Elsevier

Automatica

Volume 37, Issue 10, October 2001, Pages 1637-1645
Automatica

Brief Paper
Ripple-free conditions for lifted multirate control systems

https://doi.org/10.1016/S0005-1098(01)00116-9Get rights and content

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 satisfies 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 results.

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 composition estimates require relatively large analysis times than flow, temperature or pressure measurements. Often, in practice, it is desired to sample certain variables at a faster rate to emulate the continuous system and meet the performance specifications. Moreover, faster control moves might be necessary to achieve improved performance. 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 essentially the result of concatenation of fast-rate signals to form slow-rate signals with increased dimensionality. The concept of lifting originates from Kranc's idea (Kranc, 1957) of switch decomposition. Multirate control via lifting involves lifting signals of interest followed by controller design for the resulting lifted system. The resulting controller (referred to as the lifted controller) yields the lifted input moves which, in turn, have to be inverse lifted for implementation purposes. These techniques could provide a more practical approach since they do not assume the existence of fast-rate models as in inferential techniques for multirate control. For an elaborate discussion on lifting techniques and their applications, the reader is referred to Chen and Francis (1995). Identification of lifted systems is relatively easier than that of fast-rate models (Li, Shah, & Chen, 2001). In practice, the lifted controller is required to satisfy causality constraints (Meyer, 1990; Ravi et al., 1990; Chen & Qiu, 1994). In this work, it is shown that in addition to the causality constraints, a set of constraints exist on the gains of the lifted controller to ensure ripple-free outputs.

Intersample performance is of prime interest in sampled-data systems analysis, see, e.g., Grasselli, Jetto, and Longhi (1995), Longman and Lo (1997), Arvanitis (1998). Generalized sampled-data hold functions (GSHF) (Kabamba, 1987) have been used by Arvanitis (1998) to improve intersample performance in sampled-data systems. The focus of this work is on the presence of intersample ripples in the closed-loop outputs of multirate sampled-data systems. In this context, Grasselli et al. (1995) have presented studies on dead-beat tracking of outputs of closed-loop multirate systems while addressing the elimination of intersample ripples. Grasselli et al. (1995) arrive at a set of constraints on the multirate controller by transforming the multirate system into a periodic discrete-time system and imposing the condition of dead-beat tracking of continuous-time signals. They arrive at a set of constraints on the controller from a tracking point of view. However, this study is concerned with the analysis of sampled-data systems in the lifted framework and shows that intersample ripples can exist in the closed-loop outputs. Moreover, the problem of eliminating intersample ripples is tackled in this work from a more fundamental point of view and not only for tracking but also the regulatory problem. The main contribution of this paper is to show that intersample ripples occur due to the presence of inverse lifting and non-identical gains of the lifted system; and the existence of a set of constraints on the controller gains in order to eliminate these intersample ripples. Results are presented in this work with a primary focus on step-type setpoint and disturbance signals with the idea that these concepts can be extended to other class of reference signals. Furthermore, using the internal model principle, for step-type reference signals it is shown that the augmentation of a fast-rate integrator with the plant aids in overcoming these ripples.

The paper is organized as follows. In Section 2, we show that intersample ripples occur in lifted control systems due to non-identical gains of the lifted system. Section 3 deals with the theoretical issues concerning the constraints on the controller gains. Experimental results on a tank-level control setup are presented in the last section followed by concluding remarks and suggestions for future work.

Section snippets

Preliminaries

The lifting methodology involves the simple idea of rearranging a fast-rate signal to give rise to a slow-rate signal (lifted signal) with increased dimensionality. Lifting converts a SISO multirate system into a MIMO single-rate system. The process of moving back to the fast-rate signal from the slow-rate lifted signal is termed as inverse lifting, represented by the operator LN−1 or simply L−1. Mathematical properties of L and L−1 are discussed in detail in Chen and Francis (1995). In all the

Main result: constraints on controller gains

Multirate controller design using lifting techniques bring in causality constraints (Meyer, 1990; Ravi et al., 1990; Chen & Qiu, 1994) on the resulting controller which makes the design problem relatively more complex. In addition to the causality constraints, it is shown here that further constraints arise on the controller gains in order to avoid intersample ripples in the closed-loop outputs. We discuss this problem for a SISO system with rational sampling ratio, and the extension to MIMO

Conclusions and future work

Multirate control design in the lifted framework provides an elegant environment to apply existing control design techniques for single-rate systems. In this work, it has been shown that multirate controller design via lifting techniques involves a new set of constraints on the controller. These constraints are a result of inverse lifting and non-identical gains of the lifted system leading to closed-loop outputs with intersample ripples. Multirate controllers with conventional integrators do

Tongwen Chen received the B.Sc. degree from Tsinghua University (Beijing) in 1984, and the M.A.Sc. and Ph.D. degrees from the University of Toronto in 1988 and 1991, respectively, all in electrical engineering.

