Open access peer-reviewed chapter

Event-Triggered Static Output Feedback Simultaneous H∞ Control for a Collection of Networked Control Systems

By Sheng-Hsiung Yang and Jenq-Lang Wu

Submitted: October 29th 2015Reviewed: March 11th 2016Published: July 6th 2016

DOI: 10.5772/63020

Downloaded: 852

Abstract

This chapter considers the design of event-triggered static output feedback simultaneous H∞ controllers for a collection of networked control systems (NCSs). It is shown that conventional point-to-point wiring delayed static output feedback simultaneous H∞ controllers can be obtained by solving linear matrix inequalities (LMIs) with a linear matrix equality (LME) constraint. Based on an obtained simultaneous H∞ controller, an L2-gain event-triggered transmission policy is proposed for reducing the network usage. An illustrative example is presented to verify the obtained theoretical results.

Keywords

  • networked control systems
  • simultaneous stabilization
  • event-triggered
  • static output feedback
  • H∞ control.

1. Introduction

A networked control system (NCS) is a feedback control system with feedback loop closed through a communication network. As the signal in an NCS is exchanged via a network, the network-induced delay, packet dropout, and limited network bandwidth can degrade the control performance. Many results have been proposed for dealing with these issues [15]. In the early stages, the studies on NCSs were mainly based on periodic task models [46]. The number of data packets to be transmitted will be large as the sampling period is small. This leads to a conservative usage of network resources and possibly leads to a congested network traffic. Therefore, how to design networked feedback controllers to achieve desired performance with low network usage is an important issue in NCSs.

Recently, some sporadic task models have been presented in NCSs without degrading system performance. An important approach is the event-triggered scheme [726]. In [7], the state transmitting and the control signal updating events were triggered only if the error between the current measured state and the last transmitted state is larger than a threshold condition. In [8], event-triggered distributed NCSs with transmission delay were studied. Based on the designed event-triggered policy, an allowable upper bound of the transmission delay was derived. In [9], for distributed control systems, an implementation of event-triggering control policy in sensor-actuator network was introduced. In [10], the authors concerned with the design of event-triggered state feedback controllers for distributed NCSs with transmission delay and possible packet dropout. Under the proposed triggering policy, the tolerable packet delay and packet dropout were derived. In [11], an event-triggered control policy was developed for discrete-time control systems. In [12], under stochastic packet dropouts, an event-triggered control law for NCSs was calculated by the proposed algorithms. In [13], an event-triggered scheme was developed for uncertain NCSs under packet dropout. In [14], an event-based controller and a scheduler scheme were proposed for NCSs under limited bandwidth. The NCSs were modeled as discrete-time switched control systems. A sufficient condition for the existences of event-based controllers and schedulers was derived by the LMI optimization approach. Recently, the event-triggered scheme has been extended to Hcontrol of NCSs for achieving the disturbance attenuation performance [1521]. In [15] and [18], with considering transmission delays, event-triggered Hstate feedback controllers for NCSs were proposed. Criterion for stability and criterion for co-designing both the controller gains and the trigger parameters were derived. In [16], an event-triggered state feedback control scheme was proposed for guaranteeing finite L2-gain stability of a linear control system. In [17], an event-triggered state feedback Hcontroller for sampled-data control system was proposed. In [19], the design of event-triggered networked feedback controllers for discrete-time NCS was considered. In [20], based on Lyapunov-Krasovskii function, an event-triggered state feedback Hcontroller was derived for NCSs under time-varying delay and quantization.

All the results in [720] are derived in the assumption that the system states are available for measurement. For practical control systems, system states are often unavailable for direct measurement. In the literature, only few results have been proposed for output-based event-triggered NCSs [2226]. In [22], a dynamic output feedback event-triggered controller for NCSs was proposed for guaranteeing the asymptotic stability. In [23] and [24], by the passivity theory approach, output-based event-triggered policies were derived for guaranteeing the satisfaction of L2-gain requirements of dynamic output feedback NCSs in the presence of time-varying delays. The synthesis of controllers has not been discussed. In [25] and [26], under nonuniform sampling, new output-based event-triggered Htransmission policies were proposed of NCSs under time-varying transmission delays. Furthermore, the design of static output feedback Hcontrollers for NCSs was discussed. Conditions for the existence of Hcontrollers were presented in terms of bilinear matrix inequalities. A non-convex minimization problem must be solved to get a static output feedback Hcontroller.

