Abstract：The stable controller design problem of Networked Control System (NCS) is addressed using linear matrix inequality (LMI) technique.The generalized uncertain model of NCS is presented and a sufficient stable condition is derived.Based on this sufficient condition，a delay-dependent LMI approach for stabilizing NCS via state feedback control is presented.Numerical examples are employed to reveal that the LMI method is effective.

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Keywords：networked control system;linear matrix inequality;delay-dependent;state feedback control

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CLC：TN41，TP33 Reference Identification：A

No：1004-373X(2008)07-092-05

基于線性矩陣不等式的網(wǎng)絡(luò)控制系統(tǒng)設(shè)計方法研究

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陳曉琴

(江蘇海事職業(yè)技術(shù)學院 江蘇 南京 211170)

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摘 要：闡述了基于線性矩陣不等式的網(wǎng)絡(luò)控制系統(tǒng)的設(shè)計方法。概述中指出了網(wǎng)絡(luò)控制系統(tǒng)的不確定模型，并派生出一個充分的穩(wěn)定條件?；谶@一充分條件，提出了一種延遲依賴線性矩陣不等式方法，該方法通過狀態(tài)反饋控制來穩(wěn)定網(wǎng)絡(luò)控制系統(tǒng)。通過實例證明了線性矩陣不等式方法的有效性。

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關(guān)鍵詞：網(wǎng)絡(luò)控制系統(tǒng)；線性矩陣不等式;延遲依賴;狀態(tài)反饋控制

1 Introduction

In recent years，many researchers focused their work on analysis and synthesis of the Networked Control Systems.In general，Networked Control Systems(NCSs)，also called Networked-based Control Systems，indicate that feedback control systems in the control loops are closed through a real-time network^[1-4].In a NCS，each component of the system，such as controllers，sensors and actuators etc are connected via a real-time network，as illustrated in Figure 1.The nature of an NCS is that information (such as reference input，plant output，control signal，etc.) is exchanged among control system components (sensors，controller，actuators，etc.) by a control network.

The primary advantages of an NCS include lower-cost，reduced weight and power，simple installation and maintenance，higher reliability，ease of system diagnosis and maintenance，and increased system agility^[5].However，the insertion of the communication network in the feedback control loop presents some challenges in analyzing and synthesizing control system because the network imposes a undetermined communication delay:the network is shared by many kinds of devices and it′s not always available when one device needs to transfer data or information.Therefore，conventional control theories based on many ideal assumptions，such as synchronized control and nondelayed sensing and actuation，must be reevaluated prior to application to NCSs.There are several issues need to be addressed^[6]，which include the network-induced delay among the controller sensors，and actuators;multiple-packets transmissions of plant outputs;the case of lost packets;and the like.In those issues，the most important one is how to deal with the network-induced delay.

Fig.1 a typical NCS setup

The idea of remote controlled operation has been under investigation since 1950s.It started with Goetz and Thompson′s demonstration of their first \"master-slave\" remote control in 1954^[7] followed by Ferrell and Sheridan^[8] who focused more attention on the system control aspects.These results are regarded as the earliest research work on networked control systems.With the advances in computer，communication，control theory and the request of industry control，recently it is developed into the networked control and becoming a \"hot\" research area in control theory.A special issue is published by IEEE Control System Magazine in February 2001，and several new developments are reported in IEEE Control Systems Technology in May 2002.The following is the part of reported research results on networked control systems.

Halevi and Ray^[1] considered a continuous-time plant and discrete-time controller，and analyzed the Integrated Communication and Control System(ICCS) using a discrete time approach.They studied a clock-driven controller with mis-synchronization between plant and controller，and also took message rejection and vacant sampling into account.Nilsson^[2] analyzes NCSs in the discrete-time domain.Further modeled the network delays as constant，independently random，and random but governed by an underlying Markov chain.From there，he solved the LQG optimal control problem for the various delay models.Walsh^[3] considerd a continuous plant and a continuous controller in the control network by introducing the notion of Maximum Allowable Transfer Interval (MATI)，denoted by τ.Their goal was to find that value of τ for which the desired performance (e.g.，stability) of an NCS was guaranteed to be preserved.

Walsh and Ye^[9] presented some important issues associated with controlling systems over networks and presented relations between closed-loop stability and the size of transmission deadlines for certain types of scheduling.Further，they introduced the Try Once Discard (TOD) network protocol into Multiple-Input Multiple-Output (MIMO) networked control systems^[10]，and provided an analytic proof of global exponential stability for both the TOD protocol and the more commonly used (statically scheduled) access methods.Park^[11] presented a scheduling method for network-based control systems with three types of data(periodic data sporadic data and messages).

