Relaxed Stabilization Conditions for Interconnected Nonlinear Systems

Abstract

This paper deals with the conservatism reduction of stability and stabilization conditions for nonlinear continuous-time interconnected systems. Based on Takagi–Sugeno modeling, the interconnected system is described by a convex combination of interconnected linear systems. Line integral fuzzy Lyapunov functions are considered to develop stability and stabilizability criteria for the considered class of interconnected nonlinear systems. All analysis and design conditions are formulated in LMI terms. The effect of the coupling due to interconnection terms is also investigated and taken into account to obtain less conservative conditions. Examples are given to illustrate the merit of the obtained results.

Appendix: Proof of Corollary 1

Consider the LILF V(t). By adding its time derivative with null product (9), one obtains:

$$\begin{aligned} {\dot{V}}(t) =&\sum _{i=1}^{J} \big \lbrace {\dot{x}}_{i}^{T} P_{i}(\mu _{i}) x_{i}+x_{i}^{T} P_{i}(\mu _{i}) {\dot{x}}_{i} \\&+ 2\big (x_{i}^{T} M_{1i}+{\dot{x}}_{i}^{T} M_{2i}\big )\times \big (-{\dot{x}}_{i}+A_{i}(\mu _{i}) x_{i}+h_{i}(x)\big ) \big \rbrace \\ =&\sum _{i=1}^{J} \big \lbrace {\dot{x}}_{i}^{T} P_{i}(\mu _{i}) x_{i}+x_{i}^{T} P_{i}(\mu _{i}) {\dot{x}}_{i}-x_{i}^{T} M_{1i} {\dot{x}}_{i}-{\dot{x}}_{i}^{T} M_{1i}^{T} x_{i}\\&-{\dot{x}}_{i}^{T} Sym(M_{2i}){\dot{x}}_{i} +x_{i}^{T} Sym(M_{1i} A_{i}(\mu _{i})) x_{i} +{\dot{x}}_{i}^{T} M_{2i} A_{i}(\mu _{i}) x_{i} \\&+x_{i}^{T} (A_{i}(\mu _{i}))^{T} M_{2i}^{T} {\dot{x}}_{i}+x_{i}^{T} M_{1i}h_{i}(x)+{\dot{x}}_{i}^{T} M_{2i}^{T}h_{i}(x) \big \rbrace . \end{aligned}$$

Using (18), (19), one obtains:

$$\begin{aligned} {\dot{V}}(t) \le&\sum _{i=1}^{J} \big \lbrace {\dot{x}}_{i}^{T} P_{i}(\mu _{i}) x_{i}+x_{i}^{T} P_{i}(\mu _{i}) {\dot{x}}_{i}-x_{i}^{T} M_{1i} {\dot{x}}_{i}-{\dot{x}}_{i}^{T} M_{1i}^{T} x_{i} \nonumber \\&-{\dot{x}}_{i}^{T} Sym(M_{2i}){\dot{x}}_{i} +x_{i}^{T} Sym(M_{1i} A_{i}(\mu _{i})) x_{i}+{\dot{x}}_{i}^{T} M_{2i} A_{i}(\mu _{i}) x_{i} \nonumber \\&+x_{i}^{T} (A_{i}(\mu _{i}))^{T} M_{2i}^{T} {\dot{x}}_{i} + x_{i}^{T} M_{1i} M_{1i}^{T} x_{i}+ {\dot{x}}_{i}^{T} M_{2i} M_{2i}^{T} {\dot{x}}_{i} \nonumber \\&+2 \sum _{j=1}^{J} J \alpha _{j}^{2} x_{i}^{T} H_{ji}^{T} H_{ji} x_{i} \big \rbrace \nonumber \\ =&\sum _{i=1}^{J} \sum _{l=1}^{r_{i}} \mu _{i}^{l} \xi _{i}^{T} \varTheta _{i}^{l} \xi _{i}, \end{aligned}$$

(29)

where

$$\begin{aligned} \xi _{i} =&\begin{bmatrix} x_{i} \\ {\dot{x}}_{i} \end{bmatrix}, \nonumber \\ \varTheta _{i}^{l} =&\begin{bmatrix} Sym(M_{1i} A_{i}^{l}) +2 \sum \limits _{j=1}^{J} J \alpha _{j}^{2} H_{ji}^{T} H_{ji}+M_{1i} M_{1i}^{T} &{} * \\ P_{i}^{l}- M_{1i}^{T}+M_{2i} A_{i}^{l} &{} -Sym(M_{2i})+M_{2i} M_{2i}^{T} \end{bmatrix}. \end{aligned}$$

From (29), ${\dot{V}} < 0$ implies the asymptotic stability of the CFLSS (5) if the stability conditions of $P_{i}^{l} > 0$ and $\varTheta _{i}^{l} < 0$ are satisfied. By applying the Schur complement to $\varTheta _{i}^{l} < 0$, one has (11).