**2. Problem formulation and preliminaries**

Recently, some sporadic task models have been presented in NCSs without degrading system performance. An important approach is the event-triggered scheme [7–26]. 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 *H∞* control of NCSs for achieving the disturbance attenuation performance [15–21]. In [15] and [18], with considering transmission delays, event-triggered *H∞* state 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 *H<sup>∞</sup>* controller 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 *H∞* controller was derived for NCSs under time-varying delay and quantization.

54 Robust Control - Theoretical Models and Case Studies

All the results in [7–20] 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 eventtriggered NCSs [22–26]. 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 satisfac‐ tion of *L*2-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 *H∞* transmission policies were proposed of NCSs under time-varying transmission delays. Furthermore, the design of static output feedback *H<sup>∞</sup>* controllers for NCSs was discussed. Conditions for the existence of *H<sup>∞</sup>* controllers were presented in terms of bilinear matrix inequalities. A non-convex minimization problem must

be solved to get a static output feedback *H∞* controller.

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:

$$\begin{aligned} \dot{\mathbf{x}}(t) &= A\_j \mathbf{x}(t) + B\_{1j} \mathbf{w}(t) + B\_{2j} \boldsymbol{\mu}(t), \quad j = 1, 2, \dots, N \\ \mathbf{z}(t) &= C\_{1j} \mathbf{x}(t) + D\_{11j} \mathbf{w}(t) + D\_{12j} \boldsymbol{\mu}(t) \\ \mathbf{y}(t) &= C\_{2j} \mathbf{x}(t) \end{aligned} \tag{1}$$

where *x*(*t*)∈*R <sup>n</sup>* is the system state, *u*(*t*)∈*R <sup>m</sup>* is the control input, *z*(*t*)∈*R <sup>s</sup>* is the controlled output, *y*(*t*)∈*R* <sup>l</sup> is the measured output, *w*(*t*)∈*R <sup>r</sup>* is the exogenous input, and *Aj* , *B*<sup>1</sup> *<sup>j</sup>* , *B*<sup>2</sup> *<sup>j</sup>* , *C*<sup>1</sup> *<sup>j</sup>* , *D*<sup>11</sup> *<sup>j</sup>* , *D*<sup>12</sup> *<sup>j</sup>* , and *C*<sup>2</sup> *<sup>j</sup>* are constant matrices with appropriate dimensions. Here, for convenience, we assume *C*<sup>2</sup> *<sup>j</sup>* =*C*2, *j* = 1,2,…,*N*. Suppose that (*Aj* ,*B*<sup>2</sup> *<sup>j</sup>* ) are stabilizable and (*C*2,*Aj* ) are detectable for each *j* ∈{1, 2..., *N* }.Furthermore, assume that *γ* <sup>2</sup> *I* −*D*<sup>11</sup> *<sup>j</sup> <sup>T</sup> <sup>D</sup>*<sup>11</sup> *<sup>j</sup>* >0 for 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* + *τ<sup>i</sup>* , *i* = 1,2, …, where *τ<sup>i</sup>* ∈ *τd*min, *τd*max is the transmission delay. That is,

$$u(t) = Fy(t\_i), \\ t\_i + \tau\_i \le t < t\_{i+1} + \tau\_{i+1}, \\ i = 1, 2, \dots \tag{2}$$

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

$$\begin{aligned} \dot{\mathbf{x}}(t) &= A\_j \mathbf{x}(t) + B\_{1j} \mathbf{w}(t) + B\_{2j} FC\_2 \mathbf{x}(t\_i), \quad t\_i + \tau\_i \le t < t\_{i+1} + \tau\_{i+1}, \; j = 1, 2, \dots, N\\ \mathbf{z}(t) &= C\_{1j} \mathbf{x}(t) + D\_{11j} \mathbf{w}(t) + D\_{12} FC\_2 \mathbf{x}(t\_i) \end{aligned} \tag{3}$$

If the measured data is not critical for *L*2-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 transmis‐ sion delay, all possible closed-loop systems in (3) are internally stable and satisfy, for a given constant *γ* >0 and for any *T* >0 and *w* ∈ *L* <sup>2</sup> 0, *T* ,

$$\int\_{0}^{T} \boldsymbol{z}^{\top}(t)\boldsymbol{z}(t)\mathbf{d}t \le \boldsymbol{\gamma}\_{0}^{2} \Big|\_{0}^{T} \boldsymbol{w}^{\top}(t)\boldsymbol{w}(t)\mathbf{d}t,\text{for some } \boldsymbol{\gamma}\_{0} < \boldsymbol{\gamma} \tag{4}$$

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**

(1)

is the exogenous input, and

,*B*<sup>2</sup> *<sup>j</sup>*

*I* −*D*<sup>11</sup> *<sup>j</sup>*

) are stabilizable

*<sup>T</sup> <sup>D</sup>*<sup>11</sup> *<sup>j</sup>* >0 for

(*i* = 1,2,…) be

, *i* = 1,2,

(3)

are constant matrices with appropriate dimensions.

) and updates the control signal at time *ti* + *τ<sup>i</sup>*

= + £< + + + K (2)

 t

**2.1. Problem formulation**

output, *y*(*t*)∈*R* <sup>l</sup>

, *B*<sup>2</sup> *<sup>j</sup>*

all *j* ∈{1, 2..., *N* }.

, *C*<sup>1</sup> *<sup>j</sup>*

, *D*<sup>11</sup> *<sup>j</sup>*

receives the networked feedback data *y*(*ti*

*Aj* , *B*<sup>1</sup> *<sup>j</sup>*

and (*C*2,*Aj*

systems are:

update the control signal.

