Dynamic analysis of nonlinear control systems

A method of transient analysis which applies to autonomous and nonautonomous nonlinear systems of any order or configuration is presented. After noting that the transients which occur in such systems may be represented by an exponentially damped sinewave with time-varying damping and frequency, it is shown, by assuming the damping and frequency to be piecewise constant, that the nonlinearity in the system may be adequately characterised by an equivalent gain for transient phenomena. The equation governing the damping and frequency is then derived and shown to be nonlinear. An approximate solution of this equation is obtained, and a method is then developed for calculating the error in the approximation. Application of the theory to a bang-bang position-control system and a position-control system with resetter backlash illustrates the technique for systems with single-valued and phase-shifting nonlinearities. It is shown that the accuracy of prediction is similar to that obtained when the describing-function technique is used to predict limit cycles. Finally the results from this method are compared with those based on the describing-function characterisation, and it is found that appreciable improvement in the accuracy of prediction is obtained.