The Sensitivity of Computational Control Problems

It is well-known that many factors contribute to the accurate and efficient numerical solution of mathematical problems such as those arising in computational control system design. In simple terms these are the arithmetic of the machine on which the calculations are carried out, sensitivity (or conditioning) of the mathematical model to small changes of the data and the numerical stability of the algorithms. It happens quite often that these concepts are confused. We define these concepts and demonstrate some of the subtleties that often lead to confusion. In particular we demonstrate with several examples what may happen when a problem is modularized, i.e., split into subproblems for which computational modules are available. For three classical problems in computational control, pole placement, linear quadratic control and optimal $H_\infty$ control, we then discuss the conditioning of the problems and point out sources of difficulties. We give some ill-conditioned examples for which even numerically stable methods fail. We also stress the need for condition and error estimators that supplement the numerical algorithm and inform the user about potential or actual difficulties, and we explain what can be done to avoid these difficulties. We finally describe the current status of available condition estimators in numerical methods for control system design such as those available in the SLICOT library and we point out important areas of future research that need to be adressed