Fuzzy Networked Control Systems

Abstract

This chapter provides fuzzy feedback design methods for classes of networked control systems (NCS). The chapter is divided into three parts:

• 1. The fuzzy control designs using state feedback and observer-based feedback for linear systems connected over a common digital communication network are addressed. The network conditions including network-induced delays, data packet dropouts, and limited communication capacity due to signal quantization are taken into consideration.

• 2. The problem of robust ${\mathscr {H}}_\infty$ control for uncertain discrete-time fuzzy NCS with state quantization is studied with simultaneous consideration of network-induced delays and packet dropouts. Delay-dependent closed-loop stability condition are established. Robust ${\mathscr {H}}_\infty$ fuzzy controller is developed to guarantee the asymptotic stabilization of the NCS.

• 3. Incorporating random measurement, actuation delays, quantization and random packet dropout into the feedback fuzzy control design for NCS is treated in the third part. The main design focus is on the observer-based output feedback fuzzy controller design. Sufficient conditions for exponential stability of the resulting closed-loop fuzzy system re established.

Common to all parts is the derivation of stability analysis and control synthesis criteria on the basis of the fuzzy Lyapunov–Krasovskii functional method.

Problems

4.1

Consider a T–S fuzzy system expressed by the following If-Then rule

$$\begin{aligned} R^i: \mathrm{If} \; \theta _1(t) \; \mathrm{is} \; W^i_1 \; \mathrm{and}, \dots , \mathrm{and} \; \theta _g(t) \; \mathrm{is} \; W^i_g; \nonumber \\ \mathrm{Then} \left\{ \begin{array}{l} \dot{x} (t) = A_{i} x(t) + B_{i} u(t) + B_{\omega _{i}} \omega (t) \\ z(t) = C_{i} x(t) + D_{i} u(t) \\ \end{array} \right. \end{aligned}$$

(4.189)

where $i = 1, 2, \dots , r, r$ is the number of If-Then rules; $x(t) \in {\mathfrak {R}}^n$ and $u(t) \in R^{m}$ are the state vector and the input vector, respectively; $W^i_j (i = 1, 2 , \dots , r; j = 1, 2, \dots , g)$ are fuzzy sets; and $\theta _j(t) (j = 1, 2, \dots , g)$ represent the premise variables. Denote $\theta (t) = [\theta _1(t), \dots , \theta (t)]^T$, and assume that $\theta (t)$ is either given or a function of x(t) and does not depend on u(t). The input $\omega (t) \in L_2[0, \omega _\infty )$ denotes the exogenous disturbance signal; $z(t) \in R^p$ represents the system output; the initial condition of the system (4.146) is given by $x(t_0) = x_0 , A_{i}, B_{i}, B_{\omega _{i}}, C_{i}$ and $D_{i} (i = 1, 2, \dots , r)$ are constant matrices with compatible dimensions.

It is desired to express system (4.189) in the standard parametrized fuzzy model in terms $\mu _{i}(\theta (t)) = h_{i}(\theta (t)) \sum ^r_{i=1} h_{i}(\theta (t)) , h_{i}(\theta (t)) = \varPi ^g_{j=1} W^i_j (\theta _j(t))$, and $W^i_j (\theta _j(t))$ is the membership value of $\theta _j(t)$ in $W^i_j$.

4.2

Consider that system (4.189) is controlled over a communication network and the system state is available for feedback. The objective is to design a T–S fuzzy-model-based controller via a parallel distributed compensation (PDC) to stabilize the T–S fuzzy system (4.189). Build on the results of Problem (4.1), express the ith state feedback controller rule assuming that

• 1 The available time stamped packet to derive the premises in the system and the controller should be asynchronous.

• 2 The state error between the current sampling instant and the latest transmission instant can be calculated as $e_{k}(i_{k} h) = x(i_{k} h) - x(t_{k} h)$. Define $\eta (t) = t - i_{k} h, t \in \varOmega _{l, k}$. The transmitted state $x(t_{k} h)$ can be written as

  $$\begin{aligned} x(t_kh) = x(t - (t)) - e_{k}(i_kh), t \in \varOmega _{l, k} \end{aligned}$$

  (4.190)

Hence or otherwise, derive the closed-loop fuzzy system.

