Descriptor Form Design Methodology for Polynomial Fuzzy-Model-Based Control Systems

Abstract

This paper presents a descriptor form design methodology for polynomial fuzzy-model-based (FMB) control systems. To begin with, the closed-loop polynomial fuzzy system is cast into the descriptor form representation. Applying the commonly used Lyapunov function of the polynomial fuzzy model, the first sum-of-squares (SOS) based stabilization control design approach is proposed. Through the redundancy of descriptor form representation, polynomial fuzzy slack matrices are brought into the stabilization analysis for relaxation. Applying fuzzy slack matrices for the stabilization analysis causes the double fuzzy summation problem which can be seen as the copositivity problem. Therefore, the copositivity relaxation is applied in the stabilization control design. In addition, for the special cases that all membership functions are functions of the states being not related to the inputs, the second SOS-based stabilization control design approach is proposed by applying a novel fuzzy Lyapunov function. As the novel fuzzy Lyapunov function contains the commonly used Lyapunov function as a special case, the second design approach is guaranteed to be less conservative than the first one. Since the novel fuzzy Lyapunov function is applied, the time derivatives of membership functions are necessary to be considered in the stabilization analysis. Hence, the sector nonlinearity technique is applied to deal with the rest part of the membership function time derivative after extracting the polynomial common factors. Finally, three examples are given to shown the effectiveness and relaxation of the proposed descriptor form control design methodology.

Appendices

Appendices

1.1 The Transform Matrix

Since $\hat{\varvec{x}}(\varvec{x})$ is a vector whose entries are all monomials with the property of $\hat{\varvec{x}}(\varvec{x})=0$ iff $\varvec{x}=0$, the transformation matrix $\varvec{T}(\varvec{x})$ can always be obtained such that $\hat{{{\varvec{x}}}}({{\varvec{x}}})={{\varvec{T}}}({{\varvec{x}}}){{\varvec{x}}}$. For the case of $\hat{{{\varvec{x}}}}({{\varvec{x}}})={{\varvec{x}}}$, ${{\varvec{T}}}({{\varvec{x}}})= {{\varvec{I}}}$ is obtained. Moreover, with ${{\varvec{x}}}=[x_{1} \ x_{2}]^{T}$, the following examples can be obtained:

$$\begin{aligned}&\hat{{{\varvec{x}}}}({{\varvec{x}}})=\left[ \begin{array}{c}x^{2}_{1} \\ x^{2}_{2}\end{array}\right] \quad \text {with} \quad {{\varvec{T}}}({{\varvec{x}}}) = \left[ \begin{array}{cc}x_{1} &{} 0 \\ 0 &{} x_{2}\end{array}\right] , \end{aligned}$$

$$\begin{aligned}&\hat{{{\varvec{x}}}}({{\varvec{x}}})=\left[ \begin{array}{c}x^4_{1} \\ x^{2}_{2}\end{array}\right] \quad \text {with} \quad {{\varvec{T}}}({{\varvec{x}}}) = \left[ \begin{array}{cc}x^3_{1} &{} 0 \\ 0 &{} x_{2}\end{array}\right] , \end{aligned}$$

$$\begin{aligned}&\hat{{{\varvec{x}}}}({{\varvec{x}}})=\left[ \begin{array}{c}x^4_{1} \\ x^4_{2}\end{array}\right] \quad \text {with} \quad {{\varvec{T}}}({{\varvec{x}}}) = \left[ \begin{array}{cc}x^3_{1} &{} 0 \\ 0 &{} x^3_{2}\end{array}\right] . \end{aligned}$$

1.2 Decomposition of Membership Function Time Derivative

For the cases of all membership functions being functions of the states with no relation to the inputs (i.e. $h_{\rho }(\varvec{z})= h_{\rho }(\tilde{\varvec{x}}) \ \forall \rho$), the membership function time derivative can be represented as

$$\begin{aligned} \dot{h}_{\rho} (\tilde{\varvec{x}})&= \frac{\partial h_{\rho} (\tilde{\varvec{x}})}{\partial {{\varvec{x}}}} \dot{{{\varvec{x}}}} \\&=\frac{\partial h_{\rho} (\tilde{\varvec{x}})}{\partial {{\varvec{x}}}} \sum _{i=1}^{r}h_{i}(\tilde{\varvec{x}}) {{\varvec{A}}}_{i}({{\varvec{x}}})\hat{{{\varvec{x}}}}({{\varvec{x}}}). \end{aligned}$$

(58)

By extracting the polynomial common factors $O_{\rho} ({{\varvec{x}}})$ from $\dot{h}_{\rho} (\tilde{\varvec{x}})$, (58) can be further rewritten as:

$$\dot{h}_{\rho} (\tilde{\varvec{x}})=y_{\rho} ({{\varvec{x}}})O_{\rho} ({{\varvec{x}}}),$$

(59)

where $y_{\rho} ({{\varvec{x}}})$ is the rest part of $\dot{h}_{\rho} (\tilde{\varvec{x}})$ after extracting $O_{\rho} ({{\varvec{x}}})$. By applying the sector nonlinearity technique to $y_{\rho} ({{\varvec{x}}})$, it can be obtained that

$$y_{\rho} ({{\varvec{x}}})=\sum _{m=1}^{2} \omega _{\rho m} (\varvec{x}) C_{\rho m},$$