Consider a collect of continuous-time control systems:

2

*j*

, and *C*<sup>2</sup> *<sup>j</sup>*

…, where *τ<sup>i</sup>* ∈ *τd*min, *τd*max is the transmission delay. That is,

1 11 12 2

*j j ji*

() () () ( )

*z t C x t D w t D FC x t*

=+ +

() ()

*yt C xt*

=

, *D*<sup>12</sup> *<sup>j</sup>*

&

56 Robust Control - Theoretical Models and Case Studies

1 2 1 11 12

=+ + =

() () () ()

is the measured output, *w*(*t*)∈*R <sup>r</sup>*

Here, for convenience, we assume *C*<sup>2</sup> *<sup>j</sup>* =*C*2, *j* = 1,2,…,*N*. Suppose that (*Aj*

*zt C xt D wt D ut*

=+ +

*jj j jj j*

( ) ( ) ( ) ( ), 1,2,...,

*xt Axt B wt B ut j N*

) are detectable for each *j* ∈{1, 2..., *N* }.Furthermore, assume that *γ* <sup>2</sup>

controller node. The data transmission at the sensor node is not periodic. Let *ti*

1 1 ( ) ( ), , 12 *ii i i i u t Fy t t t t i = , ,* t

where *x*(*t*)∈*R <sup>n</sup>* is the system state, *u*(*t*)∈*R <sup>m</sup>* is the control input, *z*(*t*)∈*R <sup>s</sup>* is the controlled

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

the time that the *i*-th transmission occurs at the sensor nodes. In this case, the controller node

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

If the measured data is not critical for *L*2-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

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 transmis‐

1 22 1 1

( ) ( ) ( ) ( ), , 1,2, ,

& K

*x t A x t B w t B FC x t t t t j N*

*j j j i ii i i*

=+ + + £< + = + +

 t

t

The following Lemmas will be used later.

*Lemma 1* [30]: For any vectors *X* ,*Y* ∈*R <sup>n</sup>* and any positive definite matrix *G* ∈*R <sup>n</sup>*×*<sup>n</sup>*, the following inequality holds:

$$2X^{\top}Y \le X^{\top}GX + Y^{\top}G^{-1}Y\_{\perp \blacksquare}$$

*Lemma 2* [31]: For any given matrices *Π* <0 and *Φ* =*Φ <sup>T</sup>* , and any scalar *λ*, the following inequality holds:

$$
\Phi \Pi \Phi \leq -2\mathcal{A}\Phi - \mathcal{A}^2 \Pi^{-1} \begin{array}{c} \blacksquare \end{array}
$$

For convenience, define *xt*(*s*)= *x*(*t* + *s*), ∀*s* ∈ −*τ*max, 0 .

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

$$\dot{\mathbf{x}}(t) = A\mathbf{x}(t) + A\_d \mathbf{x}(t - \tau(t)), \ \forall t \ge 0 \tag{5}$$

with *τ*(*t*)∈ 0, *τ*max , ∀*t* ≥0. Suppose that *x*(*t*)=*ψ*(*t*), ∀*t* ∈ −*τ*max, 0 . If there exists a function

$$V: C([-\tau\_{\text{max}}, 0], R'') \to R$$

and a scalar ¿>0, such that, for all φ ∈ C([-Tmax 0], R"), V (0)≥ = | φ(0)| 2, and, along the solutions of (5),

$$\left. \frac{\mathrm{d}V(\mathbf{x}\_{\mathrm{r}})}{\mathrm{d}t} \right|\_{\mathbf{x}\_{\mathrm{r}}=\boldsymbol{\wp}} \leq -\varepsilon \left\| \boldsymbol{\wp}(\mathbf{0}) \right\|^{2},$$

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 H ್ವ controller, and then show how to derive simultaneous H ~ controller under transmission delay.

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

Define the equivalent time-varying delay

$$\pi(t) = t - t\_{\vee\prime}, \ t\_{\vee} + \pi\_{\vee} \le t < t\_{\vee\ast 1} + \pi\_{\vee \ast 1}, \ i = 1, 2, \dots, 2$$

It is clear that

$$\forall t(t) \in \left[ \begin{array}{c} \tau\_{\text{min}}, \ \tau\_{\text{max}} \end{array} \right] \forall t \ge 0, \text{and } t = 1 \text{ almost everywhere} \tag{6}$$

where Tmin =min(7;)="amin and Tmax(f;i+1=t; + T;i+)=max(f;i+1=t;} + Tamax Then, the systems in (3) can be equivalently described as

$$\begin{aligned} \dot{\mathbf{x}}(t) &= A\_j \mathbf{x}(t) + B\_{1j} \mathbf{w}(t) + B\_{2j} FC\_2 \mathbf{x}(t - \tau(t)), \quad j = 1, 2, \dots, N \\ \mathbf{z}(t) &= C\_{1j} \mathbf{x}(t) + D\_{11j} \mathbf{w}(t) + D\_{12} FC\_2 \mathbf{x}(t - \tau(t)) \end{aligned} \tag{7}$$

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 simultane ous H\_controller:

$$
\mu(t) = F\circ(t - \tau(t))\tag{8}$$

which is such that all of the possible closed-loop systems in (7) are internally stable and satisfy the condition (4) for 7(t) =[ min T max]: How to get such a delayed static output feedback simultaneous H .. controller will be discussed later.

Define the error signal:

$$e(t) = \mathbf{y}(t) - \mathbf{y}(t\_l), \mathbf{t}\_l \le t < t\_{i+1} \tag{9}$$

We have the following results.

Theorem 1: Consider the systems in (1). Suppose that the controller (8) is such that all the closedloop systems in (7) are internally stable and satisfy the condition (4). If there exist matrices P;>0, Q;>0, G;; G;; G;; G;; G;; and G4;; j=1,2,...,N, of appropriate dimensions, and scalars ; >0, j=1,2, ... ,N, satisfying the following LMIs:

