A universal servomechanism for a class of nonlinear planar systems
This paper considers a direct adaptive control methodology that extends existing results on universal stabilization to universal output tracking for a specific class of second-order systems. Specifically, the paper considers the class \(L_\delta\) of uncertain single-input, single-output systems of the form \[ \ddot y(t) = a\dot y(t) + \widehat f \bigl(t,y (t)\bigr) + bu (t) \] where \(a< - \delta\), \(\delta >0\), \(b\neq 0\), and \(f(\cdot)\) is a \(C^1\) function that satisfies a type of Lipschitz condition. Also defined is the class of reference signals, given as the Sobolev space \(R = W^{3,\infty}\), which is equivalent to those functions in \(L^\infty\) whose derivatives up to order three exist and are also \(L^\infty\). Assuming that the reference signal \(r(t) \in R\) and the output \(y(t)\) generated by a system in \(L_\delta\) are both available for feedback, the paper presents the construction of an adaptive discontinuous feedback controller that results in an asymptotic tracking error that goes to zero for any reference in \(R\) and any system in \(L_\delta\). An example is presented in which the desired reference being generated is the output signal of the Lorenz equations and the system to be controlled is the Duffing equation.
scientific article; zbMATH DE number 932150
scientific article