## Abstract

A new class of control systems termed predictive adaptive controls is developed and the performance characteristics are investigated analytically and experimentally. The concepts of signal prediction, interval control, and synthesis of the control variable by a sum of orthonormal polynomials in t are introduced and developed in relation to adaptive control. A modified least squares integral index of performance is formulated and used as the criterion for system optimization. Control of dynamic processes is subdivided into intervals of a specified length T and prediction is used to obtain estimates of future values of system error. Minimization of the index of performance leads to a family of control laws which specify the structure of the controller. The resulting control configuration is optimum in a specific mathematical sense and is readily realizable with available physical components. The adaptive capability is achieved through time-varying gains which are specific functions of the unit impulse response of the dynamic process being controlled. Predictor design is presented in terms of the classical Wiener-Lee theory, and a relationship for control interval length as a function of prediction accuracy is developed. Preliminary design of the controller is considered from the viewpoints of relative weighting of system error and control effort, control interval length T, and the number of terms needed In the orthonormal polynomial sum approximation of the control variable. A method of obtaining an engineering estimate of the latter quantity is developed and 11 lustrated by three examples, two of which are investigated experimentally. Two applications of predictive adaptive control are investigated on an analog computer. The two dynamic processes used are a first-order process whose parameter varies over a range of ten to one and a second-order process whose parameter varies in such a manner that the process is unstable at one extremum and heavily damped at the other. The results of three basic experiments which evaluate the steady-state adaptability transient response, and statistical signal response of the two systems are reported. It is found that all three aspects of system performance improve with decreasing control interval length, but that the minimum value of the interval length which can be used is limited by the accuracy of the time-varying gain and controller circuitry. Improved performance which can be achieved by increasing the relative weighting of System error and control effort, is limited by saturation considerations. Theoretical, results that .point to the need., for keeping the control interval length short to preserve stability, prediction accuracy, and loss of control doe to process parameter drift are substantiated by the experimental results. For the two systems investigated it is found that satisfactory control Is achieved If the interval length is chosen so that process parameter drift Is no more than 4% per control interval, A figure of 5% was estimated originally, : A one-term approximation of the control variable is used to control .the first-order process- and Is found to give satisfactory performance. A four-term approximation Is found to give adequate control of the second- order process whereas the three-term approximation does not. These results bear out the predictions made in the-theoretical analyses.

## Date of this Version

2-1-1961