Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 708252, 14 pages

http://dx.doi.org/10.1155/2014/708252

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

College of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, China

Received 27 December 2013; Accepted 28 March 2014; Published 5 May 2014

Academic Editor: Ge Guo

Copyright © 2014 Wen Ren and Bugong Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 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 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.

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 performance of the closed-loop system. The criteria to synthesis of the optimal 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*. is the -dimensional Euclidean space. is the set of real matrices. denotes the transpose of matrix . denotes the trace of a square matrix . denotes the set of symmetric matrices. denotes the set of positive semidefinite matrices. denotes the set of positive definite matrices. The curled inequality symbol is used to denote generalized inequality: , the matrix inequality means that , and . denotes the cardinality of set . denotes an identity matrix of appropriate order. denotes the th vector of the standard basis of . denotes the estimation operator. denotes a diagonal matrix. is used as an ellipsis for terms induced by symmetry. If and , denotes the space of all continuous functions mapping . 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: with the initial condition function , , where is the state vector, is the control input vector, is the measured output vector, is the disturbance that belongs to , is the activation function with the input vector , is the number of nonlinear activation functions, is the time delay, , , , , , , , , , , and . Assume that the activation function satisfies and belongs to a type of set as follows: which means that is sector restricted to the interval , where .

##### 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 neuron nodes to control the aforementioned DSNNM . The wireless network in the whole system can be described by a directed graph as follows:
where is the set of neuron nodes, is the set of actuators which can execute the input vector , is the set of sensors used to measure the output vector , and edge sets , , and correspond to the physical radio communication links in the wireless network. Define the following three sets: the* neighbor sensors* of , the* neighbor neurons* of , the* neighbor neurons* of , where , , are the weights of edge , , and , respectively. This implies that if can receive data directly from , if can receive data directly from , and if can receive data directly from .

The dynamic behavior of the neuron node may be represented by the following pair of nonlinear equations: where is the state of neuron node , is the state of neuron node , , is the measurement value of sensor , , is the weighted linear combination of ’s present state and exogenous input signals (from neuron nodes in or sensors in ), and is the activation function of neuron node , where . Each plant input , is a weighted linear combiner output due to neighbor neuron nodes of the actuator as follows: If each neuron node is regarded as a nonlinear dynamical subcontroller, the whole WNCN consisting of neuron nodes may act as a fully distributed RNC whose dynamic behavior may be described as where is the state vector of WNCN, , , , , and . In the above-mentioned equations, , if , if , and if . Therefore, the weight matrices , , and 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 , transmits its state information once per time frame. In the beginning, has an arbitrary initial state value and then successively receives information from its neighbors in and in each time slot of frame. After has received all the information from its neighbors, will update its state by (7). Furthermore, in a similar way, every actuator , can receive the combination of control signals from neighbor neuron nodes in and then act to system (1) by (8).

Define vectors , , and , . Then the overall closed-loop system of the DSNNM and the WNCN is described as

Consider that the performance output of the closed-loop system is described as , where , then the following definition is introduced.

*Definition 1 (see [6, 32, 33]). *Given a scalar , the closed-loop system is said to be absolutely stable with a -norm bound if there exists a distributed dynamic neural controller WNCN such that the following conditions are satisfied (Algorithm 1).(1)With zero disturbance, that is, , the zero solution of the closed-loop system is globally asymptotically stable, , .(2)Under the zero-initial condition, the performance output satisfies
Then the WNCN is said to be a controller for the DSNNM . Furthermore, if we can find a minimal to satisfy the above conditions, the WNCN is an optimal controller.

Our aim is to design the WNCN for DSNNM such that the closed-loop system satisfies the requirements and in Definition 1.

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

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

Lemma 2 (S-procedure [34]). *Let . If there exists , such that
**
then for all such that , .*

Lemma 3 (Schur complement [35]). *Consider a matrix partitioned as
**
where . If is nonsingular, the matrix is called the Schur complement of A in . Then, the following characterizations of positive definiteness or semidefiniteness of the block matrix hold:
*

Theorem 4. *Given and WNCN with parameter set , if there exist appropriate dimension matrices , , and , such that the following matrix inequality holds: **where , then the zero solution of closed-loop system is absolutely stable and the -norm constraint (11) is achieved for all nonzero .*

*Proof. *From system with , one can obtain
Assume that the is the only equilibrium of . Consider the following Lyapunov-Krasovskii functional for systems as
According to the sector bound set of , we have
where , .

Now, by defining the difference of along as and using Lemma 2 (S-procedure), we can obtain
where . By Lemma 3 (Schur complement), if (15) holds, also holds. So, if , system with , that is, system , is globally asymptotically stable, .

