Serkan Gugercin's Publications
S. Gugercin and J.-R. Li. Smith-type methods for balanced truncation of large-sparse systems. Dimension Reduction of Large-scale Systems, P. Benner, G.H. Golub, V.L. Mehrman and D.C. Sorensen editors, Springer-Verlag, Lecture Notes in Computational Science and Engineering, Vol. 45 (ISBN 3-540-24545-6), Berlin/Heidelberg, 2005. (Copyright Springer, 2005)
A.C. Antoulas, D.C. Sorensen, and S. Gugercin. A survey of model reduction methods for large-scale systems. Structured Matrices in Operator Theory, Numerical Analysis, Control, Signal and Image Processing, Contemporary Mathematics, AMS publications, 280:193-219, 2001. (Copyright AMS, 2001)
Abstract: An overview of model reduction methods and a comparison
of the resulting algorithms is presented. These approaches are divided into two
broad categories, namely SVD based and moment matching based methods. It turns
out that the approximation error in the former case behaves better globally in
frequency while in the latter case the local behavior is better.
S. Gugercin, D.C. Sorensen, and A.C. Antoulas. A modified low-rank Smith method for large-scale Lyapunov Equations. Numerical Algorithms, Vol. 32, Issue 1, pp. 27-55, January 2003. (Copyright Kluwer, 2003)
Abstract: In this note we present a modified cyclic low-rank Smith
method to compute low-rank approximations to solutions of Lyapunov equations
arising from large-scale dynamical systems. Unlike the original cyclic low-rank
Smith method introduced by Penzl in \cite{penzl}, the number of columns
required by the modified method in the approximate solution does not necessarily
increase at each step and is usually much lower than in the original cyclic
low-rank Smith method. The modified method never requires more columns than the
original one. Upper bounds are established for the errors of the low-rank
approximate solutions and also for the errors in the resulting approximate
Hankel singular values. Numerical results are given to verify the efficiency
and accuracy of the new algorithm.
S. Gugercin, A.C. Antoulas, and H.P. Zhang. An approach to identification for robust control. IEEE Transactions on Automatic Control, Volume: 48 Issue: 6, pp. 1109-1115, June 2003. (Copyright IEEE, 2003)
Abstract: Given measured data generated by
a discrete-time linear system we propose a model consisting of a linear,
time-invariant system affected by norm-bounded perturbation. Under mild
assumptions, the plants belonging to the resulting uncertain family form a
convex set. The approach depends on two key parameters: an a priori given
bound of the perturbation, and the input used to generate the data. It turns
out that the size of the uncertain family can be reduced by intersecting the
model families obtained by making use of different inputs. The model validation
problem in this identification scheme is analyzed. For a given energy level,
the invalidation problem yields the family of those models which can never be
invalidated for any possible input of fixed energy and any possible
perturbation; this leads to the intersection of all uncertain families. A consequence
of the invalidation problem is that for finite length measurements {\it not}
all models can be invalidated, using fixed-energy inputs.
S. Gugercin and A.C. Antoulas. A survey of model reduction by balanced truncation and some new results. International Journal of Control, Volume: 77 Issue: 8, pp. 748-766, 2004 (Copyright Taylor & Francis, 2004)
Abstract: Balanced truncation is one of the most common model reduction
schemes. In this note, we present a survey of balancing related model reduction
methods and their corresponding error norms, and also introduce some new
results. Five balancing methods are studied: {\bf (1)} Lyapunov balancing, {\bf
(2)} Stochastic balancing {\bf (3)} Bounded real balancing, {\bf (4)} Positive
real balancing and {\bf (5)} Frequency weighted balancing. For positive real
balancing, we introduce a multiplicative-type error bound. Moreover, for a
certain subclass of positive real systems, a modified positive-real balancing
scheme with an absolute error bound is proposed. We also develop a new
frequency-weighted balanced reduction method with a simple bound on the error
system based on the frequency domain representations of the system gramians.
