0, solving the H∞ WNCN for closed-loop system 𝒢~.
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:
min⁡Tr{𝒳kQ+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.


## 2.1. Delayed Standard Neural Network Model

Consider the following discrete-time DSNNM with input-output:(1)𝒫{x(k+1)=Ax(k)+Adx(k-d)+Bϕϕ(ε(k))x(k+1)=+Bωω(k)+Buu(k),ε(k)=Cεx(k)+Cdx(k-d)+Dϕϕ(ε(k))ε(k)=+Dωω(k)+Duu(k),y(k)=Cyx(k),
with the initial condition function x(k)=ϖ(k), ∀k∈[-d,0], where x(k)∈ℝn is the state vector, u(k)∈ℝm is the control input vector, y(k)∈ℝl is the measured output vector, ω(k)∈ℝr is the disturbance that belongs to l2[0,∞), ϕ(ε(k))∈C(ℝL;ℝL) is the activation function with the input vector ε(k)∈ℝL, L∈ℝ is the number of nonlinear activation functions, d>0 is the time delay, A∈ℝn×n, Ad∈ℝn×n, Bϕ∈ℝn×L, Bω∈ℝn×r, Bu∈ℝn×m, Cε∈ℝL×n, Cd∈ℝL×n, Dϕ∈ℝL×L, Dω∈ℝL×r, Du∈ℝL×m, and Cy∈ℝl×n. Assume that the activation function ϕ satisfies ϕ(0)=0 and belongs to a type of set Ω(K) as follows:
(2)Ω(K)≜{ϕ∣0≤ϕi(εi(k))εi(k)≤ki,i=1,…,L},
which means that ϕ is sector restricted to the interval [0,K], where K=diag⁡(k1,…,kL)≻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 ofN neuron nodes to control the aforementioned DSNNM 𝒫. The wireless network in the whole system can be described by a directed graph as follows:
(3)𝒢≜{𝒜∪𝒱∪𝒮︸vertexset,ℰℐ∪ℰ𝒞∪ℰ𝒪︸edgeset},
where 𝒱={v1,…,vN} is the set of N neuron nodes, 𝒜={a1,…,am} is the set of m actuators which can execute the input vector u(k)=[u1(k),…,um(k)]T, 𝒮={s1,…,sl} is the set of l sensors used to measure the output vector y(k)=[y1(k),…,yl(k)]T, and edge sets ℰℐ={(vi,ap)∣vi∈𝒱,ap∈𝒜}, ℰ𝒞={(vi,vj)∣vj,vi∈𝒱}, and ℰ𝒪={(sq,vi)∣sq∈𝒮,vi∈𝒱} correspond to the physical radio communication links in the wireless network. Define the following three sets:theneighbor sensors of vi, ∀i∈{1,…,N}(4)𝒮vi≜{sq∣sq∈𝒮,∃(sq,vi)∈ℰ𝒪}={sq∣wiq𝒪≠0},theneighbor neurons of ap, ∀p∈{1,…,m}(5)𝒱ap≜{vi∣vi∈𝒱,∃(vi,ap)∈ℰℐ}={vi∣wpiℐ≠0},theneighbor neurons of vi, ∀i∈{1,…,N}(6)𝒱vi≜{vj∣vj∈𝒱,∃(vj,vi)∈ℰ𝒞}={vj∣wij𝒞≠0},where wpiℐ,  wij𝒞,  wiq𝒪 are the weights of edge (vi,ap), (vj,vi), and (sq,vi), respectively. This implies that sq∈𝒮vi if vi can receive data directly from sq, vi∈𝒱ap if ap can receive data directly from vi, and vj∈𝒱vi if vi can receive data directly from vj.The dynamic behavior of the neuron nodevi may be represented by the following pair of nonlinear equations:
(7)zi(k+1)=ψi(ξi(k)),ξi(k)=wii𝒞zi(k)+∑vj∈𝒱vi‍wij𝒞zj(k)+∑sq∈𝒮vi‍wiq𝒪yq(k),
where zi(k) is the state of neuron node vi, zj(k) is the state of neuron node vj,  vj∈𝒱vi, yq(k) is the measurement value of sensor sq, sq∈𝒮vi, ξi(k) is the weighted linear combination of vi’s present state and exogenous input signals (from neuron nodes in 𝒱vi or sensors in 𝒮vi), and  ψi(·)∈Ω(K-) is the activation function of neuron node vi, where K-=diag⁡(kL+1,…,kL+N)≻0. Each plant input up(k), p∈{1,…,m} is a weighted linear combiner output due to neighbor neuron nodes of the actuator ap as follows:
(8)up(k)=∑vi∈𝒱ap‍wpiℐzi(k).