$$
\begin{bmatrix}
\boldsymbol{\Phi}\_{j} & \boldsymbol{\Sigma}\_{j} & \boldsymbol{G}\_{j}^{T} & \boldsymbol{P}\_{j}\boldsymbol{D}\_{1j} + \boldsymbol{G}\_{4j}^{T} + \boldsymbol{C}\_{1j}^{T}\boldsymbol{D}\_{11j} & \boldsymbol{\tau}\_{\max}\boldsymbol{A}\_{j}^{T}\boldsymbol{Q}\_{j} & \boldsymbol{\tau}\_{\max}\boldsymbol{G}\_{1j} \\
\ast & \boldsymbol{\Sigma}\_{j} & -\boldsymbol{G}\_{3j}^{T} & -\boldsymbol{G}\_{4j}^{T} + \boldsymbol{C}\_{2}^{T}\boldsymbol{F}^{T}\boldsymbol{D}\_{12j}^{T}\boldsymbol{D}\_{11j} & \boldsymbol{\tau}\_{\max}\boldsymbol{C}\_{2}^{T}\boldsymbol{F}^{T}\boldsymbol{B}\_{2j}^{T}\boldsymbol{Q}\_{j} & \boldsymbol{\tau}\_{\max}\boldsymbol{G}\_{2j} \\
\ast & \ast & -\boldsymbol{\varepsilon}\_{j}I & 0 & 0 & \boldsymbol{\tau}\_{\max}\boldsymbol{G}\_{3j} \\
\ast & \ast & \ast & \boldsymbol{D}\_{11j}^{T}\boldsymbol{D}\_{11j} - r^{2}I & \boldsymbol{\tau}\_{\max}\boldsymbol{B}\_{1j}^{T}\boldsymbol{Q}\_{j} & \boldsymbol{\tau}\_{\max}\boldsymbol{G}\_{4j} \\
\ast & \ast & \ast & \ast & -\boldsymbol{\tau}\_{\max}\boldsymbol{Q}\_{j} & 0 \\
\ast & \ast & \ast & \ast & \ast & \tau\_{\max}\boldsymbol{Q}\_{j}
\end{bmatrix} < 0,\tag{10}
$$

where

$$\begin{aligned} \Phi\_j &= A\_j^T P\_j + P\_j A\_j + C\_{1j}^T C\_{1j} + C\_2^T C\_2 + G\_{1j} + G\_{1j}^T \\\\ \Xi\_j &= P\_j B\_{2j} F C\_2 + C\_{1j}^T D\_{12j} F C\_2 - G\_{1j} + G\_{2j}^T \\\\ \Sigma\_j &= C\_2^T F^T D\_{12}^T D\_{12} F C\_2 - G\_{2j} - G\_{2j}^T \end{aligned}$$

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

$$\left\|e(t)\right\| < \min\_{j \in \{1, 2, \dots, N\}} \frac{1}{\sqrt{\mathcal{E}\_j}} \cdot \left\|\mathcal{y}(t)\right\|, t\_i \le t < t\_{i+1} \tag{11}$$

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

$$W\_j(\mathbf{x}(t)) = \mathbf{x}^r(t)P\_j\mathbf{x}(t) + \int\_{-\tau\_{\text{max}}}^0 \int\_{t+\sigma}^t \dot{\mathbf{x}}^r(\theta)Q\_j\dot{\mathbf{x}}(\theta)d\theta d\sigma, j = 1, 2, \dots, N.$$

Define

$$\hat{\mathbf{H}}\_{\text{d}\rangle}(\mathbf{x}(\mathbf{t}), \mathbf{x}(\mathbf{t}\_{\text{l}}), \mathbf{e}(\mathbf{t}), \mathbf{w}(\mathbf{t})) \equiv \dot{\mathbf{V}}\_{\text{l}}(\mathbf{x}(\mathbf{t})) + \mathbf{z}^{\text{T}}(\mathbf{t})\mathbf{z}(\mathbf{t}) - \boldsymbol{\gamma}^{2}\mathbf{w}^{\text{T}}(\mathbf{t})$$

$$\mathbf{w}(\mathbf{t}) + \mathbf{y}^{\text{T}}(\mathbf{t})\mathbf{y}(\mathbf{t}) - \varepsilon\_{\text{e}}\mathbf{e}^{\text{T}}(\mathbf{t})\mathbf{e}(\mathbf{t}), \mathbf{j} = 1, 2, \dots, \text{N}$$

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

$$\begin{split} \hat{H}\_{\boldsymbol{\uprho}} &= 2\boldsymbol{\upchi}^{\boldsymbol{\uprho}}(t)P\_{\boldsymbol{\uprho}}\dot{\mathbf{x}}(t) - \int\_{t-\varepsilon\_{\max}}^{t} \dot{\mathbf{x}}^{\top}(\boldsymbol{\varTheta}) \underline{\mathcal{Q}}\_{\boldsymbol{\uprho}} \dot{\mathbf{x}}(\boldsymbol{\uptheta}) \mathbf{d}\boldsymbol{\uptheta} + \boldsymbol{\uptau}\_{\max} \dot{\mathbf{x}}^{\top}(t) \underline{\mathcal{Q}}\_{\boldsymbol{\uprho}} \dot{\mathbf{x}}(t) \ + \boldsymbol{z}^{\top}(t)z(t) - \boldsymbol{\uprho}^{2} \boldsymbol{w}^{\top}(t) \boldsymbol{w}(t) \\ &+ \boldsymbol{y}^{\top}(t)\boldsymbol{y}(t) - \boldsymbol{z}\_{\boldsymbol{\uprho}} \boldsymbol{e}^{\top}(t)\mathbf{e}(t) + 2\boldsymbol{\upeta}^{\top}(t)G\_{\boldsymbol{\uprho}} \left( \mathbf{x}(t) - \mathbf{x}(t\_{i}) - \int\_{t\_{i}}^{t} \dot{\mathbf{x}}(\boldsymbol{\uptheta}) \mathbf{d}\boldsymbol{\uptheta} \right) \end{split}$$

where η(t)=[x {t) x "(t) ε " (t) w "(t)]" and G =[G+] G2; G}]" . Then,

Event-Triggered Static Output Feedback Simultaneous H∞ Control for a Collection of Networked Control Systems 61 http://dx.doi.org/10.5772/63020

$$+\mathbf{x}^{\top}(t)C\_{2}^{\top}C\_{2}\mathbf{x}(t) - \varepsilon\_{j}e^{\mathsf{T}}(t)\mathbf{e}(t) + 2\eta^{\top}(t)G\_{j}\left(\mathbf{x}(t) - \mathbf{x}(t\_{i}) - \int\_{t\_{i}}^{t} \dot{\mathbf{x}}(\theta)\mathbf{d}\theta\right) \tag{12}$$

