Upper and lower bounds for the mixed u problem have recently been developed, and this paper examines the computational aspects of these bounds. In particular a practical algorithm is developed to compute the bounds. This has been implemented as a Matlab function (m-file), and will be available shortly in a test version in conjunction with the p-Tools toolbox. The algorithm performance is very encouraging, both in terms of accuracy of the resulting bounds, and growth rate in required computation with problem size. In particular it appears that one can handle medium size problems (less than 100 perturbations) with reasonable computational reqmurements.

This paper gives a broad overview, from a LFT /µperspective, of some of the theoretical and practical issues associated with robustness in the presence of real parametric uncertainty, with a focus on computation. Recent results on the properties of µ in the mixed case are reviewed, including issues of NP completeness, continuity, computation of bounds, the equivalence ofµ and its bounds, and some direct comparisons with "Kharitonov-type" analysis methods. In addition, some advances in the computational aspects of the problem, including a new Branch and Bound algorithm, are briefly presented together with numerical results. The results of this paper strongly suggest that while the mixed µ problem may have inherently combinatoric worst-case behavior, practical algorithms with modest computational requirements can be developed for problems of medium size ( < 100 parameters) that are of engineering interest.

Robustness problems involving real parametric uncertainty can be reformulated as mixed p problems, where the block-structured uncertainty description is allowed to contain both real and complex blocks. Upper and lower bounds for the mixed p problem have recently been developed, and this paper examines the computational aspects of these bounds. In particular a practical algorithm is developed to compute the bounds. This has been implemented as a Matlab function (m-file), which is currently available in conjunction with the p-Tools toolbox. Some results from our extensive numerical experience with the algorithm are presented. The algorithm performance is very encouraging, in terms both of accuracy of the resulting bounds, and of growth rate in required computation with problem size. In particular it appears that one can handle medium-size problems (fewer than 100 perturbations) with reasonable computational requirements.

SUMMARYThe computation of the general structural singular value ( ) is NP hard, so quick solutions to medium sized problems must often be approximate. In many of the cases where the current approximate methods are unsatisfactory, improved solutions are highly desirable. It is shown that, despite their worst-case combinatorial nature, branch and bound techniques can give substantially improved solutions with only moderate computational cost.

A unified systems analysis framework is presented which includes conventional robustness analysis, model validation, and system identification as special cases and thus shows them to be instances of the same fundamental problem. A concrete version of this framework is developed for the linear case, based on a generalized structured singular value. This unification forms the basis for the use of common computational tools and and a more natural interplay between modeling, identification, and robustness analysis.

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The computation of the general structural singular value (μ) is NP hard, so quick solutions to medium sized problems must often be approximate. In many of the cases where the current approximate methods are unsatisfactory, improved solutions are highly desirable. It is shown that, despite their worst‐case combinatorial nature, branch and bound techniques can give substantially improved solutions with only moderate computational cost. © 1997 by John Wiley & Sons, Ltd.