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

Cheap control for a class of non-linear systems

“…To simplify our analysis of the solution of (16) with respect to the non-linear terms f(x) and g(x), we analyse the solution of (17) for a certain~(m)(x) by the eigenvector technique (Bass and Webber 1966). and 4>~m)(xrl, 4>h m)(x,) and 4>~m)(x) are terms about X r only, terms about x, only and remainders, respectively.…”

“…When the eigenvalues of A * are all distinct, (17) can be solved effectively by the eigenvector technique (Bass and Webber 1966). For brevity, we suppose that the eigenvalues of A * be all distinct.…”

“…As in Lee and Bien (1987), the power series method is used to solve the cheap control problem, in which a linear quadratic regulator is obtained as a first approximation and is subsequently corrected by series expansions (Lukes 1969). In analysing the power series solution, a transformation of the problem to the form for which singular and/or general perturbation theory might be applicable (Sannuti 1983) together with an eigenvector technique for solving a Lyapunov-type system of linear equations (Bass and Webber 1966) are utilized. The transformation technique is based on the appropriate selection and scaling of state variables and has been used in analysing the linear cheap control problems by Sannuti (1983) and the eigenvector technique has been used in designing non-linear optimal regulators for linear systems (Bass and Webber 1966).…”

“…In analysing the power series solution, a transformation of the problem to the form for which singular and/or general perturbation theory might be applicable (Sannuti 1983) together with an eigenvector technique for solving a Lyapunov-type system of linear equations (Bass and Webber 1966) are utilized. The transformation technique is based on the appropriate selection and scaling of state variables and has been used in analysing the linear cheap control problems by Sannuti (1983) and the eigenvector technique has been used in designing non-linear optimal regulators for linear systems (Bass and Webber 1966). By employing these techniques, a class of non-linear systems is sought for which perfect regulation is possible and a truncated control law becomes good enough to approximate the cheap control law, which is an optimal control law (Lee and Bien series solution is characterized by an eigenvector technique and conditions are presented under which the optimal cost will become zero as the small parameter multiplied with the control term goes to zero.…”

Selection of optimal inner-product controllers for oscillatory plants†

“…In such cases, the general solution leads to an open-loop control law derived from the iterative solution to a two-point boundary value problem (Leondes 1966). Explicit analytic expressions are generally very difficult to find, but for special cases the optimal control law can be found in closed-loop form (Bass and'Webber 1966, Asseo 1969). In such cases, the feedback controller utilizes non-linear feedback, which experience indicates is useful in improving, for example, step response figures of merit such as rise time and overshoot.…”