Robust Fuzzy Control

Abstract

This chapter provides an overview of the issues and problems pertaining to fuzzy logic control systems based on linear quadratic (LQ) techniques. Based on the derived fuzzy Takagi–Sugeno model, parallel distributed fuzzy LQR controller are designed to control the positions of the pitch and yaw angles in twin rotor system (TRS). The stability of the TRMS system with the proposed fuzzy controllers is discussed. Then we provide different stabilization schemes and examine their design implications. Next, we looked at robust fuzzy feedback design using state and output measurements.

Problems

6.1

Recall in Sect. 6.4, that the output-feedback control scheme was constructed based on the availability of the premise variables v(t). It is desired to investigate the case when the premise variables v(t) is unavailable. Derive your results in a parallel development to that of Sect. 6.4.

6.2

Consider a time-delay nonlinear system described by the following T–S fuzzy model:

Plant Rule j:

IF $\theta _1(t)$ is $N_{j1}$ and $\dots$ $\theta _p(t)$ is $N_{jp}$, THEN

$$\begin{aligned} \dot{x}(t)= & {} (A_j + \varDelta A_j)x(t) + (A_{1j} + \varDelta A_{1j})x(t - \sigma ) + (B_j + \varDelta B_j)u(t) + B_{1j}w(t) \\ z(t)= & {} D_j x(t) + D_{1j}u(t) \\ x(t)= & {} \varphi (t), ~~t \in [-\sigma , 0], ~i = 1, 2,\dots , k \end{aligned}$$

where $N_{jm}$ is the fuzzy set, $x(t) \in {\mathfrak {R}}^n$ is the state vector, $u(t) \in {\mathfrak {R}}^m$ is the control input. $A_j, A_{1j} \in {\mathfrak {R}}^{n \times n}, B_j \in {\mathfrak {R}}^{n \times m}, B_{1j} \in {\mathfrak {R}}^{n \times r}, D_i {\mathfrak {R}}^{q \times s}$, and $D_{1j} \in {\mathfrak {R}}^{q \times m}$. Scalar k is the number of IF-THEN rules. $\theta _1(t),\theta _2(t), \dots , \theta _p(t)$ are the premise variables independent of the input u(t). $\sigma > 0$ is a real constant representing the time delay. The matrices $\Lambda A_j$, $\Lambda A_{1j}$ and $\Lambda B_j$ denote the uncertainties in system and they are of the form

$$\begin{aligned}{}[\Lambda A_j,\Lambda A_{1j},\Lambda B_j]= & {} MF(t) [E_j, E_{1j}, E_{bj}] \end{aligned}$$

where $M, E_j, E_{1j}, E_{bj}$ are known constant matrices and F(t) is an unknown matrix function with the property $F^T(t)F(t) \le I$.

$$\begin{aligned} \dot{x}(t)= & {} \sum \limits ^k_{j=1} h_j(\theta (t)) \left[ (\bar{A}_j + \bar{A}_{1j})x(t) + \bar{B}_j u(t) - \bar{A}_{1j} \int \limits ^0_{-\sigma } \dot{x}(t+\alpha )d\alpha + B_{1j}w(t)\right] \nonumber \\ z(t)= & {} \sum \limits ^k_{j=1} h_j(\theta (t)) [D_j x(t) + D_{1j} u(t)] \end{aligned}$$

(6.110)

Given a pair of (x(t), u(t), the final output of the fuzzy system is inferred as shown in (6.110), where $h_i(\theta (t)) = \mu _i(\theta (t))/\sum _{j=1}^{k} \mu _j(\theta (t))$, $\mu _i(\theta (t)) = \prod _{m=1}^{p} N_{jm}(\theta _m(t))$ and $N_{jm}(\theta _m(t))$ is the degree of the membership of $\theta _m(t)$ in $N_{jm}$. Assume that: $\mu _j(\theta (t)) \ge 0$ for $j = 1, 2,\dots , k$ and $\sum _{j=1}^{k} \mu _j(\theta (t)) > 0$ for all t. Therefore, $h_j(\theta (t)) \ge 0$ (for $j = 1, 2,\dots k$) and $\sum _{j=1}^{k} \mu _j(\theta (t)) = 1$. Letting

$$\bar{A}_j = A_j + \varDelta A_j~~\bar{A}_{1j} = A_{1j} + \varDelta A_{1j}~~\bar{B}_j = B_j + \varDelta B_j$$

Then, the system (6.110) is expressed in the following form:

$$\begin{aligned} \dot{x}(t)= & {} \sum _{j=1}^{k} h_j(\theta (t))[\bar{A}_j x(t) + \bar{A}_{1j}x(t - \sigma ) + \bar{B}_j u(t) + B_{1j}w(t)] \nonumber \\ z(t)= & {} \sum _{j=1}^{k} h_i(\theta (t)) [D_j x(t) + D_{1j}u(t)] \end{aligned}$$

(6.111)

• 1. Use the transformation $x(t - \sigma ) = x(t) - \int _{-\sigma }^{0} \dot{x}(t - \alpha )d \alpha ,\;$ derive an alternative form of the system (6.111) can be expressed by:

  $$\begin{aligned} \dot{x}(t)= & {} \sum _{j=1}^{k} h_j(\theta (t))[(\bar{A}_j + \bar{A}_{1j})X(T) + \bar{B}_j u(t) - \bar{A}_{1j} \int _{-\sigma }^{0}\dot{x}(t + \alpha ) d \alpha + B_{1j} w(t)] \nonumber \\ z(t)= & {} \sum _{j=1}^{k} h_j(\theta (t)) [D_j x(t) + D_{1j}u(t)] \end{aligned}$$

  (6.112)

• 2. Verify that systems (6.111) and (6.112) are equivalent to each other and more specifically, they have a common solution.

