Application of Continuous-Discrete Conversion and Balanced Truncation Algorithm to the Order Reduction Problem of Unstable Systems
Received: 12 November 2024 | Revised: 7 December 2024, 19 December 2024, and 28 December 2024 | Accepted: 1 January 2025 | Online: 2 February 2025
Corresponding author: Vu Ngoc Kien
Abstract
Since the model order reduction problem was first posed, numerous order reduction algorithms have been proposed in a variety of approaches. However, the majority of these algorithms have been developed to reduce the order of stable systems. In certain practical applications, such as high-order controller design, the original system may be unstable. Consequently, there is a need for order reduction algorithms capable of reducing the order of both stable and unstable systems. The present paper focuses on introducing a Continuous-Discrete (CD) transformation-based Balanced Truncation (BT) algorithm, which has the capacity to reduce the order of both stable and unstable systems. The efficiency of the improved BT algorithm is demonstrated by the simulation results.
Keywords:
model order reduction, high-order controller, balanced truncation algorithm, stable system, unstable systemDownloads
References
B. Moore, "Principal component analysis in linear systems: Controllability, observability, and model reduction," IEEE Transactions on Automatic Control, vol. 26, no. 1, pp. 17–32, Feb. 1981.
L. Pernebo and L. Silverman, "Model reduction via balanced state space representations," IEEE Transactions on Automatic Control, vol. 27, no. 2, pp. 382–387, Apr. 1982.
K. Glover, "All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds," International Journal of Control, vol. 39, no. 6, pp. 1115–1193, Jun. 1984.
A. C. Antoulas, "Approximation of Large-Scale Dynamical Systems: An Overview," IFAC Proceedings Volumes, vol. 37, no. 11, pp. 19–28, Jul. 2004.
A. C. Antoulas, D. C. Sorensen, and S. Gugercin, "A survey of model reduction methods for large-scale systems," in Structured Matrices in Mathematics, Computer Science, and Engineering I, American Mathematical Society, 2001, pp. 193–219.
T. Stykel, "Gramian-Based Model Reduction for Descriptor Systems," Mathematics of Control, Signals and Systems, vol. 16, no. 4, pp. 297–319, Mar. 2004.
D. F. Enns, "Model reduction with balanced realizations: An error bound and a frequency weighted generalization," in The 23rd IEEE Conference on Decision and Control, Las Vegas, NV, USA, Dec. 1984, pp. 127–132.
H. Sandberg, "Model reduction of linear systems using extended balanced truncation," in 2008 American Control Conference, Seattle, WA, USA, Jun. 2008, pp. 4654–4659.
M. Heinkenschloss, D. C. Sorensen, and K. Sun, "Balanced Truncation Model Reduction for a Class of Descriptor Systems with Application to the Oseen Equations," SIAM Journal on Scientific Computing, vol. 30, no. 2, pp. 1038–1063, Jan. 2008.
T. Stykel, "Balanced truncation model reduction for semidiscretized Stokes equation," Linear Algebra and its Applications, vol. 415, no. 2, pp. 262–289, Jun. 2006.
I. P. Duff, P. Goyal, and P. Benner, "Balanced Truncation for a Special Class of Bilinear Descriptor Systems," IEEE Control Systems Letters, vol. 3, no. 3, pp. 535–540, Jul. 2019.
H. H. Bui, "The Application of LQG Balanced Truncation Algorithm to the Digital Filter Design Problem," Engineering, Technology & Applied Science Research, vol. 12, no. 6, pp. 9458–9463, Dec. 2022.
K. Mustaqim, D. K. Arif, E. Apriliani, and D. Adzkiya, "Model reduction of unstable systems using balanced truncation method and its application to shallow water equations," Journal of Physics: Conference Series, vol. 855, no. 1, Jun. 2017, Art. no. 012029.
D. Novella Rodríguez, B. Del Muro Cuéllar, O. Sename, and M. Velasco Villa, "On the stabilization of high order systems with two unstable poles plus time delay," in 2012 20th Mediterranean Conference on Control & Automation (MED), Barcelona, Spain, Jul. 2012, pp. 12–17.
J. Yang, C. S. Chen, J. a. D. Abreu-Garcia, and Y. Xu, "Model reduction of unstable systems," International Journal of Systems Science, vol. 24, no. 12, pp. 2407–2414, Dec. 1993.
H. B. Minh, C. B. Minh, and V. Sreeram, "Balanced Generalized Singular Perturbation Method for Unstable Linear Time Invariant Continuous Systems," Acta Mathematica Vietnamica, vol. 42, no. 4, pp. 615–635, Dec. 2017.
N. Mirnateghi and E. Mirnateghi, "Model reduction of unstable systems using balanced truncation," in 2013 IEEE 3rd International Conference on System Engineering and Technology, Shah Alam, Malaysia, Aug. 2013, pp. 193–196.
C. S. Hsu and D. Hou, "Reducing unstable linear control systems via real schur transformation," Electronics Letters, vol. 27, no. 11, pp. 984–986, May 1991.
S. K. Nagar and S. K. Singh, "An algorithmic approach for system decomposition and balanced realized model reduction," Journal of the Franklin Institute, vol. 341, no. 7, pp. 615–630, Nov. 2004.
A. Varga, "Model reduction software in the SLICOT library," in CACSD. Conference Proceedings. IEEE International Symposium on Computer-Aided Control System Design (Cat. No.00TH8537), Anchorage, AK, USA, Sep. 2000, pp. 249–254.
Fatmawati, R. Saragih, R. T. Bambang, and Y. Soeharyadi, "Balanced truncation for unstable infinite dimensional systems via reciprocal transformation," International Journal of Control, Automation and Systems, vol. 9, no. 2, pp. 249–257, Apr. 2011.
E. Jonckheere and L. Silverman, "A new set of invariants for linear systems–Application to reduced order compensator design," IEEE Transactions on Automatic Control, vol. 28, no. 10, pp. 953–964, Oct. 1983.
K. Zhou, G. Salomon, and E. Wu, "Balanced realization and model reduction for unstable systems," International Journal of Robust and Nonlinear Control, vol. 9, no. 3, pp. 183–198, 1999.
A. Zilouchian, "Balanced structures and model reduction of unstable systems," in IEEE Proceedings of the SOUTHEASTCON ’91, Williamsburg, VA, USA, Apr. 1991, pp. 1198–1201 vol.2.
C. Boess, N. Nichols, and A. Bunse-Gerstner, "Model reduction for discrete unstable control systems using a balanced truncation approach." University of Reading, 2010.
C. Boess, A. S. Lawless, N. K. Nichols, and A. Bunse-Gerstner, "State estimation using model order reduction for unstable systems," Computers & Fluids, vol. 46, no. 1, pp. 155–160, Jul. 2011.
A. Tamri, A. B. H. Adamou-Mitiche, and L. Mitiche, "Model order reduction, a novel method using krylov sub-spaces and genetic algorithm," Studies in Engineering and Exact Sciences, vol. 5, no. 1, pp. 525–543, Mar. 2024.
Downloads
How to Cite
License
Copyright (c) 2025 Ngo Kien Trung , Vu Thi Anh Ngoc, Nguyen Thi Tham, Vu Ngoc Kien
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain the copyright and grant the journal the right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) after its publication in ETASR with an acknowledgement of its initial publication in this journal.
