Optimal nonlinear-nonquadratic feedback control for nonlinear discrete-time systems with \(l_2\) and \(l_{\infty}\) disturbances (Q1567522)

At first a nonlinear discrete-time system with bounded input disturbances and a nonlinear-nonquadratic performance functional is considered. The performance functional is evaluated in terms of a Lyapunov function that guarantees stability and dissipativity with respect to general supply rates. This result is then specialized to linear systems with quadratic supply rates and connections are drawn to the mixed-norm \(H_2/H_{\infty}\) and mixed \(H_2\)/positivity linear-quadratic problems. Then a nonlinear discrete-time disturbance rejection control problem is stated and transformed to an optimal control problem. Sufficient conditions are obtained that characterize an optimal nonlinear feedback controller that guarantees stability and dissipativity of the closed-loop system with respect to general supply rates. This result is specialized to the linear-quadratic case to draw connections with the mixed-norm \(H_2/H_{\infty}\) linear-quadratic regulator problem and mixed \(H_2\)/positivity control problem. At last the obtained results are specialized to discrete-time nonlinear affine systems controlled by nonlinear controllers subjected to bounded exogenous disturbances that minimize a nonlinear-nonquadratic performance criterion involving a nonlinear-nonquadratic function of the state and a linear and a quadratic function of the feedback control.