Brief PaperObtaining controller parameters for a new Smith predictor using autotuning☆
Introduction
PID controllers are commonly used in practice in the process control industries. They can be tuned by using rules of thumb (Ziegler & Nichols, 1942) or formulae resulting from analytical design (Åström & Hägglund, 1984). PID controllers tuned using these conventional methods, however, may not provide satisfactory closed-loop responses when the plant considered is either a high-order plant or a plant with a long dead time. Another simple and powerful control technique is the Smith predictor. The controller can be designed as if the system were delay free. However, it was pointed out by Watanabe and Ito (1981) that these regulators cannot reject load disturbance for processes with integration. Subsequently, Åström, Hang and Lim (1994) presented a modified structure for the control of integrator and long dead time processes. This structure decouples the disturbance response from the setpoint response and thereby improves the disturbance response. Later on Mataušek and Micić (1996) proposed a structure similar to that of Åström et al. (1994) but having an additional feedback path from the difference of the plant output and the model output to the reference input for controlling integrating processes. Recently, the limitations of PID controllers controlling resonant, integrating and unstable plants in a conventional feedback structure have been studied (Kwak, Sung & Lee, 1997; Park, Sung & Lee, 1998; Atherton & Majhi, 1999a). These references use an internal feedback loop to convert the integrating or unstable process to an open-loop stable process first and then use a PID controller in the forward loop for improved setpoint and disturbance responses. Majhi and Atherton (1998b) therefore proposed a new Smith structure incorporating a similar inner feedback loop which extends its applicability to resonant, integrating and unstable processes.
Recently, relay feedback automatic tuning of SISO controllers has been studied extensively. Palmor and Blau (1994) developed an autotuning algorithm for the Smith dead time compensator using a first-order plus delay model for some stable plants. The algorithm estimates two points on the process Nyquist curve via automatically generated controlled limit cycles. The points are then used in a least-squares procedure to estimate the parameters of the model. Hang, Wang and Cao (1995) have presented methods to autotune and self-tune a modified Smith predictor by relay feedback. Both these works, however, use the describing function approximation to a relay in their analysis, which can result in errors in the estimation of the plant model parameters. Also their results are limited to stable processes only.
In this paper, a simple relay feedback autotuning method is proposed for the new Smith predictor. With prior information on the static gain, a reduced order process model in terms of a first- or second-order dynamics plus dead time (abbreviated as FOPDT and SOPDT, respectively) can be computed and used to autotune the Smith predictor from a single symmetrical relay test. Excellent performance of the autotuned Smith predictor has been substantiated by simulations for stable, integrating and unstable processes.
Section snippets
The new Smith predictor structure
The structure of the new Smith predictor (Majhi & Atherton, 1998b) for controlling stable, unstable and integrating processes is shown in Fig. 1. It has three controllers which are designed for different objectives and of the three controllers, Gc1 in the inner loop is provided to stabilise an unstable or integrating process and modify the pole locations of the transfer function of a stable process. The other two controllers, Gc and Gc2 are then used to take care of servo-tracking and
Estimation of plant model parameters
In principle, from an odd symmetrical limit cycle two unknown parameters can be found, and from an asymmetrical limit cycle four can be found. Also by doing two odd symmetrical limit cycle tests, one without hysteresis and one with, four parameters can be found but this requires extra time for the two tests. Further, estimation of four unknown parameters from an asymmetrical test or two symmetrical tests requires a nonlinear algebraic equation solver to solve the equations which may result in
Development of the autotuning formulae
The form of the main PI controller is Gc=Kp(Tis+1)/(Tis). The controllers Gc1 and Gc2 take different forms depending upon the assumed order and type of the plant transfer function model. In two cases Gc1 is taken as an ideal PD controller and the results are given for this idealised case. For practical implementation a time constant filter is used with the derivative term and if this is some 10 times smaller than the derivative time constant, which is normally possible in practice, the results
Conclusions
Simple and effective automatic tuning formulae are derived for a new Smith predictor structure assuming low-order model transfer functions with time delay for stable, unstable and integrating processes. Two important advantages of the new scheme are that the process models possess two unknowns namely, the time delay and the time constant which are easily obtainable using a single relay feedback test assuming the process steady-state gain is known and the controller parameters have very simple
Derek Atherton was born in Bradford on 21 April 1934 and is married with two sons. He received a B.Eng. from the University of Sheffield, and a Ph.D. and D.Sc. from the University of Manchester, in 1956, 1962 and 1975, respectively. He taught in Canada at McMaster University and the University of New Brunswick before taking up the position of Professor of Control Engineering at the University of Sussex in 1980. He has since served on committees of the Science and Engineering Research Council,
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Derek Atherton was born in Bradford on 21 April 1934 and is married with two sons. He received a B.Eng. from the University of Sheffield, and a Ph.D. and D.Sc. from the University of Manchester, in 1956, 1962 and 1975, respectively. He taught in Canada at McMaster University and the University of New Brunswick before taking up the position of Professor of Control Engineering at the University of Sussex in 1980. He has since served on committees of the Science and Engineering Research Council, as President of the Institute of Measurement and Control in 1990, President of the IEEE Control Systems Society in 1995, and for six years on the International Federation of Automatic Control (IFAC) Council. His major research interests are in nonlinear control theory, computer-aided control system design, simulation and target tracking. He has written three books, one of which is jointly authored, and published around 300 papers.
S. Majhi received the B.Sc.(Eng.) and the M.Sc. (Eng.) degrees from Sambalpur University, India and the D.Phil. degree from Sussex University, England in 1988, 1991, and 1999, respectively. He is currently an Assistant Professor in the Department of Electronics and Communication Engineering of the Indian Institute of Technology, Guwahati. His research interests include automatic controller tuning, identification, nonlinear, and optimal control.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor T.A. Johansen under the direction of Editor S. Skogestad.