Brief paperOn partitioned controllability of switched linear systems☆
Introduction
A switched linear system is a hybrid system which consists of several linear subsystems and a rule that orchestrates the switching among them. There are many studies on the controllability of switched linear systems. For instance, studies for low-order switched linear systems have been presented in Loparo, Aslanis, and IIajek (1987) and Xu and Antsaklis (1999). Some sufficient conditions and necessary conditions for controllability were presented in Ezzine and Haddad (1989) and Szigeti (1992) for switched linear systems under the assumption that the switching sequence is fixed. The complexity of stability and controllability of hybrid systems was addressed in Blondel and Tsitsiklis (1999) and Hu, Zhang, and Deng (2004). Sun and Zheng (2001), Sun, Ge, and Lee (2002), and Sun and Ge (2005) investigated the controllability and reachability issues for switched linear systems in detail.
Consider a switched linear system where is a piece-wise constant, right continuous mapping, called switching signal. As a particular case when there is no control input we have which is called a switched linear system without control.
The reachable set of , denoted by , is defined as: , if there exist , and , such that . (Correspondingly, for system (2), .)
Here is the trajectory of system (1) with initial point , control and switching signal . Similarly, we use to denote the trajectory of system (2).
For system (1) we define a subspace as which is the smallest subspace containing and invariant. The main result about the controllability of system (1) is the following: Theorem 1 Sun et al. (2002) For system(1), the largest reachable set from the origin is. Moreover, for any two points,. System(1)is completely controllable, if and only if,.
We call the controllable subspace of system (1). It is clear that the controllable subspace for system (2) is .
Definition 2 A sub-manifold is called a controllable sub-manifold if for any two points , .
From Theorem 1 one sees easily that the controllable subspace is a controllable sub-manifold. Moreover, it is the largest subspace, which is also a controllable sub-manifold.
Definition 3 A sub-manifold is called a control invariant sub-manifold if for any two points and , , and .
Note that if is a control invariant sub-manifold, then so is its complement . We also have (with mild revision).
Proposition 4 Cheng, Lin, and Wang (2006) is a control invariant sub-manifold.
Assume the controllable subspace, , of system (1) is not the whole space. Then becomes a zero measure set. To describe the controllability of the system over whole state space, we are interested in finding (non-subspace type of) controllable sub-manifolds in . For block diagonal systems or symmetric systems the problem has been discussed in Cheng et al. (2006). This paper investigates the same problem for more general cases. Moreover, the procedure for designing controls and switching laws is also provided.
Section snippets
Controllability of switched linear systems without control
Consider system (2). It is obvious that and are control-invariant. So we ask when is a controllable sub-manifold?
Before giving a useful sufficient condition, we need some preliminaries. Definition 5 A point is called an interior point of system (2), if 0 is an interior point of the convex cone generated by .
Controllability of switched linear systems
Consider system (1). Denote . Assume the controllable subspace of system (1), , is composed by the controllable subspaces of the switching modes. That is,
A1 Then system (1) can be expressed as where corresponds to respectively. An immediate consequence is
Lemma 9 Assumption A1 assures that,, are controllable.
For system (7), we have the following result:
Theorem 10 Consider system(7).
An illustrative example
The proof of Theorem 10 is constructive, so it can be used to construct the control. In the following example, a detailed design process of the control is depicted. Example 11 Consider the following system with , , : where Denote the controllable subspace of system (9) by , the controllable subspace of every mode of system (9) by , respectively. Then
Conclusion
This paper considered when control-invariant sub-manifolds of switched linear systems are controllable. The main controllability results of the paper consisted of two parts. First, the controllability via switching law was investigated, a sufficient condition was obtained. Then in the case that the controllable subspace is partitioned by the controllable subspaces of switching models, a necessary and sufficient condition for being a controllable sub-manifold was obtained. The proof provided
Acknowledgements
This work is supported by NNSF of China under Grant 60674022, 60736022, 60221301 and Grant SIC07010201.
