Time- and frequency-limited H₂-optimal model order reduction of bilinear control systems

ABSTRACT

In the time- and frequency-limited model order reduction, a reduced-order approximation of the original high-order model is sought to ensure superior accuracy in some desired time and frequency intervals. We first consider the time-limited ${\mathcal{H}}_2$-optimal model order reduction problem for bilinear control systems and derive first-order optimality conditions that a local optimum reduced-order model should satisfy. We then propose a heuristic algorithm that generates a reduced-order model, which tends to achieve these optimality conditions. The frequency-limited and the time-limited ${\mathcal{H}}_2$-pseudo-optimal model reduction problems are also considered wherein we restrict our focus on constructing a reduced-order model that satisfies a subset of the respective optimality conditions for the local optimum. Two new algorithms have been proposed that enforce two out of four optimality conditions on the reduced-order model upon convergence. The algorithms are tested on three numerical examples to validate the theoretical results presented in the paper. The numerical results confirm the efficacy of the proposed algorithms.

Keywords:

• ${\mathcal{H}}_2$-optimal
• bilinear systems
• frequency-limited
• model order reduction
• pseudo-optimal
• time-limited

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported in part by National Natural Science Foundation of China under Grant (Nos. 61873336, 61873335), in part by the National Key Research and Development Program (Nos. 2020YFB 1708200), in part by the Foreign Expert Program (No. 20WZ2501100) granted by the Shanghai Science and Technology Commission of Shanghai Municipality (Shanghai Administration of Foreign Experts Affairs), in part by 111 Project (No. D18003) granted by the State Administration of Foreign Experts Affairs, and in part by the Fundamental Research Funds for the Central Universities under Grant (No. FRF-BD-19-002A). M. I. Ahmad is supported by the Higher Education Commission of Pakistan under the National Research Program for Universities Project ID 10176.

Notes on contributors

Umair Zulfiqar

Umair Zulfiqar received Bachelor's degree in 2013 from University of Engineering and Technology Taxila, Pakistan, and Master's degree from National University of Sciences and Technology, Pakistan, both in Electrical Engineering. He came to Australia in 2018 on International Research Training Program of Department of Education and Training, Australia. Since 2018, he has been pursuing PhD in Electrical Engineering from University of Western Australia. His research interests include model order reduction and reduced-order controller design.

Victor Sreeram

Victor Sreeram obtained Bachelor's degree in 1981 from Bangalore University, India, Master's degree in 1983 from Madras University, India, and PhD degree from University of Victoria, British Columbia, Canada in 1990, all in Electrical Engineering. He worked as a Project Engineer in the Indian Space Research Organisation from 1983 to 1985. He joined the Department of Electrical, Electronic, and Computer Engineering, University of Western Australia in 1990 as a lecturer and he is now a Professor. He was on the editorial board of many journals including IET Control, Theory and Applications, Asian Journal of Control, Mathematical Problems in Engineering, Journal of Industrial and Management Optimization, Cogent Engineering, and Smart Grid and Renewable Energy. He was the General Chair of 3rd Australian Control Conference, Perth, November, 2013 and the Vice Chair of Australasian Universities Power Engineering Conference, Perth, WA, September, 2014. He is currently the Co-Chair of National Committee on Control Engineering (Industry 4.0), the Chair of Australian New Zealand Control Conference Steering Committee Incorporated, and the Vice President of Asian Control Association. He is now a Fellow of Institution of Engineers, Australia. His teaching and research interests are Control Systems, Signal Processing, Communications, Power Engineering and Smart Grid and Renewable Energy.

Mian Ilyas Ahmad

Mian Ilyas Ahmad received BSc in Electrical Engineering from University of Engineering & Technology Peshawar, Pakistan in 2006. He then visited Imperial College London under NUST MPhil leading to PhD fellowship and completed his PhD degree in control systems from Imperial College London, UK in 2011. He joined RCMS NUST in Jan. 2012 as an Assistant Professor in the department of Computational Engineering. In January 2013, he joined Max Planck Institute for Complex Technical Systems Magdeburg Germany under Max Planck Postdoctoral fellowship. After successfully completing three years postdoctoral fellowship he resumed his position of Assistant Professor at RCMS in December 2015. Since 2019, he is working as Associate Professor in the department of Computational Engineering at RCMS NUST. His research interests include largescale system analysis and control, model order reduction, applied linear algebra and nonlinear dynamics.

Xin Du

Xin Du received the B.S. degrees from the University of Science and Technology Beijing, Beijing, China in 2004, and the PhD degree in control engineering from Northeastern University, Shenyang in 2010. After finishing his PhD, he joined the Shanghai University in 2010, where he is currently an Associate Professor in the School of Mechatronic Engineering and Automation. During April to August 2012 and December 2013 to August 2015, he had been a Postdoc research fellow at Max-Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany. From 2016 to 2018, he served as a program officer of Bureau of International Cooperation, National Natural Science Foundation of China (NSFC). His current research interests include model order reduction, autonomous unmanned system.