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
(9)𝒦{z(k+1)=ψ(ξ(k)),ξ(k)=W𝒞z(k)+W𝒪y(k),u(k)=Wℐz(k),
where z∈ℝN is the state vector of WNCN, ψ∈C(ℝN;ℝN), ξ∈ℝN, W𝒞∈ℝN×N, W𝒪∈ℝN×l, and Wℐ∈ℝm×N. In the above-mentioned equations, ∀i∈{1,…,N}, wij𝒞=0 if vj∉𝒱vi∪{vi}, wiq𝒪=0 if sq∉𝒮vi, and wpiℐ=0 if vi∉𝒱ap. Therefore, the weight matrices W𝒞, W𝒪, and Wℐ 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 vi, i∈{1,…,N} transmits its state information once per time frame. In the beginning, vi has an arbitrary initial state value and then successively receives information from its neighbors in 𝒱vi and 𝒮vi in each time slot of frame. After vi has received all the information from its neighbors, vi will update its state by (7). Furthermore, in a similar way, every actuator ap, p∈{1,…,m} can receive the combination of control signals from neighbor neuron nodes in 𝒱ap and then act to system (1) by (8).Define vectorsx~=[xT,zT]T∈ℝn+N, ε~=[εT,ξT]T∈ℝL+N, and ϕ~=[ϕT,ψT]T∈Ω(K~),  K~=diag⁡(k1,…,kL+N)≻0. Then the overall closed-loop system 𝒢 of the DSNNM 𝒫 and the WNCN 𝒦 is described as
(10)𝒢{x~(k+1)=[ABuWℐ00]︸A~[x(k)z(k)]︸x~(k)x~(k+1)=+[Ad000]︸A~d[x(k-d)z(k-d)]︸x~(k-d)x~(k+1)=+[Bϕ00I]︸B~ϕ[ϕ(ε(k))ψ(ξ(k))]︸ϕ~(ε~(k))+[Bω0]︸B~ωω(k),ε~(k)=[CεDuWℐW𝒪CyW𝒞]︸C~ε[x(k)z(k)]︸x~(k)ε~(k)=+[Cd000]︸C~d[x(k-d)z(k-d)]︸x~(k-d)ε~(k)=+[Dϕ000]︸D~ϕ[ϕ(ε(k))ψ(ξ(k))]︸ϕ~(ε~(k))+[Dω0]︸D~ωω(k).Consider that the performance output of the closed-loop system𝒢 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 scalarγ>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 outputy~(k) satisfies
(11)∑k=0∞‍∥y~(k)∥2≤γ2∑k=0∞‍∥ω(k)∥2,∀nonzeroω(k).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 WNCN𝒦 for DSNNM 𝒫 such that the closed-loop system 𝒢 satisfies the requirements (1) and (2) in Definition 1.Algorithm 1:Given a scalar γ>0, solving the H∞ WNCN for closed-loop system 𝒢~.
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:
min⁡Tr{𝒳kQ+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.


## 3.H∞ Performance Analysis of the Closed-Loop System

In this section, we will investigate the absolute stability andH∞ 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]).
LetT0,T1,…,Tp∈𝕊n. If there exists τi≥0,  i=1,…,p such that
(12)T0-∑i=1p‍τiTi≺0
then ζTT0ζ<0 for all ζ≠0 such that ζTTiζ≤0,  i=1,…,p.Lemma 3 (Schur complement [35]).
Consider a matrixX∈𝕊n partitioned as
(13)X=[ABBTC],
where A∈𝕊k. If A is nonsingular, the matrix S=C-BTA-1B is called theSchur complement of A in X. Then, the following characterizations of positive definiteness or semidefiniteness of the block matrix X hold:
(14)(1)X≻0,iffA≻0,S≻0,(2)IfA≻0,thenX⪰0iffS⪰0.Theorem 4.
