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uncertainty in the framework of combinatorial complexity theory. In particular, it is proved that the p recognition problem with either pure real or mixed reaUcomplex uncertainty is NP-hard. This strongly suggests that it is fbtile to pursue exact methods for calculating p of general systems with pure real or mixed uncertainty for other than small problems.

We reexamine the problem of having nonconservative equations of motion arise from the use of a variational principle. In particular, a formalism is developed that allows the inclusion of fractional derivatives. This is done within the Lagrangian framework by treating the action as a Volterra series. It is then possible to derive two equations of motion, one of these is an advanced equation and the other is retarded.

This letter provides a brief explanation of echo state networks and provides a rigorous bound for guaranteeing asymptotic stability of these networks. The stability bounds presented here could aid in the design of echo state networks that would be applicable to control applications where stability is required. Index Termsecho state networks, weighted operator norms, recurrent neural networks, nonlinear systems, Lyapunov stability, robust controls

The structured singular value p measures the robustness of uncertain systems. Numerous researchers over the last decade have worked on developing efficient methods for computing p. This paper considers the complexity of calculating p with general mixed real/complex uncertainty in the framework of combinatorial complexity theory. In particular, it is proved that the 11 recognition problem with either pure real or mixed real/complex uncertainty is NP-hard. This strongly suggests that it is futile to pursue exact methods for calculating p of general systems with pure real or mixed uncertainty for other than small problems. IntroductionRobust stability and performance analysis with real parametric and dynamic uncertainties can be naturally formulated as a structured singular value (or p ) problem, where the block structured uncertainty description is allowed t o contain both real and complex blocks. I t is assumed that the reader is familar with this type of robustness analysis, as space constraints preclude covering *supported by the Fannie and John Hertz Foundation 'Chemical Engineering, 210-41, California Institute of Technology, Pasadena, CA 91125, (818)395-4186 $~l e c t r i c d Engineering, 116-81, California Institute of Technology, Pasadena, CA 91125, (818)395-4808 §TO whom correspondence should be addressed: phone (818)395-4186, fax (818)568-8743, e-mail mm@imc.caltech.edu this here. For a collection of papers describing the engineering motivation and the computational approaches, see [3] and the references contained within.In this paper we determine the computational complexity of p calculation with either pure real or mixed real/complex uncertainty. To apply computational complexity theory, we formulate p calculation as a recognition problem (a 'yes' or 'no' problem). We show that this recognition problem is NP-hard, i.e. a t least as hard as the NP-complete problems.The exact consequences of a problem being NP-complete is still a fundamental open question in the theory of computational complexity, and we refer the reader to Garey and Johnson [5] for an in depth treatment of the subject. However, it is generally accepted that a problem being NP-complete means that it cannot be computed in polynomial time in the worst case.It is important to note that being NP-complete is a property of the problem itself, not of any particular algorithm. The fact that the mixed p problem is NP-hard strongly suggests that,given any algorithm to compute p, there will be problems for which the algorithm cannot find the answer in polynomial time.The terminology of computational complexity theory is used extensively in this paper. The definitions for NP-complete, NP-hard, recognition problems, and other terms agree with those in the well-known textbooks by Garey and Johnson [5] and Papadimitriou and Steiglitz 181.The proofs are simple. First we show that indefinite quadratic programming can be cast as a p problem of "roughly" the same size. Since the recognition problem for indefinite quadratic programming is NP-complete, the p ...

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.

The robustness analysis of system performance is one of the key issues in control theory, and one approach is to reduce this problem to that of computing the structured singular value, p. When real parametric uncertainty is included, then p must be computed with respect to a block structure containing both real and complex uncertainties, and this is the situation considered here. It is shown that p is equivalent to a real eigenvalue maximization problem, and a power algorithm is developed to solve this problem. The algorithm has the property that p is (almost) always an equilibrium point of the algorithm, and that whenever the algorithm converges a lower bound for p results. Numerical experience with the algorithm is very encouraging.