On the other hand, few results have been proposed in the literature for simultaneous stabilization of NCSs. The consideration of simultaneous stabilization is important since it allows us to design highly reliable controllers that are able to accommodate possible element failures in control systems. As the signal transmitted through network, the solvability of simultaneous stabilization problem of NCSs is quite different to that of point-to-point wiring control systems. Only few results have been proposed for relevant issues [21, 27]. In [27], based on the average dwell time approach, the simultaneous stabilization for a collection of NCSs was considered. A sufficient condition for guaranteeing simultaneous stabilization was proposed. In [21], under the assumption that the network communication channel is ideal (no delay, no packet dropout, and no quantization error), we considered the design of state feedback event-triggered simultaneous Htransmission policies for a collection of NCSs. Under the proposed event-triggered transmission policies, the L2-gain stability of all the closed-loop NCSs can be guaranteeing under low network usages.

It is known that static output feedback controllers are preferred in practical applications since their implementations are much easier than dynamic output feedback controllers. However, the design of static output feedback controllers is much more difficult than dynamic ones. In this chapter, we extend our previous work [21] to static output feedback case. Furthermore, we consider the network-induced time-varying delay that has not been considered in [21]. We develop an event-triggered static output feedback simultaneous Htransmission policy for a collection of continuous-time linear NCSs under time-varying delay. It is shown that, under mild assumptions, conventional point-to-point wiring delayed static output feedback simultaneous Hcontrollers can be obtained by solving LMIs with a LME constraint. Based on the obtained static output feedback simultaneous Hcontrollers, an event-triggered transmission policy was derived for reducing network usage. Different to the results presented in [25] and [26] that only considering the design of an event-triggered Hcontroller for a single system, this chapter considers the design of a fixed event-triggered Hcontroller that is able to L2-stabilize a collection of NCSs simultaneously. By the proposed method, highly reliable NCSs that are able to accommodate possible element failures with low network usage can be designed. Even simplifying our results to the single system case, our method for designing static output feedback Hcontrollers is quite different from those in [25] and [26]. In [25] and [26], a non-convex minimization problem must be solved for getting a static output feedback Hcontroller. Moreover, the obtained controller can only guarantee uniform ultimate boundedness but not internal stability. In our approach, (simultaneous) static output feedback Hcontrollers are obtained by solving LMIs with a LME constraint. Moreover, internal stabilities of the closed-loop NCSs can be guaranteed.

2. Problem formulation and preliminaries

In this section, the problem to be solved is formulated and some preliminaries are given. For simplifying the expressions, we use the same notations x, u, w, and z to denote the states, control inputs, exogenous inputs, and the controlled outputs of all considered systems.

2.1. Problem formulation

Consider a collect of continuous-time control systems:

x˙(t)=Ajx(t)+B1jw(t)+B2ju(t),j=1,2,...,Nz(t)=C1jx(t)+D11jw(t)+D12ju(t)y(t)=C2jx(t)E1

where x(t)Rnis the system state, u(t)Rmis the control input, z(t)Rsis the controlled output, y(t)Rlis the measured output, w(t)Rris the exogenous input, and Aj, B1j, B2j, C1j, D11j, D12j, and C2jare constant matrices with appropriate dimensions. Here, for convenience, we assume C2j=C2, j = 1,2,…,N. Suppose that (Aj,B2j) are stabilizable and (C2,Aj) are detectable for each j{1,2...,N}.Furthermore, assume that γ2ID11jTD11j>0for all j{1,2...,N}.