Givenγ>0 and WNCN 𝒦 with parameter set 𝒦={W𝒪,W𝒞,Wℐ}, if there exist appropriate dimension matrices P≻0, R≻0, and T⪰0, such that the following matrix inequality holds: (15)Ξ1=[(A~TPA~-P+R+C~yTC~y)A~TPA~dA~TPB~ϕ+C~εTTK~A~TPB~ω*A~dTPA~d-RA~dTPB~ϕ+C~dTTK~A~dTPB~ω**(B~ϕTPB~ϕ+D~ϕTTK~+TK~D~ϕ-2T)B~ϕTPB~ϕ+TK~D~ω***B~ωTPB~ω-γ2I]≺0,where K~=diag⁡(k1,…,kL+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 system𝒢 with ω(k)=0, one can obtain
(16)𝒢-{x~(k+1)=A~x~(k)+A~dx~(k-d)+B~ϕϕ~(ε~(k)),ε~(k)=C~εx~(k)+C~dx~(k-d)+D~ϕϕ~(ε~(k)).
Assume that the x~(k)=0 is the only equilibrium of 𝒢-. Consider the following Lyapunov-Krasovskii functional for systems 𝒢- as
(17)V(x~(k))=x~T(k)Px~(k)+∑i=1d‍x~T(k-i)Rx~(k-i).
According to the sector bound set Ω(K~) of ϕ~, we have
(18)ϕ~i(ε~i(k))·τi·[ϕ~i(ε~i(k))-kiε~i(k)]=τiϕ~i2(ε~i(k))-τikiε~i(k)ϕ~i(ε~i(k))≤0,
where εi≥0, i=1,…,L+N.
Now, by defining the difference ofV(x~(k)) along 𝒢- as ΔV(x~(k))≜V(x~(k+1))-V(x~(k)) and using Lemma 2 (S-procedure), we can obtain
(19)ΔV(x~(k))=x~T(k+1)Px~(k+1)-x~T(k)Px~(k)+x~(k)TRx~(k)-x~T(k-d)Rx~(k-d)≤[A~x~(k)+A~dx~(k-d)+B~ϕϕ~(ε~(k))]T×P[A~x~(k)+A~dx~(k-d)+B~ϕϕ~(ε~(k))]-x~T(k)Px~(k)+x~(k)TRx~(k)-x~T(k-d)Rx~(k-d)-2∑i=1L+N‍τiϕ~i2(ε~i(k))+2∑i=1L+N‍τikiε~i(k)ϕ~i(ε~i(k))=[x~(k)x~(k-d)ϕ~(ε~(k))]T×[A~TPA~-P+RA~TPA~dA~TPB~ϕ+C~εTTK~*A~dTPA~d-RA~dTPB~ϕ+C~dTTK~**(B~ϕTPB~ϕ+D~ϕTTK~+TKD~ϕ-2T)]︸Ξ0×[x~(k)x~(k-d)ϕ~(ε~(k))],
where T=diag⁡(τ1,τ2,…,τL+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
(20)Jℓ=∑k=0ℓ‍∥y~(k)∥2-γ2∑k=0ℓ‍∥ω(k)∥2=∑k=0ℓ‍[y~T(k)y~(k)-γ2ωT(k)ω(k)].
Consider the zero initial condition V(x~(0))=0, that is
(21)V(x~(ℓ+1))=∑k=0ℓ‍[V(x~(k+1))-V(x~(k))]=∑k=0ℓ‍ΔV(x~(k))>0.
Therefore, for system 𝒢, defining vector ζ(k)=[x~T(k)x~T(k-d)ϕ~T(ε~(k))ωT(k)]T and according to (19) and (20), we have
(22)Jℓ=∑k=0ℓ‍[y~T(k)y~(k)-γ2ωT(k)ω(k)+ΔV(x~(k))]000-V(x~(ℓ+1))≤∑k=0ℓ‍[y~T(k)y~(k)-γ2ωT(k)ω(k)+ΔV(x~(k))]=∑k=0ℓ‍ζT(k)Ξ1ζ(k).
If Ξ1≺0 (15) holds, lim⁡ℓ→∞Jℓ=lim⁡ℓ→∞∑k=0ℓ‍ζT(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 conditionΞ1≺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={W𝒪,W𝒞,Wℐ}).Lemma 5 (see [27]).
There exist matricesP, Q∈𝕊++n satisfying the constraint Q=P-1 if and only if they are optimal points for the problem
(23)min⁡Tr{QP}subjectto[QIIP]⪰0,Q,P∈𝕊++n,
and the optimal cost of the problem is n.Theorem 6.