From October 1991 to April 1997, he was on faculty in the Department of Electrical and Computer Engineering at the University of Calgary, Canada. Since May 1997, he has been with the Department of Electrical and Computer Engineering at the University of Alberta, Edmonton, Canada, and is

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Tongwen Chen received the B.Sc. degree from Tsinghua University (Beijing) in 1984, and the M.A.Sc. and Ph.D. degrees from the University of Toronto in 1988 and 1991, respectively, all in electrical engineering.

From October 1991 to April 1997, he was on faculty in the Department of Electrical and Computer Engineering at the University of Calgary, Canada. Since May 1997, he has been with the Department of Electrical and Computer Engineering at the University of Alberta, Edmonton, Canada, and is presently a Professor of Electrical Engineering.

His current research interests include digital control, multirate systems, robust design, digital signal processing, process control, and their applications to industrial problems. He co-authored with B.A. Francis the book Optimal Sampled-Data Control Systems (Springer, 1995).

Dr. Chen received a University of Alberta McCalla Professorship for 2000/2001. He was an Associate Editor for IEEE Transactions on Automatic Control during 1998–2000. Currently he is a member of the Editorial Board of Dynamics of Continuous, Discrete and Impulsive Systems—Series B: Applications and Algorithms. He is a registered Professional Engineer in Alberta, Canada.

Dongguang Li was born in Jiangxi, China in 1972. He obtained his B.Sc. and M.Sc. degrees in Electrical Engineering from the University of Science and Technology of China, in 1992 and 1995, respectively. From 1996 to present he is a Ph.D. student at the Department of Chemical and Materials Engineering, University of Alberta, Canada. His research interests lie in the area of control and identification of multirate systems.

Rohit S. Patwardhan was born in Pune, India in 1972. He received his B.Tech. and M.Tech. degrees in Chemical Engineering from the Indian Institute of Technology, Bombay in 1993 and 1995, respectively. He worked as a research engineer at the Computer Aided Design Center, Bombay before joining the Ph.D. program in Process Control at the University of Alberta, Edmonton. After graduating in June 1999, he joined the Advanced Applications group at Matrikon Consulting, Edmonton as an advanced control engineer. His research interests lie in the area of model predictive control, process monitoring, system identification and optimization.

Sirish Shah received his B.Sc. degree in control engineering from Leeds University in 1971, an M.Sc. degree in automatic control from UMIST, Manchester in 1972, and a Ph.D. degree in process control (chemical engineering) from the University of Alberta in 1976. During 1977 he worked as a computer applications engineer at Esso Chemicals in Sarnia, Canada. Since 1978 he has been with the University of Alberta, where he currently holds the NSERC- Matrikon -ASRA Senior Industrial Research Chair in Computer Process Control.

In 1989, Shah was the recipient of the Albright & Wilson Americas Award of the Canadian Society for Chemical Engineering in recognition of distinguished contributions to chemical engineering. He has held visiting appointments at Oxford University and Balliol College as an SERC fellow in 1985–86, and at Kumamoto University, Japan as a senior JSPS (Japan Society for the Promotion of Science) research fellow in 1994. Shah's research over the last 25 years has focussed on the broad area of computer process control. The main area of his current research is process and performance monitoring of closed-loop control systems and industrial processes. He has published extensively in academic journals and conference proceedings and recently co-authored a textbook, Performance Assessment of Control Loops: Theory and Applications. He has been a consultant with a number of different industrial organizations.

Arun K. Tangirala was born in Secunderabad, India in 1974. In 1996, he received his Bachelors degree in Chemical Engineering from the Indian Institute of Technology, Chennai, India. He is currently a Ph.D. student at the Department of Chemical and Materials Engineering, University of Alberta, Canada. His research interests mainly include multirate controller design and performance assessment, and wavelet applications to multiscale analysis and monitoring of chemical processes.

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.

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