Optimal nonlinear feedback control derived from quartic and higher-order performance criteria

Just as minimization of quadratic performance criteria leads to linear feedback, so it is shown here that minimization of integrals containing quartic or hexadic terms in the state variables leads, respectively, to cubic or quintic feedback. This idea is extended to the minimization of integrals of arbitrarily higher order combinations of the state variables, which is desirable in order to impose inequality constraints upon the state variables. Such laws are shown to be adaptive to actuator saturation (including even bang-bang operation). These results are proved by exhibiting a closed-form solution of the corresponding Hamilton-Jacobi equation, which also provides a globally valid Liapunov function. Prior results of Kalman, Haussler, and Rekasius for linear plants appear as special cases. A new constructive procedure for computing the coefficients of the higher-order feedback terms is also presented, together with a numerical application which illustrates remarkable effectiveness in the reduction of overshoots as compared to optimal linear control.