Stabilization of nonlinear time-delay systems with input saturation via anti-windup fuzzy design

Abstract

This investigation considers stability analysis and control design for nonlinear time-delay systems subject to input saturation. An anti-windup fuzzy control approach, based on fuzzy modeling of nonlinear systems, is developed to deal with the problems of stabilization of the closed-loop system and enlargement of the domain of attraction. To facilitate the designing work, the nonlinearity of saturation is first characterized by sector conditions, which provide a basis for analysis and synthesis of the anti-windup fuzzy control scheme. Then, the Lyapunov–Krasovskii delay-independent and delay-dependent functional approaches are applied to establish sufficient conditions that ensure convergence of all admissible initial states within the domain of attraction. These conditions are formulated as a convex optimization problem with constraints provided by a set of linear matrix inequalities. Finally, numeric examples are given to validate the proposed method.

Appendix

Consider the following time-delay fuzzy systems:

$$ {\text{fuzzy}}\;{\text{plant}}\left\{ {\begin{array}{*{20}c} {\dot{x}(t) = \sum\limits_{i = 1}^{r} {\mu_{i} } (\bar{A}_{1i} x(t) + \bar{A}_{2i} x(t - h(t)) + \bar{B}_{i} u(t)),} \hfill \\ {y(t) = \sum\limits_{i = 1}^{r} {\mu_{i} } \bar{C}_{i} x(t),} \hfill \\ \end{array} } \right. $$

(43)

$$ {\text{fuzzy}}\;{\text{controller}}\left\{ {\begin{array}{*{20}c} {\dot{x}_{c} (t) = \sum\limits_{i = 1}^{r} {\mu_{i} (A_{ci} x_{c} (t) + B_{ci} y(t))} ,} \hfill \\ {u(t) = \sum\limits_{i = 1}^{r} {\mu_{i} } C_{ci} x_{c} (t),} \hfill \\ \end{array} } \right. $$

(44)

where $x(t) \in R^{n} ,\;x_{c} (t) \in R^{n} ,\;u(t) \in R^{P} ,\;y(t) \in R^{m} ,\;\bar{A}_{1i} ,\;\bar{A}_{2i} ,\;\bar{B}_{i} ,$ and $\bar{C}_{j}$ are system matrices of appropriate dimensions, and $A{}_{ci}, \, B_{ci} ,$ and $C_{ci}$ are constant matrices to be determined. Define $\xi (t) = \left[ {\begin{array}{*{20}c} {x^{T} (t)} & {x_{c}^{T} (t)} \\ \end{array} } \right]^{T} .$ The augmented system can be written as

$$ \begin{aligned} \dot{\xi }(t) & = \sum\limits_{i = 1}^{r} {\sum\limits_{j = 1}^{r} {\mu_{i} } } \mu_{j} (A_{ij} \xi (t) + A_{di} \xi (t - h(t))), \\ A_{ij} & = \left[ {\begin{array}{*{20}c} {\bar{A}_{1i} } & {\bar{B}_{i} C_{cj} } \\ {B_{ci} \bar{C}_{j} } & {A_{ci} } \\ \end{array} } \right],\quad \, A_{di} = \left[ {\begin{array}{*{20}c} {\bar{A}_{2i} } & 0 \\ 0 & 0 \\ \end{array} } \right]. \\ \end{aligned} $$

(45)

To stabilize the system, this study applies delay-independent analysis to the design of dynamic fuzzy controller. Choose the Lyapunov–Krasovskii functional as

$$ V(\xi ) = \xi^{T} (t)P\xi (t) + {\frac{1}{1 - \beta }}\int\limits_{ \, t - h(t)}^{ t} {\xi^{T} (\sigma )S\xi (\sigma ){\text{d}}\sigma ,} $$

(46)

where P and S are positive definite matrices. Differentiating $V(\xi )$ with respect to time yields

$$ \begin{aligned} \dot{V}(\xi_{t} ) & = 2\xi^{T} P\dot{\xi } + \xi^{T} S\xi /(1 - \beta ) - (1 - \dot{h})\xi^{T} (t - h)S\xi (t - h)/(1 - \beta ) \\ \, & \quad \le 2\xi^{T} P\dot{\xi } + \xi^{T} S\xi /(1 - \beta ) - \xi^{T} (t - h)S\xi (t - h). \\ \end{aligned} $$

