Adaptive stabilization of not necessarily minimum phase plants
The plant is represented by the equations \[ (1)\quad \dot x=A_ 0x+b_ 0u+A_{12}\eta,\quad x\in {\mathbb{R}}^ n,\quad \eta \in {\mathbb{R}}^ n,\quad \mu {\dot \eta}=A_{22}\eta +\mu b_ 1\dot u,\quad y=c_ 0^ Tx,y,\quad u\in {\mathbb{R}}^ 1. \] Here \(\mu >0\) is a small parameter and the coefficients in (1) are unknown. Let \(W_ 0(S)=c^*_ 0(sI-A_ 0)^{-1}b_ 0=gn(s)d(s)^{-1}\) be the transfer function of the dominant part of (1),n(s),d(s) are monic and g is known; without loss of generality \(g>0\). The adaptive regulator is taken in the form (2) \(u=- Ky,K=\gamma y.\) Let n(s) be Hurwitz and deg d(s)\(=\deg n(s)+1\). There exist \(\mu^*>0\), \(0<\alpha <1/2\), \(c_ j>0\) and \(\bar K\) such that for each \(\mu \in [0,\mu^*]\) every solution of (1), (2) which starts from the set \({\mathcal D}(\mu)=\{x,\eta,K:| x| +| K-\bar K| <c_ 1\mu^{- \alpha}\), \(| \eta | <c_ 2\mu^{-\alpha -1/2}\}\) is bounded and \(| x(t)| +| \eta (t)| \to 0\), \(K(t)\to K_{\infty}\) as \(t\to \infty\).
scientific article; zbMATH DE number 4010326