On the real-time control of time-varying linear systems

Abstract

A new class of sampled data controls for time-varying dynamic processes is developed and investigated. The notions of interval control, prediction and synthesis of the optimum control variable by a linear combination of orthonormal polynomials in time t are introduced and applied in the optimization of a modified least-squares integral index of performance. Minimization of the index of performance leads to a family of control laws which specify the controller synthesis. The resulting control configuration is optimum on a "per interval" basis and is readily realized with available physical components. The design circumvents the complex computational problems associated with the usual calculus of variations and dynamic programming approaches. All controller operations are performed in real time using analog components, and sampling and reset circuits. Controller design is discussed in terms of the available parameters: the relative weighting of system error and control effort, the control interval length, and the degree of the polynomial sum approximation of the control variable. A method of obtaining an engineering estimate of the latter quantity is developed and illustrated by an example. An analog simulation of the control of a second-order dynamic process whose parameter varies in such a manner that the process is unstable at one extremum and heavily damped at the other is presented. Experiments are performed to measure the quality of steady-state control and the transient response of the control scheme.