Computation of Impulse-Response Gramian for Interval Systems

Abstract

This paper presents an impulse response gramian for the interval systems. To reduce the higher-order interval systems, two methods used to evaluate the impulse response gramian are computed in the frequency domain. The first method is based on Routh technique with ($\alpha -\text{β}$) table to evaluate energy integrals for finding the diagonal impulse response gramian. The second method is based on the Inners technique to compute the diagonal elements of the Gramian matrix, while the off-diagonal elements are obtained by the Markov parameters. The method is computationally simple but the reduced-order stable interval systems are difficult to obtain. To overcome this problem, the reduced-order model is developed based on Kharitonov's theorem. The advantage of the proposed technique is useful for system identification and reduction technique. Illustrative examples are included and results are compared and stability of the reduced-order model is guaranteed.

Keywords:

• Impulse response gramian
• Inners technique
• Interval systems
• Markov parameters
• Routh approximation
• Kharitonov's theorem

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported under National Key Research and Development Program of China Ref. [2018YFB1305505] and the Shenzhen Science and Technology Innovation Commission Project Grant Ref. [JCYJ20170818153635759] and Science and Technology Planning Project of Guangdong Province Ref. [2017B010117009] and Guangdong Provincial Engineering Technology Research Center of Intelligent Unmanned System and Autonomous Environmental Perception.

Notes on contributors

Kranthi Kumar Deveerasetty

Kranthi Kumar Deveerasetty was born in Guntur, A P, India in 1986. He received the degree of BE from the SIR C R Reddy College of Engineering in 2008. He received MTech and PhD degrees in electrical engineering from the Indian Institute of Technology, Banaras Hindu University (IIT-BHU), Varanasi, India in 2010 and 2016, respectively. Currently, he is a post-doctoral fellow at Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences. Achieved best application paper ward at the 11th International Conference on Modelling, Identification and Control (ICMIC 2019). His areas of interest are model-order reduction, stability of interval systems, modelling and control design for UAVs, sliding mode control and image processing. Email: [email protected]

Yimin Zhou

Yimin Zhou received her PhD from the University of Oxford, UK, in 2008. Currently, she is an associate professor working at Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China. Achieved best application paper ward at the 11th International Conference on Modelling, Identification and Control (ICMIC 2019). Her research interests include automatic control, fault diagnosis, robotics, machine learning and energy management. Corresponding author. Email:[email protected]

Shyam Kamal

Shyam Kamal received the bachelor's degree in electronics and communication engineering from Gurukula Kangri Vishwavidyalaya, Haridwar, India, in 2009, and the PhD degree in systems and control engineering from IIT Bombay, India, in 2014. From 2014 to 2016, he was with the Department of Systems Design and Informatics, Kyushu Institute of Technology. He is currently an assistant professor with the Department of Electrical Engineering, IIT (BHU) Varanasi, India. He has published one monograph and 54 journal articles and conference papers. His research interests include fractional order systems, contraction analysis, and discrete and continuous higher order sliding mode control. Email: [email protected]

Shyam Krishna Nagar

S K Nagar was born in Varanasi, India, on 10 May 1955. He studied at Institute of Technology (IT) BHU and received the BTech and MTech degrees in electrical engineering in 1976 and 1978, respectively. He obtained PhD degree in electrical engineering from University of Roorkee, Roorkee, India. He joined IT, BHU in 1980 as a lecturer with the Department of Electrical Engineering and as professor in 2001 to till date. He was National Advisory Committee member for the National System Conference in 2008 (NSC2008) and Technical Program Committee member for the Student's Conference on Engineering and Systems (SCES-2014). He received Best Paper award at National System Conference (NSC-2011). He is life member of System Society of India and Fellow of Institute of Engineers. His research interests include model reduction, digital control and discrete event systems. He has supervised few PhD theses in the area of model reduction and controller design. He has published numerous papers in regular peer reviewed journals and national and international conferences. Email: [email protected]