Parametrized model reduction based on semidefinite programming

doi:10.1016/j.automatica.2013.05.022

Automatica

Volume 49, Issue 9, September 2013, Pages 2840-2844

https://doi.org/10.1016/j.automatica.2013.05.022 Get rights and content

Abstract

A parametrized model in addition to the control and state-space variables depends on time-independent design parameters, which essentially define a family of models. The goal of parametrized model reduction is to approximate this family of models. In this paper, a reduction method for linear time-invariant (LTI) parametrized models is presented, which constitutes the development of a recently proposed reduction approach. Reduced order models are computed based on the finite number of frequency response samples of the full order model. This method uses a semidefinite relaxation, while enforcing stability on the reduced order model for all values of parameters of interest. As a main theoretical statement, the relaxation gap (the ratio between upper and lower bounds) is derived, which validates the relaxation. The proposed method is flexible in adding extra constraints (e.g., passivity can be enforced on reduced order models) and modifying the objective function (e.g., frequency weights can be added to the minimization criterion). The performance of the method is validated on a numerical example.

Introduction

Model reduction is commonly used as a tool to approximate complex systems, which require excessive time and/or memory to simulate directly. LTI model reduction is a well-known problem and many methods to address it have been reported: balanced truncation (Moore, 1981), Hankel model reduction (Glover, 1984), Krylov interpolatory methods, cf. Antoulas (2005), etc. All the methods have some attractive properties and some drawbacks. Among major concerns are preservation of stability, a guarantee on approximation quality and computational efficiency; properties that not all methods can accommodate. Parametrized model reduction is a natural extension of LTI model reduction, which deals with LTI models dependent on certain design parameters. Typically, model parameters are subject to change or fine-tuning over time; therefore, it is often desirable to obtain a family of reduced order models which approximates a particular system in various parameter settings. For this purpose, Krylov-based methods are often used in various applications, such as micro-electro-mechanical systems (Rudnyi & Korvink, 2002), radio-frequency inductors (Daniel & White, 2003), interconnects (Daniel et al., 2004, Li et al., 2005, Liand et al., 2005) and general linear systems (Farle et al., 2008, Feng et al., 2009, Li et al., 2007). Krylov-based methods match the moments of the full and reduced order models at particular frequencies. This typically results in a computationally efficient algorithm, but with a local guarantee on the approximation quality. Another framework is being developed in Antoulas, Ionita, and Lefteriu (2012) and Lefteriu and Antoulas (2010), where reduced models are obtained by interpolating the frequency response of the full order model computed at a finite number of frequencies. In Sou, Megretski, and Daniel (2008) the reduced order models are also constructed from the frequency response, however, by means of semidefinite programming. This method has a few advantages in comparison to the ones mentioned above. Firstly, stability of reduced order models is guaranteed for all values of parameters. Secondly, the objective and the constraints of the semidefinite program can be adjusted according to designer’s needs. For example, frequency weights can be added to the objective function, and passivity can be enforced on reduced order models. Moreover, such modifications are performed without increasing the computational complexity. The major drawback of the method in Sou et al. (2008) is its output. Although for every fixed parameter value a stable LTI model can easily be constructed, it cannot be done for all values at once. In other words, explicit parameter-dependent models cannot be obtained.

The method from Sou et al. (2008) was further developed in Sootla (2013), but only for LTI models (the non-parametrized case). This development resulted in two algorithms: one employs a semidefinite relaxation (which is similar to the one used in Sou et al., 2008), another exploits an iterative approach (this algorithm has its roots in Sootla & Sou, 2010). The main focus of this paper is on extending both algorithms from Sootla (2013) to the parametrized case with other major contributions as follows:

•
A theoretical statement regarding the approximation quality of this relaxation is derived (i.e. a relaxation gap), which is also valid for the method in Sou et al. (2008), but was not shown earlier.
•
A method to obtain explicit parameter-dependent models is proposed, which is a combination of relaxation and iterative approaches.
•
In comparison to the method in Sou et al. (2008) the proposed relaxation approach can deliver models with a better approximation quality.

The paper is organized as follows: the problem is formulated in Section 2. The relaxation approach with the main theoretical statement is discussed in Section 3. In Section 4 the iterative approach is outlined. An estimate on the computational complexity of the method and a numerical example are presented in Section 5.

Section snippets

Preliminaries and problem formulation

Let $G (z, θ)$ be a function such that

•
the parameter $θ$ is a time-independent, real scalar in the interval $[0, π]$ ;
•
the function $G (z, θ)$ is a scalar-valued asymptotically stable discrete-time rational transfer function for every fixed $θ$ ;
•
the coefficients of the transfer function $G (z, θ)$ are real and depend smoothly on $θ$ .