Two numerical examples are illustrated to verify the efficiency of the proposed
methods.
S. Gugercin and A.C. Antoulas. Model reduction of large-scale systems by least squares. Linear Algebra and its Applications, Special Issue on Order Reduction of Large-Scale Systems, Vol. 415/2-3, pp. 290-321, 2006. (Copyright Elsevier, 2006)
Abstract: In this paper we introduce an approximation method for
model reduction of large-scale dynamical systems. This is a projection
which combines aspects of the SVD and Krylov based reduction methods. This
projection can be efficiently computed using tools from numerical analysis,
namely the rational Krylov method for the Krylov side of the projection and a
low-rank Smith type iteration to solve a Lyapunov equation for the SVD side of
the projection. For discrete time systems, the proposed approach is based on
the least squares fit of the $(r+1)^{st}$ column of a Hankel matrix to the
preceding $r$ columns, where $r$ is the order of the reduced
system. The reduced system is asymptotically stable, matches the first
$r$ Markov parameters of the full order model and minimizes a weighted $\cH_2$
error. The method is also generalized for moment matching at arbitrary
interpolation points. Application to continuous time systems is achieved via
the bilinear transformation. Numerical examples prove the effectiveness of the
approach. The proposed method is significant because it combines guaranteed
stability and moment matching, together with an optimization criterion.
S. Gugercin and K. Willcox, Krylov projection framework for Fourier model reduction. Accepted, Automatica, 2007. (Copyright Elsevier, 2007)
Abstract: This paper analyzes the Fourier model reduction (FMR) method from a rational Krylov projection framework and shows how the FMR reduced model, which has guaranteed stability and a global error bound, can be computed in a numerically efficient and robust manner. By monitoring the rank of the Krylov subspace that underlies the FMR model, the projection framework also provides an improved criterion for determining the number of Fourier coefficients that are needed, and hence the size of the resulting reduced-order model. The advantages of applying FMR in the rational Krylov projection framework are demonstrated on a simple example.
S. Gugercin, An iterative SVD-Krylov based algorithm for model reduction of large-scale dynamical systems. Accepted, Linear Algebra and its Applications, 2007.
Abstract: In this paper, we propose a model reduction algorithm for approximation of large-scale linear time-invariant dynamical systems. The method is a two-sided projection combining features of the singular value decomposition (SVD)-based and the Krylov-based model reduction techniques. While the SVD-side of the projection depends on the observability gramian, the Krylov-side is obtained via iterative rational Krylov steps. The reduced model is asymptotically stable, matches certain moments and solves a restricted $\h2$ minimization problem. We present modifications to the proposed approach for employing low-rank gramians in the reduction step and also for reducing discrete-time systems. Several numerical examples from various disciplines verify the effectiveness of the proposed approach. It performs significantly better than the {\bf q cover} \} and the {\bf least-squares} methods that have a similar projection structure to the proposed method. Also, in terms of both the $\h2$ and $\hinf$ error measures, its performance is comparable to or sometimes better than that of balanced truncation. Moreover, the method proves to be robust with respect to the perturbations due to usage of approximate gramians.
S. Gugercin, A.C. Antoulas and C.A. Beattie, H2 model reduction for large-scale linear dynamical systems. Accepted, SIAM Journal on Matrix Analysis and Applications, 2007.
Abstract: The optimal H2 model reduction problem is of great importance in the area of dynamical systems and simulation. In the literature, two independent frameworks have evolved focussing either on solution of Lyapunov equations on the one hand or interpolation of transfer functions on the other, without any apparent connection between the two approaches. In this paper, we develop a new unifying framework for the optimal H2 approximation problem using best approximation properties in the underlying Hilbert space. This new framework leads to a new set of local optimality conditions taking the form of a structured orthogonality condition. We show that the existing Lyapunov and interpolation based conditions are each equivalent to our conditions and so are equivalent to each other. Also, we provide a new elementary proof of the interpolation based condition that clarifies the importance of the mirror images of the reduced system poles. Based on the interpolation framework, we describe an iteratively corrected rational Krylov algorithm for H2 model reduction. The formulation is based on finding a reduced order model that satisfies interpolation based first-order necessary conditions for H2 optimality and results in a method that is numerically effective and suited for large-scale problems. We illustrate the performance of the method with a variety of numerical experiments and comparisons with existing methods.