Conventional control theories such as synchronized control and nondelayed sensing and actuation must be reevaluated prior to application to networked control systems.Zhang^[6] used the techniques coming from digital control systems，hybrid systems，and asynchronous systems to discuss some important issues for networked control systems over local area networks.These issues include network-induced delays among the controller，sensors and actuators;multiple-packet transmissions of plant outputs;the case of lost packets.Paper^[12] discusses the impact of network architecture on control performance in a class of networked control systems and provides design considerations related to control quality of performance as well as network quality of service.

As inducing the delay of network into a control system，we know that the loop-closed networked control system can be treated as a time-delay system with uncertainty.Control of delay systems has been a topic of recurring interest over the past decades since time delays are often the main causes for instability and poor performance of systems and encountered in various engineering systems such as chemical processes，long transmission lines in pneumatic systems，and so on^[13].Recently，the problems of robust stability analysis and robust stabilization for uncertain delay systems have been studied.In the case of uncertain systems without delay，the method based on the concepts of quadratic stability and quadratic stabilizability has been shown to be effective in dealing with these problems in both continuous and discrete contexts^[14-15].In the past few years，many authors studied the problem of designing robust controllers for uncertain systems with state delays^[16-19] and the references therein.All of the those results obtained via Riccati and Lyapunov equation approaches are independent of the size of the delays (i.e.，the time-delay is allowed to be arbitrarily large) and thus，in general，are conservative，especially in situations such as in networked control systems where delay are small.More recently，several results dependent upon the size of delay have been reported^[20-22].However，these results are somewhat conservative.An LMI-based approach is proposed^[22-24]，leading to less conservative results dependent on the size of delay.The systems under consideration therein，however，are allowed to have only one single constant state delay.

In this paper，we focus more attention on the controller synthesis problem by traditional robust stabilization problem of uncertain delay system.Based on the delay-dependent stability results of a time-delay system with uncertainty，a new LMI-based and delay-dependent method of designing a linear state feedback controller is presented.

2 Modeling Uncertain NCS

To establish the NCS model，the following assumptions are made as in ［10］.Firstly，the control law is designed without considering the presence of the network.Besides，the controller dynamics is considered continuous because the access interval of the NCS to the network is much larger than the processing period of the controller and smart sensors.Finally，the communication medium is assumed error-free.Then the dynamics of NCS can be summarized as following.

Where ‖#8226;‖ represents the Euclidean norm (i.e.the maximum singular value).

It should be mentioned that the description is different from ［10］ besides，it concerns the perturbations on [WTHX]A[WTBZ] and [WTHX]B[WTBZ].Because the time delay caused by network is only appeared in sensor-to-controller and controller-to-actuator transmission and the dynamic characteristics inside the controller and plant are not influenced by the transmission delay，they should be described apart.And the transfer delay of sensor to controller τ1 and controller to actuator τ2 can be lumped together τ=τ1+τ2for analysis purpose，so it′s natural to describe them together as in (1).And our aim is to find a stable feedback control law

Where ΔA and ΔAd are linear parametrical uncertainties with bounds.

4 Stable Networked Controller Design

In this section，based on the sufficient stable condition given in Lemma 1，we will present a stable feedback control law for the system (1).

[WTHZ]Theorem [STHZ]1[STBZ][WTBZ]:u(k)=[WTHX]K[WTBX]x(k)is a stable controller for the system (1) if there exist scalars α，β1，β2，γ，γ2，satisfying the following LMIs:

Solution:We obtain δ=0.241 4，κ=0.228 8，τ=0.016.Using Theorem 1，a stable state feedback gain matrix is [WTHX]K[WTBX]=－30.1385－6.501 3－3.401 0－30.388 8.

6 Conclusions

The problem of modeling networked control system and its robust stabilization are researched in this paper.Different from the reported approaches，physical layer design and application layer or controller design is described apart in the light of the ideal of transparence between layers.Further research reveals that the controller synthesis problem turns out to be a traditional robust stabilization problem of uncertain delay system.Based on the previous research work，a new LMI-based and delay-dependent method of designing a linear state feedback controller is presented，and the numerical examples show that this algorithm is effective.

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作者簡介 陳曉琴 女，1968年出生，湖北鄂州人，碩士，副教授。主要從事自動檢測與控制技術(shù)方向的研究。

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