Given a scalarγ>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 W𝒪,W𝒞,Wℐ∈𝒦 such that the following optimization problem: (24)min⁡⁡Tr{QP}bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb(25)s.t.ssΞ2=[-P+R0C~εT0A~TC~yT*-RC~dT0AdT0**D~ϕΣK~+ΣK~D~ϕT-2ΣD~ωΣK~B~ϕT0***-γ2IB~ωT0****-Q0*****-I]≺0,(26)s.t.ss[QIIP]⪰0,Q,P∈𝕊++n+N,where
(27)A~=[ABuWℐ00],C~ε=[CεDuWℐW𝒪CyW𝒞],
is feasible with optimal cost n+N.Proof.
InequalityΞ1≺0 (15) in Theorem 4 can be rewritten as
(28)[-P+R0C~εTTK~0*-RC~dTTK~0**D~ϕTTK~+TK~D~ϕ-2TTK~D~ω***-γ2I]+[A~TA~dTB~ϕTB~ωT]P[A~A~dB~ϕB~ωT]+[C~yT000][C~y000]≺0.
Then, using Lemma 3 (Schur complement), the inequality (28) is equivalent to
(29)[-P+R0C~εTTK~0A~TC~yT*-RC~dTTK~0A~dT0**D~ϕTTK~+TK~D~ϕ-2TTK~D~ωB~ϕT0***-γ2IB~ωT0****-Q0*****-I],≺0,
where Q=P-1. By defining
(30)S=(TK~)-1,Σ=(TK~)-1T(TK~)-1
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
(31)[-P+R0C~εT0A~TC~yT*-RC~dT0A~dT0**SD~ϕT+D~ϕS-2ΣD~ωSB~ϕT0***-γ2IB~ωT0****-Q0*****-I]≺0.
According to (30), the following equations hold:
(32)S-1=TK~,T=S-1ΣS-1.
Form (32), we know
(33)S=ΣK~.
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 givenH∞-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.Algorithm 2:Minimizing γ to solve the optimal H∞ WNCN for closed-loop system 𝒢~.
Step  1. Set k=0. Let γ-0 and γ+0 be the initial lower and upper bounds of γ, that is, γ0∈[γ-0,γ+0] where γ-0=0, γ+0 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=(γ-k+γ+k)/2.
Step  3. Use Algorithm 1 to check whether the feasible solution set Υk+1 satisfying inequality (15). If Υk+1 exists, set γ+k=γk.
Otherwise, set γ-k=γk.
Step  4. If γ+k-γ-k≤ϵ, where ϵ is the assumed calculation accuracy, set optimal H∞ performance index γ*=γ+k and exit.
Otherwise, set k=k+1, go to Step  2.


## 5. RobustH∞ Controller Design Based on Fading WNCN

Due to the large geographical nature of the closed-loop system𝒢 over a WN, a realistic distributed control design approach for WNCN should take the communication packet losses into account.According to [37], we adopt thefading 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)∈ℰ𝒪∪ℰ𝒞∪ℰI, where τ={1,…,ρ} and ρ=k(ℰ𝒪)+k(ℰ𝒞)+k(ℰI) 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 wτ𝒪,wτ𝒞,wτℐ, ∀τ=Ω(a,b) and then compacted into the following weight vector as
(34)w=[(w𝒪)T,(w𝒞)T,(wℐ)T]T=[{wτ𝒪}τ=Ω(sq,vi),{wτ𝒞}τ=Ω(vj,vi),{wτℐ}τ=Ω(vi,ap)]T,
where w𝒪∈ℝρ𝒪, w𝒞∈ℝρ𝒞, wℐ∈ℝρℐ, and w∈ℝρ. Let tk,τ denote the data packet transmitted over the τth communication link at time k. Then, aggregating all of tk,τ in a vector tk∈ℝρ, we can obtain
(35)t(k)=[Wt𝒪00Wt𝒞Wtℐ]︸Wor[Cy00IN][x(k)z(k)]=W~orx~(k),
where Wt𝒪=diag⁡(w𝒪)[eqT]q∣τ=Ω(sq,vi), eq∈ℝl, Wt𝒞=diag⁡(w𝒞)[ejT]j∣τ=Ω(vj,vi), ej∈ℝN, and Wtℐ=diag⁡(wℐ)[eiT]i∣τ=Ω(vi,ap), ei∈ℝN.(a) The wireless communication links simulated as fading channels and (b) WNCN is transformed into a robust sense.
