Robust functional observer for stabilising uncertain fuzzy systems with time-delay

Abstract

This paper presents a new technique for stabilising a Takagi–Sugeno (T-S) fuzzy system with time-delay and uncertainty. A robust fuzzy functional observer is employed to design a controller when the system states are not measurable. The model uncertainty is norm bounded, and the time-delay is time-varying but bounded. The parallel distributed compensation method is applied for defining the fuzzy functional observer to design this controller. The proposed procedure reduces the observer order to the dimension of the control input. Improved stability conditions are provided for the observer compared with the existing results of functional observer-based stabilisation of T-S fuzzy models. Lyapunov–Krasovskii functionals are used to construct delay-dependent stability conditions as linear matrix inequalities. The solution of these inequalities is used for calculating the observer parameters. The sensitivity of the estimation error to the model uncertainty is reduced by minimising the $L_2$ gain. The new design method developed is illustrated and verified using two examples.

Appendix

1.1 Proof of Lemma 1

By applying PDC controller $u=\sum _{j=1}^r \mu _j(\xi (t)) K_j x(t)$, the dynamics of closed-loop system (2) can be expressed as

$$\begin{aligned}&\sum _{i=1}^r \sum _{j=1}^r \mu _i ( \xi (t)) \mu _j ( \xi (t)) \Bigg \{ (A_i + \varDelta A_i + B_i K_j) x(t) \nonumber \\&\quad + (A_{di} + \varDelta A_{di} ) x(t-\tau (t)) \Bigg \}. \end{aligned}$$

(23)

Consider a Lyapunov–Krasovskii functional for the stability analysis of closed-loop system (23)

$$\begin{aligned} \begin{aligned} V(t)&= x^T(t) P_1 x(t) + \int _{t-\tau (t)}^{t} x^T(s) P_2 x(s) \mathrm{d}s \\&\quad +\int _{-\tau _\mathrm{M}}^{0} \int _{t+\theta }^t x^T(s) P_3 x(s) \mathrm{d}s \mathrm{d}\theta \\&\quad + \int _{-\tau _\mathrm{M}}^{-\tau _\mathrm{m}} \int _{t+\theta }^t x^T(s) P_4 x(s) \mathrm{d}s \mathrm{d}\theta . \end{aligned} \end{aligned}$$

Taking derivative along the state dynamics and applying the assumption ${\dot{\tau }}(t) \le \rho$, we can obtain

$$\begin{aligned} \begin{aligned} {\dot{V}}(t)&= 2 x^T(t) P_1 {\dot{x}}(t) + x^T(t) P_2 x(t) \\&\quad - (1-{\dot{\tau }}(t))x^T(t-\tau (t)) P_2 x(t-\tau (t)) \\&\quad + {\dot{x}}^T(t) ( \tau _\mathrm{M} P_3 + (\tau _\mathrm{M}- \tau _\mathrm{m}) P_4 ) {\dot{x}}(t) \\&\quad - \int _{t - \tau _\mathrm{M}}^t {\dot{x}}^T(s) P_3 {\dot{x}}(s) \mathrm{d}s \\&\quad - \int _{t - \tau _\mathrm{M}}^{t-\tau _\mathrm{m}} {\dot{x}}^T(s) P_4 {\dot{x}}(s) \mathrm{d}s \\&\le 2 x^T(t) P_1 {\dot{x}}(t) + x^T(t) P_2 x(t)\\&\quad - (1-\rho )x^T(t-\tau (t)) P_2 x(t-\tau (t)) \\&\quad + {\dot{x}}^T(t) ( \tau _\mathrm{M} P_3 + (\tau _\mathrm{M} - \tau _\mathrm{m}) P_4 ) {\dot{x}}(t). \end{aligned} \end{aligned}$$

(24)