In this chapter, we consider the case that the feedback loop of system (1) is closed through a real-time network, but not by the conventional point-to-point wiring. Suppose that the sensor node keeps measuring the output signal y, but not all the sampled data need to be sent to the controller node. The data transmission at the sensor node is not periodic. Let ti(i = 1,2,…) be the time that the i-th transmission occurs at the sensor nodes. In this case, the controller node receives the networked feedback data y(ti)and updates the control signal at time ti+τi, i = 1,2,…, where τi[τdmin,τdmax]is the transmission delay. That is,

u(t)=Fy(ti),ti+τit<ti+1+τi+1,i=1,2,E2

where F is the feedback gain to be designed later. With the same controller (2), the closed-loop systems are:

x˙(t)=Ajx(t)+B1jw(t)+B2jFC2x(ti),ti+τit<ti+1+τi+1, j=1,2,,Nz(t)=C1jx(t)+D11jw(t)+D12jFC2x(ti)E3

If the measured data is not critical for L2-gain stability, it will not be sent for saving the network usage. In this case, the controller node does not update the control signal. If the measured data is critical, it will be sent through the network to the controller node, and the controller will update the control signal.

Our main goal is to design an event-triggered transmission rule to determine whether the currently measured data should be sent to the controller node, such that, under the transmission delay, all possible closed-loop systems in (3) are internally stable and satisfy, for a given constant γ>0and for any T>0and wL2[0,T],

0TzT(t)z(t)dtγ020TwT(t)w(t)dt,forsomeγ0<γE4

Note that, a practical control system may have several different dynamic modes since it may have several different operating points (please see e.g., the ship steering control problem considered in [28] ). On the other hand, for achieving higher reliability of a practical control system, we may want to design a controller to accommodate possible element failures. With considering possible element failures, a control system can have several different dynamic modes (see e.g., the reliable control problem for active suspension systems considered in [29]). The problem we considered has a practical importance owing to its high applicability in designing robust and/or reliable controllers.

2.2. Preliminaries

The following Lemmas will be used later.

Lemma 1 [30]: For any vectors X,YRnand any positive definite matrix GRn×n, the following inequality holds:

2XTYXTGX+YTG1YE5

Lemma 2 [31]: For any given matrices Π<0and Φ=ΦT, and any scalar λ, the following inequality holds:

ΦΠΦ2λΦλ2Π1E6

For convenience, define xt(s)=x(t+s), s[τmax,0].

Lemma 3 (Lyapunov–Krasovskii Theorem) [32]: Consider a time-delay system:

x˙(t)=Ax(t)+Adx(tτ(t)), t0E5

with τ(t)[0,τmax],t0. Suppose that x(t)=ψ(t), t[τmax,0]. If there exists a function

V:C([τmax,0],Rn)RE8

and a scalar ε>0, such that, for all φC([τmax,0],Rn),V(φ)εφ(0)2,and, along the solutions of (5),

dV(xt)dt|xt=φεφ(0)2,E9

then the system (5) is asymptotically stable. ■

3. Main results

We first consider the design of the event-triggered transmission policy under the assumption that we have a delayed simultaneous Hcontroller, and then show how to derive simultaneous Hcontroller under transmission delay.

3.1. Event-triggered transmission policy for NCSs under time-varying delay

Define the equivalent time-varying delay

τ(t)=tti, ti+τit<ti+1+τi+1, i=1,2,.E10

It is clear that

τ(t)[τmin, τmax]t0,andτ˙=1almosteverywhereE6

where τminminiN{τi}=τdminand τmaxmaxiN{ti+1ti+τi+1}=maxiN{ti+1ti}+τdmax. Then, the systems in (3) can be equivalently described as

x˙(t)=Ajx(t)+B1jw(t)+B2jFC2x(tτ(t)),j=1,2,,Nz(t)=C1jx(t)+D11jw(t)+D12jFC2x(tτ(t))E7

To derive an event-triggered transmission policy in the presence of transmission delay, assume that, for the systems in (1), we have a conventional delayed static output feedback simultaneous Hcontroller:

u(t)=Fy(tτ(t))E8

which is such that all of the possible closed-loop systems in (7) are internally stable and satisfy the condition (4) for τ(t)[τmin, τmax]. How to get such a delayed static output feedback simultaneous Hcontroller will be discussed later.