(a)
(b)Remark 7.
W
or
∈
ℝ
ρ
×
(
N
+
l
) is arow selection matrix whose each row contains a single nonzero element which equals to a corresponding weight wτ𝒪, wτ𝒞 or wτℐ.
Next, letr(k) denote the received date from t(k) via the unreliable wireless communication links. γτ(k), τ∈{1,…,ρ} is independent and identically distributed (I.I.D) Bernoulli random variable with mean μτ=𝔼[γτ(k)] and variance στ2=𝔼[(γτ(k)-μτ)2]. γτ(k) indicates whether packet tτ(k) is successfully received by rτ(k); that is, γτ(k)=1 if packet arrives and γτ(k)=0 otherwise. If Δτ(k), τ∈{1,…,ρ} denotes an I.I.D Bernoulli random variable with zero-mean and variance στ2, γτ(k) can be transformed into a robust form such that γτ(k)=μτ+Δτ(k), where μτ is nominal value and Δτ(k) is random perturbation value. Thus, the fading channel model is described by the following bijective mapping:
(36)Γ:t(k)⟶r(k)=Γ(k)t(k)=(M+Δ(k))t(k),
where Γ(k)=diag⁡(γ1(k),…,γρ(k)), M=diag⁡(μ1,…,μρ), Δ(k)=diag⁡(Δ1(k),…,Δρ(k)), 𝔼[Δ(k)Δ(k)T]=diag⁡(σ12,…,σρ2), and 𝔼[Δ(k)]=0.
Thus, the dynamic behavior of thefading WNCN with stochastic packet losses can be described as follows:
(37)𝒦~{z(k+1)=ψ(ξ(k)),ξ(k)=Wμ𝒞z(k)+Wμ𝒪y(k)+WΔ𝒞𝒪Δ(k)t(k),u(k)=Wμℐz(k)+WΔℐΔ(k)t(k),
where WΔ𝒞𝒪=WΔ𝒞+WΔ𝒪 and
(38)Wμ𝒪=[(wμ𝒪)iq]N×l,where(wμ𝒪)iq={μτwτ𝒪,∃vi∈𝒱,sq∈𝒮,Ω(sq,vi)=τ0,else,Wμ𝒞=[(wμ𝒞)ij]N×N,where(wμ𝒞)ij={μτwτ𝒞,ifi≠j,∃vi,vj∈𝒱,Ω(vj,vi)=τwii𝒞,ifi=j,∃vi∈𝒱0,else,Wμℐ=[(wμℐ)pi]m×N,where(wμℐ)pi={μτwτℐ,∃vi∈𝒱,ap∈𝒜,Ω(vi,ap)=τ0,else,WΔ𝒪=[(wΔ𝒪)iτ]N×ρ,where(wΔ𝒪)iτ={1,∃vi∈𝒱,sq∈𝒮,Ω(sq,vi)=τ0,else,WΔ𝒞=[(wΔ𝒞)iτ]N×ρ,where(wΔ𝒞)iτ={1,∃vi,vj∈𝒱,Ω(vj,vi)=τ0,else,WΔℐ=[(wΔℐ)pτ]m×ρ,where(wΔℐ)pτ={1,∃vi∈𝒱,ap∈𝒜,Ω(vi,ap)=τ0,else.Remark 8.
Similar toWor, matrices WΔ𝒞𝒪 and WΔℐ are used to select which elements of Δ(k)t(k) are added to ξ(k) and u(k), respectively.
As shown in Figure2(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:
(39)𝒢~{x~(k+1)=A~μx~(k)+A~dx~(k-1)+B~ϕϕ~(ε~(k))x~(k+1)=+B~ωω(k)+W~1dstΔ(k)t(k),ε~(k)=C~εμx~(k)+C~dx~(k-1)+D~ϕϕ~(ε~(k))ε~(k)=+D~ωω(k)+W~2dstΔ(k)t(k),
where A~μ=[ABuWμℐ00], C~εμ=[CεDuWμℐWμ𝒪CyWμ𝒞], and W~1dst=[BuWΔℐ0], W~2dst=[DuWΔℐWΔ𝒞𝒪].Definition 9 (see [6, 32, 33]).
