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 states
\theta_{s}
and
\dot{\theta}_{s}
. No structure is assumed for
u(t)
in [1]. In this paper,
u(t)
is chosen to be an asymmetrical deadzone function that is proceeded by a
T
-second time delay. Contained within
u(t)
are three design variables: control gains
K_{\theta}
and
K_\dot{\theta}
, and asymmetry
l
. This paper shows how
\theta_{s}
and
\dot{\theta}_{s}
can be obtained from prespecified values of
K_{\theta}
and
K_{\dot{\theta}}
, and how
K_{\theta}
and
K_{\dot{\theta}}
can be obtained from prespecified values of
\theta_{s}
and
\dot{\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 between
\theta_{s}
and
\dot{\theta}_{s}
. It is shown how to construct the admissible regions for
\theta_{s}
and
\dot{\theta}_{s}
ahead of time. It is also shown how to choose the asymmetry variable
l
so 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.