Fuzzy T–S Model-Based Design of Min–Max Control for Uncertain Nonlinear Systems

Abstract

The min–max robust control synthesis for uncertain nonlinear systems is solved using Takagi–Sugeno fuzzy model and fuzzy state observer. Existence conditions are derived for the output feedback min–max control in the sense of Lyapunov asymptotic stability and formulated in terms of linear matrix inequalities. The convex optimization algorithm is used to obtain the minimum upper bound on performance and the optimum parameters of min–max controller. The closed-loop system is asymptotically stable under the worst case disturbances and uncertainty. Benchmark of inverted pendulum plant is used to demonstrate the robust performance within a much larger equilibrium region of attraction achieved by the proposed design.

Appendices

Appendix 1

Proof of Theorem 1

Consider Lyapunov function candidate

$$V = x(t)^{T} P_{1} x(t) + e(t)^{T} P_{2} e(t) = \xi^{T} P\xi ,\,\,P = \left[ {\begin{array}{*{20}c} {P_{1} } & 0 \\ 0 & {P_{2} } \\ \end{array} } \right]$$

(31)

hence $V_{1} = x(t)^{T} P_{1} x(t)$, $V_{2} = e(t)^{T} P_{2} e(t)$. Then it follows:

$$\begin{aligned} \dot{V}_{1} (x) & = \dot{x}^{T} P_{1} x + x^{T} P_{1} \dot{x} = \sum\limits_{i = 1}^{q} {\sum\limits_{j = 1}^{q} {h_{i} h_{j} [[(A_{i} - B_{i} K_{j} )x(t) + B_{i} K_{j} e(t) + \bar{D}_{i} W]^{T} P_{1} x} } \\ & \quad + x^{T} P_{1} [(A_{i} - B_{i} K_{j} )x(t) + B_{i} K_{j} e(t) + \bar{D}_{i} W]] = \\ & = \sum\limits_{i = 1}^{q} {\sum\limits_{j = 1}^{q} {h_{i} h_{j} [x^{T} (H_{ij}^{T} P_{1} + P_{1} H_{ij} )x + e^{T} (t)(B_{i} K_{j} )^{T} P_{1} x} (t) + } \\ & \quad + x^{T} (t)P_{1} B_{i} K_{j} e(t) + 2x^{T} (t)P_{1} \bar{D}_{i} W] \\ \end{aligned}$$

(32)

and

$$\dot{V}_{2} (t) = \dot{e}^{T} P_{2} e + e^{T} P_{2} \dot{e} = \sum\limits_{i = 1}^{q} {\sum\limits_{j = 1}^{q} {h_{i} h_{j} [e^{T} (\Sigma _{ij}^{T} P_{2} + P_{2}\Sigma _{ij} )e} } + 2e^{T} (t)P_{2} \bar{D}_{i} W],$$

(33)

where $H_{ij} = A_{i} - B_{i} K_{j}$. Hence

$$\begin{aligned} \dot{V} & = \dot{V}_{1} + \dot{V}_{2} = \sum\limits_{i = 1}^{q} {} \sum\limits_{j = 1}^{q} {h_{i} h_{j} } [\xi^{T} (t)(\Psi _{ij}^{T} P + P\Psi _{ij}^{{}} )\xi (t) - \\ & \quad - 2x^{T} (t)\,K_{i}^{T} B_{j}^{T} P_{1} \,x(t) + 2e^{T} (t)K_{i}^{T} B_{j}^{T} P_{1} \,x(t) + \\ & \quad + 2x^{T} (t)P_{1} \bar{D}_{i} W + 2e^{T} (t)P_{2} \bar{D}_{i} W] = \\ & = \sum\limits_{i = 1}^{q} {\sum\limits_{j = 1}^{q} {h_{i} h_{j} [\xi^{T} (t)(\Psi _{ij}^{T} P + P\Psi _{ij} )\xi (t) + 2\xi^{T} (t)P\tilde{D}_{i} W - } } \\ & \quad - 2x^{T} (t)K_{i}^{T} B_{j}^{T} P_{1} x(t) + 2e^{T} (t)K_{i}^{T} B_{j}^{T} P_{1} x(t)], \\ \end{aligned}$$

(34)

where $\Sigma _{ij} = A_{i} - G_{i} C_{j}$,$\Psi _{ij} = \left[ {\begin{array}{*{20}c} {A_{i} } & 0 \\ 0 & {\Sigma _{ij} } \\ \end{array} } \right]$, $\tilde{D}_{i} = \left[ {\begin{array}{*{20}c} {\bar{D}_{i} } \\ {\bar{D}_{i} } \\ \end{array} } \right]$.

