IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 6, JUNE 2001 991 Impulse Controllability and Observability of Rectangular Descriptor Systems João Yoshiyuki Ishihara and Marco Henrique Terra Abstract—In this note, the presence of impulsive responses in descriptor systems and how it relates to impulse controllability and impulse observ- ability is considered. It is shown that the equivalence between impulse controllability (observability) and the existence of an impulse eliminating semistate feedback (output injection) gain, although true for square descriptor systems, does not hold for rectangular descriptor systems in general. Hence, necessary and sufficient conditions for the existence of an impulse eliminating semistate feedback (output injection) gain are presented. Index Terms—Controllability, duality, observability, singular systems. I. INTRODUCTION Descriptor systems provide a more natural description of dynamical systems than the standard state-space systems [4], [12], [13], [15], [20]. They have many important applications in, for example, circuit systems [16], robotics [14], and aircraft modeling [18]. However, descriptor sys- tems may present impulsive responses and lack of uniqueness in the solutions. In engineering systems, impulses may cause degradation in performance, damage components, or even destroy the system. There- fore, impulsive behavior is undesirable and its elimination is an im- portant issue in control system design. For regular square descriptor systems it is well known that impulse controllability is equivalent to the existence of an impulse eliminating semistate feedback gain [3], [5]–[7], [12]. Also, by duality, impulse observability assures the exis- tence of an impulse eliminating output injection gain. Many character- izations of impulse controllability and impulse observability have been provided for regular descriptor systems during the last decade [2], [3], [6], [7], [12], [19]. A simple sufficient condition for impulse uncon- trollability and impulse unobservability is given in [11]. Extensions for more general nonregular and rectangular descriptor systems have been considered more recently [8]–[10]. In particular, [10] presents a rank condition of impulse observability for regular descriptor systems that remains valid for arbitrary descriptor systems. Although many equiva- lent test conditions of impulse controllability (observability) for regular descriptor systems still remain valid for general rectangular descriptor systems, in this article we show that the equivalence between impulse controllability (observability) and the existence of an impulse elimi- nating semistate feedback (output injection) gain is lost. Hence, nec- essary and sufficient conditions for the existence of an impulse elim- inating semistate feedback (output injection) gain are presented. An interesting feature for general nonsquare systems is that these condi- tions are not dual. This paper is organized as follows. Section II con- tains basic definitions and some tests for impulse controllability (ob- servability); in particular, we extend the tests of impulse-free regular systems to the general case and present an impulse controllability con- dition valid for the general case. In Section III, we present necessary and sufficient conditions for the existence of an impulse eliminating semistate feedback (output injection) gain. The conclusion is given in Section IV. Manuscript received April 25, 2000; revised November 10, 2000. Recom- mended by Associate Editor L. Dai. This work was supported by FAPESP, São Paulo State University Council, under Grant 98/12113-2. The authors are with the Electrical Engineering Department—EESC, Univer- sity of São Paulo, 1465 CP 359, CEP 13560-970, São Carlos, São Paulo, Brazil. (e-mail: ishihara@sel.eesc.sc.usp.br; terra@sel.eesc.sc.usp.br). Publisher Item Identifier S 0018-9286(01)05114-5. II. IMPULSE-FREE SYSTEMS,IMPULSE CONTROLLABILITY AND OBSERVABILITY To determine and compare the rank conditions for the existence of semistate feedback and output injection matrices in Section III, a brief discussion about descriptor systems is presented. We consider linear, time-invariant systems described by differential equations of the form : (1) (2) where ; ; ; ; . It is assumed that and are admissible, i.e., they are such that there exists at least one trajectory satisfying (1) [10]. System is called a descriptor system, generalized state-space system, or singular system. If and , system is called regular. Allowing distributions as possible forcing functions and solu- tions [9], and using Laplace transform, it is easy to show the fol- lowing solvability characterizations (for simplicity, no new notation for the Laplace transform of is introduced): S1) the pair is admissible if and only if [10] - - S2) there exists an admissible for every initial condition if and only if - S3) for , we can always find an admissible initial condition (at least we can take the trivial solution); S4) for , every initial condition is admissible if and only if - Solvability characterizations regarding unicity or other classes of inputs and initial conditions are also possible (see, e.g., [8] and [9]). When it is convenient, we can assume without loss of generality that is in the SVD coordinate system [3] and where is a nonsingular matrix with rank , and and are orthog- onal matrices. In the following, we consider the non existence of impulsive re- sponses and its characterization. Definition 1: System (1) with , or the pair , is impulse-free if all free solutions are smooth in the time interval 0018–9286/01$10.00 © 2001 IEEE