Brief paperFinite-time stabilization and control for a class of nonlinear Hamiltonian descriptor systems with application to affine nonlinear descriptor systems☆
Introduction
Descriptor systems are often referred to as implicit systems, which have wide applications in many practical systems, such as electric circuit systems, chemical process, economy systems, and so on (Dai, 1989). In the last three decades, linear descriptor systems have received a lot of attention and many significant results on their stabilization and control have been obtained, for example, see Dai (1989), Izumi, Yoshiyuki, Atsumi, and Nobuhide (1997), Lin and Chen (1999), Xu and Yang (2000), Zhu, Zhang, Cheng, and Feng (2007), Hill and Mareels (1990), Fridman and Shaked (2002) and references therein. On contrary, nonlinear descriptor systems (NDSs) have received much less attention due to difficulty involved, and as a result, there are few works on NDSs reported in literature (e.g. Hong and Jiang, 2006, Huang et al., 2005, Izumi et al., 1997, Sun and Wang, 2010). For nonlinear differential-algebraic systems, the stabilization and output regulation were considered by the feedback linearization approach in Liu and Ho (2004) and Zhu and Cheng (2001), respectively; meanwhile, the stabilization and robust stabilization problems were discussed using the Hamiltonian function method in Liu, Li, and Wu (2006). In Sun and Wang (2010), the stabilization and control problems were investigated for a class of nonlinear Hamiltonian descriptor systems (NHDSs).
The concept of finite-time stability arises naturally in time-optimal control. A classical example is double integrator with bang–bang time-optimal feedback control (Athans & Falb, 1966). In recent years, finite-time stabilization has received a great deal of attention and a number of results have been obtained in a series of works (Bhat and Bernstein, 1998, Hong and Jiang, 2006, Huang et al., 2005, Qian and Li, 2005, Wang and Feng, 2008, Zhang et al., 2012). It was demonstrated that finite-time stable systems possess not only faster convergence but also better robustness and disturbance attenuation properties (Bhat & Bernstein, 1998). In Qian and Li (2005), a finite-time stable output feedback controller was constructed for a class of planar systems by developing a nonsmooth observer and modifying a power integrator technique. Using backstepping and input-to-state stability techniques, a nonsmooth finite-time stable partial-state feedback controller was designed for nonlinear systems with parametric and dynamic uncertainties in Hong and Jiang (2006). For uncertain nonlinear systems that are dominated by a lower-triangular system, the global finite-time stabilization problem was investigated by Hölder continuous state feedback in Huang et al. (2005) and the dynamic gain control approach and state transformation in Zhang et al. (2012), respectively. In Wang and Feng (2008), finite-time stabilization of nonlinear Port-Controlled Hamiltonian (PCH) systems was studied by energy shaping plus damping injection techniques. It should be noted, to the authors’ best knowledge, that there are few results on the issue of finite-time stabilization of nonlinear Hamiltonian descriptor systems.
On the other hand, it has been shown that the PCH system has some advantages in control designs (Fujimoto et al., 2003, Ortega et al., 2005, Shen et al., 2003, Wang and Feng, 2008). The Hamiltonian function, the sum of potential energy and kinetic energy in physical systems, is a good Lyapunov function candidate for many physical systems. Due to this and its nice structural properties with clear physical meaning, the PCH system has been extensively used in practical control systems, such as multimachine power systems (Ortega et al., 2005, Shen et al., 2003).
In this paper, we take advantages of the PCH system to study the finite-time stabilization and finite-time control problems of NDSs, and to set up a new way, called energy-based approach, to study NDSs. To this end, we first investigate the finite-time stabilization and finite-time control problems of NHDSs, and then apply the results obtained for NHDSs to study the affine NDSs. This is the actual motivation of this paper.
In this paper, we study the finite-time stabilization and finite-time control problems of a class of NHDSs with application to NDSs. First, a strictly dissipative NHDS is obtained by a suitable state feedback. And then the strictly dissipative system is transformed into a strictly dissipative Hamiltonian differential-algebraic system. With the equivalent differential-algebraic form, the finite-time stabilization problem is discussed for the NHDSs, and the settling time of the closed loop system is also estimated. Using the obtained stabilization result, a finite-time controller is designed for the NHDSs with external disturbances. Finally, the obtained results for the NHDS are applied to affine NDSs. Simulation results show the effectiveness of the controllers proposed in this paper.
The remainder of this paper is organized as follows. In Section 2, we present the preliminaries and study the equivalent transformation of NHDSs. Section 3 presents the main results of this paper, that is finite-time stabilization of NHDSs and finite-time control of NHDSs with external disturbances. In Section 4, we apply the results obtained for NHDSs to affine NDSs. Section 5 gives an illustrative example, which is followed by the conclusion in Section 6.
Section snippets
Preliminaries and equivalent transformation of NHDSs
To study finite-time stabilization of NHDSs, we recall the following lemmas and definition first.
