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Inversion Error, Condition Number, And Approximate Inverses Of Uncertain Matrices
 Inverses of Uncertain Matrices. Linear Algebra and its Applications
, 2000
"... The classical condition number is a very rough measure of the effect of perturbations on the inverse of a square matrix. First, it assumes the perturbation is infinitesimally small. Second, it does not take into account the perturbation structure (e.g., Vandermonde). Similarly, the classical notion ..."
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The classical condition number is a very rough measure of the effect of perturbations on the inverse of a square matrix. First, it assumes the perturbation is infinitesimally small. Second, it does not take into account the perturbation structure (e.g., Vandermonde). Similarly, the classical notion of inverse of a matrix neglects the possibility of large, structured perturbations. We define a new quantity, the structured maximal inversion error, that takes into account both structure and non necessarily small perturbation size. When the perturbation is infinitesimal, we obtain a "structured condition number". We introduce the notion of approximate inverse, as a matrix that best approximates the inverse of a matrix with structured perturbations, when the perturbation varies in a given range. For a wide class of perturbation structures, we show how to use (convex) semidefinite programming to compute bounds on on the structured maximal inversion error and structured condition number, and compute an approximate inverse. The results are exact when the perturbation is "unstructured"we then obtain an analytic expression for the approximate inverse. When the perturbation is unstructured and additive, we recover the classical condition number; the approximate inverse is the operator related to the Total Least Squares (orthogonal regression) problem.
Algorithms and Software for LMI Problems in Control
 IEEE Control Systems Magazine
, 1997
"... this article is to provide an overview of the state of the art of ..."
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this article is to provide an overview of the state of the art of
Constrained optimal fitting of threedimensional vector patterns
 IEEE Transactions on Robotics and Automation
, 1998
"... Abstract — This paper addresses the problem of finding whether a given set of threedimensional (3D) vectors (the object) can be brought to match a second set of vectors (the template) by means of an affine motion, minimizing a measure of the mismatch error and satisfying an assigned set of geometr ..."
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Abstract — This paper addresses the problem of finding whether a given set of threedimensional (3D) vectors (the object) can be brought to match a second set of vectors (the template) by means of an affine motion, minimizing a measure of the mismatch error and satisfying an assigned set of geometrical constraints. This problem is encountered in many applications of computer vision, robotics, and manufacturing processes, and has been tackled by several authors in the unconstrained case. Spherical, ellipsoidal and polyhedral constraints are here introduced in the problem, and a solution scheme based on an efficient convex optimization algorithm is proposed. An example of application of the proposed methodology to a manufacturing tolerancing problem is also provided. Index Terms—Absolute orientation, automated tolerancing, computer vision, convex optimization, least squares, visual servo control.
LMI Optimization for Nonstandard Riccati Equations Arising in Stochastic Control
, 1996
"... We consider coupled Riccati equations that arise in the optimal control of jump linear systems. We show how to reliably solve these equations using convex optimization over linear matrix inequalities (LMIs). The results extend to other nonstandard Riccati equations, that arise e.g. in the optimal co ..."
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We consider coupled Riccati equations that arise in the optimal control of jump linear systems. We show how to reliably solve these equations using convex optimization over linear matrix inequalities (LMIs). The results extend to other nonstandard Riccati equations, that arise e.g. in the optimal control of linear systems subject to statedependent multiplicative noise. Some nonstandard Riccati equations (such as those connected to linear systems subject to both state and controldependent multiplicative noise) are not amenable to the method. We show that we can still use LMI optimization to compute the optimal control law for the underlying control problem, without solving the Riccati equation. Keywords: Jump linear system, multiplicative noise, nonstandard Riccati equation, Linear Matrix Inequality. Research supported in part by DRET under contract 92017BC14. y Author to whom correspondence should be sent. Internet: elghaoui@ensta.fr 1 Notation. For a real matrix A, A ? 0 (r...
Robust Multiobjective Lti Control Design For Systems With Structured Perturbations
, 1996
"... . We consider a multiobjective robust controller synthesis problem for an LTI system subject to structured perturbations. Our design specifications include robust stability, robust performance (H 2 norm) bounds and timedomain bounds (output and command input peak). We derive sufficient conditions, ..."
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. We consider a multiobjective robust controller synthesis problem for an LTI system subject to structured perturbations. Our design specifications include robust stability, robust performance (H 2 norm) bounds and timedomain bounds (output and command input peak). We derive sufficient conditions, based on a single quadratic Lyapunov function, for the existence of an LTI controller such that the closedloop system satisfies all specifications simultaneously. These nonconvex conditions can be numerically checked with an efficient LMIbased heuristic. We illustrate our results on a threeparameter mass/spring system. Keywords. Multiobjective robust control, mixed H 2 /H1 , timedomain specification, structured perturbation, quadratic Lyapunov function, linear matrix inequality. 1. INTRODUCTION In a multiobjective robust control problem, one seeks a control law such that the closedloop system satisfies a number of objectives (specifications) despite unkwnonbut bounded perturbations. I...
Inversion error, condition number, and
, 2000
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LQ Control without Riccati Equations: Deterministic Systems
, 1999
"... We study a deterministic linearquadratic (LQ) control problem over an infinite horizon, and develop a general approach to the problem based on semidefinite programming (SDP) and related duality analysis. This approach allows the control cost matrix R to be nonnegative (semidefinite), a case that ..."
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We study a deterministic linearquadratic (LQ) control problem over an infinite horizon, and develop a general approach to the problem based on semidefinite programming (SDP) and related duality analysis. This approach allows the control cost matrix R to be nonnegative (semidefinite), a case that is beyond the scope of the classical approach based on Riccati equations. We show that the complementary duality condition of the SDP is necessary and sufficient for the existence of an optimal LQ control. Moreover, when the complementary duality does hold, an optimal state feedback control is constructed explicitly in terms of the solution to the semidefinite program. On the other hand, when the complementary duality fails, the LQ problem has no attainable optimal solution, and we develop an fflapproximation scheme that achieves asymptotic optimality.
Robust Solutions To LeastSquares Problems With Uncertain Data
, 1997
"... We consider leastsquares problems where the coefficient matrices A, b are unknown but bounded. 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 interprete ..."
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We consider leastsquares problems where the coefficient matrices A, b are unknown but bounded. 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.