Optimal design of adaptive tracking controllers for nonlinear systems (Q1371654)

For the general nonlinear system \[ \begin{aligned} \dot x & =f(x)+ F(x)\theta + g(x) u(t)\\ y & =h(x) \end{aligned} \] where \(\theta\) is a vector parameter, one considers the problem of constructing an adaptive controller \[ \begin{aligned} u & =\alpha(t,e, \widehat \theta) \\ \widehat \theta & = \Gamma\tau (t,e,\widehat \theta) \end{aligned} \] where \(e=x-x_r\) is the tracking error; here \(x_r(t)\) is the so-called ``limit-regime'' of the system when \(y(x_r(t)) \equiv y_r(t)\), the signal to be tracked. To solve the problem, two rather strong assumptions are made: existence of the above mentioned limit regime and existence of a control Liapunov function in the sense of Artstein and Sontag i.e. a Liapunov function for the free system \(\dot x=f(x)\) which is disturbed by the control terms but still preserves some properties that ensure stability of the controlled system (more precisely of its limit regime) by a proper choice of the controller.