Define the error signal:

e(t)=y(t)y(ti),tit<ti+1E9

We have the following results.

Theorem 1: Consider the systems in (1). Suppose that the controller (8) is such that all the closed-loop systems in (7) are internally stable and satisfy the condition (4). If there exist matrices Pj>0, Qj>0, G1j, G2j, G3j, and G4j, j=1,2,…,N, of appropriate dimensions, and scalars εj>0, j=1,2,…,N, satisfying the following LMIs:

[ΦjΞjG3jTPjB1j+G4jT+C1jTD11jτmaxAjTQjτmaxG1j*ΣjG3jTG4jT+C2TFTD12jTD11jτmaxC2TFTB2jTQjτmaxG2j**εjI00τmaxG3j***D11jTD11jr2IτmaxB1jTQjτmaxG4j****τmaxQj0*****τmaxQj]<0,E10

where

Φj=AjTPj+PjAj+C1jTC1j+C2TC2+G1j+G1jT,E16
Ξj=PjB2jFC2+C1jTD12jFC2G1j+G2jT,E17
Σj=C2TFTD12jTD12jFC2G2jG2jT,E18

then all the networked closed-loop systems in (7) are internally stable and satisfy the condition (4) if the following condition holds:

e(t)<minj{1,2,...,N}1εjy(t),tit<ti+1E11

Proof: For the systems in (7), choose the candidate storage functions:

Vj(x(t))=xT(t)Pjx(t)+τmax0t+σtx˙T(θ)Qjx˙(θ)dθdσ,j=1,2,,N.E11

Define

H^dj(x(t),x(ti),e(t),w(t))V˙j(x(t))+zT(t)z(t)γ2wT(t)w(t)+yT(t)y(t)εjeT(t)e(t),j=1,2,,N.E21

Along the solutions of the j-th system, we have

H^dj=2xT(t)Pjx˙(t)tτmaxtx˙T(θ)Qjx˙(θ)dθ+τmaxx˙T(t)Qjx˙(t) +zT(t)z(t)γ2wT(t)w(t)+yT(t)y(t)εjeT(t)e(t)+2ηT(t)Gj(x(t)x(ti)titx˙(θ)dθ)E22

where η(t)=[xT(t)xT(ti)eT(t)wT(t)]Tand Gj=[G1jTG2jTG3jTG4jT]T. Then,

H^dj=2xT(t)Pj(Ajx(t)+B1jw(t)+B2jFC2x(tτ(t)))+xT(t)C1jTC1jx(t)E24

+2xT(t)C1jTD12jFC2x(tτ(t))+xT(tτ(t))C2TFTD12jTD12jFC2x(tτ(t))E24
+wT(t)D11jTD11jw(t)+2xT(t)C1jTD11jw(t)+2xT(tτ(t))C2TFTD12jTD11jw(t)E25
r2wT(t)w(t)tτmaxtx˙T(θ)Qjx˙(θ)dθE26
+τmaxxT(t)AjTQjAjx(t)+τmaxwT(t)B1jTQjB1jw(t)E27
+τmaxxT(tτ(t))C2TFTB2jTQB2jFC2x(tτ(t))E28
+2τmaxxT(t)AjTQjB1jw(t)+2τmaxxT(t)AjTQjB2jFC2x(tτ(t))E29
+2τmaxxT(tτ(t))C2TFTB2jTQjB1jw(t)E30
+xT(t)C2TC2x(t)εjeT(t)e(t)+2ηT(t)Gj(x(t)x(ti)titx˙(θ)dθ)E12

