Adaptive Fuzzy Observer-Based Fault-Tolerant Control for Takagi–Sugeno Descriptor Nonlinear Systems with Time Delay

Abstract

This paper investigates the problems of state/fault estimation and active fault-tolerant control (AFTC) design for time-delay descriptor fuzzy systems subject to external disturbances and actuator faults. Using Takagi–Sugeno fuzzy models, an adaptive fuzzy observer is proposed to achieve system state and actuator fault estimation simultaneously. According to Lyapunov functional method, design and analysis conditions of the resulting closed-loop delayed descriptor system are formulated in terms of linear matrices inequalities (LMIs). Observer and controller gains are computed by solving a set of LMIs in only one step and then used to both estimate the unmeasured states and actuator faults in AFTC context. Numerical examples are provided to show the merit and the conservativeness of the proposed approach in comparison with the existing methods by considering various types of actuator faults.

Appendices

Appendix A: Proof of Theorem 1

Consider the following Lyapunov–Krasovskii functional:

$$\begin{aligned} V(t)= & {} (Ex(t))^\mathrm{T} P_1 x(t) + \int _{t-h}^{t} x^\mathrm{T}(s)Q_1 x(s)\,\mathrm {d}s + e_x^\mathrm{T}(t)P_2 e_x(t) \nonumber \\&+ \int _{t-h}^{t} e_x^\mathrm{T}(s)Q_2 e_x(s)\,\mathrm {d}s \nonumber \\&+\, \frac{1}{\sigma } e_f^\mathrm{T}(t)\Gamma ^{-1} e_f(t) + h \int _{-h}^{0} \int _{t+\theta }^{t}(E\dot{x}(s))^\mathrm{T} Z_1 (E\dot{x}(s)) \,\mathrm {d}s\,\mathrm {d}\theta \nonumber \\&+\, h \int _{-h}^{0} \int _{t+\theta }^{t} \dot{e}_{x}^\mathrm{T}(s) Z_2 \dot{e}_x(s) \,\mathrm {d}s\,\mathrm {d}\theta \end{aligned}$$

(37)

The time derivative of V(t) is given by:

$$\begin{aligned} \dot{V}(t)= & {} (E\dot{x}(t))^\mathrm{T} P_1 x(t) + (Ex(t))^\mathrm{T} P_1 \dot{x}(t)+ x^\mathrm{T}(t)Q_1 x(t) \nonumber \\&-\, x^\mathrm{T}(t-h)Q_1 x(t-h) +2 \dot{e}_x^\mathrm{T}(t)P_2 e_x(t)\nonumber \\&+\, e_x^ T(t) Q_2 e_x(t)- e_x^ T(t-h) Q_2 e_x(t-h)+ \frac{2}{\sigma } e_f^\mathrm{T}(t) \Gamma ^{-1} \dot{e}_f(t) \nonumber \\&+\, h^2[(E\dot{x}(t))^\mathrm{T} Z_1 (E\dot{x}(s))]\nonumber \\&-\,h\int _{t-h}^{t}(E\dot{x}(s))^\mathrm{T} Z_1 (E\dot{x}(s)) \,\mathrm {d}s\ + h^2 [\dot{e}_{x}^\mathrm{T}(s) Z_2 \dot{e}_x(s)]\nonumber \\&-\,h\int _{t-h}^{t} \dot{e}_{x}^\mathrm{T}(s) Z_2 \dot{e}_x(s) \,\mathrm {d}s\ \end{aligned}$$

(38)

By using Lemma 1, we have:

$$\begin{aligned} \frac{2}{\sigma }e_f^\mathrm{T}(t)\Gamma ^{-1} \dot{f}_a(t)\leqslant & {} \frac{1}{\sigma \mu } e_f^\mathrm{T}(t)M e_f(t)+ \frac{\mu }{\sigma } \dot{f}_a^\mathrm{T}(t)\Gamma ^{-1} M^{-1}\Gamma ^{-1} \dot{f}_a(t) \nonumber \\ \frac{2}{\sigma }e_f^\mathrm{T}(t)\Gamma ^{-1} \dot{f}_a(t)\leqslant & {} \frac{1}{\sigma \mu } e_f^\mathrm{T}(t)M e_f(t)+ \delta \end{aligned}$$

(39)

where

$$\begin{aligned} \delta =\frac{\mu }{\sigma }f^2_{1\mathrm{max}} \lambda _\mathrm{max}(\Gamma ^{-1} M^{-1}\Gamma ^{-1}) \end{aligned}$$

(40)

