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Control System Analysis And Synthesis Via Linear Matrix Inequalities
, 1993
"... A wide variety of problems in systems and control theory can be cast or recast as convex problems that involve linear matrix inequalities (LMIs). For a few very special cases there are "analytical solutions" to these problems, but in general they can be solved numerically very efficiently. In many c ..."
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Cited by 13 (1 self)
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A wide variety of problems in systems and control theory can be cast or recast as convex problems that involve linear matrix inequalities (LMIs). For a few very special cases there are "analytical solutions" to these problems, but in general they can be solved numerically very efficiently. In many cases the inequalities have the form of simultaneous Lyapunov or algebraic Riccati inequalities; such problems can be solved in a time that is comparable to the time required to solve the same number of Lyapunov or Algebraic Riccati equations. Therefore the computational cost of extending current control theory that is based on the solution of algebraic Riccati equations to a theory based on the solution of (multiple, simultaneous) Lyapunov or Riccati inequalities is modest. Examples include: multicriterion LQG, synthesis of linear state feedback for multiple or nonlinear plants ("multimodel control"), optimal transfer matrix realization, norm scaling, synthesis of multipliers for Popovlike...
The Generalized Gradient At A Multiple Eigenvalue
, 1995
"... : When a symmetric, positive, isomorphism between a reflexive Banach space (that is densely and compactly embedded in a Hilbert space) and its dual varies smoothly over a Banach space, its eigenvalues vary in a Lipschitz manner. We calculate the generalized gradient of the extreme eigenvalues at an ..."
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Cited by 11 (5 self)
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: When a symmetric, positive, isomorphism between a reflexive Banach space (that is densely and compactly embedded in a Hilbert space) and its dual varies smoothly over a Banach space, its eigenvalues vary in a Lipschitz manner. We calculate the generalized gradient of the extreme eigenvalues at an arbitrary crossing. We apply this to the generalized gradient, with respect to a coefficient in an elliptic operator, of (i) the gap between the operator's first two eigenvalues, and (ii) the distance from a prescribed value to the spectrum of the operator. 1. The Result Intent on studying the variation in the eigenvalues of an elliptic operator under variations in its measurable coefficients we work in the context of a Gelfand triple. That is, a reflexive Banach space V , a Hilbert space H, and a continuous, injective imbedding i : V ! H with dense image. The adjoint of i is consequently a continuous, injective imbedding i 0 : H ! V 0 with image dense in V 0 , the dual of V . We reca...
Optimizing Eigenvalues of Symmetric Definite Pencils
 in Proceedings of the 1994 American Control Conference
, 1994
"... We consider the following quasiconvex optimization problem: minimize the largest eigenvalue of a symmetric definite matrix pencil depending on parameters. A new form of optimality conditions is given, emphasizing a complementarity condition on primal and dual matrices. Newton's method is then applie ..."
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Cited by 8 (0 self)
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We consider the following quasiconvex optimization problem: minimize the largest eigenvalue of a symmetric definite matrix pencil depending on parameters. A new form of optimality conditions is given, emphasizing a complementarity condition on primal and dual matrices. Newton's method is then applied to these conditions to give a new quadratically convergent interiorpoint method which works well in practice. The algorithm is closely related to primaldual interiorpoint methods for semidefinite programming. 1. Introduction Many matrix inequality problems in control can be cast in the form: minimize the maximum eigenvalue of the Hermitian definite pencil (A(x); B(x)), w.r.t. a parameter vector x, subject to positive definite constraints on B(x) and sometimes also on other Hermitian matrix functions of x. The maximum eigenvalue is a quasiconvex function of the pencil elements and therefore of the parameter vector x if A, B depend affinely on x. This quasiconvexity reduces to convexity i...
The longstep method of analytic centers for fractional problems
 Mathematical Programming
, 1997
"... We develop a longstep surfacefollowing version of the method of analytic centers for the fractionallinear problem min {t0  t0B(x) − A(x) ∈ H, B(x) ∈ K, x ∈ G}, where H is a closed convex domain, K is a convex cone contained in the recessive cone of H, G is a convex domain and B(·), A(·) are a ..."
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Cited by 6 (1 self)
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We develop a longstep surfacefollowing version of the method of analytic centers for the fractionallinear problem min {t0  t0B(x) − A(x) ∈ H, B(x) ∈ K, x ∈ G}, where H is a closed convex domain, K is a convex cone contained in the recessive cone of H, G is a convex domain and B(·), A(·) are affine mappings. Tracing a twodimensional surface of analytic centers rather than the usual path of centers allows to skip the initial “centering ” phase of the pathfollowing scheme. The proposed longstep policy of tracing the surface fits the best known overall polynomialtime complexity bounds for the method and, at the same time, seems to be more attractive computationally than the shortstep policy, which was previously the only one giving good complexity bounds. 1
Range of the first three eigenvalues of the planar Dirichlet Laplacian
"... We conduct extensive numerical experiments aimed at nding the admissible range of the ratios of the rst three eigenvalues of a planar Dirichlet Laplacian. The results improve the previously known theoretical estimates of M Ashbaugh and R Benguria. We also prove some properties of a maximizer o ..."
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Cited by 2 (1 self)
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We conduct extensive numerical experiments aimed at nding the admissible range of the ratios of the rst three eigenvalues of a planar Dirichlet Laplacian. The results improve the previously known theoretical estimates of M Ashbaugh and R Benguria. We also prove some properties of a maximizer of the ratio 3 = 1 .