 This paper was not presented at any IFAC metting. This paper was recommended for publication in revised form by Associate Editor Tor Arne Johansen under the direction of Editor Sigurd Skogestad. * Corresponding author. Fax: #1-780-492-2881. E-mail address: sirish.shah@ualberta.ca (S.L. Shah). Automatica 37 (2001) 1637}1645 Brief Paper Ripple-free conditions for lifted multirate control systems  A. K. Tangirala, D. Li, R. S. Patwardhan, S. L. Shah*, T. Chen Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Canada AB T6G 2G6 Matrikon Consulting Co., Advanced Applications Group, Edmonton, Canada AB T5N 4A3 Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada AB T6G 2G7 Received 9 November 1999; revised 26 February 2000; received in "nal form 5 February 2001 Abstract Measurements in chemical processes are often unavailable at a uniform rate due to constraints on the sampling rates of process variables. Situations such as these and others give rise to a set of multirate signals comprising a multirate system. Control of multirate systems is appealing and challenging from a theoretical and practical point of view. Multirate control design in the lifting framework consists of lifting the system and subsequently designing a controller for the single-rate lifted system. In this work, it is shown that under certain conditions, intersample ripples arise in the outputs of closed-loop multirate systems. The process output can be guaranteed to be ripple-free if the controller satis"es certain constraints. Further, it is shown that the presence of an integrator in the plant aids in eliminating these intersample ripples. Experimental evaluations are presented in support of these theoretical re- sults.  2001 Elsevier Science Ltd. All rights reserved. Keywords: Multirate; Digital control; Lifting; Tracking; Ripples 1. Introduction Multirate systems are commonly encountered in chemical processes due to constraints on the sampling rates of certain physical variables. A classical example of such systems is a distillation column where the composi- tion estimates require relatively large analysis times than #ow, temperature or pressure measurements. Often, in practice, it is desired to sample certain variables at a fas- ter rate to emulate the continuous system and meet the performance speci"cations. Moreover, faster control moves might be necessary to achieve improved perfor- mance. Under such conditions, it could be expected that multirate controllers can yield better performance than single-rate controllers. For instance, these controllers would be preferable to single-rate controllers because of the extra degrees of freedom they allow in manipulating control variables. Factors such as these and several others have motivated researchers over four decades (Kranc, 1957; Kalman & Bertram, 1959; Meyer & Burrus, 1975; Crochiere & Rabiner, 1983; Ravi, Khargonekar, Minto, & Nett, 1990; Chen & Francis, 1991) to develop techniques to handle these complex yet appealing systems. Issues dealing with design of optimal multirate control are also discussed in Araki and Yamamoto (1986), Meyer and Burrus (1975), Ravi et al. (1990). The analysis and design of multirate control systems involve mainly two approaches, namely, (i) the periodic discrete-time modelling approach and (ii) the lifting methodology. This paper adopts the latter approach for the following reasons. The main advantage of the lifting approach is that it is conceptually simple and enables a convenient analysis of stability and performance issues of multirate control systems. In addition, the lifting framework translates a MR system into a linear time- invariant (LTI) system, whereas the former approach results in a time-varying system. Clearly, it is simpler to analyze LTI systems because of the rich framework of theory that exists in this area (Khargonekar, Poolla, & Tannenbaum, 1985; Araki & Yamamoto, 1986; Chen & Qiu, 1994; Sagfors & Toivonen, 1998). Lifting techniques (Khargonekar et al., 1985) are essen- tially the result of concatenation of fast-rate signals to form slow-rate signals with increased dimensionality. 0005-1098/01/$-see front matter  2001 Elsevier Science Ltd. All rights reserved. PII:S0005-1098(01)00116-9