Given scalarγ>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 outputy~(k) satisfies
(40)∑k=0∞‍𝔼{∥y~(k)∥2}≤γ2∑k=0∞‍𝔼{∥ω(k)∥2},aaaaaaaaaaaaaaaaaaa∀nonzeroω(k)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 scalarγ>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, Wμ𝒪, Wμ𝒞, Wμℐ, and scalar θi,  i=1,…,ρ, such that the following optimization problem: (41)min⁡Tr{QP}hhhhbbbbbbbhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh(42)s.t.Ξ3=[-P+R+Π0(C~εμ)T0A~μTC~yT*-RC~dT0A~dT0**D~ϕΣK~+ΣK~D~ϕT-2ΣD~ωΣK~B~ϕT0***-γ2IB~ωT0****-Q0*****-I]≺0,(43)θi≥σi2(W~1dst)iTP(W~1dst)i,∀i=1,…,ρ,(44)[QIIP]⪰0,Q,P∈𝕊++n+N,where Π=(W~or)TΘW~or, Θ=diag⁡(θ1,…,θρ), and θi=σi2(W~1dst)iTP(W~1dst)i, (W~1dst)i denote the ith column of the matrix W~1dst, is feasible with optimal cost n+N.Proof.
Consider a Lyapunov candidate as follows:(45)V[M(k)]=Tr[M(k)P].
The difference of V[M(k)] along the trajectory of stochastic closed-loop system 𝒢~ with ω(k)=0 is given by
(46)ΔV[M(k)]=V[M(k+1)]-V[M(k)]=Tr[M(k+1)P]-Tr[M(k)P]=Tr{𝔼[x~T(k+1)Px~(k+1)-x~T(k)Px~(k)]}=𝔼[x~T(k+1)Px~(k+1)-x~T(k)Px~(k)].
By considering Δ(k) with 𝔼[Δ(k)]=0 and 𝔼[Δ(k)ΔT(k)]=diag⁡(σ12,…,σρ2) is independent from x~(k) and ϕ~(ε~(k)) and using Lemma 2 (S-procedure) one can obtain
(47)ΔV[M(k)]=[x~(k)x~(k-d)ϕ~(ε~(k))]T×[(A~μTPA~μ-P+R+Π)A~μTPA~dA~μTPB~ϕ+(C~εμ)TTK~*A~dTPA~d-RA~dTPB~ϕ+C~dTTK~**(B~ϕTPB~ϕ+D~ϕTTK~+TK~D~ϕ-2T)]︸Ξ~0×[x~(k)x~(k-d)ϕ~(ε~(k))],
where T=diag⁡(τ1,τ2,…,τL+N), K~=diag⁡(k1,…,kL+N), Π=(W~or)TΘW~or, Θ=diag⁡(θ1,…,θρ), θi=σi2(W~1dst)iTP(W~1dst)i, (W~1dst)i denote the ith column of the matrix W~1dst.
According to (20)–(22), ∀ℓ>0, we have
(48)𝔼{Jℓ}=∑k=0ℓ‍𝔼{∥y~(k)∥2}-γ2∑k=0ℓ‍𝔼{∥ω(k)∥2}=∑k=0ℓ‍𝔼{y~T(k)y~(k)-γ2ωT(k)ω(k)}≤∑k=0ℓ‍ζT(k)Ξ~1ζ(k),
where(49)Ξ~1=[(A~μTPA~μ-P+R+C~yTC~y+Π)A~μTPA~dA~μTPB~ϕ+(C~εμ)TTK~A~μTPB~ω*A~dTPA~d-RA~dTPB~ϕ+C~dTTK~A~dTPB~ω**(B~ϕTPB~ϕ+D~ϕTTK~+TK~D~ϕ-2T)B~ϕTPB~ϕ+TK~D~ω***B~ωTPB~ω-γ2I.].Since Ξ~0 is the principal minor of Ξ~1, if Ξ~1≺0, then Ξ~0≺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 Ξ~1≺0 holds, lim⁡ℓ→∞𝔼{Jℓ}≤lim⁡ℓ→∞∑k=0ℓ‍ζT(k)Ξ~1ζ(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 Ξ~1≺0 is equivalent to optimization problems (41)–(44) described in Theorem 10. This completes the proof.As in the previous section, we present Algorithms3 and 4.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. If there exists an initial feasible solution set Υ0={P,Q,R,Σ,Wμ𝒪,Wμ𝒞,Wμℐ} 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,Wμ𝒪,Wμ𝒞,Wμℐ} by solving the following LMI problem:
min⁡Tr{𝒳kQ+𝒴kP} s.t. (42)–(44).
