Optimization of discrete-time systems with combined modulation

In the past a great deal of attention has been given to discrete-time systems containing either pulse-width or pulse-amplitude modulators. This paper presents general necessary conditions that a sequence of pulse amplitudes and pulse widths must satisfy to minimize a function of the states at the sampling instants plus a function of the states between the sampling instants. These necessary conditions will also apply to systems with either pulse-amplitude or pulse-width modulation done by simply deleting various parts of the general conditions. The results of this paper are applicable to the design of discrete controllers.