High-gain robust adaptive controllers for multivariable systems
The time-varying linear system \[ (1)\quad dx/dt=({\mathcal A}-k(t){\mathcal D})x(t),\quad t\geq 0 \] is considered. Here \({\mathcal A}\), \({\mathcal D}\in {\mathbb{R}}^{n^ 2}\), \(k(t)\in{\mathcal K}\), where \({\mathcal K}\) is a set of all piecewise continuous functions monotonically non-decreasing, \(k(t)\to +\infty\) as \(t\to \infty\). The following conditions are equivalent: (i) Re \(\lambda_ j({\mathcal D})>0\), \(j=1,...,n\), where \(\lambda_ j({\mathcal D})\) are eigenvalues of \({\mathcal D},\) (ii) Re \(\lambda_ j[{\mathcal A}-k(t){\mathcal D}]\to -\infty\), \(j=1,...,n,\) (iii) For some \(M>0\), \(\omega\) (\(\cdot)\in {\mathcal K}:\| \Phi (t,t_ 0)\| \leq Me^{-\omega (t_ 0)(t-t_ 0)}\), for all \(t\geq t_ 0\geq 0\), where \(\Phi (t,t_ 0)\) is the evolution matrix of (1). Let the following conditions be satisfied: (a) There exist \(k^*\), \(\epsilon >0\) such that \(\lambda_ j[{\mathcal A}-k{\mathcal D}]<-\epsilon\) for all \(k>k^*,\) (b) If \({\mathfrak O}\in \{\lambda_ j({\mathcal D})\}\) then \({\mathfrak O}\) is semisimple. Then (1) is exponentially stable for every k(\(\cdot)\in {\mathcal K}\). Moreover, there exists \(M'>0\) such that for all \(\epsilon >0\) and all k(\(\cdot)\in {\mathcal K}\) we can find \(T=T(\epsilon,k(\cdot))\) such that for all \(t\geq t_ 0\geq T:\| \Phi (t,t_ 0)\| \leq M'e^{-\epsilon (t- t_ 0)}.\) This result can be used for the construction of the adaptive regulator \(u(\cdot)=\psi [\sigma (\cdot)]\sigma (t)\) of the system \[ (2)\quad dx/dt=Ax+Bu,\quad \sigma =Cx. \] The matrices A,B,C are unknown, \(A,D=BC\) satisfy the conditions (a),(b); u(t), \(\sigma (t)\in {\mathbb{R}}^ 1\). The robustness of this adaptive controller with respect to small nonlinear perturbations of (2) is proved. Reviewer's remark. For stabilization problems of minimum phase objects, the adaptive regulators of the above type were widely used in the Russian literature, see, e.g., the book [\textit{V. N. Fomin}, \textit{A. L. Fradkov}, and \textit{V. A. Yakubovich}, Adaptive control of dynamical objects (1981; Zbl 0522.93002) (Russian), Ch. 7 and references to Ch. 7].
scientific article; zbMATH DE number 4029396