From the definition of maxy it is clear that max≥t-t; as t =[t; + T;, t;i++ + T;i]). As a result,

$$-\int\_{t-\varepsilon\_{\text{max}}}^{t} \dot{\mathbf{x}}^{\top}(\theta) \underline{Q}\_{\prime} \dot{\mathbf{x}}(\theta) \mathbf{d}\theta \le -\int\_{t\_{i}}^{t} \dot{\mathbf{x}}^{\top}(\theta) \underline{Q}\_{\prime} \dot{\mathbf{x}}(\theta) \mathbf{d}\theta. \tag{13}$$

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

$$
\hat{H}\_{\phi} \leq \eta^{\top}(t) \begin{bmatrix}
 \Phi\_{j} & P\_{j}B\_{2}F\_{i}C\_{2} + C\_{1}^{\top}D\_{12}f^{\top}\_{j}F\_{2} - G\_{ij} + G\_{j}^{\top} & G\_{j}^{\top} & P\_{j}B\_{j} + G\_{ij}^{\top} + G\_{i}^{\top}D\_{11} \\
 \Phi\_{j} & \Phi\_{j} & -G\_{j}^{\top} & -G\_{j}^{\top} & -G\_{ij}^{\top} + C\_{2}^{\top}F^{\top}D\_{12}^{\top}D\_{11} \\
 + & \* & -\tau\_{j}I & 0 \\
 + & \* & \* & D\_{11}^{\top}D\_{11} - \tau^{2}I
\end{bmatrix} \eta(t)
$$

$$
+\tau\_{\text{max}}\mathbf{x}^{\top}(t)A\_{j}^{\top}Q\_{j}A\_{j}\mathbf{x}(t) + \tau\_{\text{max}}\mathbf{w}^{\top}(t)B\_{1j}^{\top}Q\_{j}B\_{1j}\eta(t)
$$

$$
+\tau\_{\text{max}}\mathbf{x}^{\top}(t-\tau(t))C\_{2}^{\top}F^{\top}B\_{j}^{\top}Q\_{j}B\_{2j}FC\_{2}\mathbf{x}(t-\tau(t))
$$

$$
+2\tau\_{\text{max}}\mathbf{x}^{\top}(t)A\_{j}^{\top}Q\_{j}B\_{1j}\eta(t) + 2\tau\_{\text{max}}\mathbf{x}^{\top}(t)A\_{j}^{\top}Q\_{j}B\_{2j}FC\_{2}\mathbf{x}(t-\tau(t))
$$

$$
+2\tau\_{\text{max}}\mathbf{x}^{\top}(t-\tau(t))C\_{2}^{\top}F^{\top}B\_{j}^{\top}Q\_{j}B\_{1j}\eta(t) + \tau\_{\text{max}}\$$

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

$$\mathbf{H}\_{\textsf{d}}(\mathbf{x}(\mathbf{t}), \mathbf{x}(\mathbf{t}\_{\textsf{l}}), \mathbf{e}(\mathbf{t}), \mathbf{w}(\mathbf{t})) < \text{for all } \mathfrak{n}(\mathbf{t}) \neq \mathbf{0}$$

That is, under (11),

$$\dot{\mathbf{V}}\_{\cdot}(\mathbf{x}(\mathbf{t})) + \mathbf{z}^{\top}(\mathbf{t})\mathbf{z}(\mathbf{t}) - \gamma^{2}\mathbf{w}^{\top}(\mathbf{t})\mathbf{w}(\mathbf{t}) < 0,\\ \eta(\mathbf{t}) \neq 0 \tag{15}$$

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

$$\dot{V}\_{\cdot}(\mathbf{x}(t)) < -z^{\mathsf{T}}(t)z(t) \le 0, \forall \mathbf{x}(t) \ne 0.$$

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

$$\|e(t)\| \ge \eta \cdot \min\_{j \in \{1, 2, \dots, N\}} \frac{1}{\sqrt{\varepsilon\_j}} \cdot \|\nu(t)\|.$$

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

#### 3.2. Synthesis of static output feedback delayed simultaneous H \_ controllers

In this subsection, we introduce how to derive a conventional delayed simultaneous static output feedback H .. controller (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 1 and T. if there exist matrices S >0, Q>0, T1;, T2;, T3;, j=1,2,...,N, and matrices M and L of appropriate dimensions, satisfying the following LMIs and LME :

$$\begin{bmatrix} \Lambda\_{j} & \mathcal{L}\_{j} & B\_{lj} + T\_{3j}^{T} + S\mathcal{C}\_{1j}^{T}D\_{1ij} & \tau\_{\text{max}}\Delta\mathcal{A}\_{j}^{T} & S\mathcal{C}\_{1j}^{T} & \tau\_{\text{max}}T\_{1j} \\ \ast & -T\_{2j} - T\_{2j}^{T} & -T\_{3j}^{T} + C\_{2}^{T}L^{T}D\_{1ij}^{T}D\_{1ij} & \tau\_{\text{max}}C\_{2}^{T}L^{T}B\_{2j}^{T} & C\_{2}^{T}L^{T}D\_{1ij}^{T} & \tau\_{\text{max}}T\_{2j} \\ \ast & \ast & D\_{11j}^{T}D\_{11j} - r^{2}I & \tau\_{\text{max}}B\_{1j}^{T} & 0 & \tau\_{\text{max}}T\_{3j} \\ \ast & \ast & \ast & -\tau\_{\text{max}}Q^{-1} & 0 & 0 \\ \ast & \ast & \ast & \ast & -I & 0 \\ \ast & \ast & \ast & \ast & \ast & \tau\_{\text{max}}\left(-2\lambda S + \lambda^{2}Q^{-1}\right) \end{bmatrix} < 0 \tag{16}$$

$$\text{MC}\_2 = \text{C}\_2\text{S} \tag{17}$$

where 11=S A + A S + T ; + T ; and ; = B ; L C = T ; + T ; + T ; + T ; + T ; + T ; ; + T ; ; + T ; ; + T ; ; + T ; ; + T ; ; + T ; ; + T ; ; + T ; ; + T ; ; ; ; ; ; ; ; ; F = L M - is a simultaneous H .. controller for the systems in (1).

