Approximation and Control of Large-Scale Dynamical Systems

Research funded under NSF-0505971 and NSF-0645347
PIs for NSF-0505971: Chris Beattie and Serkan Gugercin
PI for NSF-0645347: Serkan Gugercin

• Summary
• SVD-based Approximation Methods
• Krylov-based Optimal Approximation Methods
• SVD-Krylov based Approximation Methods
• Krylov-based Controller Reduction for Large-Scale Systems

Summary

Dynamical systems are a principal tool in modeling and control of many physical phenomena, such as heat transfer and temperature control in various media, signal propagation and interference in electric circuits, wave propagation and vibration suppression in large structures, and behavior of micro-electro-mechanical systems (MEMS). Direct numerical simulation of the associated models has been one of the few available means for studying the complex underlying physical phenomena. However, the ever increasing need for improved accuracy requires the inclusion of ever more detail in the modeling stage, leading inevitably to ever larger-scale, ever more complex dynamical systems.  Simulations in such large-scale settings lead in turn to unmanageably large demands on computational resources, which is the main motivation for system approximation (model reduction). The goal is to produce a low dimensional system that has the same response characteristics as the original system. Low dimensionality means far less storage and far less evaluation time. The resulting reduced order model can be used to replace the original system as a component in a larger simulation or it might be used to develop a simpler and faster controller suitable for real time applications.

In this research, we examine dynamical systems described by sets of coupled differential or difference equations. The complexity of the dynamical system is the number of state variables, i.e. the number of differential or difference equations needed to describe them. The complexity is called the "dimension" and usually denoted by "n". In the setting that I am working in, “n” may vary from 10² to 10⁶ (or higher). 

Model reduction should be applied in such a way that

1.       The approximation error is small

2.       Critical system featues, such as stability or second-order structure, are retained

3.       The model reduction algorithm is computationally efficient, numerically stable and robust with minimal application-specific tuning required.

Goals 1 and 2 are generally met through the underlying theory behind a reduction scheme. Meeting the third goal for large-scale settings requires algorithmic development and implementation. One class of methods, based on the singular value decomposition (SVD), includes Balanced Truncation. Even though the SVD-based methods have nice system theoretic properties, i.e., meet goals 1 and 2 above, they require dense matrix factorizations, and therefore they are applicable to only small-to-medium scale problems where " n < 10⁴  ". A different class of methods, based on Krylov subspace projection, includes the Lanczos, Arnoldi and rational Krylov methods. Unlike the SVD-based methods, only matrix-vector multiplications are needed. These methods are numerically reliable and can be implemented iteratively, hence they satisfy goal 3. The Krylov based methods can be effectively applied to the cases where " n >>10⁴ ". However, they lack an error bound. The current research trend in the area of model reduction is to connect SVD and Krylov based approximation methods. These methods aim at combining the theoretical features of the SVD based methods such as stability and global error bounds, with the efficient numerical implementation of the moment matching based methods.

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SVD-based Approximation Methods

Our research in this category focuses on developing new methods and theoretical analysis for balancing related model reduction methods, such as Lyapunov Balancing, Positive Real Balancing, and Frequency Weighted Balancing. Related publications are listed below:

1. S. Gugercin and J-R. Li, Smith-Type Methods for Balanced Truncation of Large Sparse Systems,  In P. Benner, G. Golub, V. Mehrmann, and D. Sorensen, editors, Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin/Heidelberg,  Germany, 2005.

2. 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.

3. 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.

4. S. Gugercin, A. C. Antoulas and N. Bedrossian , Approximation of the International Space Station 1R and 12A flex models, Proceedings of the 40th IEEE Conference on Decision and Control, December 2001.

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Krylov-based Approximation Methods

The goals here are to provide error expressions for Krylov based model reduction and to devise an optimal shift selection so that the resulting model is optimal in a certain norm measure. We have developed efffetive shift selection srategies and algorithms for Krylov-based optimal H₂ approximation. Moreover, effect of employing inexact solves in Krylov-based model reduction is being investigated with resulting perturbation analysis and backward error estimates. We also study structure preserving Krylov-based model reduction of second-order systems. Related publications are listed below:

1. S. Gugercin, A.C. Antoulas and C.A. Beattie, Rational Krylov methods for optimal H2 model reduction, submitted to SIMAX, 2006.