By using (23) and substituting (22), (17) and (39) into Eq. (38), one can obtain:

$$\begin{aligned} \dot{V}(t)\le & {} x(t)^\mathrm{T} [P_1^\mathrm{T}(A_i-B K_{i})+(A_i^\mathrm{T}- K_{i}^\mathrm{T} B ^\mathrm{T})P_1 +Q_1] x(t) \nonumber \\&+\,2 x(t)^\mathrm{T} P_1^\mathrm{T} A_{hi}x(t-h)+2x(t)^\mathrm{T} P_1^\mathrm{T} B K_{i} e_x(t) + 2x(t)^\mathrm{T} P_1^\mathrm{T} F_i e_f(t) \nonumber \\&- x^\mathrm{T}(t-h)Q_1 x(t-h) + e_x^\mathrm{T}(t)[ P_2 (TA_i-L_{1i}C) \nonumber \\&+(TA_i-L_{1i}C)^\mathrm{T} P_2 + Q_2] e_x(t)+2e_x^\mathrm{T}(t)P_2(TA_{hi} -L_{2i} C) e_x(t-h)\nonumber \\&-\, e_x^ T(t-h) Q_2 e_x(t-h) + \frac{1}{\sigma \mu } e_f^\mathrm{T}(t)M e_f(t)+ \delta \nonumber \\&+\, h^2[(E\dot{x}(t))^\mathrm{T} Z_1 (E\dot{x}(s))] -h\int _{t-h}^{t}(E\dot{x}(s))^\mathrm{T} Z_1 (E\dot{x}(s)) \,\mathrm {d}s \nonumber \\&+\, h^2 [\dot{e}_{x}^\mathrm{T}(s) Z_2 \dot{e}_x(s)]-h\int _{t-h}^{t} \dot{e}_{x}^\mathrm{T}(s) Z_2 \dot{e}_x(s) \,\mathrm {d}s\ \end{aligned}$$

(41)

Applying Jessen’s inequality [9] to deal with the cross product items, we obtain

$$\begin{aligned}&-h\int _{t-h}^{t}(E\dot{x}(s))^\mathrm{T} Z_1 (E\dot{x}(s)) \,\mathrm {d}s\ \le \left[ \begin{array}{cc} Ex(t)\\ Ex(t-h) \end{array} \right] ^\mathrm{T} \left[ \begin{array}{cc} -Z_1 &{} Z_{1} \\ * &{} -Z_{1} \end{array} \right] \left[ \begin{array}{cc} Ex(t)\\ Ex(t-h) \end{array} \right] \nonumber \\ \end{aligned}$$

(42)

$$\begin{aligned}&\le \left[ \begin{array}{cc} x(t)\\ x(t-h) \end{array} \right] ^\mathrm{T} \left[ \begin{array}{cc} -E^\mathrm{T} Z_1 E &{} E^\mathrm{T} Z_{1} E \\ * &{} -E^\mathrm{T} Z_{1} E \end{array} \right] \left[ \begin{array}{cc} x(t)\\ x(t-h) \end{array} \right] \end{aligned}$$

(43)

$$\begin{aligned}&-h\int _{t-h}^{t} \dot{e}_{x}^\mathrm{T}(s) Z_2 \dot{e}_x(s) \,\mathrm {d}s\ \le \left[ \begin{array}{cc} e_x(t)\\ e_x(t-h) \end{array} \right] ^\mathrm{T} \left[ \begin{array}{cc} -Z_2 &{} Z_{2} \\ * &{} -Z_{2} \end{array} \right] \left[ \begin{array}{cc} e_x(t)\\ e_x(t-h) \end{array} \right] \end{aligned}$$

(44)

Noting the extended state vector as follows:

$$\begin{aligned} \xi (t)= \left[ \begin{array}{ccccc} x^\intercal (t)&x^\intercal (t-h)&e_x^\mathrm{T}(t)&e_x^\mathrm{T}(t-h)&e^\mathrm{T}_f(t) \end{array} \right] ^\mathrm{T} \end{aligned}$$

(45)

Then, we can write :

$$\begin{aligned} \dot{V}(t)\le \xi ^ T \phi _{i}^{11} \xi (t) + h^2[(E\dot{x}(t))^\mathrm{T} Z_1 (E\dot{x}(s))] + h^2 [\dot{e}_{x}^\mathrm{T}(s) Z_2 \dot{e}_x(s)]+\delta \end{aligned}$$

(46)

where

$$\begin{aligned} \phi _{i}^{11} =\left[ \begin{array}{ccccc} \varphi _{i}^{11} &{} \varphi _{i}^{12} &{} P_1^\mathrm{T} B K_{i} &{} 0 &{} P_1^\mathrm{T} F_i \\ *&{} -(Q_1+E^\mathrm{T} Z_1 E) &{} 0 &{} 0 &{} 0 \\ *&{} *&{} \varphi _{i}^{33} &{} \varphi _{i}^{34} &{} \varphi _{i}^{35} \\ *&{} *&{} *&{} -(Q_2+ Z_2) &{} \varphi _{i}^{45} \\ *&{} *&{} *&{} *&{} \varphi _{i}^{55} \end{array} \right] \end{aligned}$$

