LFT-free μ-analysis of LTI/LPTV systems

Abstract

μ-analysis has for many years been the de facto standard robustness analysis tool in a wide variety of control applications. To use μ-analysis, the uncertain system must be cast in the form of a linear fractional transformation (LFT). Once it is appropriately transformed, a variety of algorithms are available to calculate the upper and the lower bounds on μ. Several difficulties arise during this process. Firstly, the uncertainties in the system must appear as polynomial fractions - if they do not, then some approximation steps must be applied in order to write the system as an LFT. Secondly, symbolic manipulations are required in general in order to obtain linear fractional transformation automatically. Finally, for problems involving real parametric uncertainties calculating the lower bound is a well known NP(Non-deterministic Polynomialtime)-hard problem. Therefore, there will always be some conservatism introduced in the bound or else the computation time will increase exponentially with the number of uncertain parameters. We present an efficient algorithm to overcome these issues by combining a randomisation approach and a geometric interpretation of the robustness analysis problem. The uncertainty is re-defined by a subtraction between the uncertain system and the nominal system. Thus the procedure does not require that the uncertain parameters are actually decoupled from the system (as with LFT's) but only requires the evaluation of the difference between the nominal system and the perturbed system. Here, we illustrate the application of the proposed approach to the robustness analysis of uncertain linear periodically time-varying systems, and in particular to magnetic torquer controlled spacecraft attitude dynamics.