The stabilization of symmetric circulant composite systems with input saturation (Q2725433)

The authors discuss the following symmetric circulant composite system with input saturation NEWLINE\[NEWLINE\Sigma:\dot x=Ax=B\sigma(u),\tag{*}NEWLINE\]NEWLINE where \(A=scl[A_1,A_2,\dots,A_N]\), \(B=scl[B_1,B_2,\dots,B_n]\) are symmetric circulant matrices, \(A_i\in \mathbb{R}^{n\times n}\), \(B_i\in \mathbb{R}^{n\times m}\), \(\sigma: \mathbb{R}^{Nm}\to \mathbb{R}^{Nm}\) is a saturation function, \(x=(x^T_1,x_2^T, \dots, x_N^T)^T\), \(u=(u_1^T, u^T_2, \dots,u_N^T)^T\) denote the state-vector and control input vector, respectively, where \(x_i\in\mathbb{R}^n\), \(u_i\in\mathbb{R}^m\), \(i=1, \dots,N\).NEWLINENEWLINENEWLINEApplying properties of saturation functions, some known results on stabilization and a theorem on the relationship between the coefficient matrices \(P,Q,R\) of Riccati algebraic equations proved by \textit{Wang Qinglin} [Acta Autom. Sin. 22, 111-114 (1996; Zbl 0846.93047)], a necessary and sufficient condition for system (*) to be semiglobally stabilized by linear feedback control or to be globally stabilized by nonlinear feedback control is derived.NEWLINENEWLINENEWLINEIt is also shown that the linear feedback control law can be obtained by solving several algebraic Riccati equations of order equal to each isolated subsystems of the system (*). An illustrative example is also discussed in detail.