From the definition of τmax, it is clear that τmaxttias t[ti+τi, ti+1+τi+1). As a result,

tτmaxtx˙T(θ)Qjx˙(θ)dθtitx˙T(θ)Qjx˙(θ)dθ.E13

By (12), (13), and the Jensen integral inequality [33], we can show that

H^djηT(t)[ΦjPjB2jFC2+C1jTD12jFC2G1j+G2jTG3jTPjB1j+G4jT+C1jTD11j*C2TFTD12jTD12jFC2G2jG2jTG3jTG4jT+C2TFTD12jTD11j**εjI0***D11jTD11jr2I]η(t)E13
+τmaxxT(t)AjTQjAjx(t)+τmaxwT(t)B1jTQjB1jw(t)E34
+τmaxxT(tτ(t))C2TFTB2jTQjB2jFC2x(tτ(t))E35
+2τmaxxT(t)AjTQjB1jw(t)+2τmaxxT(t)AjTQjB2jFC2x(tτ(t))E36
+2τmaxxT(tτ(t))C2TFTB2jTQjB1jw(t)+τmaxηT(t)GjQj1GjTη(t)E14

Then, by Schur complement and after some manipulations, it can be proved that if (10) holds, we have

H^dj(x(t),x(ti),e(t),w(t))<0forallη(t)0I71
.

That is, under (11),

V˙j(x(t))+zT(t)z(t)γ2wT(t)w(t)<0,η(t)0E15

This shows that the j-th closed-loop system in (7) satisfies condition (4). To prove the internal stability, by letting w(t)=0in (15) yields (note that j can be any number belonging to {1,2,…,N})

V˙j(x(t))<zT(t)z(t)0,x(t)0.E38

That is, the j-th closed-loop system is internally stable. Note that j can be any number belonging to {1,2,…,N}. The above proof shows that all the closed-loop systems are internally stable and satisfy condition (4). ■

Remark 1: Note that condition (11) is checked at the sensor node but not the controller node. In practice, the transmission event is triggered by the condition

e(t)ηminj{1,2,...,N}1εjy(t)E39

for some constant 0<η<1. In general we set ηnear to 1. ■

3.2. Synthesis of static output feedback delayed simultaneous Hcontrollers

In this subsection, we introduce how to derive a conventional delayed simultaneous static output feedback Hcontroller (8) such that all of the closed-loop systems (7) are internally stable and satisfy the condition (4). We have the following results.

Lemma 4: Consider the systems in (1). For given positive scalars λand τmax, if there exist matrices S>0, Q>0, T1j, T2j, T3j, j=1,2,…,N, and matrices Mand Lof appropriate dimensions, satisfying the following LMIs and LME :

[ΛjζjB1j+T3jT+SC1jTD11jτmaxSAjTSC1jTτmaxT1j*T2jT2jTT3jT+C2TLTD12jTD11jτmaxC2TLTB2jTC2TLTD12jTτmaxT2j**D11jTD11jr2IτmaxB1jT0τmaxT3j***τmaxQ100****I0*****τmax(2λS+λ2Q1)]<0E16
MC2=C2SE17

where Λj=SAjT+AjS+T1j+T1jTand ζj=B2jLC2T1j+T2jT, then the feedback law (8) with F=LM1is a simultaneous Hcontroller for the systems in (1).

Proof: Let P=S1. Choose a candidate storage function

V(x(t))=xT(t)Px(t)+τmax0t+σtx˙T(θ)Qx˙(θ)dθdσ,E42

and define

Hdj(x(t),w(t))V˙(x(t))+(C1jx(t)+D11jw(t)+D12ju(t))T(C1jx(t)+D11jw(t)+D12ju(t))γ2wT(t)w(t),j=1,2,...,N.E43

