Stability analysis of interconnected complex nonlinear systems using the Lyapunov and Finsler property

Abstract

This study focuses on the question of the stability analysis of complex interconnected nonlinear systems using the property of Lyapunov and Finsler. The main idea is to minimize the effect of interconnections between the subsystems, for that, we use the Lyapunov function and the H∞ control, then applying Finsler’s lemma to release the conditions of stability, the independent matrices allow to obtain less conservative results. The proposed control approach is formulated in a minimization problem and derived in terms of linear matrix inequalities (LMIs) whose resolution yields the decentralized control gain matrices. All the developed results are tested on two representative examples and compared with some recent previous ones.

Appendix

1.1 Decentralized State Feedback H_∞ Control

In this Appendix, we verify the inequality (19) used in section 3, with:

$$ {\varphi}_i=\sum \limits_{j=1,j\ne i}^N{A}_{ij}{x}_j\left(t-{\eta}_{ij}(t)\right) $$

(A.1)

$$ {\displaystyle \begin{array}{c}{\varphi}_i^T{\varphi}_i={\left(\sum \limits_{j=1,j\ne i}^N{A}_{ij}{x}_j\left(t-{\eta}_{ij}(t)\right)\right)}^T\sum \limits_{j=1,j\ne i}^N{A}_{ij}{x}_j\left(t-{\eta}_{ij}(t)\right)\\ {}\kern1.44em =\sum \limits_{j=1,j\ne i}^N\sum \limits_{\begin{array}{c}l=1\\ {}l\ne i,l\ne j\end{array}}^N\left({x}_j^T\left(t-{\eta}_{ij}(t)\right){A}_{ij}^T{A}_{il}{x}_l\left(t-{\eta}_{ij}(t)\right)+{x}_l^T\left(t-{\eta}_{ij}(t)\right){A}_{il}^T{A}_{ij}{x}_j\left(t-{\eta}_{ij}(t)\right)\right)\\ {}\kern1.32em +\sum \limits_{j=1,j\ne i}^N{x}_j^T\left(t-{\eta}_{ij}(t)\right){A}_{ij}^T{A}_{ij}{x}_j\left(t-{\eta}_{ij}(t)\right)\\ {}\kern1.32em =\sum \limits_{j=1,j\ne i}^N\Big[{x}_j^T\left(t-{\eta}_{ij}(t)\right){A}_{ij}^T{A}_{ij}{x}_j\left(t-{\eta}_{ij}(t)\right)\\ {}\kern1.2em +\sum \limits_{\begin{array}{c}l=1\\ {}l\ne i,l\ne j\end{array}}^N\left({x}_j^T\left(t-{\eta}_{ij}(t)\right){A}_{ij}^T{A}_{il}{x}_l\left(t-{\eta}_{ij}(t)\right)+{x}_l^T\left(t-{\eta}_{ij}(t)\right){A}_{il}^T{A}_{ij}{x}_j\left(t-{\eta}_{ij}(t)\right)\right)\end{array}} $$

(A.2)

Apply the lemma of the square matrix, we have:

$$ {\displaystyle \begin{array}{c}\sum \limits_{\begin{array}{c}\kern0.6em l=1\\ {}l\ne i,l\ne j\end{array}}^N\left({x}_j^T\left(t-{\eta}_{ij}(t)\right){A}_{ij}^T{A}_{il}{x}_l\left(t-{\eta}_{ij}(t)\right)+{x}_l^T\left(t-{\eta}_{ij}(t)\right){A}_{il}^T{A}_{ij}{x}_j\left(t-{\eta}_{ij}(t)\right)\right)\\ {}\kern0.72em \le \sum \limits_{\begin{array}{c}\kern0.6em l=1\\ {}l\ne i,l\ne j\end{array}}^N{\left[{A}_{ij}{x}_j\left(t-{\eta}_{ij}(t)\right)\right]}^T\left[{A}_{ij}{x}_j\left(t-{\eta}_{ij}(t)\right)\right]+{\left[{A}_{il}{x}_l\left(t-{\eta}_{ij}(t)\right)\right]}^T\left[{A}_{il}{x}_l\left(t-{\eta}_{ij}(t)\right)\right]\kern1.92em \\ {}\kern0.6em =\left(N-2\right){\left[{A}_{ij}{x}_j\left(t-{\eta}_{ij}(t)\right)\right]}^T{A}_{ij}{x}_j\left(t-{\eta}_{ij}(t)\right)+\sum \limits_{\begin{array}{c}\kern0.6em l=1\\ {}l\ne i,l\ne j\end{array}}^N{\left[{A}_{il}{x}_l\left(t-{\eta}_{ij}(t)\right)\right]}^T\left[{A}_{il}{x}_l\left(t-{\eta}_{ij}(t)\right)\right]\\ {}\kern1.44em \end{array}} $$

(A.3)

Then

$$ {\varphi}_i^T{\varphi}_i\le \sum \limits_{j=1,j\ne i}^N\left(\left(N-1\right){x}_j^T\left(t-{\eta}_{ij}(t)\right){A}_{ij}^T{A}_{ij}{x}_j\left(t-{n}_{ij}(t)\right)+\sum \limits_{\begin{array}{c}l=1\\ {}l\ne i,l\ne j\end{array}}^N{x}_l^T\left(t-{\eta}_{ij}(t)\right){A}_{ij}^T{A}_{il}{x}_l\left(t-{\eta}_{ij}(t)\right)\right) $$

(A.4)

Since

$$ \sum \limits_{j=1,j\ne i}^N\left({\psi}_{ij}+\sum \limits_{\begin{array}{l}\kern0.6em l=1\\ {}l\ne i,l\ne j\end{array}}^N{\psi}_{il}\right)=\left(N-1\right)\sum \limits_{j=1,j\ne i}^N{\psi}_{ij} $$

(A.5)

Inequality (A.4) can be rewritten as follows:

$$ {\varphi}_i^T{\varphi}_i\le \left(2N-3\right)\sum \limits_{j=1,j\ne i}^N{x}_j^T\left(t-{\eta}_{ij}(t)\right){A}_{ij}^T{A}_{ij}{x}_j\left(t-{\eta}_{ij}(t)\right) $$

(A.6)

Finally, the inequality (24) is verified.