Stable Controller Design for T–S Fuzzy Control Systems with Piecewise Multi-linear Interpolations into Membership Functions

Abstract

This paper focuses on stabilization of T–S fuzzy control systems. We use the information of the premises of the T–S fuzzy control systems to reduce the conservativeness of the stabilization conditions. First, the membership functions (MFs) in the premises are approximated with their piecewise multi-linear interpolations. In this way, different types of MFs can be tackled in a unified approach. We use the errors between the T–S fuzzy systems and the interpolated systems as feedbacks to ensure that the errors tend to zero. Then, we design stable controllers for the fuzzy control systems based on the obtained systems with piecewise multi-linear interpolations and express our results as a group of linear matrix inequalities. It is proved that when the MFs are both single-variate and multi-variate, our results can stabilize the T–S fuzzy control systems. Finally, several simulation examples are utilized to illustrate the merits of the proposed method with both PDC and non-PDC in this paper.

Appendix

1.1 Proof of Theorem 1 and Theorem 2

Proof

Let $P_1$ and $P_2$ be positive definite matrices. Then, we define

$$\begin{aligned} V(t)&= \begin{pmatrix} \hat{x}^T&e^T \end{pmatrix} {\hbox{diag}}(P_1,P_2) \begin{pmatrix} \hat{x}\\ e \end{pmatrix}\\&=\hat{x}^T P_1 \hat{x}+e^TP_2e, \end{aligned}$$

which is a Lyapunov function of the augmented system with state vector$\begin{pmatrix} \hat{x}\\ e \end{pmatrix}$, where $\hat{x}$ and e are short for $\hat{x}(t)$ and e(t) as stated in Remark 1.

$\dot{V}(t)$, the derivation of the Lyapunov function V(t), is formulated in (11).

$$\begin{aligned} \begin{aligned} \dot{V}&=\frac{{\hbox{d}}V}{{\hbox{d}}t}\\ &=\dot{\hat{x}}^T P_1 \hat{x}+\hat{x}^T P_1 \dot{\hat{x}} + \dot{e}^TP_2e + e^TP_2\dot{e}\\ &=\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})\hat{x}\right. \\&\left. +\sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})e-Ce\right) ^TP_{1}\hat{x}\\&+\hat{x}^{T}P_1\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})\hat{x}\right. \\&\left. +\sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})e-Ce\right) \\&+\left( \sum _{i=1}^{p}\sum _{j=1}^{q} (g_{ij}-\bar{g}_{ij}-\varDelta g_{ij})(A_i+B_iG_j)x-Ce\right) ^T P_2 e\\&+e^T P_2 \left( \sum _{i=1}^{p}\sum _{j=1}^{q} (g_{ij}-\bar{g}_{ij}-\varDelta g_{ij})(A_i+B_iG_j)x -Ce\right) \\ &=\hat{x}^T \left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})^{T}\right) P_{1}\hat{x}\\&+e^{T}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})^{T}-C^{T}\right) P_{1}\hat{x}\\&+\hat{x}^{T}P_{1}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})\right) \hat{x}\\&+\hat{x}^{T}P_{1}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})-C\right) e\\&+x^{T}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(g_{ij}-\bar{g}_{ij}-\varDelta g_{ij})(A_{i}+B_{i}G_{j})^{T}\right) P_{2}e\\&+e^{T}P_{2}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(g_{ij}-\bar{g}_{ij}-\varDelta g_{ij})(A_{i}+B_{i}G_{j})\right) x\\&+e^{T}C^{T}P_{2}e+eP_{2}Ce^{T}\\ &=\hat{x}^{T}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})^{T}P_{1}\right. \\&\left. +P_{1}\sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})\right) \hat{x}\\&+e^{T}(C^{T}P_{2}+P_{2}C)e\\&+e^{T}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})^{T}-C^{T}\right) P_{1}\hat{x}\\ &=\hat{x}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})^{T}P_{1}\right. \\&\left. +P_{1}\sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})\right) \hat{x}\\&+e^{T}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})^{T}-C^{T}\right) P_{1}\hat{x}\\&+\hat{x}P_{1}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})-C\right) e \\&+x^{T}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(g_{ij}-\bar{g}_{ij}-\varDelta g_{ij})(A_{i}+B_{i}G_{j})^{T}\right) P_{2}e\\&+e^{T}P_{2}(\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(g_{ij}-\bar{g}_{ij}-\varDelta g_{ij})(A_{i}+B_{i}G_{j})\right) x\\&+e^{T}(C^{T}P_{2}+P_{2}C)e \end{aligned} \end{aligned}$$

(11)

As $\varDelta g_{ij}=\varDelta _{ij}sgn(e^{T}P_{2}(A_{i}+B_{i}G_{j})x)$, then the last term but one of (11),

$$\begin{aligned} e^{T}P_{2}\left( \left( \sum _{i=1}^{p}\sum _{j=1}^{q}(g_{ij}-\bar{g}_{ij}-\varDelta g_{ij})(A_{i}+B_{i}G_{j})\right) x\right. , \end{aligned}$$

can be proved to be semi-negative definite as shown in (12).

