. We consider least-squares problems where the coefficient matrices A; b are unknown-butbounded. We minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution, and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomial-time using semidefinite programming (SDP). We also consider the case when A; b are rational functions of an unknown-but-bounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worst-case residual. We provide numerical examples, including one from robust identification and one from robust interpolation. Key Words. Least-squares, uncertainty, robustness, second-order cone...

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.

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We show that some basic linear control design problems are NP-hard, implying that, unless P=NP, they cannot be solved by polynomial time algorithms. The problems that we consider include simultaneous stabilization by output feedback, stabilization by state or output feedback in the presence of bounds on the elements of the gain matrix, and decentralized control. These results are obtained by first showing that checking the existence of a stable matrix in an interval family of matrices is NP-hard.

We show that the boundedness of the set of all products of a given pair Sigma of rational matrices is undecidable. Furthermore, we show that the joint (or generalized) spectral radius #(#) is not computable because testing whether #(#)61 is an undecidable problem. As a consequence, the robust stability of linear systems under time-varying perturbations is undecidable, and the same is true for the stability of a simple class of hybrid systems. We also discuss some connections with the so-called "finiteness conjecture". Our results are based on a simple reduction from the emptiness problem for probabilistic finite automata, which is known to be undecidable.

this paper. This method is described in detail in [15]. We can briefly describe the method as formulating the least squares problem as a bilinear optimization, and then iteratively holding one set of variables constant while the others are optimized, so that each optimization is linear. We use their method in our experiments, because it has good convergence properties and is easy to implement. For the problem they consider, Shum et al. state that a random starting point is sufficient to produce a good final solution. However, their experiments on this point cannot be used to draw conclusions for the problem of determining 3-D structure from a sequence of 2-D images. 3 A Novel Algorithm

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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.

. Perturbation bounds in numerical linear algebra are traditionally derived and expressed using norms. Norm bounds cannot reflect the scaling or sparsity of a problem and its perturbation, and so can be unduly weak. If the problem data and its perturbation are measured componentwise, much smaller and more revealing bounds can be obtained. A survey is given of componentwise perturbation theory in numerical linear algebra, covering linear systems, the matrix inverse, matrix factorizations, the least squares problem, and the eigenvalue and singular value problems. Most of the results described have been published in the last five years. Our hero is the intrepid, yet sensitive matrix A. Our villain is E, who keeps perturbing A. When A is perturbed he puts on a crumpled hat: e A = A+E. G. W. Stewart and J.-G. Sun, Matrix Perturbation Theory (1990) 1. Introduction Matrix analysis would not have developed into the vast subject it is today without the concept of representing a matrix by ...

Abstract. We prove that it is NP-hard to compute the exact componentwise bounds on solutions of all the linear systems which can be obtained from a given linear system with a nonsingular matrix by perturbing all the data independently of each other within prescribed tolerances. (1)