• 3. Investigate the robust stabilization and robust $H_\infty$ control, which depend on the size of the time delay.

• 4. Develop appropriate methods for designing state feedback fuzzy controllers for (6.112) which ensure either robust stability and/or robust performance in an $H_\infty$ sense for any constant time delay $0 \le \sigma \le \sigma ^*$.

6.3

Consider a continuous fuzzy plant composed of the following two rules: If $x_1$ is $M_{1j}$, then $\dot{x}(t) A_j x(t) + A_{1j}x(t - \sigma )$, $j = 1, 2$.

$$\begin{aligned} A_1= & {} \left[ \begin{array}{cc} -2 &{} 0 \\ 0 &{} -0.9 \end{array}\right] , A_{11} = \left[ \begin{array}{cc} -1 &{} 0 \\ -1 &{} -1 \end{array}\right] \\ A_2= & {} \left[ \begin{array}{cc} -1.5 &{} 1 \\ 0 &{} -0.75 \end{array}\right] , A_{21} = \left[ \begin{array}{cc} -1 &{} 0 \\ 1 &{} -0.85 \end{array}\right] \end{aligned}$$

Analyze the system and examine the developed results of the foregoing problem

6.4

A model for truck trailer system with time-delay is described by

$$\begin{aligned} \dot{x}_1(t)= & {} -a\frac{v\bar{t}}{(L+\varDelta L(t))t_0}x_1(t) - (1-a)\frac{v\bar{t}}{(L+\varDelta L(t))t_0}x_1(t-\sigma ) + \frac{v\bar{t}}{(l+\varDelta l(t))t_0}u(t)+w(t)\\ \dot{x}_2(t)= & {} \frac{v\bar{t}}{(L+\varDelta L(t))t_0}x_1(t) + (1-a)\frac{v\bar{t}}{(L+\varDelta L(t))t_0}x_1(t-\sigma )\\ \dot{x}_3(t)= & {} \frac{v\bar{t}}{t_0}\sin (x_2(t))\frac{v\bar{t}}{2(L+\varDelta L(t))t_0}x_1(t) + (1-a)\frac{v\bar{t}}{2(L+\varDelta L(t))t_0}x_1(t-\sigma ) \end{aligned}$$

It is desired to design a fuzzy ${\mathscr {H}}_\infty$ controller where $a = 0.7$, $v = -1.0$, $\bar{t} = 2.0$, $t_0 = 0.5$, $L = 5.5$, $l = 2.8$, $-0.2619 \le \varDelta L \le 0.2895$, and $-0.1333 \le \varDelta l \le 0.1474$.

6.5

Consider the following switched T–S fuzzy models.

Plant Rule:

$R^i_{\sigma }$ : If $z_1(t)\mathrm{is} \varOmega ^i_{\sigma 1} , z_2(t)$ is $\varOmega ^i_{\sigma 2}, \ldots , z_n(t)$ is $\varOmega ^i_{\sigma n}$,

Then

$$\begin{aligned} \dot{x}(t)= & {} A_{\sigma i}x(t) + B_{\sigma i} u_{\sigma }(t) + C_{\sigma i}\omega (t), \nonumber \\ y(t)= & {} D_{\sigma i}x(t) + E_{\sigma i}\omega (t), \end{aligned}$$

(6.113)

where ${\mathfrak {R}}^i_{\sigma }$ denotes the ith fuzzy inference rule; $i = 1, 2, \ldots , r_{\sigma }, r_{\sigma }$ is the number of inference rules; $\sigma = \sigma (t) : [0,+\infty ) \rightarrow M = {1, 2, \ldots , N}$ is a switching signal; $\varOmega ^i_{ \sigma n}$ represents the fuzzy set; n is the number of state variables; z(t) is the premise variables; x(t) is the system state; y(t) is the system output; $u_{\sigma }(t)$ is the control input; $\omega (t)$ is the exogenous disturbance input; $A_{\sigma i}, B_{\sigma i}, C_{\sigma i}, D_{\sigma i}$ and $E_{\sigma i}$ are appropriate dimensions constant matrices.

Employ singleton fuzzification, product inference engine and center average defuzzification to derive the global model in terms of the membership function $\mu ^i_{\sigma p}(z_p)$.

6.6

Building on system (6.113) and the derived results in Problem 6.5, develop a fuzzy state observer to reconstruct the state of system (6.113). Based thereon, design a state feedback controller using the parallel-distributed compensation (PDC) technology. Derive an expression for the augmented switched fuzzy system whose state is given by

$$\begin{aligned} \tilde{x}= \left[ \begin{array}{c} \hat{x}(t)\\ x(t) - \hat{x}(t) \end{array}\right] , \end{aligned}$$