$$\begin{aligned}&e^{T}P_{2}\left( \left( \sum _{i=1}^{p}\sum _{j=1}^{q}(g_{ij}-\bar{g}_{ij}-\varDelta g_{ij})(A_{i}+B_{i}G_{j})\right) x\right. \\&\quad \le \sum _{i=1}^{p}\sum _{j=1}^{q}|g_{ij}-\bar{g}_{ij}|\cdot |e^{T}P_{2}(A_{i}+B_{i}G_{j}))x|\\&\qquad -\sum _{i=1}^{p}\sum _{j=1}^{q}e^{T}P_{2}\varDelta g_{ij}(A_{i}+B_{i}G_{j}))x\\&\quad =\sum _{i=1}^{p}\sum _{j=1}^{q}|g_{ij}-\bar{g}_{ij}|\cdot |e^{T}P_{2}(A_{i}+B_{i}G_{j}))x|\\&\qquad -\sum _{i=1}^{p}\sum _{j=1}^{q}\varDelta _{ij}sgn(e^{T}P_{2}(A_{i}+B_{i}G_{j})x)(e^{T}P_{2}(A_{i}+B_{i}G_{j})x)\\&\quad =\sum _{i=1}^{p}\sum _{j=1}^{q}(|g_{ij}-\bar{g}_{ij}|-\varDelta g_{ij})|e^{T}P_{2}(A_{i}+B_{i}G_{j}))x|\\&\quad \le 0. \end{aligned}$$

(12)

Let $z=P_{1}e$, $\hat{z}=P_{1} \hat{x}$, $C=-P_{1}$, $P_{2}=\gamma P_{1}$ and $X=P_{1}^{-1}$. Utilizing (11) and (12), we can obtain an estimation of the upper bound of $\dot{V}$, which is deduced in (13) in detail.

$$\begin{aligned} \dot{V} \le&\hat{x}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})^{T}P_{1}\right. \\&\left. +\,P_{1}\sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})\right) \hat{x}\\&+\,e^{T}(-2\gamma P_{1})e+e^{T}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})^{T}+P_{1}^{T}\right) P_{1}\hat{x}\\&+\hat{x}P_{1}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})+P_{1}\right) e\\ &=\hat{z}^{T}X\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})^{T}X^{-1}\right. \\&\left. +X^{-1}\sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})\right) X\hat{z}\\&+z^{T}X(-2\gamma X^{-2})Xz\\&+\,z^{T}X\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})^{T}+X^{-1}\right) X^{-1}X\hat{z}\\&+\hat{z}XX^{-1}\left( \sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}+B_{i}G_{j})+X^{-1}\right) Xz\\ &=\hat{z}^{T}\sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(XA_{i}^{T}+A_{i}X+B_{i}N{j}+N_{j}^{T}B_{i}^{T})\hat{z}+(-2\gamma z^{T}z)\\&+z^{T}\sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(XA_{i}^{T}+{N_{j}^{T}B_{i}^{T}}+I)\hat{z}\\&+\hat{z}^{T}\sum _{i=1}^{p}\sum _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(A_{i}X+B_{i}N_{j})z\\ &=\begin{pmatrix} \hat{z}\\ z \end{pmatrix}^{T} \begin{pmatrix} \sum \limits _{i=1}^{p}\sum \limits _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij}) (\begin{array}{c}XA_{i}^{T}+A_{i}X+\\ B_{i}N_{j}+N_{j}^{T}B_{i}^{T}\end{array})&{}*\\ \sum \limits _{i=1}^{p}\sum \limits _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(XA_{i}^{T}+N_{j}^{T}B_{i}^{T})+I&{}-2\gamma I \end{pmatrix} \begin{pmatrix} \hat{z}\\ z \end{pmatrix} \end{aligned}$$

(13)

As a result, $\dot{V}< 0$ if

$$\begin{aligned} \begin{pmatrix} \sum \limits _{i=1}^{p}\sum \limits _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij}) \left( \begin{array}{c}XA_{i}^{T}+A_{i}X+\\ B_{i}N_{j}+N_{j}^{T}B_{i}^{T}\end{array}\right) &{}*\\ \sum \limits _{i=1}^{p}\sum \limits _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij})(XA_{i}^{T}+N_{j}^{T}B_{i}^{T})+I&{}-2\gamma I \end{pmatrix}<0 \end{aligned}$$

(14)

at every points of state vector x. Notice that (14) is equivalent to

$$\begin{aligned} \sum \limits _{i=1}^{p}\sum \limits _{j=1}^{q}(\bar{g}_{ij}+\varDelta g_{ij}) \begin{pmatrix} XA_{i}^{T}+A_{i}X+\\ B_{i}N_{j}+N_{j}^{T}B_{i}^{T}&{}*\\ XA_{i}^{T}+N_{j}^{T}B_{i}^{T}&{}0 \end{pmatrix} < \begin{pmatrix} 0&{}*\\ I&{}-2\gamma I \end{pmatrix}. \end{aligned}$$

(15)

In (15), the left side is a linear combination of some matrices, where $\bar{g}_{ij}+\varDelta g_{ij}$ are piecewise multi-linear functions. According to Lemma 1, (15) holds at every point if and only if it is valid at all linear interpolation end points. As a consequence, the T–S fuzzy control system (3) is stabilized if (8) is valid at every interpolation endpoints. $\square$