Analysis of dual-rate inferential control systems

doi:10.1016/S0005-1098(01)00295-3

Automatica

Volume 38, Issue 6, June 2002, Pages 1053-1059

https://doi.org/10.1016/S0005-1098(01)00295-3 Get rights and content

Abstract

For a dual-rate control system where the output sampling interval is an integer multiple of the control interval, we propose a model-based inferential control scheme which uses a fast-rate model to estimate the intersample outputs and then supply them to a controller at the fast rate. Comparing such an inferential controller with the corresponding fast single-rate controller, we conclude that the former is no better in disturbance rejection capability; however, in the presence of model-plant mismatch, the former is advantageous in stability robustness of the closed-loop system.

Introduction

Multirate systems are common in the chemical process industry because many property or quality variables are not available fast enough: for example, in a polymer reactor, the composition, density or molecular weight distribution measurements typically take several minutes of analysis time, while the control signals can be adjusted at relatively fast rates, the only limitation being the load on the distributed computer control system. In such cases, it makes sense to configure the control systems so that several sample rates co-exist to achieve better tradeoff between performance and implementation cost. This paper is concerned with a dual-rate case where the output is measured at a relatively slow rate, whereas the control signal is adjusted faster.

Research on multirate systems started in the early 1950s. The first important work was Kranc's switch decomposition (Kranc, 1957); this was further developed into the lifting technique by Friedland (1960), Khargonekar, Poolla, and Tannenbaum (1985). The lifting technique converts a periodic discrete-time system into a time-invariant system and is now one of the main tools for studying multirate systems. The advantages of lifting are that it preserves norms of the signals and that the lifted systems are linear time invariant (LTI). We will use lifting for our subsequent analysis in this paper.

In dual-rate systems where the output sampling period is an integer multiple of the control period, it is possible to identify fast single-rate models based on multirate input–output data (Li, Shah, & Chen, 2001a). In this paper, we propose a simple and practical control scheme, referred to as dual-rate inferential control, in which the fast single-rate models are used to estimate the missing output samples at the fast rate, and then single-rate control algorithms are implemented at the fast rate. We comment that such a dual-rate inferential control scheme has been applied to an industrial continuous catalytic reforming (CCR) unit for octane quality control (Li, Shah, Chen, & Qi, 2001b), resulting in significant reduction in octane quality variance; the industrial partner involved was Shell Canada. In this work, we will focus on performance and stability robustness analysis of such an inferential control scheme.

In the process control literature, multirate systems have also been studied a great deal: Lu and Fisher (1988) studied the intersample output estimation and inferential control for dual-rate systems and developed a method for output estimation based on the past control and output measurements; Guilandoust, Morris, & Tham 1986, Guilandoust, Morris, & Tham 1988 considered more general multirate systems with primary outputs sampled at a slow rate and showed that the intersample outputs can be estimated using a fast sampled secondary output; Lee and Morari (1992) developed a state-space framework called the generalized inferential control scheme for multirate systems and discussed various $LQG / H_{2}$ optimal design techniques; and finally, Gudi, Shah, and Gray (1993) developed an enhanced observability estimation method for multirate processes.

In this paper, we focus on inferential control systems using fast sampled models. The contribution of the paper is as follows:

•
We propose a simple and practical inferential control scheme for dual-rate systems using fast single-rate models, and develop a lifted framework for analysis of performance and stability robustness.
•
We compare the proposed inferential controller with the corresponding fast single-rate controller, and conclude that the former is no better in disturbance rejection capability; however, surprisingly, the former is advantageous in stability robustness in the presence of model–plant mismatch (MPM).

Briefly, this paper is organized as follows: In Section 2 we introduce the dual-rate inferential control scheme. In Section 3 we look at performance of the inferential control scheme in the absence of MPM. In Section 4 we discuss stability robustness of inferential control systems in the presence of MPM. In Section 5 we discuss some extension to the result given in Section 4.

Section snippets

Dual-rate inferential control scheme

First, let us consider a single-input, single-output single-rate control system shown in Fig. 1, where P_c is a continuous-time LTI plant and K a digital controller. The two systems P_c and K are interfaced by the A/D and D/A converters, modeled by S_f, the ideal sampler, and H_f, the zero-order hold (ZOH), respectively, both operating with the fast period T. This is a single-rate sampled-data control system which involves two exogenous signals, the discrete-time reference r(k) and the

Nominal performance

Consider the dual-rate inferential control system in Fig. 4. Assume in this section that there is no MPM; thus $P ̂ =P$ . If furthermore there is no disturbance in the system, i.e., d(k)=0, then $y ̂ (k)=y(k)$ and hence v(k)≡y(k). Thus the dual-rate system is equivalent to the single-rate system in Fig. 2; we conclude

•
Without MPM, closed-loop stability of the dual-rate system in Fig. 4 is equivalent to that of the single-rate system in Fig. 2.
•
Without MPM and disturbance, the tracking performance (y

Stability robustness

In this section we assume that there is MPM in the dual-rate system in Fig. 4, and hence $P ̂ ≠P$ . We will study issues related to stability robustness. We treat P as uncertain and P̂ as the nominal plant model; we assume a standard additive uncertainty model (Doyle, Francis, & Tannenbaum, 1992), i.e., P belongs to the uncertainty class given by ${P ̂ + Δ W_{1} : || Δ ||_{∞} <1}.$ The MPM is represented by ΔW₁, where Δ is the perturbation, assumed to be stable and LTI, with norm less than 1 (normalized), and W₁ is a

Concluding remarks

In the preceding section, we studied stability robustness of the dual-rate inferential control system assuming an additive uncertainty model. We point out here that similar result holds if a multiplicative uncertainty model is used; in this case P belongs to the class ${(I+ Δ W_{1}) P ̂ : || Δ ||_{∞} <1}.$ Similar to the condition in (9) in Proposition 2, the robust stability condition for this case is $|| W ̄_{1} P ̂ (I+ K ̄ P ̂)^{−1} K ̄ R_{1} ||_{∞} <1.$ Based on this, we can make the same conclusion that the dual-rate inferential control

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.

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From 1991 to 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. He held visiting positions at the Hong Kong University of Science and Technology and Tsinghua University.

His current research interests include process control, multirate systems, robust control, digital signal processing, 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 an Associate Editor for Systems & Control Letters and 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 received his B.Sc. and M.Sc. degrees in 1992 and 1995, respectively, from the Department of Automation, University of Science and Technology of China, and his Ph.D. degree in process control (Chemical Engineering) from the University of Alberta in 2001. He is now working as an advanced process control engineer with Honeywell, Calgary. The main areas of his current research are model-based predictive control and system identification of multirate systems.

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 1985–1986, and at Kumamoto University, Japan as a senior JSPS (Japan Society for the Promotion of Science) research fellow in 1994. The main area of his current research is process and performance monitoring of closed-loop control systems and industrial processes. He has 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.

^☆: This paper was not presented at any IFAC Meeting. 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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Brief PaperAnalysis of dual-rate inferential control systems☆

Abstract

Introduction

Section snippets

Dual-rate inferential control scheme

Nominal performance

Stability robustness

Concluding remarks

Chemical Engineering Science

Optimal sampled-data control systems

Feedback control theory

Control of uncertain sampled-data systems

Brief Paper
Analysis of dual-rate inferential control systems☆