Define

μ(t)=[x(t)x(tτ(t))w(t)],Tj=[PT1jPPT2jPT3jP]E44

Then, along the trajectories of the j-th system,

Hdj=2xT(t)Px˙(t)+zT(t)z(t)γ2wT(t)w(t)tτmaxtx˙T(θ)Qx˙(θ)dθ+τmaxx˙T(t)Qx˙(t)+2μT(t)Tj(x(t)x(tτ(t))tτ(t)tx˙(θ)dθ)E45
=2xT(t)P(Ajx(t)+B1jw(t)+B2jFC2x(tτ(t)))+xT(t)C1jTC1jx(t)E46
+2xT(t)C1jTD12jFC2x(tτ(t))+xT(tτ(t))C2TFTD12jTD12jFC2x(tτ(t))E47
+wT(t)D11jTD11jw(t)+2xT(t)C1jTD11jw(t)+2xT(tτ(t))C2jTFTD12jTD11jw(t)E48
r2wT(t)w(t)tτmaxtx˙T(θ)Qx˙(θ)dθ+τmaxxT(t)AjTQAjx(t)E49
+τmaxwT(t)B1jTQB1jw(t)+τmaxxT(tτ(t))C2TFTB2jTQB2jFC2x(tτ(t))E50
+2τmaxxT(t)AjTQB1jw(t)+2τmaxxT(t)AjTQB2jFC2x(tτ(t))E51
+2τmaxxT(tτ(t))C2TFTB2jTQB1jw(t)+2μT(t)Tj(x(t)x(tτ(t))tτ(t)tx˙(θ)dθ)E18

By Lemma 1 and the Jensen integral inequality [33], we can show that

2μT(t)Tjtτ(t)tx˙(θ)dθτmaxμT(t)TjQ1TjTμ(t)+tτ(t)tx˙T(θ)Qx˙(θ)dθE19

As a result,

Hdj2xT(t)P(Ajx(t)+B1jw(t)+B2jFC2x(tτ(t)))+xT(t)C1jTC1jx(t)E19
+2xT(t)C1jTD12jFC2x(tτ(t))+xT(tτ(t))C2TFTD12jTD12jFC2x(tτ(t))E55
+wT(t)D11jTD11jw(t)+2xT(t)C1jTD11jw(t)+2xT(tτ(t))C2TFTD12jTD11jw(t)E56
r2wT(t)w(t)tτ(t)tx˙T(θ)Qx˙(θ)dθE57
+τmaxxT(t)AjTQAjx(t)+τmaxwT(t)B1jTQB1jw(t)E58
+τmaxxT(tτ(t))C2TFTB2jTQB2jFC2x(tτ(t))+2τmaxxT(t)AjTQB1jw(t)E59
+2τmaxxT(t)AjTQB2jFC2x(tτ(t))+2τmaxxT(tτ(t))C2TFTB2jTQB1jw(t)E60
+2μT(t)Tjx(t)2μT(t)Tjx(tτ(t))+τmaxμT(t)TjQ1TjTμ(t)+tτ(t)tx˙T(θ)Qx˙(θ)dθE61
=μT(t)[ΘjPB2jFC2+C1jTD12jFC2PT1jP+PT2jTPPB1j+PT3jT+C1jTD11j*C2TFTD12jTD12jFC2PT2jPPT2jTPPT3jT+C2TFTD12jTD11j**D11jTD11jr2I]μ(t)E62
+τmaxxT(t)AjTQAjx(t)+τmaxwT(t)B1jTQB1jw(t)E63
+τmaxxT(tτ(t))C2TFTB2jTQB2jFC2x(tτ(t))+2τmaxxT(t)AjTQB1jw(t)E64
+2τmaxxT(t)AjTQB2jFC2x(tτ(t))+2τmaxxT(tτ(t))C2TFTB2jTQB1jw(t)E65
+τmaxμT(t)TjQ1TjTμ(t)E66
μT(t)Ωjμ(t)E67

where

Θj=PAj+AjTP+C1jTC1j+PT1jP+PT1jTPE68

and

Ωj=[ΘjPB2jFC2+C1jTD12jFC2PT1jP+PT2jTPPB1j+PT3jT+C1jTD11j*C2TFTD12jTD12jFC2PT2jPPT2jTPPT3jT+C2TFTD12jTD11j**D11jTD11jr2I]E69