Proof: Let P = S -1. Choose a candidate storage function

$$V(\mathbf{x}(t)) = \mathbf{x}^r(t)P\mathbf{x}(t) + \int\_{-r\_{\text{max}}}^0 \int\_{t\ast\sigma}^r \dot{\mathbf{x}}^r(\theta)\underline{Q}\dot{\mathbf{x}}(\theta)d\theta d\sigma$$

and define

$$H\_{\neq}(\mathbf{x}(t),\mathbf{w}(t)) \equiv \dot{V}(\mathbf{x}(t)) + (C\_{1j}\mathbf{x}(t) + D\_{11j}\mathbf{w}(t) + D\_{12}\boldsymbol{\mu}(t))^{\uparrow}(C\_{1j}\mathbf{x}(t) + D\_{11j}\mathbf{w}(t) + D\_{12j}\boldsymbol{\mu}(t))$$

$$-\gamma^{2}\mathbf{w}^{\uparrow}(t)\mathbf{w}(t), \quad j = 1,2,...,N.$$

Define

$$\mu(t) = \begin{bmatrix} \mathbf{x}(t) \\ \mathbf{x}(t - \tau(t)) \\ \mathbf{w}(t) \end{bmatrix}, T\_j = \begin{bmatrix} PT\_{1\_j}P \\ PT\_{2\_j}P \\ T\_{3\_j}P \end{bmatrix}$$

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

t) 1

$$+2\pi\_{\text{max}}\mathbf{x}^{\top}(t-\tau(t))C\_{2}^{\top}F^{\top}B\_{2/}^{\top}QB\_{1/}\mathbf{w}(t) + 2\boldsymbol{\mu}^{\top}(t)T\_{j}\left(\mathbf{x}(t) - \mathbf{x}(t-\tau(t)) - \int\_{t-\tau(t)}^{t} \dot{\mathbf{x}}(\theta)d\theta\right) \tag{18}$$

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

$$-2\mu^{\mathbb{T}}(t)T\_j\int\_{t-\varepsilon(t)}^t \dot{\mathbf{x}}(\theta)\mathbf{d}\theta \le \mathfrak{r}\_{\text{max}}\mu^{\mathbb{T}}(t)T\_j\underline{Q}^{-1}T\_j^{\mathbb{T}}\mu(t) + \int\_{t-\varepsilon(t)}^t \dot{\mathbf{x}}^{\mathbb{T}}(\theta)\underline{Q}\dot{\mathbf{x}}(\theta)\mathbf{d}\theta\tag{19}$$

As a result,

$$\begin{split} R\_{\mathcal{A}} \le 2\dot{\mathbf{x}}^T(t)P\left(A\_j\mathbf{x}(t) + B\_{jj}\mathbf{w}(t) + B\_{2j}FC\_{\mathbf{x}}\mathbf{x}(t-\tau(t))\right) + \mathbf{x}^T(t)C\_{\mathbf{x}}^\top C\_{\mathbf{x}}\mathbf{w}(t) \\\\ + 2\mathbf{x}^T(t)C\_{\mathbf{x}}^\top D\_{\mathbf{x},j}FC\_{\mathbf{x}}\mathbf{x}(t-\tau(t)) + \mathbf{x}^T(t-\tau(t))C\_{\mathbf{x}}^\top F^T D\_{\mathbf{x},j}^\top D\_{\mathbf{x},j}FC\_{\mathbf{x}}\mathbf{x}(t-\tau(t)) \\\\ + \mathbf{w}^T(t)D\_{\mathbf{l},j}^\top D\_{\mathbf{l},j}\mathbf{w}(t) + 2\mathbf{x}^T(t)C\_{\mathbf{l},j}^\top D\_{\mathbf{l},j}\mathbf{w}(t) + 2\mathbf{x}^T(t-\tau(t))C\_{\mathbf{x}}^\top F^T D\_{\mathbf{x},j}^\top D\_{\mathbf{l},j}\mathbf{w}(t) \\\\ - \tau^2\mathbf{w}^T(t)\mathbf{n}(t) - \int\_{t-\tau(t)}^t \dot{\mathbf{x}}^T(\theta)Q\dot{\mathbf{x}}(\theta)d\theta \\\\ + \tau\_{\text{max}}\mathbf{x}^T(t)A\_j^\top Q\_xA\_\mathbf{x}(t) + \tau\_{\text{max}}\mathbf{x}^T(t)B\_{j,j}^\top QB\_{\mathbf{x},j}\mathbf{w}(t) \\\\ + \tau\_{\text{max}}\mathbf{x}^T(t-\tau(t))C\_j^\top F^T B\_{\mathbf{x},j}^\top QB\_{\mathbf{x},j}FC\_{\mathbf{x}}\mathbf{x}(t-\tau(t)) + 2\tau\_{\text{max}}\mathbf{x}^T(t)A\_j^\top QB\_{\mathbf{x},j}\mathbf{w}$$

$$+\boldsymbol{\tau}\_{\max} \mathbf{x}^{\sf T}(t) A\_{\sf f}^{\sf T} Q A\_{\sf f} \mathbf{x}(t) + \boldsymbol{\tau}\_{\max} \mathbf{w}^{\sf T}(t) B\_{\sf f}^{\sf T} Q B\_{\sf f} \mathbf{w}(t)$$