By (23) and the assumption of (3), it can be shown that

$$\begin{aligned} \begin{array}{ll} & {\dot{x}}^T(t) ( \tau _\mathrm{M} P_3 + (\tau _\mathrm{M} - \tau _\mathrm{m}) P_4 ) {\dot{x}}(t) \\ & \quad = \displaystyle \sum _{i=1}^r \sum _{j=1}^r \mu _i ( \xi (t)) \mu _j ( \xi (t)) \left\{ {\dot{x}}^T(t) \varLambda \begin{bmatrix}A_i + B_iK_j & A_{di}\end{bmatrix} {\bar{x}}(t) \right. \\ & \left. \quad \quad +{\dot{x}}^T(t) \begin{bmatrix}\varLambda R_i & \varLambda R_{di} \end{bmatrix} \begin{bmatrix}U_i(t)S_i & 0 \\ 0 & U_{di}(t)S_{di} \end{bmatrix} {\bar{x}}(t) -\tau _\mathrm{m} {\dot{x}}^T(t) P_4 {\dot{x}}(t) \right\} \\ & \quad \le \displaystyle \sum _{i=1}^r \sum _{j=1}^r \mu _i ( \xi (t)) \mu _j ( \xi (t)) \left\{ {\dot{x}}^T(t) \varLambda \begin{bmatrix}A_i + B_iK_j & A_{di}\end{bmatrix} {\bar{x}}(t) \right. \\ & \quad \quad + {\dot{x}}^T(t) \begin{bmatrix}\varLambda R_i & \varLambda R_{di} \end{bmatrix} \begin{bmatrix}\varLambda R_i & \varLambda R_{di} \end{bmatrix}^T {\dot{x}}(t) +{\bar{x}}^T(t) \begin{bmatrix}S_i^T S_i & 0 \\ 0 & S_{di}^T S_{di} \end{bmatrix} {\bar{x}}(t) \\ & \left. \quad \quad -\tau _\mathrm{m} {\dot{x}}^T(t) P_4 {\dot{x}}(t) \right\} {,} \end{array} \end{aligned}$$

(25)

where ${\bar{x}}^T(t) =\begin{bmatrix}x^T(t)&x^T(t-\tau (t)) \end{bmatrix}$ and $\varLambda = \tau _\mathrm{M}(P_3 + P_4)$. It can also be shown that

$$\begin{aligned} \begin{aligned}&2 x^T(t) P_1 {\dot{x}}(t) \\&\quad \le \sum _{i=1}^r \sum _{j=1}^r \mu _i ( \xi (t)) \mu _j ( \xi (t)) {\bar{x}}^T(t) \left\{ \begin{bmatrix} P_1(A_i +B_iK_j) + (A_i + B_iK_j)^TP_1&P_1 A_{di} \\ \star&0 \end{bmatrix} \right. \\&\qquad \left. +\begin{bmatrix}P_1R_i&P_1 R_{di} \\ 0&0\end{bmatrix} \begin{bmatrix}(P_1R_i)^T&0\\ (P_1 R_{di})^T&0\end{bmatrix} + \begin{bmatrix}S_i^TS_i&0 \\ 0&S_{di}^TS_{di}\end{bmatrix} \right\} {\bar{x}}(t){.} \end{aligned} \end{aligned}$$

(26)

Therefore, using (24), (25) and (26) we get

$$\begin{aligned} \begin{aligned} {\dot{V}}(t)&\le \sum _{i=1}^r \sum _{j=1}^r \mu _i ( \xi (t)) \mu _j ( \xi (t)) \eta ^T(t) \left( \begin{bmatrix} \begin{matrix}P_2 +P_1A_i + A_i^TP_1 \\ + P_1B_iK_j +K_j^TB_i^TP_1\end{matrix}&P_1A_{di}&\frac{1}{2}(A_i + B_iK_j)^T \varLambda \\ \star&-(1-\rho )P_2&\frac{1}{2}A_{di}^T \varLambda \\ \star&\star&-\tau _\mathrm{m} P_4\end{bmatrix} \right. \\&\quad \left. +2 \begin{bmatrix}S_i^T S_i&0&0 \\ 0&S_{di}^T S_{di}&0 \\ 0&0&0\end{bmatrix} +\begin{bmatrix}P_1R_i&P_1 R_{di}&0&0\\ 0&0&0&0 \\0&0&\varLambda R_i&\varLambda R_{di}\end{bmatrix} \begin{bmatrix}(P_1R_i)^T&0&0 \\ (P_1 R_{di})^T&0&0 \\ 0&0&R_i^T \varLambda \\ 0&0&R_{di}^T \varLambda \end{bmatrix} \right) \eta (t), \end{aligned} \end{aligned}$$

where $\eta ^T(t) =\begin{bmatrix}x^T(t)&x^T(t-\tau (t))&{\dot{x}}^T(t) \end{bmatrix}$. As a consequence, an asymptotic stability condition of the fuzzy system can be given as