Next, the following local checking function:

$$\phi (t) = \dot{V} + u^{T} u - W^{T} W$$

(35)

is constructed. Substitution of (34) into the above expression yields

$$\begin{aligned} \varphi (t) & = \sum\limits_{i = 1}^{q} {} \sum\limits_{j = 1}^{q} {h_{i} h_{j} } [\xi^{T} \left[ {\begin{array}{*{20}c} {A_{i}^{T} P_{1} + P_{1} A_{i} } & 0 \\ 0 & {\Sigma _{ij}^{T} P_{2} + P_{2}\Sigma _{ij} } \\ \end{array} } \right]\xi + \\ & \quad + 2u^{T} (t)\,B_{i}^{{}} P_{1} \,x(t) + 2x^{T} (t)\,P_{1} \,\bar{D}_{i} \,W + \\ & \quad + 2e^{T} (t)P_{2} \bar{D}_{i} W + u^{T} u - W^{T} W] \\ \end{aligned}$$

(36)

Thus maximization of (36) about $W$ yields

$$W^{*} = \sum\limits_{i = 1}^{q} {h_{i} [\bar{D}_{i}^{T} P_{1}^{T} x(t) + } \bar{D}_{i}^{T} P_{2}^{T} e(t)]\,\, = \sum\limits_{i = 1}^{q} {h_{i} } \tilde{D}_{i}^{T} P_{{}}^{T} \xi (t)$$

(37)

Because of $\frac{{\partial^{2} \phi (t)}}{{\partial W^{2} }} = - 2 < 0$, (37) represents the parametric expression of the worst case disturbance [24–26] for system (16)–(17). Substitution of $W^{ * }$ into (36) gives:

$$\begin{aligned} \mathop {\hbox{max} }\limits_{W} \varphi (t) & = \sum\limits_{i = 1}^{q} {} \sum\limits_{j = 1}^{q} {h_{i} h_{j} } [\xi^{T} \left[ {\begin{array}{*{20}c} {A_{i}^{T} P_{1} + P_{1} A_{i} } & 0 \\ 0 & {\Sigma _{ij}^{T} P_{2} + P_{2}\Sigma _{ij} } \\ \end{array} } \right]\xi + \\ & \quad + 2x^{T} (t)\,P_{1} B_{i}^{{}} \,u(t) + 2x^{T} (t)\,P_{1} \,\bar{D}_{i} \,W + \\ & \quad + \xi^{T} (t)P\tilde{D}_{i} \tilde{D}_{j}^{T} P\xi (t) + u^{T} (t)u(t) \\ & \quad + 2e^{T} (t)P_{2} \bar{D}_{i} W + u^{T} u - W^{T} W] \\ & = \sum\limits_{i = 1}^{q} {\sum\limits_{j = 1}^{q} {h_{i} h_{j} } } [\xi^{T} (t)(\Psi _{ij}^{T} P + P\Psi _{ij}^{{}} )\xi (t) + \\ & \quad + 2x^{T} (t)P_{1} B_{i} u(t) + u^{T} (t)u(t) + \xi^{T} (t)P\tilde{D}_{i} \tilde{D}_{j}^{T} P\xi (t) \\ \end{aligned}$$

(38)

Minimization of the above expression about $u$ yields

$$u^{*} = - \sum\limits_{i = 1}^{q} {h_{i} } B_{i}^{T} P_{1} x(t)\,\, = - \sum\limits_{i = 1}^{q} {h_{i} } (B_{i}^{T} P_{1} \tilde{x}(t) + B_{i}^{T} P_{1} e(t)).$$

(39)

And for $\frac{{\partial^{2} (\mathop {\hbox{max} }\limits_{W} \phi (t))}}{{\partial u^{2} }} = I > 0$, the (39) is the parametric expression of min–max controller for system (16)–(17) apparently. Next, the substitution of $u^{ * }$ into (38) gives

$$\begin{aligned} \mathop {\hbox{min} }\limits_{u} \,\mathop {\hbox{max} }\limits_{W} \varphi (t) & = \sum\limits_{i = 1}^{q} {} \sum\limits_{j = 1}^{q} {h_{i} h_{j} } [\xi^{T} (t)(\Psi _{ij}^{T} P + P\Psi _{ij}^{{}} )\xi (t) - x^{T} (t)\,P_{1}^{T} B_{i} \,B_{j}^{T} P_{1} x(t) + \\ & \quad + \xi^{T} (t)P\tilde{D}_{i} \tilde{D}_{j}^{T} P\xi (t)] \cdots = \sum\limits_{i = 1}^{q} {\sum\limits_{j = 1}^{q} {h_{i} h_{j} \xi^{T} (t)[\Psi _{ij}^{T} P + P\Psi _{ij} - P^{T} \tilde{B}_{i} \tilde{B}_{j}^{T} P + P\tilde{D}_{i} \tilde{D}_{j}^{T} P]\xi (t),} } \\ \end{aligned}$$

where $\tilde{B}_{i} = \left[ {\begin{array}{*{20}c} {B_{i} } \\ 0 \\ \end{array} } \right]$. Further, let it be denoted

$$\mathop {\hbox{min} }\limits_{u} \mathop {\hbox{max} }\limits_{W} \phi (t) = - \xi^{T} Q\xi (t)$$

