Research Papers

Mixed Additive/Multiplicative H Model Reduction

[+] Author and Article Information
Yin-Lam Chow

e-mail: yldick.chow@gmail.com

Yue-Bing Hu

e-mail: yuebinghu@gmail.com
Department of Mechanical Engineering,
The University of Hong Kong,
Hong Kong, China

Xianwei Li

Research Institute of Intelligent Control and Systems,
Harbin Institute of Technology,
Heilongjiang, Harbin 150080, China
e-mail: lixianwei1985@gmail.com

Andreas Kominek

Institute of Control Systems,
Hamburg University of Technology,
Hamburg 21073, Germany
e-mail: andreas.kominek@tu-harburg.de

James Lam

Department of Mechanical Engineering,
The University of Hong Kong,
Hong Kong, China
e-mail: james.lam@hku.hk

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received July 22, 2012; final manuscript received March 22, 2013; published online May 23, 2013. Assoc. Editor: Won-jong Kim.

J. Dyn. Sys., Meas., Control 135(5), 051005 (May 23, 2013) (10 pages) Paper No: DS-12-1233; doi: 10.1115/1.4024111 History: Received July 22, 2012; Revised March 22, 2013

In this paper, the problem of mixed additive/multiplicative model reduction for stable linear continuous-time systems is studied. To deal with nonsquare or nonminimum phase systems, the multiplicative error bound is constructed using spectral factorization technique. By virtue of the bounded real lemma and the projection lemma, a linear matrix inequality approach is proposed for mixed additive/multiplicative H model reduction, which can be implemented by the well-known cone complementary linearization method. Finally, two numerical examples are provided to demonstrate the effectiveness and advantages of the obtained results.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Zhou, K., and Doyle, J. C., 1998, Essentials of Robust Control, Prentice-Hall, NJ.
Zhou, K., Doyle, J. C., and Glover, K., 1995, Robust and Optimal Control, Prentice-Hall, Englewood Cliffs, NJ.
Obinata, G., and Anderson, B. D. O., 2000, Model Reduction for Control System Design, Springer-Verlag, London.
Moore, B. C., 1981, “Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction,” IEEE Trans. Auto. Contr., 26(1), pp. 17–32. [CrossRef]
Green, M., and Limbeer, D. J. N., 2012, Linear Robust Control, Dover Publications, Mineola, NY.
Glover, K., 1984, “All Optimal Hankel-Norm Approximations of Linear Multivariable Systems and Their L-Error Bounds,” Int. J. Contr., 39(6), pp. 1115–1193. [CrossRef]
Desai, U. B., and Pal, D., 1984, “A Transformation Approach to Stochastic Model Reduction,” IEEE Trans. Auto. Contr., 29(12), pp. 1097–1100. [CrossRef]
Green, M., 1988, “A Relative Error Bound for Balanced Stochastic Truncation,” IEEE Trans. Auto. Contr., 33(10), pp. 961–965. [CrossRef]
Wang, W., and Safonov, M. G., 1992, “Multiplicative-Error Bound for Balanced Stochastic Truncation Model Reduction,” IEEE Trans. Auto. Contr., 37(8), pp. 1265–1267. [CrossRef]
Matson, J. B., Lam, J., Anderson, B. D. O., and James, B., 1993, “Multiplicative Hankel Norm Approximation of Linear Multivariable System,” Int. J. Contr., 58, pp. 129–167. [CrossRef]
Grigoriadis, K. M., 1995, “Optimal H Model Reduction via Linear Matrix Inequalities: Continuous and Discrete-Time Cases,” Sys. Contr. Lett., 26, pp. 321–333. [CrossRef]