Step  4. Substitute Υk+1 into matrix Ξ~1. If inequality Ξ~1≺0 holds, stop the algorithm. Otherwise, set k=k+1, 𝒳k=P,
𝒴k=Q and go to Step  2.Algorithm 4:Minimizing γ to solve the optimal H∞  fading WNCN with unreliable communication links for closed-loop system 𝒢~.
Step  1. Set k=0. Let γ-0 and γ+0 be the initial lower and upper bounds of γ, that is, γ0∈[γ-0,γ+0] where γ-0=0, γ+0 can be
assigned an arbitrarily sufficiently large value to make inequalities (42)–(44) have initial feasible solution set
Υ0={P,Q,R,Σ,Wμ𝒪,Wμ𝒞,Wμℐ}.
Step  2. At k≥0, computer γk=(γ-k+γ+k)/2.
Step  3. Use Algorithm 3 to check whether the feasible solution set Υk+1 satisfying Ξ~1≺0. If Υk+1 exists, set γ+k=γk.
Otherwise, set γ-k=γk.
Step  4. If γ+k-γ-k≤ϵ, where ϵ is the assumed calculation accuracy, set optimal H∞ performance index γ*=γ+k and exit.
Otherwise, set k=k+1, go to Step  2.


## 6. Numerical Simulation

Consider the following nonlinear system [38]:
(50)x1(k+1)=-x12(k)+0.3x2(k)+0.1x12(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).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=[-0.50.30.11], Ad=[0.05-0.100], Bϕ=[10], Bu=[1000.5], Cε=[10], Cd=[-0.10.2], Dϕ=0, Du=01×2, Cy=[0.60], Bω=[0.50.1], Dω=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 Figure3. 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:(51)P=[3.6031-1.0546-0.0825-0.08250.00470.0047-0.0617-0.0617-1.054611.33981.25461.2546-1.1665-1.1665-1.1256-1.1256-0.08251.25461.16100.0840-0.0821-0.0821-0.0785-0.0785-0.08251.25460.08401.1610-0.0821-0.0821-0.0785-0.07850.0047-1.1665-0.0821-0.08211.14950.07250.07080.07080.0047-1.1665-0.0821-0.08210.07251.14950.07080.0708-0.0617-1.1256-0.0785-0.07850.07080.07081.14540.0684-0.0617-1.1256-0.0785-0.07850.07080.07080.06841.1454],Q=[0.29310.0501-0.0192-0.01920.03780.03780.05450.05450.05010.1789-0.1421-0.14210.13590.13590.13420.1342-0.0192-0.14210.99290.0644-0.0579-0.0579-0.0576-0.0576-0.0192-0.14210.06440.9929-0.0579-0.0579-0.0576-0.05760.03780.1359-0.0579-0.05790.98840.05990.05940.05940.03780.1359-0.0579-0.05790.05990.98840.05940.05940.05450.1342-0.0576-0.05760.05940.05940.98900.06050.05450.1342-0.0576-0.05760.05940.05940.06050.9890],R=[1.1713-0.11820.01220.0122-0.0342-0.0342-0.1895-0.189500.5427-0.0197-0.0197-0.0580-0.05800.04050.0405000.7390-0.71830.00780.0078-0.0614-0.06140000.17380.06570.0657-0.5153-0.515300000.7494-0.68790.07130.0713000000.29730.34470.34470000000.7422-0.708900000000.2198],Σ=diag⁡(0.5313,1.2461,0.1357,0.4233,1.4452,0.7652,1.2891),Wμ𝒞=[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],Wμ𝒪=[-1.3588-1.35880000],Wμℐ=[0000-0.0646-0.06460000-0.2989-0.2989].Figure 3
A discrete-time DSNNM synthesized by a WNCN with 6 wireless neuron nodes.Figure4 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 underH∞ controller WNCN with γ*=0.7921.


## 7. Conclusions

A novel wireless networkedH∞ 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.



---

*Source: 708252-2014-05-05.xml*