Stability analysis of a class of time-varying systems with state feedback control (Q2785172)

The authors consider the single input and single output linear time-varying discrete system NEWLINE\[NEWLINE\begin{multlined} y(k)=\sum^n_{i=1} \left( \sum^m_{j=1} \bigl(a^*_{ij}+ \Delta a_{ij}(k) \bigr)f_i (k)\right)y(k-i)+\\ \sum^n_{i=1} \left(\sum^m_{i=1} \bigl(b^*_{ij}+ \Delta b_{ij}(k)\bigr) f_i (k)\right) u(k-i)+ v(k),\end{multlined} \tag{1}NEWLINE\]NEWLINE where \(u(k)\), \(y(k)\) and \(v(k)\) denote the input, output and the bounded perturbation of the system, respectively, \(a^*_{ij}, b^*_{ij}\) are unknown constants, and \(f_j(k)\), \(j=1, \dots, m\), are known piecewise continuous bounded functions.NEWLINENEWLINENEWLINEHere, the unmodelled uncertainty is small and a usual condition expressing the controllability of the corresponding exact system of (1), given by \textit{G. C. Goodwin} and \textit{K. S. Sin} [Adaptive filtering, prediction and control, Englewood Cliffs, NJ: Prentice-Hall (1984; Zbl 0653.93001)], is assumed.NEWLINENEWLINENEWLINEApplying the least square parameter estimation algorithm and a state feedback control scheme, a new adaptive state feedback controller for (1) is obtained. Sufficient conditions for the global stability and the robustness of the respective closed-loop system are also established.