Next, for , define
Consider the zero initial condition , that is
Therefore, for system , defining vector and according to (19) and (20), we have
If (15) holds, . Thus, and the -norm constraint (11) is achieved. This completes the proof.

#### 4. Controller Design Based on WNCN

In the previous stage, the matrix inequality condition (15) is not an LMI, which cannot be solved by LMI tools. In what follows, we first convert the matrix inequality condition (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 ).

Lemma 5 (see [27]). *There exist matrices , satisfying the constraint if and only if they are optimal points for the problem
**
and the optimal cost of the problem is .*

Theorem 6. *Given a scalar , the closed-loop system is said to be absolutely stabilizable by using a WNCN and the -norm constraint (11) is achieved for all nonzero if there exist appropriate dimension matrices , , , , and such that the following optimization problem: **where
**
is feasible with optimal cost .*

*Proof. *Inequality (15) in Theorem 4 can be rewritten as
Then, using Lemma 3 (Schur complement), the inequality (28) is equivalent to
where . By defining
and pre- and postmultiplying the left-hand side matrix of (29) by , respectively, the inequality (29) is equivalent to
According to (30), the following equations hold:
Form (32), we know
Substituting (33) into (31), one can obtain (25). By using Lemma 5, the nonconvex constraint 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 -norm bound of the closed-loop system. In what follows, we give the Algorithm 2 based on the bisection method to design WNCN for the optimal control problem: min s.t. (15), , , .

#### 5. Robust Controller Design Based on Fading WNCN

Due to the large geographical nature of the closed-loop system 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 , , where and is the total number of wireless links in WNCN, to concisely enumerate all links in the network. Therefore, the weights of links can be mapped to , and then compacted into the following weight vector as
where , , , and . Let denote the data packet transmitted over the th communication link at time . Then, aggregating all of in a vector , we can obtain
where , , , , and , .

*Remark 7. * is a* row selection matrix* whose each row contains a single nonzero element which equals to a corresponding weight , or .

Next, let denote the received date from via the unreliable wireless communication links. , is independent and identically distributed (I.I.D) Bernoulli random variable with mean and variance . indicates whether packet is successfully received by ; that is, if packet arrives and otherwise. If , denotes an I.I.D Bernoulli random variable with zero-mean and variance , can be transformed into a robust form such that , where is nominal value and is random perturbation value. Thus, the fading channel model is described by the following bijective mapping:
where , , , , and .

Thus, the dynamic behavior of the* fading* WNCN with stochastic packet losses can be described as follows:
where and

*Remark 8. *Similar to , matrices and are used to select which elements of are added to and , 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 . Consider the following stochastic closed-loop system:
where , , and , .

*Definition 9 (see [6, 32, 33]). *Given scalar and , the state correlation matrix, as , the stochastic closed-loop system is said to be absolutely stable in mean-square with a -norm bound , if there exists a distributed dynamic neural controller WNCN such that the following conditions are satisfied.(1)With zero disturbance, that is, , , , and .(2)Under the zero-initial condition, the performance output satisfies
Then the WNCN is said to be a robust controller for the DSNNM . Furthermore, if we can find a minimal to satisfy the above conditions, the WNCN is an optimal robust controller.

Theorem 10. *Given a scalar , the stochastic closed-loop system is said to be absolutely stabilizable in mean-square by using a fading WNCN and the -norm constraint (40) is achieved for all nonzero if there exist appropriate dimension matrices , , , , , , , and scalar , , such that the following optimization problem: **where , , and , denote the th column of the matrix , is feasible with optimal cost .*

*Proof. *Consider a Lyapunov candidate as follows:
The difference of along the trajectory of stochastic closed-loop system with is given by
By considering with and is independent from and and using Lemma 2 (S-procedure) one can obtain
where , , , , , denote the th column of the matrix .

According to (20)–(22), , we have
whereSince is the principal minor of , if , then such that converges to zeros as for system with , , and . Further, if holds, . Thus, and the -norm constraint (40) is achieved. Similar to the proof process of Theorem 6, by using Lemma 3 (Schur complement), the inequality 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.

#### 6. Numerical Simulation

Consider the following nonlinear system [38]:

According to [39, 40], when we consider the disturbance , the nonlinear system (50) can be transformed into the discrete-time DSNNM (1), where , , , , , , , , , , , and .

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) and variance . For , Algorithm 4 can be solved by CVX, a package for specifying and solving convex programs [41]. Then, we obtain the minimum optimal performance index , the solutions of (41)–(44), and the interconnection weight matrix parameters of WNCN as follows: