Optimum nonlinear bang-bang control systems with complex roots: I - System synthesis

ALMOST ALL OF THE WORK published on the synthesis of optimum nonlinear bang-bang or relay control systems has dealt with the case where the controlled system is characterized by real roots. For such systems there have been numerous papers devoted to the derivation of the optimum controller. Outside of Bushaw's work1,2 there has not been much in the literature on the explicit definition of the optimum nonlinear controller in the phase plane, or phase space, in the case where the controlled system is characterized by complex roots. Even in the case of Bushaw's work, the solution for the controller is presented only for regulator-type systems and his solution is not applicable to servomechanism-type systems. These points are discussed in more detail in the first part of this paper. There have been other very fine contributions to the literature on this problem.
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However, in these contributions the question of the complete optimum nonlinear controller as defined in the phase plane or space is still left unsolved.