Model reduction using LMIs

Helmersson, Anders

doi:10.1109/cdc.1994.411635

Cited by 41 publications

(27 citation statements)

References 5 publications

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“…The example is a model reduction algorithm from [22] where semidefinite programming is used to reduce the order of a linear time-invariant (LTI) system. A short description of the algorithm now follows.…”

Section: Computational Resultsmentioning

confidence: 99%

Low-rank exploitation in semidefinite programming for control

Falkeborn

Löfberg

Hansson

2010

2010 IEEE International Symposium on Computer-Aided Control System Design

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Many control related problems can be cast as semidefinite programs but, even though there exist polynomial time algorithms and good publicly available solvers, the time it takes to solve these problems can be long. Something many of these problems have in common, is that some of the variables enter as matrix valued variables. This leads to a low-rank structure in the basis matrices which can be exploited when forming the Newton equations. In this paper, we describe the how this can be done, and show how our code can be used when using SDPT3. The idea behind this is old and is implemented in LMI Lab, but we show that when using a modern algorithm, the computational time can be reduced. Finally, we describe how the modeling language YALMIP is changed in such a way that our code can be interfaced using standard YALMIP commands, which greatly simplifies for the user. Keywords: Semidefinite programming, Structure exploitationLowrank exploitation in semidefinite programming for control Rikard Falkeborn, Johan Löfberg and Anders HanssonAbstract-Many control related problems can be cast as semidefinite programs but, even though there exist polynomial time algorithms and good publicly available solvers, the time it takes to solve these problems can be long. Something many of these problems have in common, is that some of the variables enter as matrix valued variables. This leads to a low-rank structure in the basis matrices which can be exploited when forming the Newton equations. In this paper, we describe the how this can be done, and show how our code can be used when using SDPT3. The idea behind this is old and is implemented in LMI Lab, but we show that when using a modern algorithm, the computational time can be reduced. Finally, we describe how the modeling language YALMIP is changed in such a way that our code can be interfaced using standard YALMIP commands, which greatly simplifies for the user.

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Section: Computational Resultsmentioning

confidence: 99%

Low-rank exploitation in semidefinite programming for control

Falkeborn

Löfberg

Hansson

2010

2010 IEEE International Symposium on Computer-Aided Control System Design

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“…The optimal MOR solution can be obtained by solving a feasible point of LMIs feasible set coupling with a rank constraint. The method is proposed in [6] in terms of time domain response of the error system and [5] extends the result to the ∞ MOR problem in the state space. Reference [7] extends the results to the stochastic case.…”

Section: Problem Formulationmentioning

confidence: 99%

“…The suboptimal errors can be obtained as well as the corresponding positive matrices , by means of our algorithm. The reduced state space can be evaluated according to (6)- (8) (Theorem 1 in [5]), and the obtained reduced system is stable.…”

Section: Lagrangian Multipliermentioning

confidence: 99%

“…It is a difficult work to list all the references since we have many references in this field. The ∞ MOR problem has been studied by many researchers such as [4][5][6][7][8][9][10][11][12][13] and references therein. It mainly consists of finding a lowerorder model̂of the original system such that the ∞ error ‖ −̂‖ is small.…”

Section: Introductionmentioning

confidence: 99%

“…LMI is also a powerful tool in control and systems theory and was widely used in output feedback stabilization, reducedorder ∞ synthesis, synthesis with constant scaling [15], and so on. In [6], an LMI approach was followed for the 2 Mathematical Problems in Engineering based on linearization [17], alternating projections [18,19], trace minimization methods that solve the problem by solving a related convex problem [20], augmented Lagrangian methods (ALF) [21], sequential semidefinite programming [22], and so on. Aside from [21,22], these methods did not give the proof of convergence and the challenge still remains to find the numerical schemes with verifiable local/global convergence.…”

Section: Introductionmentioning

confidence: 99%

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An Augmented Lagrangian Method for the Optimal H_∞ Model Order Reduction Problem

Yang

Zhao

2017

Mathematical Problems in Engineering

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This paper treats the computational method of the optimal ∞ model order reduction (MOR) problem of linear time-invariant (LTI) systems. Optimal solution of MOR problem of LTI systems can be obtained by solving the LMIs feasibility coupling with a rank inequality constraint, which makes the solutions much harder to be obtained. In this paper, we show that the rank inequality constraint can be formulated as a linear rank function equality constraint. Properties of the linear rank function are discussed. We present an iterative algorithm based on augmented Lagrangian method by replacing the rank inequality with the linear rank function. Convergence analysis of the algorithm is given, which is distinct to the now available heuristic methods. Numerical experiments for the MOR problems of continuous LTI system illustrate the practicality of our method.

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L norm simultaneous system approximation

Kavranoglu

Bettayeb

Anjum

1996

Int. J. Robust Nonlinear Control

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In this paper the L. nonn optimal simultaneous system approximation problem is studied. Various LMI based approaches to solve the simultaneous L. norm system approximation problem are developed. Computational schemes based on LMI based optimization techniques are derived. Examples are given to illustratethe results. KEYWORDS optimal (weighted) L_ (simultaneous) approximation; LMI basedconvex optimization INTRODUCTIONH_ norm and associated optimization problems in control, filtering and estimation have been proven to be very meaningful. A vast amount of literature on robust control and H _ optimal control has been developed since theearly 1980s. 1 It is often desirable to have a low order nominal plant since the degree of the controllers obtained by modem control design schemes such as LQG and H_ optimal control theory will be the same as the plant degree. The problem of model reduction has always attracted much attention and many model reduction schemes have been reported in the literature. The 11.11_ is widely accepted to be the most relevant measure for approximation error for robustness considerations.The L_ simultaneous model approximation problem is defined below. Definition JGiven G1(s), G 2(s), ... , GN(s) all in RL_ and weights Wi(s) E RH., find Gr(s) E RL_ with specified number of stable and unstable poles such that is minimized.

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Model reduction using LMIs

Cited by 41 publications

References 5 publications

Low-rank exploitation in semidefinite programming for control

Low-rank exploitation in semidefinite programming for control

An Augmented Lagrangian Method for the Optimal H_∞ Model Order Reduction Problem

L norm simultaneous system approximation

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Model reduction using LMIs

Cited by 41 publications

References 5 publications

Low-rank exploitation in semidefinite programming for control

Low-rank exploitation in semidefinite programming for control

An Augmented Lagrangian Method for the Optimal H∞ Model Order Reduction Problem

L norm simultaneous system approximation

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An Augmented Lagrangian Method for the Optimal H_∞ Model Order Reduction Problem