A class of nonlinear \(H_\infty\) controller design via state feedback for disturbance attenuation. (Q2783674)

A class of disturbance attenuation approach is proposed for affine nonlinear systems NEWLINE\[NEWLINE \dot x = f(x) + \sum_{i=0}^lg_{w_i}(x)w_i+g(x)u,\quad x = h(x) + d(x)u, NEWLINE\]NEWLINE where \(x\in \mathbb R^n\) stands for the state, \(u\in\mathbb R^m\) is the local control input, \(w_i\in\mathbb R^r\) are the square-integrable exogenous disturbances, \(z\in \mathbb R^q\) is an output to be regulated, \(f(x)\), \(g(x)\) are smooth vector fields in \(\mathbb R^n\) and \(\mathbb R^m\), \(g_{w_i}(x)\) is a matrix for weight of exogenous disturbances, \(h(x)\), \(d(x)\) are all known smooth mapping vector functions of appropriate dimensions, and satisfying \(f(0) = 0\), \(h(0) = 0\). The problem of nonlinear \(H_\infty\)-control problem for periodic disturbance attenuation is considered. A methodology for designing nonlinear state feedback controllers that ensure local stability and a prescribed bound for the closed-loop system is developed. The proposed design methodology involves the explicit construction of the control law on a Lyapunov function of system, which avoids the need for solving Hamilton-Jacobi-Isaacs equations.