On-off limit-cycle controllers for reaction-jet-controlled systems

Analytical expressions have been obtained by Mendel [1] for performance cost functions that describe the behavior of a reaction-jet-controlled system during an on-off limit cycle. The cost functions are expressed in terms of the amplitudes of the limit-cycle switch statestheta_{s}anddot{theta}_{s}. No structure is assumed foru(t)in [1]. In this paper,u(t)is chosen to be an asymmetrical deadzone function that is proceeded by aT-second time delay. Contained withinu(t)are three design variables: control gainsK_{theta}andK_dot{theta}, and asymmetryl. This paper shows howtheta_{s}anddot{theta}_{s}can be obtained from prespecified values ofK_{theta}andK_{dot{theta}}, and howK_{theta}andK_{dot{theta}}can be obtained from prespecified values oftheta_{s}anddot{theta}_{s}; thus, it provides one with the means for utilizing the results in [1] for a specified controller. In addition, conditions are obtained for the existence of an on-off limit cycle. These conditions are interpreted in the phase plane; that is to say, each condition is viewed as a phase-plane constraint or constraints betweentheta_{s}anddot{theta}_{s}. It is shown how to construct the admissible regions fortheta_{s}anddot{theta}_{s}ahead of time. It is also shown how to choose the asymmetry variablelso that limit-cycle existence is assured. All of the results are brought together first as a synthesis procedure, and then as an analysis procedure. Details of both procedures are given. This paper provides the control system designer with new tools for: 1) better understanding the limitations of a fixed controller for the reaction-jet-controlled vehicle; 2) more efficient ways to analyze choices of the controller's parameters on system performance; and 3) more efficient ways to choose these parameters.