Event-Triggered Static Output Feedback Simultaneous H∞ Control for a Collection of Networked Control Systems http://dx.doi.org/10.5772/63020 65

$$\begin{aligned} \left. + \tau\_{\text{max}} \mathbf{x}^{\top}(t - \tau(t)) C\_{2}^{T} F^{T} B\_{z\_{j}}^{T} Q B\_{z\_{j}} F C\_{2} \mathbf{x}(t - \tau(t)) + 2 \tau\_{\text{max}} \mathbf{x}^{\top}(t) A\_{j}^{T} Q B\_{1\_{j}} \mathbf{w}(t) \right| \\\\ \left. + 2 \tau\_{\text{max}} \mathbf{x}^{\top}(t) A\_{j}^{T} Q B\_{z\_{j}} F C\_{2} \mathbf{x}(t - \tau(t)) + 2 \tau\_{\text{max}} \mathbf{x}^{\top}(t - \tau(t)) C\_{2}^{T} F^{T} B\_{2\_{j}}^{T} Q B\_{1\_{j}} \mathbf{w}(t) \right| \\\\ \left. + \tau\_{\text{max}} \boldsymbol{\mu}^{\top}(t) T\_{j} \underline{Q}^{-1} T\_{j}^{\top} \boldsymbol{\mu}(t) \\\\ \equiv \boldsymbol{\mu}^{\top}(t) \boldsymbol{\Omega}\_{\boldsymbol{\mu}} \boldsymbol{\mu}(t) \end{aligned}$$

where

max 2 21 ( ( ) ) 2 ( ( )) ( ) 2 ( ) ( ) ( ( )) ( ) *<sup>t</sup> <sup>T</sup> T TT <sup>T</sup> j j <sup>j</sup> t t x t t C F B QB w t t T x t x t t x d*


1

( 1 22 ) 1 1 2 ( ) ( ) ( ) ( ( )) ( ) ( ) *<sup>T</sup> T T H x t P A x t B w t B FC x t t x t C C x t dj <sup>j</sup> <sup>j</sup> <sup>j</sup> j j* £ + + -+

1 12 2 2 12 12 2 2 () ( ( )) ( ( )) ( ( )) *T T <sup>T</sup> TTT j j j j* + *x t C D FC x t t x t t C F D D FC x t t* - +-

11 11 1 11 2 12 11 ( ) ( ) 2 ( ) ( ) 2 ( ( )) ( ) *T T T T <sup>T</sup> TTT w t D D wt x tC D wt x t t C F D D wt j j j j j j* + + +-

( ) ( ) ( ) ( ) ( )d *<sup>t</sup> T T t t r w t w t x Qx* t

max max 1 1 () () () () *T T T T j j j j* + +

max 2 222 max <sup>1</sup> ( ( )) ( ( )) 2 ( ) ( ) *<sup>T</sup> T TT T T j j j j* + -

max 2 2 max 2 21 2 () ( ( )) 2 ( ( )) ( ) *T T <sup>T</sup> T TT j j j j* +

max ( ) 2 ( ) ( ) 2 ( ) ( ( )) ( ) ( ) ( ) ( )d *<sup>t</sup> T T TT T <sup>j</sup> <sup>j</sup> j j t t t T x t t T x t t t T Q T t x Qx*

> 2 2 1 12 2 1 2 1 3 1 11 2 12 12 2 2 2 3 2 12 11

*T T T T*

*jj j j j j j*

() \* ( )

*j j jj j j j j jj*

é ù Q + - + ++ ê ú <sup>=</sup> - - -+


max max 1 1 () () () () *T T T T j j j j* + +

 t*x t A QA x t w t B QB w t*

*PB FC C D FC PT P PT P PB PT C D t C F D D FC PT P PT P PT C F D D t*

 t m

*T TTT T T TTT*


 t

 *x t A QB FC x t t x t t C F B QB w t* -+ tt

 *x t t C F B QB FC x t t x t A QB w t* - + t

 t*x t A QA x t w t B QB w t*


q

 t  m


m

max ( ) ( ) 2 ( ) ( )d ( ) ( ) ( ) ( )d *t t <sup>T</sup> TT T <sup>j</sup> j j t t t t tT x t T Q T t x Qx*

 m

t

2

t

 t

 m

\* \*

t

t

t

m

m

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

t

As a result,

 t

64 Robust Control - Theoretical Models and Case Studies

m

t

 qqt t  qq

t

t

 t

t


t

 qq

> t

 t

1

 m

> t

11 11

*D D rI*

*T j j*


 q

2

 qq

> m


 q qq

$$\Theta\_{\slash} = P A\_{\slash} + A\_{\slash}^{\overline{r}} P + C\_{1\slash}^{\overline{r}} C\_{1\slash} + P T\_{1\slash} P + P T\_{1\slash}^{\overline{r}} P$$

and

T T T T j 2 j 2 1j 12 j 2 1j 2 j 1j 3j 1j 11j TT T T T TT T j 2 12 j 12 j 2 2 j 2 j 3j 2 12 j 11j T 2 11j 11j TT T max j j max j 2 j 2 max j 1j TT T T max 2 2 j 2 j 2 max 2 PB FC C D FC PT P PT P PB PT C D \* C F D D FC PT P PT P PT C F D D \* \* D D rI A QA A QB FC A QB \* C F B QB FC C F é ù Q + - + ++ ê ú W = - - -+ - ë û tt t +t t T T 1 T 2 j 1j max j j T max 1j 1j B QB TQ T \* \* B QB é ù ê ú + t <sup>t</sup> ë û

