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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 ..."
Abstract

Cited by 149 (13 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...
Robust Least Squares and Applications
"... We consider leastsquares problems where the coefficient matrices A, b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder cone programming (SOCP), yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be int ..."
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Cited by 2 (0 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 (SOCP), 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 per turbation 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.
A Fast Algorithm for the Computation of an Upper Bound on the µnorm
, 1999
"... A fast algorithm for the computation of the optimally frequencydependent scaled H1norm of a finite dimensional LTI system is presented. It is well known that this quantity is an upper bound to the "¯norm"; furthermore, it was recently shown to play a special role in the context of slowly timevary ..."
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Cited by 1 (1 self)
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A fast algorithm for the computation of the optimally frequencydependent scaled H1norm of a finite dimensional LTI system is presented. It is well known that this quantity is an upper bound to the "¯norm"; furthermore, it was recently shown to play a special role in the context of slowly timevarying uncertainty. Numerical experimentation suggests that the algorithm generally converges quadratically. 1 Introduction In the context of robust control analysis and synthesis a quantity of great interest is the structured singular value norm, or ¯norm, of the system. Consider a feedback connection of a continuoustime system with real coefficients as in Figure 1. Let P (s) = C(sI \Gamma A) \Gamma1 B be an m \Theta m stable transfer This paper has already appeared, in slightly different form, in the Proceedings of the 1996 IFAC World Congress [13]. y Research supported in part by NSF's Engineering Research Center Program, under grant NSFDCDR8803012. z Supported by the Belgian ...