Lemma 1 where and are all real numbers.Jensen’s Inequality (Hong & Jiang, 2006)
In Lemma 1, let and , then we obtain the following inequality (Wang & Feng, 2008)
From the inequality (2), we also obtain the following inequality
Lemma 2 Consider a dynamic systemIf there exist a real number and a Bhat & Bernstein, 1998
Main results
In this section, we discuss the finite-time stabilization and finite-time control problems of the system (9). With the equivalent form in Section 2, we first study the finite-time stabilization problem of the system (9), and then discuss the finite-time control problem.
Application to affine nonlinear descriptor systems
In this section, we utilize the results obtained in Section 3 to investigate the finite-time stabilization and finite-time control problems of affine NDSs. First, an equivalent Hamiltonian form is obtained, and then we use the equivalent Hamiltonian form to design a finite-time stabilization controller and a finite-time controller respectively.
To study the finite-time stabilization and finite-time control of affine NDSs, we recall the following lemmas.
Lemma 5 An arbitrary systemWang et al., 2003
An illustrative example
In this section, we give an illustrative example to show how to use Corollary 1 to design a finite-time controller for nonlinear circuit system.
Consider the nonlinear circuit system in Fig. 1, where the capacitance and the inductance are controlled by electric charge and magnetic flux , respectively; and their characteristic are represented by ; the volt–ampere characteristic of the nonlinear resistance is represented by , and is disturbance signal.
Conclusion
In this paper, the finite-time stabilization and finite-time control problems have been investigated for a class of NHDSs. It is shown that the nonlinear Hamiltonian descriptor system can be transformed into an equivalent nonlinear dissipative differential-algebraic form by using the structural characteristic of the system and a suitable state feedback. Using the feedback dissipative form, a finite-time stabilization controller is developed first, and a finite-time controller is then
Acknowledgments
The authors are grateful to the associate editor and reviewers for their constructive comments based on which the paper is greatly improved.
Liying Sun received her B.E. degree in Automatic Control from Dalian Marine College, China in 1989, her M.S. degree from School of Mathematics and System Sciences and her Ph.D degree from School of Control Science and Engineering from Shandong University, China in 2000 and 2010 respectively. She has been with School of Mathematics Science, University of Jinan, China since 2000, and she is currently an associate professor. From March 2012 to September 2012, she was a visiting scholar in
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Liying Sun received her B.E. degree in Automatic Control from Dalian Marine College, China in 1989, her M.S. degree from School of Mathematics and System Sciences and her Ph.D degree from School of Control Science and Engineering from Shandong University, China in 2000 and 2010 respectively. She has been with School of Mathematics Science, University of Jinan, China since 2000, and she is currently an associate professor. From March 2012 to September 2012, she was a visiting scholar in Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong. Her main research interests include nonlinear descriptor systems and Hamiltonian systems.
Gang Feng received his B.E. and M.E. degrees in Automatic Control from Nanjing Aeronautical Institute, China in 1982 and in 1984 respectively, and his Ph.D. degree in Electrical Engineering from the University of Melbourne, Australia in 1992.
He has been with City University of Hong Kong since 2000 where he is now Chair Professor of Mechatronic Engineering. He was lecturer/senior lecturer at School of Electrical Engineering, University of New South Wales, Australia, 1992–1999. He was awarded an Alexander von Humboldt Fellowship in 1997–1998, and the IEEE Transactions on Fuzzy Systems Outstanding Paper Award in 2007. His current research interests include hybrid systems and control, intelligent systems and control, and networked systems and control.
He is an IEEE Fellow, an associate editor of IEEE Trans. Fuzzy Systems, and was an associate editor of IEEE Trans. Automatic Control, IEEE Trans. Systems, Man & Cybernetics, Part C, Mechatronics, and Journal of Control Theory and Applications.
Yuzhen Wang graduated from Tai’an Teachers College in 1986, received his M.S. degree from Shandong University of Science & Technology in 1995 and his Ph.D. degree from the Institute of Systems Science, Chinese Academy of Sciences in 2001. Since 2003, he is a professor with the School of Control Science and Engineering, Shandong University, China, and now the Dean of the School of Control Science and Engineering, Shandong University. From 2001 to 2003, he worked as a Postdoctoral Fellow in Tsinghua University, Beijing, China. From March 2004 to June 2004, from February 2006 to May 2006 and from November 2008 to January 2009, he visited City University of Hong Kong as a Research Fellow. From September 2004 to May 2005, he worked as a visiting Research Fellow at the National University of Singapore. His research interests include nonlinear control systems, Hamiltonian systems and Boolean networks. He received the Prize of Guan Zhaozhi in 2002, the Prize of Huawei from the Chinese Academy of Sciences in 2001, the Prize of Natural Science from Chinese Education Ministry in 2005, and the National Prize of Natural Science of China in 2008. Currently, he is an associate editor for IMA Journal of Mathematical Control and Information, and a Technical Committee member of IFAC (TC2.3).
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This work is supported by the National Nature Science Foundation of China (61374002, 61374065, 61034007) and the Research Grants Council, University Grants Committee, Hong Kong, China (CityU113311). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Alessandro Astolfi under the direction of Editor Andrew R. Teel.
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