The authors would like to acknowledge the anonymous reviewers and the Associate Editor for their accurate reading and useful suggestions.
Yupeng Qiao received her bachelor degree and master degree in Science from the Harbin Institute of Technology in 2003 and 2005, respectively, and Ph.D. from Institute of Systems Science, AMSS, CAS in 2008. Her research interests include control of nonlinear and hybrid systems.
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2015, Nonlinear Analysis: Hybrid SystemsCitation Excerpt :However, a drawback of most of the existing proposals based on Lyapunov stability analysis is that the modes should share a common equilibrium point. On the other hand, the second point of view uses the controllability (or stabilizability) conditions to derive directly the switching law (see [13–17]); therefore the results are not restricted to systems that share a common equilibrium point, and they can be applied regardless the stability properties of the modes. This point of view generally leads to state-dependent switching laws, the contribution of this paper use this methodology.
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2015, AutomaticaCitation Excerpt :In particular, adaptive control has proven its great capability in compensating for non-switched nonlinearly parameterized systems involving inherent nonlinearity on the basis of a parameter separation technique (Lin & Qian, 2002a,b). On the other hand, the study of switched systems has been extensively investigated in the last decade (Cao & Morse, 2010; Colaneri, Geromel, & Astolfi, 2008; Girard, Pola, & Tabuada, 2010; Goebel, Sanfelice, & Teel, 2009; Mancilla-Aguilar & García, 2006; Qiao & Cheng, 2009), and the rapidly developing area of intelligent control, such as robotic, mechatronic and mechanical systems, gene regulatory networks, switching power converters, is an important source of motivation for this study (Hespanha, 2003; Lin & Antsaklis, 2009; Mojica-Nava, Quijano, Rakoto-Ravalontsalama, & Gauthier, 2010; Serres, Vivalda, & Riedinger, 2011). Meanwhile, many methodologies such as single Lyapunov function, multiple Lyapunov functions (MLFs), average dwell-time have been proposed based on some specified classes of switching laws in the study of switched systems (Branicky, 1998; Han, Ge, & Lee, 2010; Liberzon, 2003; Liberzon & Morse, 1999; Long & Zhao, 2012, 2014b).
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2014, Applied Mathematics and ComputationStructural controllability of switched linear systems
2013, AutomaticaCitation Excerpt :Besides, switched linear systems also have promising applications in control of mechanical systems, aircrafts, satellites and swarming robots. Driven by its importance in both theoretical research and practical applications, the switched linear system has attracted considerable attention during the last decade; see e.g., Ji, Lin, and Lee (2009), Liberzon (2003), Lin and Antsakis (2007), Qiao and Cheng (2009), Sun, Ge, and Lee (2002), Xie and Wang (2003). Much work has been done on the controllability of switched linear systems.
Global stabilization for a class of switched nonlinear feedforward systems
2011, Systems and Control LettersCitation Excerpt :Stability is the most important issue in the study of switched systems (see, e.g., [3–8]). This issue is very difficult to handle due to the hybrid nature of switched systems operation [9–14]. Stability under an arbitrary switching is a very desirable system property [15].
Yupeng Qiao received her bachelor degree and master degree in Science from the Harbin Institute of Technology in 2003 and 2005, respectively, and Ph.D. from Institute of Systems Science, AMSS, CAS in 2008. Her research interests include control of nonlinear and hybrid systems.
Daizhan Cheng graduated from Tsinghua University in 1970, received M.S. from Graduate School, Chinese Academy of Sciences in 1981, Ph.D. from Washington University, St. Louis, in 1985. Since 1990, he has been a professor with Institute of Systems Science, AMSS, CAS. His research interests include nonlinear systems, hybrid systems and numerical method for system control. He is an IEEE Fellow and IFAC Fellow.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Maria Elena Valcher under the direction of Editor Roberto Tempo.