(47)

Denote

$$\begin{aligned} \phi _{i}=\left[ \begin{array}{ccc} \phi _{i}^{11} &{} \phi _{i}^{12} &{} \phi _{i}^{13} \\ *&{} -(h^2 P_2^{-1} Z_2 P_2^{-1})^{-1} &{} 0 \\ *&{} *&{} -(h^2 P_1^{-1} Z_1 P_1^{-T})^{-1} \\ \end{array} \right] \end{aligned}$$

(48)

where

$$\begin{aligned} ({\phi _{i}^{12}})^\mathrm{T}= & {} \left[ \begin{array}{ccccc} 0 &{} 0 &{} P_2 (T A_i-L_{1i} C) &{} P_2(T A_{hi}-L_{2i} C) &{} P_2(TF_i) \\ \end{array} \right] \\ ({\phi _{i}^{13}})^\mathrm{T}= & {} \left[ \begin{array}{ccccc} P_1^\mathrm{T}(A_i-B K_{i}) &{} P_1^\mathrm{T} A_{hi} &{} P_1^\mathrm{T} B K_{i} &{} 0 &{} P_1^\mathrm{T} F_i \\ \end{array} \right] \end{aligned}$$

By using Schur complement, inequality (25) is equivalent to $\xi ^ T \phi _{i}^{11} \xi (t) + h^2[(E\dot{x}(t))^\mathrm{T} Z_1 (E\dot{x}(s))] + h^2 [\dot{e}_{x}^\mathrm{T}(s) Z_2 \dot{e}_x(s)] < 0$.

If condition (25) holds, it follows from (41) that

$$\begin{aligned} \dot{V}(t) \le -\zeta \Vert \xi (t) \Vert ^2 + \delta \end{aligned}$$

(49)

where $\zeta =\lambda _\mathrm{min}(-\phi _i)$

It follows that $\dot{V}(t) \leqslant 0$ for $\zeta \Vert \xi (t) \Vert ^2 > \delta$, and according to Lyapunov stability theory, $\xi (t)$ will converge to a small set $\Psi = \{ \xi (t) / \Vert \xi (t) \Vert ^2 \le \frac{\delta }{\zeta } \}$ ; thus, $\xi (t)$ is uniformly bounded.

The proof is completed.

Appendix B: Proof of Theorem 2

We can write inequality (26) in this form

$$\begin{aligned} \varLambda _{i}=\left[ \begin{array}{ccc} \varLambda _{i}^{11} &{} \varLambda _{i}^{12} &{} \varLambda _{i}^{13} \\ *&{} \varLambda _{i}^{22} &{} \varLambda _{i}^{23} \\ *&{} *&{} \varLambda _{i}^{33} \\ \end{array} \right] < 0 \end{aligned}$$

(50)

where

$$\begin{aligned} \varLambda _{i}^{11}= & {} \left[ \begin{array}{cc} \mathrm{sym}(P_1^\mathrm{T} A_{i}-P_1^\mathrm{T} B K_{i})+Q_1-E^\mathrm{T} Z_1 E &{} P_1^\mathrm{T} A_{hi}+E^\mathrm{T} Z_1 E \\ *&{} -(Q_1+E^\mathrm{T} Z_1 E) \\ \end{array} \right] \\ \varLambda _{i}^{12}= & {} \left[ \begin{array}{cccc} P_1^\mathrm{T} B K_{i} &{} 0 &{} P_1^\mathrm{T} F_i &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ \end{array} \right] , \, \varLambda _{i}^{13}= \left[ \begin{array}{c} (A_i-B K_{i})^\mathrm{T} P_1 \\ A_{hi}^\mathrm{T} P_1 \end{array} \right] \end{aligned}$$

$$\begin{aligned} \varLambda _{i}^{22}= & {} \left[ \begin{array}{cccc} \mathrm{sym}(P_2 H_1 A_i-P_2 L_{1i} C) +Q_2-Z_2 &{} P_2(H_1 A_{hi}- L_{2i} C)+Z_2 &{} -\frac{1}{\sigma }(A_{hi}^\mathrm{T} H_1^\mathrm{T} P_2 - C^\mathrm{T} Y_{2i}^\mathrm{T})H_1 F_i &{} (H_1 A_{i}-L_{1i}C)^\mathrm{T} P_2 \\ *&{} -(Q_2+ Z_2) &{} -\frac{1}{\sigma }(A_{hi}^\mathrm{T} H_1^\mathrm{T} P_2 -C^\mathrm{T} Y_{2i}^\mathrm{T})H_1 F_i &{} (H_1 A_{hi}-L_{2i}C)^\mathrm{T} P_2 \\ *&{} *&{} -\frac{1}{\sigma }(H_1 F_{i})^\mathrm{T} P_2 (H_1 F_{i})+\frac{1}{\sigma \mu }M &{} (H_1 F_i)^\mathrm{T} P_2 \\ *&{} *&{} *&{} -P_2(h^2 Z_2)^{-1} P_2 \\ \end{array} \right] \\ \varLambda _{i}^{33}= & {} \left[ \begin{array}{c} -P_1^\mathrm{T}(h^2 Z_1)^{-1} P_1 \\ \end{array} \right] \end{aligned}$$

