Hybrid control of \(L_2\)-gain analysis incorporated with adaptive control for a class of nonlinear systems (Q2725078)

One considers an adaptive/nonlinear control of a large scale controlled system described by NEWLINE\[NEWLINE\begin{aligned} \dot x_i & =x_{i+1}+ \gamma_i^T(x_1, \dots,x_i) \theta,\;1\leq i\leq n-1\\ \dot x_n & =\gamma_0 (x)+\gamma_n^T (x) \theta+ \bigl(\beta_0 (x)+\beta^T (x)\theta \bigr)u(t)+ \overline\eta(x,t)\\ y & =x_{\mathbf{1}}\end{aligned}NEWLINE\]NEWLINE where \(x_i\) are real scalars, \(u(t)\) is a scalar control, \(y\) a scalar output and \(\theta\) is the \(n\)-dimensional vector of parameters; \(\overline\eta (x,t)\) is a bounded perturbation. The control strategy is developed in order to ensure NEWLINE\[NEWLINE\int^T_0 \bigl|y(t)\bigr |^2 dt\leq\gamma^2 \int^T_0\bigl|\theta(t)\bigr|^2dt+ \overline N_0NEWLINE\]NEWLINE and is designed step by step for \(i=1,\dots,n\) using quadratic Lyapunov functions.