(60)

where

$$\begin{aligned}&C_{\rho 1}= \max \limits _{\varvec{x}\in D_{\text {op}}} \ y_{\rho} ({{\varvec{x}}}), \quad C_{\rho 2}= \min \limits _{\varvec{x}\in D_{\text {op}}} \ y_{\rho} ({{\varvec{x}}}) \\&\omega _{\rho 1} (\varvec{x}) =\frac{y_{\rho} ({{\varvec{x}}})-C_{\rho 2}}{C_{\rho 1}-C_{\rho 2}} ,\quad \omega _{\rho 2} (\varvec{x}) =\frac{C_{\rho 1}-y_{\rho} ({{\varvec{x}}})}{C_{\rho 1}-C_{\rho 2}} \end{aligned}$$

with the following properties:

$$\omega _{\rho m} (\varvec{x}) > 0, \ \sum _{m=1}^{2} \omega _{\rho m} (\varvec{x}) =1.$$

By substituting (60) into (59), the decomposition of the membership function time derivative can be implemented as

$$\dot{h}_{\rho} (\tilde{\varvec{x}})=\sum _{m=1}^{2} \omega _{\rho m} (\varvec{x}) \mu _{\rho m}({{\varvec{x}}})$$

with $\mu _{\rho m}({{\varvec{x}}})= C_{\rho m}O_{\rho} ({{\varvec{x}}})$.

For example, for $\hat{{{\varvec{x}}}}({{\varvec{x}}})=\varvec{x}=[x_{1} \ x_{2}]^{T}$, consider a polynomial fuzzy model (2) of three rules with the following system matrices and membership functions:

$$\begin{aligned}&A_{1}=\left[ \begin{array}{cc}x_{1}^{2}+x_{2}^{2} &{} x_{1}+x_{2} \\ 0.25 &{} 0.25\end{array}\right] , \ A_{2}=\left[ \begin{array}{cc}3x_{1}x_{2} &{} -x_{1}+x_{2} \\ 0.25 &{} 0.25\end{array}\right] , \\&A_3=\left[ \begin{array}{cc}-1+x_{1}+x_{1}^{2} &{} -4 \\ 0.25 &{} 0.25\end{array}\right] \\&B_{1}=\left[ \begin{array}{c}1 \\ 0\end{array}\right] , \ B_{2}=\left[ \begin{array}{c}8 \\ 0\end{array}\right] , \ B_3=\left[ \begin{array}{c}x_{2}^{2} \\ 0\end{array}\right] \\&h_{1}(x_{2})=\frac{1+\sin(x_{2})}{3}, \ h_{2}(x_{2})= h_3(x_{2})=\frac{2-\sin(x_{2})}{6}. \end{aligned}$$

The time derivative of $h_{1}(x_{2})$ can be obtained as

$$\begin{aligned} \dot{h}_{1}(x_{2})&= \frac{\partial h_{1} (x_{2})}{\partial x_{2}} \dot{x}_{2} \\&=\frac{\partial h_{1} (x_{2})}{\partial x_{2}} \sum _{i=1}^{r}h_{i}(x_{2}) {{\varvec{A}}}^{2}_{i}({{\varvec{x}}})\hat{{{\varvec{x}}}}({{\varvec{x}}}) \\&=\frac{\cos(x_{2})}{3} \times 0.25 \times (x_{1}+x_{2}). \end{aligned}$$

Rewrite $\dot{h}_{1}(x_{2})$ into the form of (59), it can be obtained that

$$O_{1}({{\varvec{x}}})=\frac{x_{1}+x_{2}}{12}, \ y_{1}({{\varvec{x}}})=\cos(x_{2}).$$

Assume that the polynomial fuzzy model are with the following operation domain:

$$D_{op}=\{{{\varvec{x}}}: -\pi \le x_{k} \le \pi , \ k=1, \ 2\}.$$

By applying the sector nonlinearity technique to $y_{1}({{\varvec{x}}})$, it can be obtained that

$$y_{1}({{\varvec{x}}})=\sum _{m=1}^{2} \omega _{1 m} (\varvec{x}) C_{1 m},$$

where

$$\begin{aligned} &C_{1 1}= \max \limits _{\varvec{x}\in D_{op}} \ y_{1}({{\varvec{x}}})=1, \quad C_{1 2}= \min \limits _{\varvec{x}\in D_{op}} \ y_{1}({{\varvec{x}}})=-1 \\&\omega _{1 1} (\varvec{x}) =\frac{y_{1}({{\varvec{x}}})-C_{1 2}}{C_{1 1}-C_{1 2}} ,\quad \omega _{1 2} (\varvec{x}) =\frac{C_{1 1}-y_{1}({{\varvec{x}}})}{C_{1 1}-C_{1 2}}. \end{aligned}$$

Finally, the decomposition of $\dot{h}_{1}(x_{2})$ can be implemented as

$$\dot{h}_{1}(\tilde{\varvec{x}})=\sum _{m=1}^{2} \omega _{1 m} (\varvec{x}) \mu _{1 m}({{\varvec{x}}})$$

with

$$\begin{aligned}&\mu _{1 1}({{\varvec{x}}})=C_{11}O_{1}({{\varvec{x}}})=\frac{x_{1}+x_{2}}{12}, \\&\mu _{1 2}({{\varvec{x}}})=C_{12}O_{1}({{\varvec{x}}})=-\frac{x_{1}+x_{2}}{12}. \end{aligned}$$