(47)

Substituting Eq. 45 into Eq. 47 gives

$$ \begin{aligned} \dot{V}(\xi ) & \le \sum\limits_{i = 1}^{r} {\mu_{i}^{2} } \xi^{T} \left( {A_{ii}^{T} P + PA_{ii} + {\frac{1}{1 - \beta }}S + PA_{di} S^{ - 1} A_{di}^{T} P} \right)\xi \\ & \quad + \sum\limits_{i < j} {\mu_{i} } \mu_{j} \xi^{T} ((A_{ij} + A_{ji} )^{T} P + P(A_{ij} + A_{ji} ) + {\frac{2}{1 - \beta }}S \\ & \quad + P(A_{di} + A_{dj} )(2S)^{ - 1} (A_{di} + A_{dj} )^{T} P)\xi . \\ \end{aligned} $$

(48)

Consequently, the stability condition can be written as

$$ \begin{aligned} \bar{M}_{ii} & = \left[ {\begin{array}{*{20}c} {A_{ii}^{T} P + PA_{ii} + S/(1 - \beta )} & {PA_{di} } \\ * & { - S} \\ \end{array} } \right] < 0,\quad {\text{for}}\;i = 1,2, \ldots ,r; \\ \bar{M}_{ij} & = \left[ {\begin{array}{*{20}c} {(A_{ij} + A_{ji} )^{T} P + P(A_{ij} + A_{ji} ) + 2S/(1 - \beta )} & {P(A_{di} + A_{dj} )} \\ * & { - 2S} \\ \end{array} } \right] < 0,\quad {\text{for}}\,1 \le i < j \le r. \\ \end{aligned} $$

(49)

Let the partition and the inverse matrices of P be

$$ P = \left[ {\begin{array}{*{20}c} {P_{11} } & {P_{12} } \\ {P_{12}^{T} } & {P_{22} } \\ \end{array} } \right],\quad P^{ - 1} = \left[ {\begin{array}{*{20}c} {X_{11} } & {X_{12} } \\ {X_{12}^{T} } & {X_{22} } \\ \end{array} } \right] = X. $$

(50)

Since $P^{ - 1} P = I,$ it follows that $X_{12} P_{12}^{T} = I - X_{11} P_{11} .$ Define

$$ F_{1} = \left[ {\begin{array}{*{20}c} {X_{11} } & I \\ {X_{12}^{T} } & 0 \\ \end{array} } \right],\quad F_{2} = \left[ {\begin{array}{*{20}c} I & {P_{11} } \\ 0 & {P_{12}^{T} } \\ \end{array} } \right]. $$

(51)

Then, it follows that $PF_{1} = F_{2}$ and

$$ \, F_{1}^{T} PF_{1} = \left[ {\begin{array}{*{20}c} {X_{11} } & I \\ I & {P_{11} } \\ \end{array} } \right] > 0. $$

(52)

Pre- and post-multiplying $\bar{M}_{ii}$ by the matrices ${\text{diag}}(F_{1}^{T} , \, I)$ and ${\text{diag}}(F_{1} , \, I)$ lead to

$$ \left[ {\begin{array}{*{20}c} {F_{1}^{T} A_{ii}^{T} F_{2} + F_{2}^{T} A_{ii} F_{1} + F_{1}^{T} SF_{1} /(1 - \beta )} & {F_{2}^{T} A_{di} } \\ * & { - S} \\ \end{array} } \right] < 0. $$

(53)

Choose $S = {\text{diag}}(S_{1} , \, S_{2} ).$ By proper manipulation, the condition (53) is equivalent to

$$ \left[ {\begin{array}{*{20}c} {(1,1)} & {(1,2)} & {(1,3)} & 0 \\ * & {(2,2)} & {(2,3)} & 0 \\ * & * & { - S_{1} } & 0 \\ * & * & * & { - S_{2} } \\ \end{array} } \right] < 0, $$

