Design and Analog-Computer Analysis of an Optimum Third-Order Nonlinear Servomechanism

Great interest has been shown recently in deliberately nonlinear control systems, including a class of programmed control systems. The design objective which distinguishes these systems is minimization of the response time for step inputs by proper automatic timing of relay operations. For a second-order relay servomechanism, the required switching relation is expressed by a curve in a two-dimensional phase plane which can be realized by a one-variable function generator in combination with linear elements. For a third-order system, the switching relation is expressed by a surface in a three-dimensional phase space and requires a two-variable function generator for its realization. The switching surface is computed in this paper for an idealized third-order positioning servomechanism having an output member characterized completely by its moment of inertia and a torque which varies linearly with time between two limits; small-signal nonlinearities such as backlash or relay threshold are neglected. An electro-optical two-variable function generator is employed in an analog-computer study of this system. For purposes of comparison, two alternative modes of control also are examined. The system using programmed control shows the expected superiority for step inputs, the advantage being greatest for small step magnitudes. For sinusoidal or random inputs, programmed control is superior when the input amplitude and/or frequency are low but exhibits some anomalous behavior for large amplitudes and/or frequencies, which result in inferior performance. Parameter tolerances for programmed control systems are somewhat less severe than anticipated.