Modelling linear systems for pulsewidth-modulated control

An approximate linear model is developed for a linear process in which the control signal can assume only two values,
U_{\max}
and
U_{\min}
, and which is controlled by varying the fraction δ
n
of a sampling cycle of duration
T
that the control is at
U_{\max}
. The dynamic equations are of the form
w_{n+1} = \Phi(T)w_{n} + \bar{\Gamma}(T)r_{n}
where w
n
is related to the average state x
n
over one cycle, and
r_{n} = \delta_{n} - \delta_{s}
where δ
s
is the steady-state value of δ
n
required to maintain a desired average state x
d
. The system error
e(nT)
at sampling intervals is related to these variables by an equation of the form
e(nT) = M_{1}w_{n} + M_{2}\delta_{s} + M_{3}b
, where
b
is 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.