Refereed Conference Proceedings:
S. Gugercin and A.C. Antoulas. On the assignment of eigenvalues in LTI systems. Proceedings of the 38th IEEE Conference on Decision and Control, Vol. 1, pp. 486, Phoenix, Arizona, December 1999. (Copyright IEEE, 1999)
S. Gugercin and A.C. Antoulas. On consistency and model validation for systems with parameter uncertainty. Proceedings of SYSID2000, Santa Barbara, California, June 2000. (Copyright IFAC, 2000)
S. Gugercin and A.C. Antoulas. A comparative study of 7 algorithms for model reduction. Proceedings of the 39th IEEE Conference on Decision and Control, Vol. 3, pp. 2367-2372, Sydney, Australia, December 2000. (Copyright IEEE, 2000)
S. Gugercin, A.C. Antoulas, N. Bedrossian. Approximation of International Space Station 1R and 12A Models. Proceedings of the 40th IEEE Conference on Decision and Control, Vol. 3, pp. 1515-1516, Orlando, Florida, December 2001. (Copyright IEEE, 2001)
A.C. Antoulas and S. Gugercin. A new approach to model reduction which preserves stability and passivity. Proceedings of the 41st IEEE Conference on Decision and Control, Vol. 3, pp. 2544-2545, Las Vegas, Nevada, December 2002. (Copyright IEEE, 2002)
S. Gugercin and A.C. Antoulas. A survey of balancing methods for model reduction. Proceedings of European Control Conference 2003, Cambridge, UK,September 2003.
S. Gugercin and A.C. Antoulas. A time-limited balanced reduction method. Proceedings of the 42nd IEEE Conference on Decision and Control, Vol. 5, pp. 5250-5253, December 2003. (Copyright IEEE, 2003)
S. Gugercin and A.C. Antoulas. An H2 error expression for the Lanczos procedure. Proceedings of the 42nd IEEE Conference on Decision and Control, Vol. 2, 1869-1872, December 2003. (Copyright IEEE, 2003)
C.A. Beattie, S. Gugercin, A.C. Antoulas and E. Gildin. Controller reduction by Krylov projection methods. Proceedings of the 16th International Symposium on Mathematical Theory of Networks and Systems, July 2004.
S. Gugercin, A.C. Antoulas, C.A. Beattie, and E. Gildin. Krylov-based controller reduction for large-scale systems. Proceedings of the 43rd IEEE Conference on Decision and Control, Vol. 3, pp. 3074-3077, December 2004. (Copyright IEEE, 2004)
S. Gugercin. An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems. Proceedings of the 44th IEEE Conference on Decision and Control, pp. 5905-5910, December 2005. (Copyright IEEE, 2005)
C.A. Beattie and S. Gugercin. Krylov-based model reduction of second-order systems with proportional damping. Proceedings of the 44th IEEE Conference on Decision and Control, pp. 2278-2283, December 2005. (Copyright IEEE, 2005)
C.A. Beattie, J. Borggaard, S. Gugercin and T. Iliescu. A domain decomposition approach to POD. Proceedings of the 45th IEEE Conference on Decision and Control, pp. 6750-6756, December 2006. (Copyright IEEE, 2006)
C.A. Beattie and S. Gugercin. Inexact solves in Krylov-based model reduction. Proceedings of the 45th IEEE Conference on Decision and Control, pp. 3405-3411, December 2006. (Copyright IEEE, 2006)
S. Gugercin, A. C. Antoulas and Christopher A. Beattie. A rational Krylov Iteration for Optimal H2 Model Reduction. Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, pp. 1665-1667, July 2006.