Modelling linear systems for pulsewidth-modulated control
Bernard Friedland · IEEE Transactions on Automatic Control · 1976
An approximate linear model is developed for a linear process in which the control signal can assume only two values,U_{\max}andU_{\min}, and which is controlled by varying the fraction δnof a sampling cycle of durationTthat the control is atU_{\max}. The dynamic equations are of the formw_{n+1} = \Phi(T)w_{n} + \bar{\Gamma}(T)r_{n}where wnis related to the average state xnover one cycle, andr_{n} = \delta_{n} - \delta_{s}where δsis the steady-state value of δnrequired to maintain a desired average state xd. The system errore(nT)at sampling intervals is related to these variables by an equation of the forme(nT) = M_{1}w_{n} + M_{2}\delta_{s} + M_{3}b, wherebis a bias vector. These relations may be used to design a linear control system by well-known techniques for discrete-time systems. The method is illustrated by the design of a third-order process which could represent a temperature control problem. Simulation results are given for a design that includes a Kalman filter for estimating the inaccessible states.