$$\begin{aligned} \begin{aligned}&\begin{bmatrix} \begin{matrix}P_2 +P_1A_i + A_i^TP_1 \\ + P_1B_iK_j +K_j^TB_i^TP_1\end{matrix}&P_1A_{di}&\frac{1}{2}(A_i + B_iK_j)^T \varLambda \\ \star&-(1-\rho )P_2&\frac{1}{2}A_{di}^T \varLambda \\ \star&\star&-\tau _\mathrm{m} P_4\end{bmatrix} \\&\quad +2 \begin{bmatrix}S_i^T S_i&0&0 \\ 0&S_{di}^T S_{di}&0 \\ 0&0&0\end{bmatrix}&\\&\quad +\begin{bmatrix}P_1R_i&P_1 R_{di}&0&0\\ 0&0&0&0 \\0&0&\varLambda R_i&\varLambda R_{di}\end{bmatrix} \begin{bmatrix}(P_1R_i)^T&0&0 \\ (P_1 R_{di})^T&0&0 \\ 0&0&R_i^T \varLambda \\ 0&0&R_{di}^T \varLambda \end{bmatrix}<0.&\end{aligned}\end{aligned}$$

(27)

By the Schur complement, (27) can be expressed as

$$\begin{aligned} \begin{bmatrix} \begin{matrix}P_2 +P_1A_i + A_i^TP_1 \\ + P_1B_iK_j +K_j^TB_i^TP_1\end{matrix}&P_1A_{di}&\frac{1}{2}(A_i^T + K_j^TB_i^T)\varLambda&P_1R_i&P_{1}R_{di}&0&0&S_i^T&0 \\ \star&-(1-\rho ) \bar{P_2}&\frac{1}{2} A_{di}^T \varLambda&0&0&0&0&0&S_{di}^T \\ \star&\star&-\tau _\mathrm{m} P_4&0&0&\varLambda R_i&\varLambda R_{di}&0&0 \\ \star&\star&\star&-I&0&0&0&0&0 \\ \star&\star&\star&\star&-I&0&0&0&0 \\ \star&\star&\star&\star&\star&-I&0&0&0 \\ \star&\star&\star&\star&\star&\star&-I&0&0 \\ \star&\star&\star&\star&\star&\star&\star&-\frac{1}{2}I&0 \\ \star&\star&\star&\star&\star&\star&\star&\star&-\frac{1}{2}I \\ \end{bmatrix} < 0. \end{aligned}$$

(28)

Considering $P_3={\bar{\sigma }}_1 P_1$ and $P_4 = {\bar{\sigma }}_2 P_1$, and pre-multiplying and post-multiplying (28) by block-diagonal matrix

$$\begin{aligned} \text {diag}(\begin{matrix}P_1^{-1}&P_1^{-1}&P_1^{-1}&I&I&I&I&I&I\end{matrix}), \end{aligned}$$

we obtain

$$\begin{aligned} \begin{bmatrix} \begin{matrix}{\bar{P}}_2 +A_i{\bar{P}}_1 \\ + {\bar{P}}_1A_i^T + B_i {\bar{Y}}_j \\ +{\bar{Y}}_j^TB_i^T\end{matrix}&A_{di} {\bar{P}}_1&\frac{1}{2}\kappa ( {\bar{P}}_1A_i^T + {\bar{Y}}_j^TB_i^T)&R_i&R_{di}&0&0&{\bar{P}}_1S_i^T&0 \\ \star&-(1-\rho ) \bar{P_2}&\frac{1}{2}\kappa {\bar{P}}_1 A_{di}^T&0&0&0&0&0&{\bar{P}}_1S_{di}^T \\ \star&\star&-\tau _\mathrm{m} \bar{\sigma _2} {\bar{P}}_1&0&0&\kappa R_i&\kappa R_{di}&0&0 \\ \star&\star&\star&-I&0&0&0&0&0 \\ \star&\star&\star&\star&-I&0&0&0&0 \\ \star&\star&\star&\star&\star&-I&0&0&0 \\ \star&\star&\star&\star&\star&\star&-I&0&0 \\ \star&\star&\star&\star&\star&\star&\star&-\frac{1}{2}I&0 \\ \star&\star&\star&\star&\star&\star&\star&\star&-\frac{1}{2}I \\ \end{bmatrix} < 0, \end{aligned}$$

where ${\bar{P}}_1 = P_1^{-1}$, $\bar{P_2}=\bar{P_1} P_2 \bar{P_1}$, ${\bar{Y}}_j= K_j\bar{P_1}$, $\kappa =\tau _\mathrm{M} ({\bar{\sigma }}_1 + {\bar{\sigma }}_2)$ with some given scalars ${\bar{\sigma }}_1$ and ${\bar{\sigma }}_2$.