IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 1, JANUARY 2009 3 Inclusion Principle for Descriptor Systems Delin Chu, Yuzo Ohta, Senior Member, IEEE, and Dragoslav D. ˇ Siljak, Life Fellow, IEEE Abstract—The purpose of this paper is to propose an expan- sion-contraction framework for linear constant descriptor systems within the inclusion principle for dynamic systems. Our primary objective is to provide an explicit characterization of the expan- sion process whereby a given descriptor system is expanded into the larger space where all its solutions are reproducible by the ex- panded descriptor system if appropriate initial conditions are se- lected. When a control law is formulated in the expanded space, the proposed characterizations provide contractibility conditions for implementation of the control law in the original system. A full freedom is provided for selecting appropriate matrices in the pro- posed expansion-contraction control scheme. In particular, the de- rived theoretical framework serves as a flexible environment for expansion-contraction control design of descriptor systems under overlapping information structure constraints. Index Terms—Contractibility, contractions, controllability at in- finity, descriptor systems, expansions, inclusion principle. I. INTRODUCTION I N recent years, the inclusion principle for dynamic sys- tems [1], [2] has emerged as a flexible and powerful math- ematical framework for comparing properties and performance of systems with different dimensions. The principle has found applications in a wide variety of theoretical and practical sit- uations involving model-reduction, large dynamic systems, op- timal control, parallel computations, inclusions of dynamic con- trollers and observers, expert systems and decentralized control of hybrid, mechanical and electrical systems, control of seg- mented telescope and platoons of vehicles in the air and on the ground, as well as the analysis of the finite word-length [3]–[24]. Recently, the development of the inclusion principle for con- tinuous, discrete and stochastic systems [11]–[21] has been fo- cused on formulating conditions for expansions and contrac- tions of control systems, which can help resolve outstanding the- oretical and practical aspects of the principle in building control systems under overlapping information structure constraints. Descriptor systems, which are also called singular systems or generalized state space systems, appear as models in as diverse areas as electrical circuits and multibody systems, chemical en- gineering and economic systems, mechanical structure and bi- ological systems [25]–[34]. Motivated by this wide-spread use of descriptor models, we formulate in this paper the inclusion Manuscript received July 22, 2006; revised April 10, 2007, November 18, 2007, and April 18, 2008. Current version published January 14, 2009. Recom- mended by Associate Editor M. Fujita. D. Chu is with the Department of Mathematics, National University of Sin- gapore, Singapore 117543 (e-mail: matchudl@nus.edu.sg). Y. Ohta is with the Department of Computer and Systems Engineering, Grad- uate School of Engineering, Kobe University, Kobe 657-8501, Japan (e-mail: tcs.y.ohta@people.kobe-u.ac.jp). D. D. ˇ Siljak is with the Department of Electrical Engineering, Santa Clara University, Santa Clara, CA 95053-0569 USA (e-mail: dsiljak@scu.edu). Digital Object Identifier 10.1109/TAC.2008.2009482 principle for expansion and contraction of descriptor systems including contractibility of control laws. In deriving explicit al- gebraic characterizations of the expansion-contraction process, we shall define canonical forms for descriptor systems within the inclusion framework, which generalize canonical forms ob- tained for linear time-invariant systems [22]. Theoretical re- sults established in this work reveal that the expansion-contrac- tion and contractibility in the inclusion principle for descriptor systems are much more complicated than those for standard linear time-invariant systems. In particular, the expanded system cannot be generally expressed in terms of the original system. Furthermore, a control law for the expanded system, which is contractible to a control law for the original systems, cannot be always expressed in terms of the given control law for the orig- inal system. These difficulties can be overcome only if either the original system is controllable at infinity, or the set of the order of the poles of the expanded system at infinity contains the set of the order of the poles of the original system at infinity. The paper is organized as follows. In the next section, we de- rive the inclusion principle for descriptor systems. Section III contains our main results. Some concluding remarks are pro- vided in Section IV. Proofs of a number of lemmas and theo- rems are given in Appendix. II. INCLUSION AND CONTRACTIBILITY Consider a pair of descriptor systems (1) and (2) where , , are the state, input and output of system at time , and , , are those of , and , , , , , , , are constant matrices. For descriptor systems and , unique solutions are guaranteed to exist if and only if the pencils and are regular, i.e., and do not vanish identically. For regular systems and , in order to have standard continuous solutions, the inputs and should be sufficiently smooth, that is, and must belong to some suitable function spaces [34], [41], say, and , respectively. Otherwise, if and are not sufficiently smooth, then the impulses may arise in the re- sponses of systems and even if these two systems are reg- ular. For this reason, descriptor systems are considerably more difficult to analyze and control than the standard linear time-in- variant systems. 0018-9286/$25.00 © 2009 IEEE Authorized licensed use limited to: Santa Clara University. Downloaded on July 17, 2009 at 15:02 from IEEE Xplore. Restrictions apply.