Numerical approximation of the \(H_ \infty\) norm of nonlinear systems
The systems under consideration are described by models of the kind \[ \begin{aligned} x(t) & = f(x(t), w(t)),\quad x(0)= x_0,\\ z(t) & = h(x(t)),\qquad t\geq 0,\end{aligned} \] where \(x(t)\in \mathbb{R}^n\), \(w(t)\in W\subseteq \mathbb{R}^m\), \(0\in W\), \(z(t)\in \mathbb{R}^p\), \(h(0)= 0\), \(f(0, 0)= 0\). Inputs are controls \(w\in L_2([0, T], W)\) and states are \(x(t) = \gamma_w(t) x_0\). In applications, \(w\) is a disturbance input whose effect on a given output \(z\) is to be measured by, for example, using the \(H_\infty\)-norm of the input-output mapping \(G\) given by \[ (Gw)(t)= h(\gamma_w(t) x_0),\quad x(0)= 0. \] A numerical technique for computing the \(H_\infty\)-norm is proposed. It involves a sequence of steps which entails the solution of a first-order nonlinear partial differential inequality, if the original system is nonlinear. The numerical method is based on the finite difference numerical scheme (Capuzzo Dolcetta, Falcone, Kushner, Dupuis). Convergence of the numerical method is proved using weak solution methods. The discrete version of the PDI corresponds to an infinite-horizon problem for a controlled Markov chain. Results of numerical experiments and impressive figures carried out on SUN4 and CONNECTION machines are presented. The techniques of local velocity and the acceleration method of Capuzzo Dolcetta and Falcone, respectively, are also illustrated in the examples.
scientific article; zbMATH DE number 806693
scientific article