(54)

where

$$ \begin{aligned} (1,1) & = X_{11} \bar{A}_{1i}^{T} + X_{12} C_{ci}^{T} \bar{B}_{i}^{T} + \bar{A}_{1i} X_{11} + \bar{B}_{i} C_{ci} X_{12}^{T} \\ & \quad + (X_{11} S_{1} X_{11} + X_{12} S_{2} X_{12}^{T} )/(1 - \beta ), \\ (1,2) & = X_{11} (\bar{A}_{1i}^{T} P_{11} + \bar{C}_{i}^{T} B_{ci}^{T} P_{12}^{T} ) + X_{12} (C_{ci}^{T} \bar{B}_{i}^{T} P_{11} + A_{ci}^{T} P_{12}^{T} ) + \bar{A}_{1i} \\ & \quad + X_{11} S_{1} /(1 - \beta ), \\ (2,2) & = \bar{A}_{1i}^{T} P_{11} + \bar{C}_{i}^{T} B_{ci}^{T} P_{12}^{T} + P_{11} \bar{A}_{1i} + P_{12} B_{ci} \bar{C}_{i} + S_{1} /(1 - \beta ), \\ (1,3) & = \bar{A}_{2i} , \\ (2,3) & = P_{11} \bar{A}_{2i} . \\ \end{aligned} $$

(55)

Define the following variables:

$$ \begin{aligned} \hat{A}_{i} & = X_{11} \bar{A}_{1i}^{T} P_{11} + X_{12} A_{ci}^{T} P_{12}^{T} + \hat{C}_{i}^{T} \bar{B}_{i}^{T} P_{11} + X_{11} \bar{C}_{i}^{T} \hat{B}_{i}^{T} , \\ \hat{B}_{i} & = P_{12} B_{ci} ,\quad \hat{C}_{i} = C_{ci} X_{12}^{T} , \, \\ \hat{S}_{1} & = X_{11} S_{1} ,\quad \hat{S}{}_{2} = X_{12} S_{2} X_{12}^{T} /(1 - \beta ). \\ \end{aligned} $$

(56)

By using Schur complement, Eq. 54 can be rewritten as

$$ \left[ {\begin{array}{*{20}c} {\Upomega_{i1} } & {\hat{S}_{1} } & {\Upomega_{i2} } & {\bar{A}_{2i} } & 0 \\ * & { - (1 - \beta )S_{1} } & 0 & 0 & 0 \\ * & * & {\Upomega_{i3} } & {P_{11} \bar{A}_{2i} } & 0 \\ * & * & * & { - S_{1} } & 0 \\ * & * & * & * & { - S_{2} } \\ \end{array} } \right] < 0,\quad {\text{ for}}\; \, i = 1,2, \ldots ,r, $$

(57)

where

$$ \begin{aligned} \Upomega_{i1} & = \bar{A}_{1i} X_{11} + X_{11} \bar{A}_{1i}^{T} + \bar{B}_{i} \hat{C}_{i} + \hat{C}_{i}^{T} \bar{B}_{i}^{T} + \hat{S}_{2} , \\ \Upomega_{i2} & = \bar{A}_{1i} + \hat{A}_{i} + \hat{S}_{1} /(1 - \beta ), \\ \Upomega_{i3} & = P_{11} \bar{A}_{1i} + \bar{A}_{1i}^{P} P_{11} + \bar{C}_{i}^{T} \hat{B}_{i}^{T} + \hat{B}_{i} \bar{C}_{i} + S_{1} /(1 - \beta ). \\ \end{aligned} $$

(58)

The same calculation is performed to deal with the inequality of $\bar{M}_{ij} .$ That is, computing ${\text{diag}}(F_{1}^{T} , \, I) \times \bar{M}_{ij} \times {\text{diag}}(F_{1} , \, I)$ results in

$$ \left[ {\begin{array}{*{20}c} {\overline{(1, \, 1)} } & {\overline{(1, \, 2)} } & {(\bar{A}_{2i} + \bar{A}_{2j} )} & 0 \\ * & {\overline{(2, \, 2)} } & {P{}_{11}(\bar{A}_{2i} + \bar{A}_{2j} )} & 0 \\ * & * & { - 2S_{1} } & 0 \\ * & * & * & { - 2S_{2} } \\ \end{array} } \right] < 0, $$