2. S. Gugercin and K. Willcox, Krylov projection framework for Fourier model reduction, submitted to Automatica, 2006.

3. C.A. Beattie and S. Gugercin, Inexact solves in Krylov-based model reduction, accepted to appear in the Proceedings of the 45th IEEE Conference on Decision and Control, December 2006.

4. 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, December 2005.

5. S. Gugercin and A.C. Antoulas, An H2 error expression for the Lanczos procedure, Proceedings of the 42nd IEEE Conference on Decision and Control, December 2003.

6. A.C.Antoulas, D. C. Sorensen, and S. Gugercin, A survey of model reduction methods for large scale systems , Contemporary Mathematics, AMS Publications, 280: 193-219, 2001

7. S. Gugercin and A.C. Antoulas, A comparative study of 7 model reduction algorithms, Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, December 2000.

   The example below illustrates the performance of the Krylov-based optimal H₂ model reduction method introduced in the first publication above, Rational Krylov methods for optimal H2 model reduction [GAB06]. The full-order model (due to Peter Benner) results from a semi-discretized heat transfer problem for optimal cooling of steel profiles. This problem arises during a cooling process in a rolling mill when different steps in the production process require different temperatures of the raw material. To achieve high throughput, one seeks to reduce the temperature as fast as possible to the required level before entering the next production phase. But the cooling process is realized by spraying cooling fluids on the surface and must be controlled so that material properties, such as durability or porosity, stay within given quality standards. The problem is modeled as boundary control of a two dimensional heat equation. A finite element discretization results in a descriptor system of order n = 79841. We apply the Iterative Rational Krylov Algorithm (IRKA) of [GAB06] and reduce the order to r = 6. Amplitude Bode plots of the full model (H(s)) and the reduced-order model (H_r(s)) are shown in the figure below. The output response of the reduced model H_r(s) is virtually indistinguishable from the full-order model H(s) in the frequency range considered. The important observation of this example is the following: Until now, the shift selection for the rational Krylov algorithm was ad hoc. Here, for a system of order n = 79841, our algorithm (IRKA) effectively and efficiently searches for an optimal shift selection and, consequently yields, at least, a locally optimalreduced order model with no tuning or user intervention.

    

    

   Figure-1

   
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SVD-Krylov based Methods

In SVD-Krylov based model reduction, the goal is to combine the best features of each category. Our work under this category includes the iterative SVD-Krylov based model reduction method (Gugercin), the least-squares model reduction method (Gugercin/Antoulas) and the Modified Smith’s Method (Gugercin/Sorensen/Antoulas).

1. S. Gugercin, An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems, submitted to Linear Algebra and its Applications, 2006.

2. S. Gugercin and A.C. Antoulas, Model reduction of large scale systems by least squares, to appear in  Linear Algebra and its Applications, Special Issue on Order Reduction of Large-scale Systems, 2004

3. 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.

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Kryloy-based Controller Reduction for Large-scale Systems

Given a stabilizing large-scale controller, the goal here is to devise efficient controller reduction algorithms which will yield stabilizing and performance preserving reduced-order controllers. We have developed (Gugercin/Antoulas/Beattie) a Krylov-based controller reduction technique which can be efficiently applied to systems of very large order. Unlike the general approach, frequency weighted balancing based controller reduction, no Lyapunov equations need to be solved. Moreover, there is a guaranteed matching of the full-order closed loop system. 

Related publications and presentations are listed below:

1. 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, December 2004.
2. S. Gugercin, A.C. Antoulas and C.A. Beattie, Krylov-based controller reduction for large-scale systems, SIAM Conference on Computational Science and Engineering, February 2005.

The example below shows the result for a system of dimension n=2000. The plant is unstable and an observed-based state-feedback stabilizing controller of the same order has been designed. We reduce the order of the controller to r=20 using our approach. Figure-2 compares the Bode plots of the full-order closed systems T(s) with that of the reduced closed loop system Tr(s) (upper), and the Bodel plots of the full-order controller with that of the reduced order controller Kr(s) (lower). On the other hand, Figure-3 shows the time responses of T(s) and Tr(s) to a unit impulse (upper) and to sin(4t). Both figures illustrates almost a perfect match.