Grigoriadis, K. M., 1997, “L2 and L2 − L Model Reduction via Linear Matrix Inequalities,” Int. J. Contr., 68, pp. 485–498. [CrossRef]
Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V., 1994, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA.
Abbas, H. S., and Werner, H., 2011, “Frequency-Weighted Discrete-Time LPV Model Reduction Using Structurally Balanced Truncation,” IEEE T. Contr. Syst. Tech., 19(1), pp. 140–147. [CrossRef]
Gao, H., Lam, J., and Wang, C., 2006, “Model Simplification for Switched Hybrid Systems,” Sys. Contr. Lett., 55, pp. 1015–1021. [CrossRef]
Gao, H., Lam, J., Wang, C., and Wang, Q., 2004, “Hankel Norm Approximation of Linear Systems With Time-Varying Delay: Continuous and Discrete Cases,” Int. J. Contr., 77(17), pp. 1503–1520. [CrossRef]
Gao, H., Lam, J., Wang, C., and Xu, S., 2004, “H Model Reduction for Discrete Time-Delay Systems: Delay-Independent and Dependent Approaches,” Int. J. Contr., 77(4), pp. 321–335. [CrossRef]
Ghafoor, A., and Sreeram, V., 2008, “A Survey/Review of Frequency-Weighted Balanced Model Reduction Techniques,” ASME J. Dyn. Sys., Meas., Control, 130(6), p. 061004. [CrossRef]
Ghafoor, A., Sreeram, V., and Treasure, R., 2007, “Frequency Weighted Model Reduction Technique Retaining Hankel Singular Values,” Asian J. Contr., 9(1), pp. 50–56. [CrossRef]
Huang, X.-X., Yan, W.-Y., and Teo, K.-L., 2001, “H2 Near-Optimal Model Reduction,” IEEE Trans. Auto. Contr., 46(8), pp. 1279–1284. [CrossRef]
Li, P., Lam, J., Wang, Z., and Date, P., 2011, “Positivity-Preserving H Model Reduction for Positive Systems,” Automatica, 47(7), pp. 1504–1511. [CrossRef]
Louca, L. S., Stein, J. L., and Hulbert, G. M., 2010, “Energy-Based Model Reduction Methodology for Automated Modeling,” ASME J. Dyn. Sys., Meas., Control, 132(6), p. 061202. [CrossRef]
Wang, Q., Lam, J., Xu, S., and Zhang, L., 2006, “Delay-Dependent γ-Suboptimal H Model Reduction for Neutral Systems With Time-Varying Delays,” ASME J. Dyn. Sys., Meas., Control, 128(2), pp. 394–399. [CrossRef]
Wang, S.-G., and Wang, B., 2010, “Parameterization Theory of Balanced Truncation Method for Any Evenly Distributed RC Interconnect Circuits and Transmission Lines,” Automatica, 46(3), pp. 625–628. [CrossRef]
Wu, L., and Ho, D. W. C., 2009, “Reduced-Order L2 − L Filtering for a Class Nonlinear Switched Stochastic Systems,” IET Contr. Theory Appl., 3(5), pp. 493–508. [CrossRef]
Wu, L., Shi, P., Gao, H., and Wang, C., 2006, “H Model Reduction for Two-Dimensional Discrete State-Delayed Systems,” IEE Pr. Vis. Image Sig. Process., 153(6), pp. 769–784. [CrossRef]
Wu, L., Shi, P., Gao, H., and Wang, J., 2009, “H Model Reduction for Linear Parameter-Varying Systems With Distributed Delay,” Int. J. Contr., 82(3), pp. 408–422. [CrossRef]
Wu, L., Su, X., Shi, P., and Qiu, J., 2011, “Model Approximation for Discrete-Time State-Delay Systems in the T-S Fuzzy Framework,” IEEE Trans. Fuzzy Syst., 19(2), pp. 366–378. [CrossRef]
Wu, L., and Zheng, W.-X., 2009, “Weighted H Model Reduction for Linear Switched Systems With Time-Varying Delay,” Automatica, 45(1), pp. 186–193. [CrossRef]
Xu, S., and Chen, T., 2003, “H Model Reduction in the Stochastic Framework,” SIAM J. Contr. Optimiz., 42(4), pp. 1293–1309. [CrossRef]
Xu, S., and Lam, J., 2003, “H Model Reduction for Discrete-Time Singular Systems,” Sys. Contr. Lett., 48, pp. 121–133. [CrossRef]
