Study on stability of nonlinear closed-loop control systems based on generalized frequency response function matrices (Q2725390)

Based on the representation of generalized frequency response function matrices (GFRFM), a closed-loop stability criterion is proposed for the polynomial class of nonlinear MIMO control systems NEWLINE\[NEWLINE\sum_{n=1}^N \sum_{p_1=0}^N\dots\sum_{p_n=0}^N \left(a_{n,p_1,\dots,p_n}\prod_{i=1}^{n} D^{p_i}y(t)+ c_{n,p_1,\dots,p_n}\prod_{i=1}^{n} D^{p_i}u(t)\right) +NEWLINE\]NEWLINE NEWLINE\[NEWLINE+\sum_{n=1}^{N-1}\sum_{q=1}^{N-n} \sum_{p_1=0}^M\dots\sum_{p_{n+q}=0}^M b_{n,q,p_1,\dots,p_{n+q}}\prod_{i=1}^{n}\prod_{k=n+1}^{N}D^{p_i}y(t) D^{p_k}u(t) =0, NEWLINE\]NEWLINE where \(D\) is the differential operator, \(M\) is the maximum of differential order, and \(N\) is the maximum of multiple degree, \(a's\), \(b's\), \(c's\) are coefficient matrices, \(u\in\mathbb R^r\), \(y\in\mathbb R^m\) are the input and output of the system, respectively. Due to not considering the problem of GFRFM's power series convergence of the nonlinear closed-loop, the criterion is very simple and practical. Numerous simulation examples are offered to illustrate the efficiency of this criterion.