KYPD: a solver for semidefinite programs derived from the Kalman-Yakubovich-Popov lemma

Proceedings of the 45th IEEE Conference on Decision and Control

2006

Semidefinite programs and especially those derived from the Kalman-YakubovichPopov lemma are quite common in control applications. KYPD is a dedicated solver for KYP-SDPs. It solves the optimization problem via the dual SDP. The solver is iterative. In each step a Hessian is formed and a linear system of equations is solved. The calculations can be performed much faster if we utilize sparsity and low rank structure. We show how to transform a dense optimization problem into a sparse one with low rank structure. A customized calculation of the Hessian is presented and investigated.Keywords: Semidefinite programming, Kalman-Yakubovich-Popov lemma, low rank Utilizing low rank properties when solving KYP-SDPs Janne Harju, Ragnar Wallin and Anders HanssonAbstract-Semidefinite programs and especially those derived from the Kalman-Yakubovich-Popov lemma are quite common in control applications. KYPD is a dedicated solver for KYP-SDPs. It solves the optimization problem via the dual SDP. The solver is iterative. In each step a Hessian is formed and a linear system of equations is solved. The calculations can be performed much faster if we utilize sparsity and low rank structure. We show how to transform a dense optimization problem into a sparse one with low rank structure. A customized calculation of the Hessian is presented and investigated.

Section: Reduction Of the Number Of Dual Variablesmentioning

confidence: 99%

“…The solver KYPD (Wallin and Hansson, 2004) is a dedicated solver for KYP-SDPs and utilizes the special stucture in the problem. In this paper we will exploit sparseness and low rank properties in order to solve KYP-SDPs efficiently.…”

Section: Introductionmentioning

confidence: 99%

Utilizing low rank properties when solving KYP-SDPs

Harju

Wallin

Proceedings of the 45th IEEE Conference on Decision and Control

2006

“…We remark that the optimization problem considered is not within the class of problems considered in e.g. Wallin and Hansson [2004] and Gillberg and Hansson [2003].…”

Section: Wherementioning

confidence: 99%

Structure exploitation in Semi-Definite Programs for Systems Analysis

Johansson

IFAC Proceedings Volumes

2008

A wide variety of problems involving analysis of systems can be rewritten as a semidefinite program. When solving these problems optimization algorithms are used. Large size makes the problems unsolvable in practice and computationally more effective solvers are needed. This paper investigates how to exploit structure and problem knowledge in some control applications. It is shown that inexact search directions are useful to reduce the computational burden and that operator formalism can be utilized to derive tailored calculations. A wide variety of problems involving analysis of systems can be rewritten as a semidefinite program. When solving these problems optimization algorithms are used. Large size makes the problems unsolvable in practice and computationally more effective solvers are needed. This paper investigates how to exploit structure and problem knowledge in some control applications. It is shown that inexact search directions are useful to reduce the computational burden and that operator formalism can be utilized to derive tailored calculations. Linköping, Sweden (e-mail: hansson@isy.liu.se). NyckelordAbstract: A wide variety of problems involving analysis of systems can be rewritten as a semidefinite program. When solving these problems optimization algorithms are used. Large size makes the problems unsolvable in practice and computationally more effective solvers are needed. This paper investigates how to exploit structure and problem knowledge in some control applications. It is shown that inexact search directions are useful to reduce the computational burden and that operator formalism can be utilized to derive tailored calculations.

“…Assume that any uncertainty ∆ u ∈ ∆ u satisfies the IQC given in (15). If there exists a multiplier Π( jω) and a γ such that the following frequency-domain inequality holds …”

Section: Formulation Of the Perturbed Stability Margin Problemmentioning

confidence: 99%

“…The software used for the calculation of the lower bound were the IQC-toolbox [7] with the semidefinite program solver SDPT3 [13]. The computation time for the lower bound was reduced substantially (by a factor of 8) with the use of the dedicated solver for KYP problems KYPD ( [15]). …”

Section: Demonstration Of the Stability Margin Analysismentioning

confidence: 99%

Formulation of the stability margins clearance criterion as a convex optimization problem

Papageorgiou

Falkeborn

IFAC Proceedings Volumes

2009

This paper presents the formulation of a flight clearance criterion as a convex optimization problem. The criterion is the stability margins criterion which is specified as an allowable phase and gain margin of a certain loop transfer function. The satisfaction of the criterion amounts to the Nichols plot of the loop transfer function being outside a specified trapezoidal region. It was shown previously that the exclusion condition from an ellipsoidal region is implied by using the generalized stability margin b P C and its calculation was performed frequency-wise by solving a sequence of convex optimization problems. In this paper we formulate the calculation of a lower bound on b P C as a convex optimization problem using Integral Quadratic Constraints (IQCs) and avoid the gridding procedure in the frequency domain. Furthermore, we formulate the problem of obtaining a lower bound on the perturbed stability margin, which is defined as the worst-case b P C over variations in real uncertain parameters. Abstract: This paper presents the formulation of a flight clearance criterion as a convex optimization problem. The criterion is the stability margins criterion which is specified as an allowable phase and gain margin of a certain loop transfer function. The satisfaction of the criterion amounts to the Nichols plot of the loop transfer function being outside a specified trapezoidal region. It was shown previously that the exclusion condition from an ellipsoidal region is implied by using the generalized stability margin b PC and its calculation was performed frequency-wise by solving a sequence of convex optimization problems. In this paper we formulate the calculation of a lower bound on b PC as a convex optimization problem using Integral Quadratic Constraints (IQCs) and avoid the gridding procedure in the frequency domain. Furthermore, we formulate the problem of obtaining a lower bound on the perturbed stability margin, which is defined as the worst-case b PC over variations in real uncertain parameters.