(40)

Notice that if the following inequality is satisfied

$$\Psi _{ij}^{T} P + P\Psi _{ij} - P^{T} \tilde{B}_{i} \tilde{B}_{j}^{T} P + P\tilde{D}_{i} \tilde{D}_{j}^{T} P < 0 ,$$

(41)

then $Q > 0$ holds true. Substitution of (37) and (39) into (34) yields

$$\dot{V} = \sum\limits_{i = 1}^{q} {\sum\limits_{j = 1}^{q} {h_{i} h_{j} \xi^{T} \left( t \right)\left[ {\Psi _{ij}^{T} P + P\Psi _{ij} - 2P^{T} \tilde{B}_{i} \tilde{B}_{j}^{T} P + 2P\tilde{D}_{i} \tilde{D}_{j}^{T} P} \right]} } \xi \left( t \right)$$

In turn, if $\tilde{D}_{i} \tilde{D}_{j}^{T} - \tilde{B}_{i} \tilde{B}_{j}^{T} < 0$ and (39) hold, then $\dot{V} < 0$ obviously. Thus the closed-loop system (16)–(17) is asymptotically stable and $\xi (\infty ) = 0$. Pre- and post- multiplication of both sides of (41) by $diag(P_{1}^{ - 1} ,P_{2}^{ - 1} )$, and then application of Schur’s complement yields

$$\left[ {\begin{array}{*{20}c} {XA_{i}^{T} + A_{i}^{{}} X +\Gamma } & 0 \\ 0 & {YA_{i}^{T} + A_{i}^{{}} Y - Z - Z^{T} + D_{i} D_{j} } \\ \end{array} } \right] < 0$$

(42)

Apparently, the above expression is equivalent to (18)–(19) in Theorem 1. Now the integral of (42) is calculated; after some appropriate transpose, to give:

$$\begin{aligned} \mathop {\hbox{min} }\limits_{u} \mathop {\hbox{max} }\limits_{v,\omega } J(u,W) & = \mathop {\hbox{min} }\limits_{u} \mathop {\hbox{max} }\limits_{v,\omega } \int_{0}^{\infty } {(\xi^{T} Q\xi + u^{T} u - W^{T} W)dt} \le - \int_{0}^{\infty } {\dot{V}dt} \\ & = V(\xi (0)) - V(\xi (\infty )) = \xi (0)^{T} P\xi (0) \\ \end{aligned}$$

(43)

Therefore, due to Assumption 2 and via considering the expected value of the performance cost, it follows

$$\bar{J} = E\left\{ J \right\} \le E\left\{ {\xi_{0}^{T} P\xi_{0} } \right\} = Trace(P) .$$

(44)

Nonetheless, notice the initial state of a plant system can hardly be accurately measured in real-world practice.

Appendix 2

Proof of Theorem 2

From the proof of Theorem 1 it is seen if the inequality

$$\Psi _{ij}^{T} P + P\Psi _{ij} - 2P^{T} \tilde{B}_{i} \tilde{B}_{j}^{T} P + 2P\tilde{D}_{i} \tilde{D}_{j}^{T} P < 0 ,$$

(45)

is satisfied then apparently $\dot{V} < 0$. That is, the closed-loop system (16)–(17) is asymptotically stable and $\xi (\infty ) = 0$. Further, if $\tilde{D}_{i} \tilde{D}_{j}^{T} -$ $\tilde{B}_{i} \tilde{B}_{j}^{T} \ge 0$, then

$$\Psi _{ij}^{T} P + P\Psi _{ij} - P^{T} \tilde{B}_{i} \tilde{B}_{j}^{T} P + P\tilde{D}_{i} \tilde{D}_{j}^{T} P < 0 ,$$

(46)

and thus $Q > 0$ is guaranteed. After pre- and post- multiplication of both sides of (23) by $diag(P_{1}^{ - 1} ,P_{2}^{ - 1} )$, the application of Schur’s complement yields

$$\left[ {\begin{array}{*{20}c} {XA_{i}^{T} + A_{i}^{{}} X + 2\Gamma } & 0 \\ 0 & {YA_{i}^{T} + A_{i}^{{}} Y - Z - Z^{T} + 2D_{i} D_{j} } \\ \end{array} } \right] < 0 .$$

(47)

As seen, the above expression is equivalent to (21). Now recall the proof of Theorem 1. After the appropriate transposing and then calculating the integral, the investigation of expected value of performance index yields

$$\bar{J} = E\left\{ J \right\} \le E\left\{ {\xi_{0}^{T} P\xi_{0} } \right\} = Trace(P) .$$

(48)