4.3

In what follows, we supplement the initial condition of the state x(t) on $[t_0 - \bar{\eta }, t_0]$ as $x(t_0 + \theta ) = \phi (\theta ), \theta \in [- \bar{\eta } , 0]$, with $\phi (0) = x_0$, where $\phi (\theta )$ is a continuous function on $[t_0 - \bar{\eta } , t_0]$ and $\bar{\eta }$ is given by $\eta (t) = t - i_{k} h, t \in \varOmega _{l, k}$ such that

$$\begin{aligned} \dot{\eta }(t) = 1 , 0 < \tau _{t_{k+l}} \le \eta (t) \le h + \bar{\tau } \triangleq \bar{\eta } , t \in \varOmega _{l, k} \end{aligned}$$

(4.191)

where h and $\bar{\tau }$ are the sampling period and the allowable upper network-induced delay bound, respectively.

In view of the above consideration and making use of Problems (4.2) and (4.3), the objective is to design a state feedback controller of the type derive in the previous problem such that

• i The closed-loop fuzzy system in the absence of disturbance $\omega (t) \equiv 0$ is asymptotically stable;

• ii Under the zero initial state condition, the ${\mathscr {H}}_\infty$ performance $\Vert z(t) \Vert < \gamma \Vert \omega (t) \Vert$ is guaranteed for any nonzero $\omega (t) \in L_2[0, \infty )$ and a prescribed $\gamma > 0$.

Use the Lyapunov–Krasovskii functional candidate as

$$\begin{aligned} V (t , x_t)= & {} V_1(t , x_t) + V_2(t , x_t) \end{aligned}$$

(4.192)

$$\begin{aligned} V_1(t, x_t)= & {} x^T (t) P_1 x(t) + \int ^t_{t- \bar{\eta }} x^T (v) P_2 x(v) dv + \int ^t_{t- \bar{\eta }} \int ^t_s \dot{x}^T (v)T \dot{x} (v) dv ds \end{aligned}$$

(4.193)

$$\begin{aligned} V_2(t, x_t)= & {} (\bar{\eta } - \eta (t)) \{[x^T (t) - x^T (sk)] R_1[x(t) - x(sk)] + \int ^t_{sk} \dot{x}^T (v) R_2 \dot{x} (v) dv\} \qquad \quad \end{aligned}$$

(4.194)

and $T> 0, P_{i} > 0$ and $R_{i} > 0 (i = 1 , 2), s_{k} = i_{k} h + \tau _{t_{k}+l}, t \in \varOmega _{l, k}$. Derive sufficient conditions for stability.

4.4

Extend the results arrived at in Problem (4.4) to design the state feedback controller that achieves the desired objectives. Comment on the results.

4.5

Consider, the following system

$$\begin{aligned} \left\{ \begin{array}{l} \dot{x}_1 = x_2 \\ \dot{x}_2 = -0.01 x_1 - 0.67 x^3_1 + \omega + u \end{array} \right. \end{aligned}$$

(4.195)

where $x_1 \in [-1, 1]$ and $\omega = 0.2 \sin (2 \pi t) \exp (-t)$ is an external disturbance. Apply the results attained in Problems (4.1), (4.2), (4.3) and (4.4) and outline some useful conclusions.

4.6

Consider the following switched nonlinear system:

$$\begin{aligned} \dot{x}(t) = \sum ^{m}_{p=1} \delta _{p}(\sigma (t))f_{p}(x(t), t), x(t_0) = x_0, t \ge t_0 \end{aligned}$$

(4.196)

where $x(t) \in {\mathfrak {R}}^n$ is the state vector, and $x_0$ and $t_0 \ge 0$ denote the initial state and initial time, respectively; $\sigma (t)$ is a switching signal which is a piecewise constant function from the right of time and takes its values in the finite set $S = \{1, \dots , m \}$, where $m > 1$ is the number of subsystems. $f_{p} : {\mathfrak {R}}^n \times R \rightarrow {\mathfrak {R}}^n$ are smooth functions for any $\sigma (t) = p \in S$. Moreover, all the subsystems in system (4.196) may be unstable. Also, for a switching sequence, $0< t_1< \dots< t_{k}< t_{k+1} < \dots , \sigma (t)$ may be either autonomous or controlled. When $t \in [t_{k}, t_{k+1})$, we say $\sigma (t_{k})$th mode is active, that is, the indication functions $\delta _{p}(\sigma (t))$ satisfy

$$\begin{aligned} \delta _{p}(\sigma (t)) = \left\{ \begin{array}{ll} 1, &{} \quad \mathrm{if}\; \sigma (t) = p \\ 0, &{} \quad \mathrm{otherwise}. \\ \end{array} \right. \end{aligned}$$

