The robust \(H_\infty\) state feedback control of nonlinear discrete-time system with uncertainty (Q2785206)

The authors study a problem about robustness for the uncertain discrete-time nonlinear system: NEWLINE\[NEWLINE \begin{matrix} x_{k+1}=f(x_k)+\Delta f(x_k)+[g_1(x_k)+\Delta g_1(x_k)]u_k+g_2(x_k)w_k,\\ z_k=h(x_k)+l(x_k)u_k,\end{matrix}NEWLINE\]NEWLINE where \(x_k\) is the state vector, \(z_k\) is the output vector, \(u_k\) is the control, \(w_k\) is the exogenous disturbance with a bounded \(l_2\)-norm, and \(\Delta f\) and \(\Delta g_1\) are the perturbations of \(f\) and \(g\), respectively. They give a sufficient condition, which is an extension of Lin and Byrnes' results, for the existence of an \(H_{\infty}\) state feedback controller which guarantees the robust stability and a disturbance attenuation level, and which can be obtained in terms of solving the Hamilton-Jacobi-Isaacs equation.