Synthesis of quasi-stationary optimum nonlinear control systems: Part I ¿ Synthesis considerations

TYPE I SYSTEMS are defined as systems that give optimum time response (i.e., that reduce system error and its derivatives to zero in minimum time) by using maximum control effort. The number of torque reversals was shown by Bogner1 to be (n ¿ 1) for nth-order systems whose characteristic roots are all real and distinct and whose initial conditions are on the switching surface. The switching criterion that gives optimum system response is found to be unique. The plant or controlled system will be assumed to be described by a linear differential equation and, therefore, outside of a nonlinear-controller element, linear theory is fully applicable. Systems which operate on this principle are, sometimes, called piece-wise linear systems. The name is derived from the fact that the error behavior (Appendix) can still be described by a linear differential equation from one switching time to the next if the driving force is completely known. Phase-plane and phase-space concepts will be used throughout the analysis. This paper will be devoted to the optimization method of second- and third-order systems which gives best system response for stationary-class systems as well as for quasi-stationary-class systems, and a good response for properly restricted nonstationary-type systems (Appendix).