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

$$
\hat{\Omega}\_{j} = \begin{bmatrix}
\Psi\_{j} & \delta\_{j} & \boldsymbol{P}\boldsymbol{B}\_{1j} + \boldsymbol{C}\_{1j}^{T}\boldsymbol{D}\_{11j} & \boldsymbol{\tau}\_{\max}\boldsymbol{A}\_{j}^{T} & \boldsymbol{C}\_{1j}^{T} & \boldsymbol{\tau}\_{\max}\boldsymbol{P}\boldsymbol{T}\_{1j}\boldsymbol{P} \\
\ast & -\boldsymbol{P}\boldsymbol{T}\_{2j}\boldsymbol{P} - \boldsymbol{P}\boldsymbol{T}\_{2j}^{T}\boldsymbol{P} & -\boldsymbol{P}\boldsymbol{T}\_{3j}^{T} + \boldsymbol{C}\_{2}^{T}\boldsymbol{F}^{T}\boldsymbol{D}\_{12}\boldsymbol{D}\_{11j} & \boldsymbol{\tau}\_{\max}\boldsymbol{C}\_{2}^{T}\boldsymbol{F}^{T}\boldsymbol{B}\_{2j} & \boldsymbol{C}\_{2}^{T}\boldsymbol{F}^{T}\boldsymbol{D}\_{12}^{T} & \boldsymbol{\tau}\_{\max}\boldsymbol{P}\boldsymbol{T}\_{2}\boldsymbol{P} \\
\ast & \boldsymbol{\ast} & \boldsymbol{D}\_{11j}^{T}\boldsymbol{D}\_{11j} - \boldsymbol{r}^{2}\boldsymbol{I} & \boldsymbol{\tau}\_{\max}\boldsymbol{B}\_{1j}^{T} & \boldsymbol{0} & \boldsymbol{\tau}\_{\max}\boldsymbol{T}\_{3}\boldsymbol{P} \\
\ast & \ast & \ast & -\boldsymbol{\tau}\_{\max}\boldsymbol{Q}^{-1} & \boldsymbol{0} & \boldsymbol{0} \\
\ast & \ast & \ast & \ast & -\boldsymbol{I} & \boldsymbol{0} \\
\ast & \ast & \ast & \ast & -\boldsymbol{\tau}\_{\max}\boldsymbol{Q} \\
\ast & \ast & \ast & \ast & \ast & -\boldsymbol{\tau}\_{\max}\boldsymbol{Q}
\end{bmatrix}
$$

with

$$\begin{aligned} \boldsymbol{\Psi}\_{\boldsymbol{\restriction}} &= \mathbf{P} \mathbf{A}\_{\boldsymbol{\restriction}} + \mathbf{A}\_{\boldsymbol{\restriction}}^{\top} \mathbf{P} + \mathbf{P} \mathbf{T}\_{\boldsymbol{\mathbbm{1}}\boldsymbol{\rrbracket}} \mathbf{P} + \mathbf{P} \mathbf{T}\_{\boldsymbol{\mathbbm{1}}\boldsymbol{\rrbracket}}^{\top} \mathbf{P} \\ \boldsymbol{\delta}\_{\boldsymbol{\restriction}} &= \mathbf{P} \mathbf{B}\_{\boldsymbol{\mathbbm{1}}\boldsymbol{\rrbracket}} \mathbf{F} \mathbf{C}\_{\boldsymbol{\mathbbm{2}}} - \mathbf{P} \mathbf{T}\_{\boldsymbol{\mathbbm{1}}\boldsymbol{\rrbracket}} \mathbf{P} + \mathbf{P} \mathbf{T}\_{\boldsymbol{\mathbbm{2}}\boldsymbol{\rrbracket}}^{\top} \mathbf{P} \end{aligned}$$

Moreover, *Ω* ^ *<sup>j</sup>* <0 if and only if *Ω*˜ *<sup>j</sup>* <0, where *Ω*˜ *j* is the matrix obtained by pre- and postmultiplying *Ω* ^ *j* by *diag*{*SSIIIS*}:


By Lemma 2, it follows that *Ω*˜ *<sup>j</sup>* <0 (and then *Ω<sup>j</sup>* <0) if (16) and (17) hold. This proves that the feedback law (8) with *F* = *L M* <sup>−</sup><sup>1</sup> is a simultaneous static output feedback *H∞* controller 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:

$$\begin{aligned} \dot{\mathbf{x}}(t) &= A\_{\,\circ} \mathbf{x}(t) + B\_{\,\circ} \mathbf{w}(t) + B\_{\,\circ} \boldsymbol{\mu}(t) \\ \mathbf{z}(t) &= C\_{\,\circ} \mathbf{x}(t) + D\_{\,1\,\circ} \mathbf{w}(t) + D\_{\,1\,\circ} \boldsymbol{\mu}(t), \, j = 1, 2, 3 \\ \mathbf{y}(t) &= C\_{\,\circ} \mathbf{x}(t) \end{aligned} \tag{20}$$

where

$$\begin{aligned} A\_{1} &= \begin{bmatrix} 0.211 & -1.471 & -0.361 \\ -0.585 & -1.683 & 0.729 \\ -1.811 & 0.64 & -2.287 \end{bmatrix}, B\_{11} = \begin{bmatrix} 0.696 \\ 0.385 \\ 0.176 \end{bmatrix}, B\_{21} = \begin{bmatrix} -1.824 \\ -1.182 \\ 2.564 \end{bmatrix}, \\ C\_{11} &= \begin{bmatrix} 0.686 & -0.421 & -2.211 \end{bmatrix}, D\_{11} = 1.164, D\_{121} = 0.665 \\ C\_{2} &= \begin{bmatrix} 0.657 & 0.265 & -1.288 \\ -0.439 & 0.336 & -0.246 \end{bmatrix}, \end{aligned}$$

Event-Triggered Static Output Feedback Simultaneous H∞ Control for a Collection of Networked Control Systems http://dx.doi.org/10.5772/63020 67

$$\begin{aligned} A\_{2} &= \begin{bmatrix} -0.406 & -1.525 & 0.321 \\ 0.625 & -1.145 & 1.239 \\ -4.185 & 1.212 & -1.431 \end{bmatrix}, B\_{12} = \begin{bmatrix} -0.559 \\ 0.47 \\ -0.679 \end{bmatrix}, B\_{22} = \begin{bmatrix} -0.591 \\ 1.521 \\ 2.351 \end{bmatrix}, \\\ C\_{12} &= \begin{bmatrix} -0.829 & 0.451 & 2.395 \end{bmatrix}, D\_{112} = -1.523, D\_{122} = -0.414, \\\ C\_{2} &= \begin{bmatrix} 0.657 & 0.265 & -1.288 \\ -0.439 & 0.336 & -0.246 \end{bmatrix}, \end{aligned}$$

