A tutorial introduction to the complex structured singular value (µ) is presented, with an emphasis on the mathematical aspects of µ. The µ-based methods discussed here have been useful for analyzing the performance and robustness properties of linear feedback systems. Several tests

In this second report on system analysis via integral quadratic constraints, the theory is refined compared to Part I [6], to cover a number of additional cases. The report is split into two halfs, denoted Part IIa and Part IIb. Unbounded operators are treated by encapsulating them in a feedback loop, that has bounded closed loop gain. A general theorem for well-posedness of such feedback loops is given. A concept of "fading memory" is introduced and plays an important role in the study of exponential stability. It is also shown how system performance can be studied with restrictions on the class of input signals. In particular, for sinusodal inputs, we compute bounds on high order harmonics in the system response.

The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fields. We begin with a brief introduction to models of computation, the concepts of undecidability, polynomial time algorithms, NP-completeness, and the implications of intractability results. We then survey a number of problems that arise in systems and control theory, some of them classical, some of them related to current research. We discuss them from the point of view of computational complexity and also point out many open problems. In particular, we consider problems related to stability or stabilizability of linear systems with parametric uncertainty, robust control, time-varying linear systems, nonlinear and hybrid systems, and stochastic optimal control.

Optimization problems involving eigenvalues arise in many applications. Let x be a vector of real parameters and let A(x) be a continuously differentiable symmetric matrix function of x. We consider a particular problem which occurs frequently: the minimization of the maximum eigenvalue of A(x), subject to linear constraints and bounds on x. The eigenvalues of A(x) are not differentiable at points x where they coalesce, so the optimization problem is said to be nonsmooth. Furthermore, it is typically the case that the optimization objective tends to make eigenvalues coalesce at a solution point. There are three main purposes of the paper. The first is to present a clear and self-contained derivation of the Clarke generalized gradient of the max eigenvalue function in terms of a "dual matrix". The second purpose is to describe a new algorithm, based on the ideas of a previous paper by the author (SIAM J. Matrix Anal. Appl. 9 (1988) 256-268), which is suitable for solving l...

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Abstract-The structured singular value p measures the robustness of uncertain systems. Numerous researehers over the last decade have worked on developing efficient methods for computing p. This paper considers the complexity of calculating p with general mixed dcomplex 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. I.

It is shown that many system stability and robustness problems can be reduced to the question of when there is a quadratic Lyapunov function of a certain structure which establishes stability of x = Ax for some appropriate A. The existence of such a Lyapunov function can be determined by solving a convex program. We present several numerical methods for these optimization problems. A simple numerical example is given.

We consider linear systems with unspecified parameters that lie between given upper and lower bounds. Except for a few special cases, the computation of many quantities of interest for such systems can be performed only through an exhaustive search in parameter space. We present a general branch and bound algorithm that implements this search in a systematic manner and apply it to computing the minimum stability degree. 1 Introduction 1.1 Notation R (C) denotes the set of real (complex) numbers. For c 2 C, Re c is the real part of c. The set of n \Theta n matrices with real (complex) entries is denoted R n\Thetan (C n\Thetan ). P T stands for the transpose of P , and P , the complex conjugate transpose. I denotes the identity matrix, with size determined from context. For a matrix P 2 R n\Thetan (or C n\Thetan ), i (P ); 1 i n denotes the ith eigenvalue of P (with no particular ordering). oe max (P ) denotes the maximum singular value (or spectral norm) of P , define...

A wide variety of problems in control system theory fall within the class of parameterized Linear Matrix Inequalities (LMIs), that is, LMIs whose coefficients are functions of a parameter conned to a compact set. Such problems, though convex, involve an innite set of LMI constraints, hence are inherently difficult to solve numerically. This paper investigates relaxations of parameterized LMI problems into standard LMI problems using techniques relying on directional convexity concepts. An in-depth discussion of the impacts of the proposed techniques in quadratic programming, Lyapunov-based stability and performance analysis, µ analysis and Linear Parameter Varying control is provided. Illustrative examples are given to demonstrate the usefulness and practicality of the approach.

Aircraft conflict detection and resolution is currently attracting the interest of many air transportation service providers and is concerned with the following question: Given a set of airborne aircraft and their intended trajectories, what control strategy should be followed by the pilots and the air traffic service provider to prevent the aircraft from coming too close to each other? This paper addresses this problem by presenting a distributed air-ground architecture, whereby each aircraft proposes its desired heading while a centralized air traffic control architecture resolves any conflict arising between the aircraft involved in the conflict, while minimizing the deviation between desired and conflict-free heading for each aircraft. The resolution architecture relies on a combination of convex programming and randomized searches: It is shown that aversion of the planar, multi-aircraft conflict resolution problem that accounts for all possible crossing patterns among aircraft might be recast as a nonconvex, quadratically constrained quadratic program. For this type of problem, there exist efficient numerical relaxations, based on semidefinite programming, that provide lower bounds