2094 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 12, DECEMBER 2005 Preservation of Controllability of Single-Input Time-Varying Linear Systems Under Sampling Gera Weiss Abstract—We show that a time-varying controllable continuous-time system with real analytic coefficients, when sampled in almost any rate, yields a completely controllable discrete-time system. Index Terms—Analytically varying systems, linear control systems, preservation of controllability, sampling. I. INTRODUCTION AND MAIN RESULT In this note, we study preservation of controllability under zero-hold uniform sampling for time-varying single-input linear systems with real analytic coefficients. Sampling and controllability are key notions in modern con- trol-theory (see, e.g., [1]). In particular, the problem of preservation of controllability under sampling attracted attention since the pioneering work of Kalman et al. [5]. Extensions to multi-input systems [3], [6] and to nonlinear systems [8] were established. Different sampling models, such as multirate-sampling [7] and generalized sampled-data hold functions [4] were also considered. This work contributes to this line of research by investigating the problem of preservation of controllability for time-varying systems with real analytic coefficients under zero-hold uniform sampling. A formal description of the main result follows. Consider an -dimensional single-input time-varying control system with real analytic coefficients (1) where and are time-varying matrix and -dimensional column vector, respectively. All the coefficients in these matrices are assumed to be real analytic in . The sampling control strategy that we consider is to sample the state at prescribed equidistributed times and to hold the control constant in periods between samplings. Let be the length of the sampling interval and let be the control value at the th time step. For the discrete set of times where sampling occur, say , , we get a discrete-time system of the form (2) where (3) and is the fundamental matrix solution associated to . Manuscript received June 1, 2005; revised July 14, 2005. Recommended by Associate Editor E. Jonckheere. This work was supported by grants from the Israel Science Foundation and from the Information Society Technologies Pro- gram of the European Commission. The author is with the Department of Computer Science and Applied Math- ematics, The Weizmann Institute of Science, Rehovot 76100, Israel (e-mail: gera.weiss@weizmann.ac.il). Digital Object Identifier 10.1109/TAC.2005.860276 Generally speaking, a system is said to be controllable if, at all times, there exists an input that drives any initial state to any final state. The following definitions give the precise notions for continuous-time and discrete-time systems that we use in this note. Definition I.1: The continuous-time control system (1) is said to be controllable if for every and every pair of states, , there is a time and a control such that if then . Definition I.2: The discrete-time control system (2) is said to be completely controllable (in time steps) if for every and every pair of states, , there are control values such that if then . In the statement of the main result that follows, the term “almost every sampling period” means except for a countable set of sampling periods. Theorem I.3: If the system (1) is controllable then, for almost every sampling period , the sampled-data system (2) is completely controllable. II. PROOF OF THE MAIN RESULT The proof of Theorem I.3 is presented in two steps: First, a statement about real analytic curves is given as Proposition II.1 and proved using some intermediate claims. Then, the proof of the theorem is derived as a corollary of that proposition. Proposition II.1: Let be a real analytic curve. Assume that there is an uncountable set such that, for every , there exists such that the vectors arelinearlydependent.Thenthereexistsaproperlinearsubspace such that . Claim II.2: Let be a curve satisfying the conditions of Proposition II.1. Then there exists such that the function vanishes for every . Proof: For a curve satisfying the conditions of Proposition II.1, we have a map from an uncountable set to a countable set. Since the union of a countable number of countable sets is a countable set, there must be whose preimage is uncountable. For this , the real analytic function has an uncountable zero set, therefore it is identically zero. Claim II.3: Let be a real analytic curve. The th derivative at zero, , of the function defined in the pre- ceding claim is given by where is and denotes the th derivative of at zero. 0018-9286/$20.00 © 2005 IEEE