Consider the following symmetric matrix:

$$\begin{aligned} \mathbb {Z}= \left[ \begin{array}{ccc} \mathbb {Z}_{11} &{} 0 &{} 0\\ 0 &{} \mathbb {Z}_{22} &{} 0 \\ 0 &{} 0 &{} \mathbb {Z}_{33} \\ \end{array} \right] \end{aligned}$$

where $\mathbb {Z}_{11} =\hbox {diag}(P_1^{-T},P_1^{-T}), \mathbb {Z}_{22} = \hbox {diag}(P_1^{-T},I,I,I)$ and $\mathbb {Z}_{33} =P_1^{-T}$

We can transform inequality (50) by pre- and post-multiplying it by $\mathbb {Z}$, and we obtain this form:

$$\begin{aligned} \left[ \begin{array}{ccc} \mathbb {Z}_{11} \varLambda _{i}^{11} \mathbb {Z}_{11}^\mathrm{T} &{} \mathbb {Z}_{11} \varLambda _{i}^{12} \mathbb {Z}_{22}^\mathrm{T} &{} \mathbb {Z}_{11} \varLambda _{i}^{13} \mathbb {Z}_{33}^\mathrm{T}\\ *&{} \mathbb {Z}_{22} \varLambda _{i}^{22} \mathbb {Z}_{22}^\mathrm{T} &{} \mathbb {Z}_{22} \varLambda _{i}^{23} \mathbb {Z}_{33}^\mathrm{T} \\ *&{} *&{} \mathbb {Z}_{33} \varLambda _{i}^{33} \mathbb {Z}_{33}^\mathrm{T} \end{array} \right] <0 \end{aligned}$$

(51)

By using Lemma 2, we obtain the following inequalities:

$$\begin{aligned} -P_2(h^2 Z_2)^{-1} P_2\le & {} -2 \lambda _1 P_2+\lambda _1^2 h^2 Z_2 \end{aligned}$$

(52)

$$\begin{aligned} \mathbb {Z}_{33} \varLambda _{i}^{33} \mathbb {Z}_{33}^\mathrm{T}= & {} -P_1^{-T}(h^2 \widetilde{Z}_1)^{-1} P_1^{-1} \le -\lambda _2 (P_1^{-T}+P_1^{-1}) + \lambda _2^{2} h^2 \widetilde{Z}_1\qquad \quad \end{aligned}$$

(53)

$$\begin{aligned} \mathbb {Z}_{22} \varLambda _{i}^{22} \mathbb {Z}_{22}^\mathrm{T}\le & {} -\lambda _3 (\mathbb {Z}_{22}+\mathbb {Z}_{22}^\mathrm{T}) - \lambda _3^{2} (\varLambda _{i}^{22})^{-1} \end{aligned}$$

(54)

By applying Schur complement, we obtain the following inequality:

$$\begin{aligned} \left[ \begin{array}{cccc} \mathbb {Z}_{11} \varLambda _{i}^{11} \mathbb {Z}_{11}^\mathrm{T} &{} \mathbb {Z}_{11} \varLambda _{i}^{12} \mathbb {Z}_{22}^\mathrm{T} &{} \mathbb {Z}_{11} \varLambda _{i}^{13} \mathbb {X}_{33}^\mathrm{T} &{} 0 \\ * &{} -\lambda _3 (\mathbb {Z}_{22}+\mathbb {Z}_{22}^\mathrm{T})&{} \mathbb {Z}_{22} \varLambda _{i}^{23} \mathbb {Z}_{33}^\mathrm{T} &{} \lambda _3 I \\ * &{} * &{} \mathbb {Z}_{33} \varLambda _{i}^{33} \mathbb {Z}_{33}^\mathrm{T} &{} 0 \\ * &{} * &{} * &{} \varLambda _{i}^{22} \\ \end{array} \right] < 0 \end{aligned}$$

(55)

By posing $X_1=P_1^{-1}, X_2=P_2, \widetilde{Z}_1=P_1^{-1}Z_1 P_1^{-T}, \widetilde{Q}_1=P_1^{-T}Q_1 P_1^{-1}, Y_{1i}=P_2 L_{1i}, Y_{2i}=P_2 L_{2i}$ and $W_i=K_{i} P^{-1}_1$, we obtain inequality (30).

The proof is completed.