By noting (17) and the Schur complement, we know that Ωj<0if Ω^j<0, where

+[τmaxAjTQAjτmaxAjTQB2jFC2τmaxAjTQB1j*τmaxC2TFTB2jTQB2jFC2τmaxC2TFTB2jTQB1j**τmaxB1jTQB1j]+τmaxTjQ1TjT.E70

with

Ψj=PAj+AjTP+PT1jP+PT1jTP,δj=PB2jFC2PT1jP+PT2jTPE72

Moreover, Ω^j<0if and only if Ω˜j<0, where Ω˜jis the matrix obtained by pre- and post-multiplying Ω^jby diag{SSIIIS}:

Ω^j=[ΨjδjPB1j+PT3jT+C1jTD11jτmaxAjTC1jTτmaxPT1jP*PT2jPPT2jTPPT3jT+C2TFTD12jTD11jτmaxC2TFTB2jTC2TFTD12jTτmaxPT2jP**D11jTD11jr2IτmaxB1jT0τmaxT3jP***τmaxQ100****I0*****τmaxQ],E71

By Lemma 2, it follows that Ω˜j<0(and then Ωj<0) if (16) and (17) hold. This proves that the feedback law (8) with F=LM1is a simultaneous static output feedback Hcontroller for all the systems in (1). ■

4. An illustrative example

Suppose that a control system operates at three different operating points. The dynamics at these operating points are different. Suppose that it behaves in the following three possible modes:

x˙(t)=Ajx(t)+B1jw(t)+B2ju(t)z(t)=C1jx(t)+D11jw(t)+D12ju(t)y(t)=C2x(t), j=1,2,3E20

where

A1=[0.211-1.471-0.361-0.585-1.6830.729-1.8110.64-2.287], B11=[0.6960.3850.176], B21=[-1.824-1.1822.564], C11=[0.686-0.421-2.211], D111=1.164, D121=0.665C2=[0.6570.265-1.288-0.4390.336-0.246],E20

A2=[-0.406-1.525 0.3210.625-1.1451.239-4.1851.212-1.431], B12=[-0.5590.47-0.679], B22=[-0.5911.5212.351], C12=[-0.8290.4512.395], D112=-1.523, D122=-0.414, C2=[0.6570.265-1.288-0.4390.336-0.246],E77
A3=[-0.121-1.2350.2610.625-0.7330.639-2.1060.551.147], B13=[0.5090.585-0.776], B23=[-0.8241.2023.514],C13=[0.686-0.421-2.211], D113=1.164, D123=0.569,C2=[0.6570.265-1.288-0.4390.336-0.246].E78

We want to design a static output feedback event-triggered Hcontroller that is able to L2-stabilize the system at all the three possible operating points with γ=7. Suppose that the minimal and maximal transmission delays are τdmin=0.1msand τdmax=0.45ms, respectively. We first need to derive a conventional simultaneous static output feedback Hcontroller for all the modes in (20) and then, based on the obtained controller, we can obtain an event-triggered transmission policy.

Given λ=0.6and τmax=0.1 s, by solving (16) and (17) we can get a simultaneous Hcontroller

u(t)=Fy(tτ(t))=[0.885-1.559]y(tτ(t)).E79

With this controller, by solving (10) we can get solutions:

P1=[112.141-30.286-9.24-30.286113.67514.086-9.2414.08647.207]>0,P2=[60.909-1.9578.043-1.95742.793-1.258.043-1.2571.935]>0,E80
P3=[129.678-14.921-18.771-14.92163.544-18.135-18.771-18.13540.175]>0,Q1=[ 297.0174-97.161142.8020-97.1611345.488858.558042.802058.5580134.3278]>0,E82
Q2=[157.511114.49876.604614.4987120.861411.68336.604611.6833117.3282]>0,Q3=[282.727-17.226-12.768-17.22661.7053-10.87-12.768-10.87103.349]>0,E84
G11=[-217.2794119.904311.819539.805212.3064-28.920446.2486-53.3922-72.2522],G21=[217.253-42.00410.02919.61973.2420.249-88.03131.144156.995],E86
G31=[-9.2414.08647.20760.91-1.95742.793],G41=[-40.5049.75216.069],E88
G12=[-134.79435.38665.79773.976-46.104-60.232118.174-34.092-46.002],G22=[170.595-39.806-49.551-7.74656.8317.2-21.12416.814140.097],E90
G32=[-9.240214.086447.207060.9096-1.957242.7932],G42=[-7.001-8.08921.297],E92
G13=[-150.59938.683-6.69469.354-43.986-33.385126.071-47.435-130.298],G23=[258.559-32.327-49.367-40.33258.95115.119-142.39935.247141.214],E94
G33=[-9.2414.08647.20760.91-1.95742.793],G43=[-78.3961.26945.971],E96
ε1=38.2561,ε2=72.7127,andε3=72.8613E96