The algorithms presented in the sequel can be extended to a vector-valued $θ$ . The restriction to the interval $[0, π]$ can always be achieved by a linear variable change, if $θ$ belongs to any

Relaxation approach and error bounds

Let $a_{i j}, b_{i j}$ be real scalars and $a (ω) = \sum_{i = - k}^{k} \sum_{j = 0}^{n_{r}} a_{i j} e^{- i j ω} cos (j θ)$ $b (ω) = \sum_{i = - k}^{k} \sum_{j = 0}^{n_{r}} b_{i j} e^{- i j ω} cos (j θ) .$ Substituting $a (ω)$ and $b (ω)$ into (3) instead of $q (ω) φ^{\sim} (ω)$ and $p (ω) φ^{\sim} (ω)$ correspondingly yields a semidefinite relaxation. The constant $n_{r}$ in (4) should be equal to $2 n$ in order to perform this relaxation; however, the notation $n_{r}$ is kept to simplify the presentation. The relaxation results in the following program: $min_{γ > 0, a_{i j}, b_{i j} \in R} γ subject to:$ $Re (a (ω)) > 0 \forall ω \in {[0, π]}^{2}$ $| G (ω) a (ω) - b (ω) | < γ Re (a (ω)) \forall ω \in {[0, π]}^{2} .$ The

Iterative approach and explicit parameter-dependent models

Recall the optimization problem (3). Lemma 1 shows that in optimality

φ

is equal to

q

and this property is used to derive Algorithm 2. Here, the gridding relaxation for the norm constraint is employed as well, with the same considerations about the choice of the grid. Note that (3) slightly differs from (8); this is done for convenience only and the constraints in these programs are equivalent. In order to guarantee the stability of

p / q

, the functions

1 / φ^{v}

should be stable for all

θ

and

v

. It

Complexity and a numerical example

The proposed methods are implemented using yalmip (Löfberg, 2004) and sedumi (Sturm, 1999). There are two main contributors to complexity of the methods: computation of frequency response samples and the semidefinite program. The total cost of the algorithm does not exceed (cf. Sootla, 2012): $O (l^{3} N) + O (k^{4} n^{4} N^{2.5} + N^{3.5})$ where $l$ is the order of the original model with respect to $z, k$ is the order of the reduced model with respect to $z, n$ is the order of the reduced model with respect to the parameter

Conclusion

The main contribution of this paper is a model reduction method for parametrized LTI systems. It is based on the semidefinite programming and matching of frequency response samples. This paper extends the results of Sootla (2013) to parametrized models, as well as compliments the results of Sou et al. (2008), by providing a theoretical guarantee on the approximation quality, explicit parameter-dependent models and an improved approximation quality of the reduced order models. Further

Aivar Sootla received a Ph.D. degree at the Department of Automatic Control at Lund University in 2012. Since 2012 he has been a postdoctoral research associate at Imperial College London, UK.

His research interests include model reduction, applications of convex optimization in control theory, optimal control, reinforcement learning and applications of control theory in systems and synthetic biology.

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Cited by (0)

Aivar Sootla received a Ph.D. degree at the Department of Automatic Control at Lund University in 2012. Since 2012 he has been a postdoctoral research associate at Imperial College London, UK.

Kin Cheong Sou received a Ph.D. degree in Electrical Engineering and Computer Science at Massachusetts Institute of Technology in 2008.

He was a postdoctoral researcher at Lund University from 2008 to 2010, and Royal Institute of Technology (KTH), Stockholm from 2010 to 2013.

In 2013 Sou has joined the Department of Mathematical Sciences at Chalmers University of Technology and University of Gothenburg, Gothenburg. His research interests include power system cyber-security analysis, environment aware building and community, convex/non-convex optimization and model reduction for dynamical systems.

Anders Rantzer received a Ph.D. degree in optimization and systems theory in 1991 from the Royal Institute of Technology (KTH), Stockholm, Sweden. After postdoctoral positions at KTH and at IMA, University of Minnesota, he joined the Department of Automatic Control at Lund University in 1993. He was appointed as a professor of Automatic Control in Lund 1999. During the academic year of 2004/05 he was a visiting associate faculty member at Caltech. Since 2008 he coordinates the Linnaeus center LCCC at Lund University. For the period 2013–15 he is also appointed as the chairman of the Swedish Scientific Council for Natural and Engineering Sciences.

Rantzer has been the associate editor of IEEE Transactions on Automatic Control and several other journals. He is a winner of the SIAM Student Paper Competition, the IFAC Congress Young Author Price and the IET Premium Award for the best article in IEE Proceedings—Control Theory & Applications during 2006. He is a fellow of IEEE and a member of the Royal Swedish Academy of Engineering Sciences.

His research interests include modeling, analysis and synthesis of control systems, with particular attention to uncertainty, optimization and distributed control.

^☆: This work was performed during the Ph.D. studies of Aivar Sootla and funded through the LCCC—Lund Center for Control of Complex Engineering Systems by the Swedish Research Council. The material in this paper was partially presented at the Swedish Control Meeting, June 2010, Lund, Sweden. This paper was recommended for publication in revised form by Associate Editor Mario Sznaier under the direction of Editor Roberto Tempo.

¹: Tel.: +44 0 20 7594 2835; fax: +44 0 20 7594 9817.

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Automatica

Brief paperParametrized model reduction based on semidefinite programming☆

Abstract

Introduction

Section snippets

Preliminaries and problem formulation

Relaxation approach and error bounds

Iterative approach and explicit parameter-dependent models

Complexity and a numerical example

Conclusion

On two-variable rational interpolation

Linear Algebra and its Applications

Multiparameter moment matching model reduction approach for generating geometrically parameterized interconnect performance models

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems

Positive trigonometric polynomials and signal processing applications

Multi-parameter polynomial order reduction of linear finite element models

Mathematical and Computer Modelling of Dynamical Systems

Factorization of multivariate positive Laurent polynomials

Journal of Approximation Theory

All optimal Hankel-norm approximations of linear multivariable systems and their L∞-error bounds

International Journal of Control

A new approach to modeling multiport systems from frequency-domain data

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems

Brief paper
Parametrized model reduction based on semidefinite programming☆

All optimal Hankel-norm approximations of linear multivariable systems and their $L_{\infty}$ -error bounds