(4.197)

It is desired to cast the switched nonlinear system (4.196) into a fuzzy formalism using appropriate fuzzy rules,

4.7

Consider the fuzzy version of the switched nonlinear system (4.196), derived in Problem (4.6). Let $\lambda> 0, \eta _{p} > 1, 0< \mu < 1$ satisfying $0< \mu < \{(c_{p} \eta _{p}/\eta _{p} + 1) \}$, and $\tau ^{*} > 0$ be given constants. Recall the results presented in the Appendix of the book. Show that if there exists a set of matrices $P_{p}> 0, Q_{p} > 0,$ and $G_{p}, p \in S$ such that $\forall i \in R, p \ne q, \forall (p \times q) \in S \times S$

$$\begin{aligned}&A^T_{pi} P_{p} + P_{p} A_{pi} + G^T_{p} + G_{p} - Q_{p} - \lambda P_{p} + \frac{\varepsilon _{p} c_{p}}{\tau ^{*}} \le 0 \end{aligned}$$

(4.198)

$$\begin{aligned}&\left[ \begin{array}{cc} \frac{\varepsilon _{p}}{\tau ^{*}} P_{p} &{} G_{p} \\ G^T_{p} &{} -\bigl ( \frac{\varepsilon _{p} \mu ^{2}}{\tau ^{*} \eta _{p}} \bigr ) P_{p} + \mu Q_{p} \\ \end{array} \right] \ge 0 \end{aligned}$$

(4.199)

$$\begin{aligned}&P_{p} - \eta _{p} \mu P_q \le 0 \end{aligned}$$

(4.200)

where $\varepsilon _{p} = (1/\sqrt{(c^{2}_{p} /4) - (\mu ^{2}/\eta _{p})})$ arctan $h((\eta _{p} - 1) \sqrt{(c^{2}_{p} /4) - (\mu ^{2}/\eta _{p})})(c_{pk} /2)(\eta _{p} + 1) - 2\mu$, then, the system is GAS for any switching signal with ADT satisfying

$$\begin{aligned} \tau ^{*} \le \tau _{a} \le \frac{1\mathrm{n}\mu }{\lambda }. \end{aligned}$$

(4.201)

Comment on the result.

4.8

Consider, the switched nonlinear system composed of the following two subsystems:

$$\begin{aligned} \sum _1= & {} \left\{ \begin{array}{l} \dot{x}_1(t) = -6.48x_1(t) - 7.32 \frac{1}{1+e{-(x_1+4)}} x_1(t) \\ \quad + 4.98x_2(t) + 5.52 \frac{1}{1+e^{-(x_1+4)}} x_2(t) \\ \dot{x}_2(t) = -5.48x_1(t) - 6.12 \frac{1}{1+e{-(x_1+4)}} x_1(t) \\ \quad + 4.23x_2(t) + 4.62 \frac{1}{1+e^{-(x_1+4)}} x_2(t) \\ \dot{x}_1(t) = -5.77x_1(t) + 3.36 \frac{1}{1+e{-(x_1+4)}} x_1(t) \\ \quad - 6.82 x_2(t) + 3.96 \frac{1}{1+e^{-(x_1+4)}} x_2(t) \\ \dot{x}_1(t) = 7.52 x_1(t) + 4.36 \frac{1}{1+e{-(x_1+4)}} x_1(t) \\ \quad - 8.92x_2(t) - 5.16 \frac{1}{1+e^{-(x_1+4)}} x_2(t). \end{array} \right. \end{aligned}$$

Simple check can show that the subsystems are all unstable.

We are interested in designing a kind of switching signal $\sigma (t)$ with ADT property to asymptotically stabilize the system. Make use of the results in the previous problems.