Pergamon 0005-1098(94)E0039-K Automatica, Vol. 30, No. 12, pp. 1975-1981, 1994 Elsevier Science Ltd Printed in Great Britain 0005-1098/94 $7.00 +0.00 Brief Paper Self-tuning PID Control: a Multivariable Derivation and Application* RUBIYAH YUSOF,t SIGERU OMATU~ and MARZUKI KHALIDt Key Words--Multivariable self-tuning PID controller; fixed-tuned multivariable PID controller; parameter estimation; temperature control; microcomputer application. Abstract--In this paper, a multivariable self-tuning con- troller with a proportional plus integral plus derivative (PID) structure is derived. The algorithm features a combination of the self-tuning property, in which the controller parameters are tuned automatically on-line, and also the structure of a multivariable PID controller, making it more favourable for use in industry. The algorithm is applied to a microcomputer based multi-input multi-output (MIMO) furnace. Some experiments are conducted to observe the ability of the controller in the temperature control of MIMO furnace under set-point changes and its relative robustness as compared with a fixed-tuned multivariable PID (FTMPID) controller. The experimental results prove that the controller is capable of giving a good control result for the process. 1. Introduction SIGNIFICANTPROGRESS in the area of multivariable self-tuning control (STC) can be observed over the last few years. The approach of Borrison (1979) in extending the minimum variance strategy of /lstrOm and Wittenmark (1977, 1989) into a multivariable case is a stepping stone to the developments of multivariable STC theories. In the sequel, Koivo (1980) extended the single-input single-output (SISO) STC proposed by Clarke and Gawthrop (1975, 1979) into a multivariable case. To this end, various multivariable STC schemes have been reported (Scattolini, 1986; Bayoumi et al., 1981 etc.). Although the volume of literature devoted to the theories of SISO and multivariable STC is now considerable, the number of applications of these controllers in the process control industries is still discouraging. One main reason contributing to this state of affairs is the domination of PID controllers due to the simplicity of their structures. However, PID controllers have a major drawback in that the controllers may need to be retuned if the systems are subjected to some kind of disturbances to achieve optimum performance. In this light, current developments in the area of STCs are toward combining the ability to adjust the * Received 14 July 1992; revised 19 February 1993; revised 15 July 1993; revised 26 October 1993; received in final form 7 March 1994. The original version of this paper was presented at the 12th IFAC World Congress which was held in Sydney, Australia, during 18-23 July 1993. The published proceedings of this IFAC Meeting may be ordered from Elsevier Science Limited, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, U.K. This paper was recommended for publication in revised form by Associate Editor I. M. Y. Marcels under the direction of Editor C. C. Hang. Corresponding author Dr Rubiyah Yusof. Fax +60 3 2934844. / Department of Control Engineering, Faculty of Electrical Engineering, Universiti Teknologi Malaysia, Jalan Semarak, 54100 Kuala Lumpur, Malaysia. S Department of Information Science & Intelligence Systems, Faculty of Engineering, University of Tokushima, Japan. controllers parameters automatically on-line and the simplicity of the PID controllers structures. The possibility of incorporating the flexibility of STC and the simplicity of PID controller structures in the multivari- able case, has been realised by Tanttu (1987). Based on the work by Peltomaa and Koivo (1983), Tanttu (1987) presented the tuning parameters of PID controller in the form of deterministic auto-regressive moving average (DARMA) model. In this method, the PID controller parameters are tuned on-line via the recursive parameter estimation scheme. Jones and Porter (1987) also presented an algorithm of this type in which the step response matrices of the system are calculated using recursive identifiers. However, both of these methods require a matrix inversion calculation in every sampling interval which is computation- ally rather laborious. This paper highlights a derivation of a multivariable self-tuning PID controller (MSTPID) and its application to a temperature control problem. The multivariable controller is an extension of the SISO self-tuning PID controller of Cameron and Seborg (1983) in which the STC is orientated to have a PID-like structure. The computational requirement of the algorithm is relatively modest. The algorithm uses a model following a rational matrix in the form of a right matrix-fraction description in which the denominator is the prefilter matrix for the process output. The PID-like structure is obtained by including an integrator in the multivariable STC control law and an appropriate selection of the order of the prefilter matrix. A stability analysis of the algorithm (Yusof and Omatu, 1992) shows that a stable closed loop can be obtained with the appropriate choice of prefilter matrix. The algorithm is applied to a microcomputer-based multi-channel miniature model in- dustrial furnace. The furnace is used in the plastic moulding industry for transporting plastic beads via a corkscrew-like mechanism to a pressure chamber to be moulded. The transportation process is non-trivial if the viscidity of the beads is uneven due to uneven distribution of heat. The melted plastic beads have to be at the correct temperature in order to obtain a perfect mould. Thus, the precision of the temperature control of the furnace is an important factor. It is important to consider the furnace as multivariable due to some couplings and heat interactions between the channels. The heat interaction is partly due to the propagation of temperature variation of the furnace in the direction of heat flow. Therefore, some heat from one channel of the furnace will affect the other. Some experiments are conducted to observe the ability of the controller in the temperature control of MIMO furnace under set-point changes and its relative robustness as compared with a fixed-tuned multivariable PID (FTMPID) controller. Experimental results prove that the controller is able to give a good control result for the process. 2. Derivation of the control law 2.1. Multivariable self-tuning controller. In deriving the multivariable STC law, we consider a controlled auto-regressive moving average (CARMA) model of the AUTO 30:12-K 1975