Optimization of pulse frequency modulated control systems via modified maximum principle

The problem of finding the optimum control function for the pulse frequency modulated (PFM) system is considered in this paper. In the PFM systems discussed here, the control function consists of a series of standard pulses. The optimization procedure consists of determining the polarity and positions of the pulses which make up the control function. The performance index is assumed to be a linear combination of the final values of the state variables. This does not exclude the problem of optimizing a system with respect to an integral, provided that the integrand is linear with respect to the state variables, but not necessarily with respect to the control function. In the PFM systems considered, the control function is fixed for a period of time following the initiation of each pulse. This fact precludes the direct application of the existing standard optimizing techniques. The Modified Maximum Principle is presented. It is based on Pontryagin's Maximum Principle and is applicable to open-loop systems with linear plants with fixed operating time. The Modified Maximum Principle is valid for systems with and without final value constraints on the state variables.