Properties of optimal stochastic control systems with dead-time

Abstract

Structural, stability and sensitivity properties of optimal stochastic control systems for dead-time, stable minimum phase as well as non-minimum phase processes are presented. The processes are described by rational transfer functions plus dead-times and the disturbances by rational spectral densities. It is shown that although the frequency domain design techniques guarantee asymptotically stable systems for given process and disturbance models, many of the designs might be practically unstable. Necessary and sufficient conditions that must be imposed on the design to assure practically stable optimal systems are derived. The uncertainties in the parameters and in the structure of the process model are measured by means of an ignorance function. Sufficient conditions in terms of the ignorance function, which guarantee stable design and by means of which the bounds of the uncertainties for a given design may be estimated, are stated. Conditions under which the optimal designs possess attractive relative stability properties, namely gain and phase margins of at least 2 and 60°, respectively, are stated, too. It is further shown that any optimal controller, for the type of processes discussed in this paper, may be separated into a primary controller and into a dead-time compensator where the latter is completely independent of the cost and the disturbance properties. Such a decomposition gives excellent insight into the role of the cost and the disturbance in the design. When low order process and disturbance models are used, the conventional PI and PID control laws coupled with the dead-time compensator emerge.