On the optimization of linear, time-variant, multivariable control systems using the contraction mapping principle

Abstract

The problem of optimizing a linear, time-variant, multivariable control system with quadratic cost functional is first phrased in the context of functional analysis. It is shown that the optimum control is given implicitly as the solution of a matrix integral equation. After the fixed-point contraction mapping theorem is invoked, an iterative method for solving this equation is developed and conditions for its convergence to a unique optimum are derived. Techniques for transforming the system operator are discussed so that, when convergence of the original sequence of iterations cannot be assured, that of the transformed system can. However, it is shown specifically that there is always a finite optimization interval for which the procedure may be used. Bounds are also given for the errors, in the sense of norms, between the control aftern iterations and its ultimate value and between the cost functional aftern iterations and its ultimate value. These bounds are used to decide when to terminate the sequence. Solutions of the iterative scheme using a hybrid computer in parallel and serial modes are discussed and the delays inherent in both methods calculated. It is concluded that the method can be used to track an optimum control system, which drifts from optimum because of parameter variations, with little delay and particularly when the optimization interval is extrapolated only a little into the future. Comparison of the proposed scheme with the steepest-descent approach developed by Balakrishnan shows that the present scheme requires one-third of the computations per step and, therefore, may converge more quickly.