(59)

where

$$ \begin{aligned} \overline{(1,1)} & = X_{11} (\bar{A}_{1i} + \bar{A}_{1j} )^{T} + (\bar{A}_{1i} + \bar{A}_{1j} )X_{11} + X_{12} (\bar{B}_{i} C_{cj} + \bar{B}_{j} C_{ci} )^{T} \\ & \quad + (\bar{B}_{i} C_{cj} + \bar{B}_{j} C_{ci} )X_{12}^{T} + 2X_{11} S_{1} X_{11} /(1 - \beta ) + 2X_{12} S_{2} X_{12}^{T} /(1 - \beta ), \\ \overline{(1,2)} & = X_{11} (\bar{A}_{1i} + \bar{A}_{1j} )^{T} P_{11} + X_{11} (B_{ci} \bar{C}_{j} + B_{cj} \bar{C}_{i} )^{T} P_{12}^{T} \\ & \quad + X_{12} (\bar{B}_{i} C_{cj} + \bar{B}_{j} C_{ci} )^{T} P_{11} + X_{12} (A_{ci} + A_{cj} )^{T} P_{12}^{T} + \bar{A}_{1i} + \bar{A}_{1j} \\ & \quad + 2X_{11} S_{1} /(1 - \beta ), \\ \overline{(1,3)} & = (\bar{A}_{1i} + \bar{A}_{1j} )^{T} P_{11} + P_{11} (\bar{A}_{1i} + \bar{A}_{1j} ) + (B_{ci} \bar{C}_{j} + B_{cj} \bar{C}_{i} )^{T} P_{12}^{T} \\ & \quad + P_{12} (B_{ci} \bar{C}_{j} + B_{cj} \bar{C}_{i} ) + 2S_{1} /(1 - \beta ). \\ \end{aligned} $$

(60)

In the simulation, it is noticed that $\bar{C}_{i} = \bar{C}_{j}$ and $\bar{B}_{i} = \bar{B}_{j} .$ Due to these conditions, Eq. 59 can be modified as

$$ \left[ {\begin{array}{*{20}c} {\Upxi_{ij}^{1} } & {\hat{S}_{1} } & {\Upxi_{ij}^{2} } & {(\bar{A}_{2i} + \bar{A}_{2j} )} & 0 \\ * & {0.5(1 - \beta )S_{1} } & 0 & 0 & 0 \\ * & * & {\Upxi_{ij}^{3} } & {P_{11} (\bar{A}_{2i} + \bar{A}_{2j} )} & 0 \\ * & * & * & { - 2S_{1} } & 0 \\ * & * & * & * & { - 2S_{2} } \\ \end{array} } \right] < 0, \, 1 \le i < j \le r, $$

(61)

where

$$ \begin{aligned} \Upxi_{ij}^{1} & = X_{11} (\bar{A}_{1i} + \bar{A}_{1j} )^{T} + (\bar{A}_{1i} + \bar{A}_{1j} )X_{11} + \hat{C}_{j}^{T} \bar{B}_{i}^{T} + \hat{C}_{i}^{T} \bar{B}_{j}^{T} + \bar{B}_{i} \hat{C}_{j} \\ & \quad + \bar{B}_{j} \hat{C}_{i} + 2\hat{S}_{2} , \\ \Upxi_{ij}^{2} & = \hat{A}_{i} + \hat{A}_{j} + \bar{A}_{1i} + \bar{A}_{1j} + 2\hat{S}_{1} /(1 - \beta ), \\ \Upxi_{ij}^{3} & = P_{11} (\bar{A}_{1i} + \bar{A}_{1j} ) + (\bar{A}_{1i} + \bar{A}_{1j} )^{T} P_{11} + \bar{C}_{j}^{T} \hat{B}_{i}^{T} + \bar{C}_{i}^{T} \hat{B}_{j}^{T} + \hat{B}_{i} \bar{C}_{j} \\ & \quad + B_{j} \bar{C}_{i} + 2S_{1} /(1 - \beta ). \\ \end{aligned} $$

(62)

In summary, given the solution of the LMIs (52), (57), and (61), the dynamic fuzzy controller (44) is constructed via the following procedures:

• 1. Solve $X_{12}$ from $\hat{S}_{2}$ as defined in Eq. 56.

• 2. Compute $P_{12}$ using the relation $X_{12} P_{12}^{T} = I - X_{11} P_{11} .$

• 3. Determine $B_{ci} ,C_{ci} ,$ and $A_{ci}$ sequentially from the obtained solutions of $X_{11} ,P_{11} ,X_{12} ,$ and $P_{12} .$