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70
Robust Solutions To LeastSquares Problems With Uncertain Data
, 1997
"... . We consider leastsquares problems where the coefficient matrices A; b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpret ..."
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Cited by 200 (14 self)
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. We consider leastsquares problems where the coefficient matrices A; b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder 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 polynomialtime using semidefinite programming (SDP). We also consider the case when A; b are rational functions of an unknownbutbounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worstcase residual. We provide numerical examples, including one from robust identification and one from robust interpolation. Key Words. Leastsquares, uncertainty, robustness, secondorder cone...
A Survey of Computational Complexity Results in Systems and Control
, 2000
"... 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 fi ..."
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Cited by 188 (21 self)
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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, NPcompleteness, 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, timevarying linear systems, nonlinear and hybrid systems, and stochastic optimal control.
NPhardness of some linear control design problems
, 1997
"... We show that some basic linear control design problems are NPhard, 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 bound ..."
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Cited by 85 (2 self)
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We show that some basic linear control design problems are NPhard, 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 NPhard.
The Boundedness of All Products of a Pair of Matrices is Undecidable
, 2000
"... 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 stabil ..."
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Cited by 71 (17 self)
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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 timevarying 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 socalled "finiteness conjecture". Our results are based on a simple reduction from the emptiness problem for probabilistic finite automata, which is known to be undecidable.
Linear Fitting with Missing Data for StructurefromMotion
 Computer Vision and Image Understanding
, 1997
"... 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 ..."
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Cited by 61 (6 self)
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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 3D structure from a sequence of 2D images. 3 A Novel Algorithm
Randomized algorithms for probabilistic robustness with real and complex structured uncertainty
 IEEE Trans. Autom. Control
, 2000
"... Abstract—In recent years, there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems. Unlike classical worst case methods, these algorithms provide probabilistic estimates assessing, for instance, if a certain design specification ..."
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Cited by 43 (10 self)
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Abstract—In recent years, there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems. Unlike classical worst case methods, these algorithms provide probabilistic estimates assessing, for instance, if a certain design specification is met with a given probability. One of the advantages of this approach is that the robustness margins can be often increased by a considerable amount, at the expense of a small risk. In this sense, randomized algorithms may be used by the control engineer together with standard worst case methods to obtain additional useful information. The applicability of these probabilistic methods to robust control is presently limited by the fact that the sample generation is feasible only in very special cases which include systems affected by real parametric uncertainty bounded in rectangles or spheres. Sampling in more general uncertainty sets is generally performed through overbounding, at the expense of an exponential rejection rate. In this paper, randomized algorithms for stability and performance of linear time invariant uncertain systems described by a general1 configuration are studied. In particular, efficient polynomialtime algorithms for uncertainty structures 1 consisting of an arbitrary number of full complex blocks and uncertain parameters are developed. Index Terms—Random matrices, randomized algorithms, robust control, uncertainty. I.
Computational complexity of calculation
 IEEE Trans. Autom. Control
, 1994
"... AbstractThe 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 ..."
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Cited by 41 (10 self)
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AbstractThe 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 NPhard. 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.
A Survey of Componentwise Perturbation Theory in Numerical Linear Algebra
 in Mathematics of Computation 19431993: A Half Century of Computational Mathematics
, 1994
"... . 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 an ..."
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Cited by 23 (0 self)
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. 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 ...