According to Theorem 1 and Remark 1, the event-triggered policy is (let η=0.99):

e(t)ηminj{1,2,3}1εjy(t)=0.1116y(t)E21

With the triggering condition (21), the sensor node can determine whether the currently measured data must be transmitted. If the currently measured data is such that condition (21) is violated, it will be discarded for reducing network usage. If the measured data is such that condition (21) holds, it will be sent to the controller node for updating the control signal.

By simulation, for guaranteeing the simultaneous L2–gain stability, the number of transmission events at the sensor node of the first system is 64 in the first 10 s. The average inter-transmitting time is 0.1563 s. The number of transmission events at the sensor node of the second system is 585. The average inter-transmitting time is 0.0171 s. The number of transmission events at the sensor node of the third system is 595. The average inter-transmitting time is 0.0168 s. Figures 13 are the responses of the event-triggered and non-event-triggered closed-loop systems under the same initial condition x(0)=[1-11]Tand the same exogenous disturbance w(t)=(3sin(8t)+2cos(5t))×e0.5t(shown in Figure 4). It is clear that the proposed event-triggered policy guarantees simultaneous L2–gain stability under low network usages. Moreover, it can be seen that the responses of closed-loop systems controlled by the event-triggered controller and the non-event-triggered controller are almost the same. That is, the obtained event-triggered controller, in a very low network usage rate, can perform almost the same control performance as the conventional non-event-triggered controller. A low network usage rate will in general lead to a good quality of network service.

Figure 1.

Responses of the first closed-loop NCS.

Figure 2.

Responses of the second closed-loop NCS.

Figure 3.

Responses of the third closed-loop NCS.

Figure 4.

Disturbance input.

5. Conclusions

In this chapter, we develop an event-triggered static output feedback simultaneous Htransmission policy for NCSs under time-varying transmission delay. With the proposed method, we do not need to switch controllers or event-triggered policies for an NCS with several different operating points. Moreover, the reliability of NCSs can be improved as possible element failures can be accommodated. The implementation of the obtained event-triggered simultaneous Hcontroller is easy as it is in the static output feedback framework. One weakness of our result is that the conditions for the existence of static output feedback simultaneous Hcontrollers are represented in terms of LMIs with a LME constraint. Standard LMI tools cannot be directly applied to find the solutions. Possible issues for further study include finding less conservative event-triggered transmission policies, considering the possibility of packet dropouts, and relaxing the continuous monitoring requirement at the sensor node by replacing the event-triggered scheme with a self-triggered one.

Acknowledgments

This work was supported by the National Science Council of the Republic of China under Grant NSC 101-2221-E-019-037.

Nomenclatures

Rnreal vector of dimension n.

Rn×mreal n×mmatrix.

∥⋅∥the Euclidean vector norm.

MT(resp., M1) the transpose (resp., inverse) of matrix M.

M>0(resp., M0) the matrix M is positive definite (resp., positive semidefinite).

M=[AB*C]the symbol * denotes the symmetric terms in a symmetric matrix

Ithe identity matrix of appropriate dimension.

diag{}the block diagonal matrix.

min z()the minimum value of z().

© 2016 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Sheng-Hsiung Yang and Jenq-Lang Wu (July 6th 2016). Event-Triggered Static Output Feedback Simultaneous H∞ Control for a Collection of Networked Control Systems, Robust Control - Theoretical Models and Case Studies, Moises Rivas López and Wendy Flores-Fuentes, IntechOpen, DOI: 10.5772/63020. Available from:

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