T T

T

1 3 1 11 max 1 max 1

11 11 max 1 max 3 1 max

*D D rI B T Q*

*j j j j*

*<sup>j</sup>* <0 (and then *Ω<sup>j</sup>* <0) if (16) and (17) hold. This proves that the

is a simultaneous static output feedback *H∞* controller for all


is the matrix obtained by pre- and post-

tt

*I*

max

t

 t

 t

*SQS*

(20)

j j j 1j 1j

PA A P PT P PT P PB FC PT P PT P

*j*

*j j j j jj j j j T T TTT T TT T T T jj j j j j j j*

<sup>é</sup> <sup>Y</sup> + + <sup>ù</sup> <sup>ê</sup> <sup>ú</sup> -- -+ <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> - <sup>ú</sup>

2 2 3 2 12 11 max 2 2 2 12 max 2

*T T T SC F D D SC F B SC F D T*

t

*T T T T*

t

t

j 2j 2 1j 2j

Y= + + + d= - +

*<sup>j</sup>* <0, where *Ω*˜

2

\*\* \* 0 0 \*\* \* \* 0

W = <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> - <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> - <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> - <sup>ú</sup> <sup>ë</sup> <sup>û</sup>

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

> 1 2 1 11 12

=+ + =

1 11 21

*A B B*

0.211 -1.471 -0.361 0.696 -1.824 -0.585 -1.683 0.729 , 0.385 , -1.182 , -1.811 0.64 -2.287 0.176 2.564 0.686 -0.421 -2.211 , 1.164, 0.665

ë û ëû ë û

<sup>é</sup> ù éù é ù <sup>ê</sup> ú êú ê ú <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>ê</sup>

11 111 121

= = =

*C D D*

( ) ( ) ( ) ( ), 1,2,3

() () () ()

*xt Axt B wt B ut zt C xt D wt D ut j*

=+ +

*jj j jj j*

2

[ ]

0.657 0.265 -1.288

<sup>=</sup> , .336 -0.246 é ù ê ú ë û

() ()

*yt Cxt*

=


&

2

ê

*C*

*T T*

*S S S S B T SC D SA SC T*

\* \* 0

\*\* \* \* \*

Moreover, *Ω*

multiplying *Ω*

*j*

%

^

^ *j*

\*

By Lemma 2, it follows that *Ω*˜

feedback law (8) with *F* = *L M* <sup>−</sup><sup>1</sup>

**4. An illustrative example**

the systems in (1). ■

modes:

where

*<sup>j</sup>* <0 if and only if *Ω*˜

66 Robust Control - Theoretical Models and Case Studies

d

by *diag*{*SSIIIS*}:

$$\begin{aligned} A\_3 &= \begin{bmatrix} -0.121 & -1.235 & 0.261 \\ 0.625 & -0.733 & 0.639 \\ -2.106 & 0.55 & 1.147 \end{bmatrix}, B\_{13} = \begin{bmatrix} 0.509 \\ 0.5885 \\ -0.776 \end{bmatrix}, B\_{23} = \begin{bmatrix} -0.824 \\ 1.202 \\ 3.514 \end{bmatrix}, \\\ C\_{13} &= \begin{bmatrix} 0.686 & -0.421 & -2.211 \end{bmatrix}, D\_{113} = 1.164, D\_{123} = 0.569, \\\ C\_2 &= \begin{bmatrix} 0.657 & 0.265 & -1.288 \\ -0.439 & 0.336 & -0.246 \end{bmatrix}. \end{aligned}$$

We want to design a static output feedback event-triggered *H<sup>∞</sup>* controller that is able to *L*2 stabilize the system at all the three possible operating points with *γ* =7. Suppose that the minimal and maximal transmission delays are *τd*min =0.1ms and *τd*max =0.45ms, respectively. We first need to derive a conventional simultaneous static output feedback *H<sup>∞</sup>* controller for all the modes in (20) and then, based on the obtained controller, we can obtain an event-triggered transmission policy.

Given *λ* =0.6 and *τ*max =0.1 *s*, by solving (16) and (17) we can get a simultaneous *H∞* controller

$$u(t) = F\chi(t - \tau(t)) = \begin{bmatrix} 0.88 \ \mathbf{S} & -1.559 \end{bmatrix} \chi(t - \tau(t))$$

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

$$\mathbf{P}\_1 = \begin{bmatrix} 112.141 & \text{-30.286} & \text{-9.24} \\ \text{-30.286} & 113.675 & \text{14.086} \\ \text{-9.24} & \text{14.086} & \text{47.207} \end{bmatrix} > 0 \quad \mathbf{P}\_2 = \begin{bmatrix} 60.909 & \text{-1.957} & \text{8.043} \\ \text{-1.957} & \text{42.793} & \text{-1.25} \\ \text{8.043} & \text{-1.25} & \text{71.935} \end{bmatrix} > 0$$

$$\mathbf{P}\_3 = \begin{bmatrix} 129.678 & \text{-14.921} & \text{-18.771} \\ \text{-14.921} & 63.544 & \text{-18.135} \\ \text{-18.771} & \text{-18.135} & \text{40.175} \end{bmatrix} > 0 \quad \mathbf{Q}\_1 = \begin{bmatrix} 297.0174 & \text{-97.1611} & \text{42.8020} \\ \text{-97.1611} & \text{345.4888} & \text{58.5580} \\ \text{42.8020} & \text{58.5580} & \text{134.3278} \end{bmatrix} > 0$$

Q 0

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

$$\left\|\mathbf{e}(\mathbf{t})\right\| \ge \eta \cdot \min\_{\left\|\mathbf{e}(1,2,3)\right\|} \frac{1}{\sqrt{\mathbb{E}\_{\parallel}}} \cdot \left\|\mathbf{y}(\mathbf{t})\right\| = 0.1116 \left\|\mathbf{y}(\mathbf{t})\right\|\tag{21}$$

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 1-3 are the responses of the event-triggered and non-event-triggered closed-loop systems under the same initial condition x(0)=[1 -1 1]+ and the same exogenous disturbance w(t)=(3sin(8t) + 2cos(5t))×e구55(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 eventtriggered 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.

Event-Triggered Static Output Feedback Simultaneous H∞ Control for a Collection of Networked Control Systems http://dx.doi.org/10.5772/63020 71

**Figure 4.** Disturbance input.