Xu, S., Lam, J., Huang, S., and Yang, C., 2001, “H Model Reduction for Linear Time-Delay Systems: Continuous-Time Case,” Int. J. Contr., 74(11), pp. 1062–1074. [CrossRef]
Zadegan, A. H., and Zilouchian, A., 2005, “Model Reduction of Large-Scale Discrete Plants With Specified Frequency Domain Balanced Structure,” ASME J. Dyn. Sys., Meas., Control, 127(3), pp. 486–498. [CrossRef]
Zhang, L., Boukas, E.-K., and Shi, P., 2009, “H Model Reduction for Discrete-Time Markov Jump Linear Systems With Partially Known Transition Probability,” Int. J. Contr., 82(2), pp. 343–351. [CrossRef]
Zhang, L., Boukas, E.-K., and Shi, P., 2009, “μ-Dependent Model Reduction for Uncertain Discrete-Time Switched Linear Systems With Average Dwell Time,” Int. J. Contr., 82(2), pp. 378–388. [CrossRef]
Zhang, L., Huang, B., and Lam, J., 2003, “H Model Reduction of Markovian Jump Linear Systems,” Sys. Contr. Lett., 50, pp. 103–118. [CrossRef]
Zhang, L., Shi, P., Boukas, E.-K., and Wang, C., 2008, “H Model Reduction for Uncertain Switched Linear Discrete-Time Systems,” Automatica, 44, pp. 2944–2949. [CrossRef]
Glover, K., 1986, “Multiplicative Approximation of Linear Multivariable Systems With L Error Bounds,” Proceedings of the 1986 American Control Conference, pp. 1705–1709.
Gahinet, P., and Apkarian, P., 1994, “A Linear Matrix Inequality Approach to H Control,” Int. J. Robust Nonlin. Appr., 4(4), pp. 421–448. [CrossRef]
Iwasaki, T., and Skelton, R. E., 1994, “All Controllers for the General H Control Problem: LMI Existence Conditions and State Space Formulas,” Automatica, 30, pp. 1307–1317. [CrossRef]
Ghaoui, L. E., Oustry, F., and AitRami, M., 1997, “A Cone Complementarity Linearization Algorithm for Static Output-Feedback and Related Problems,” IEEE Trans. Auto. Contr., 42, pp. 1171–1176. [CrossRef]
Löfberg, J., 2004, “YALMIP: A Toolbox for Modeling and Optimization in MATLAB,” Proceedings of the 2004 IEEE International Symposium on Computer Aided Control System Design, pp. 284–289. [CrossRef]
Toh, K. C., Todd, M. J., and Tütüncü, R. H., 1999, “SDPT3—A Matlab Software Package for Semidefinite Programming,” Optimiz. Meth. Soft., 11(1), pp. 545–581. [CrossRef]
Safonov, M. G., and Chiang, R. Y., 1988, “Model Reduction for Robust Control: A Schur Relative-Error Method,” Proceedings of the 1988 American Control Conference, pp. 1685–1690.


Grahic Jump Location
Fig. 1

Block-diagram representation of mixed model reduction scheme

Grahic Jump Location
Fig. 2

Comparison of maximum singular values of the original system and the reduced-order systems in Example 1

Grahic Jump Location
Fig. 3

Comparison of additive errors in Example 1

Grahic Jump Location
Fig. 4

Comparison of multiplicative errors in Example 1

Grahic Jump Location
Fig. 5

The Pareto curve between 1/αadd and 1/αmul of Theorem 1 in Example 1

Grahic Jump Location
Fig. 6

Comparison of magnitude responses of the original system and the reduced-order systems in Example 2

Grahic Jump Location
Fig. 7

Comparison of phase responses of the original system and the reduced-order systems in Example 2

Grahic Jump Location
Fig. 8

Comparison of multiplicative error magnitude responses in Example 2




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In