A continuous stirred tank reactor operating under digital computer control has been designed and built, and a comparison made between digital control methods that employ recent advances in estimation theory, and an equivalent classical analog control system. The digital control method required process identification, optimal control, and Kalman filtering for its implementation. The unknown parameters in the nonlinear mathematical model describing the process were identified by the method of quasi-linearization. This technique not only linearizes the non-linear system equations but also provides a sequence of functions that in general converge rapidly to the solution of the true non-linear equations. The optimal controller is believed to be original and comprises an adaptation of the quasi-linear algorithm. The controller calculates, at preset time intervals, those changes in the desired values of supervisory feedback controllers such that a weighted sum of squared errors at the end of the time interval is minimised. The predicted value of the state vector obtained in this calculation was then used in a fixed gain formulation of the Kalman filter. Simulation studies verified the viability of both the identification and optimal control methods, which were then applied to the process. The classical controller was designed using either the method of Ziegler and Nicholl or that of Bode, which utilised the frequency response. Experimental work showed good agreement with the simulation studies. However, it was found that the Kalman filter was unnecessary because of the essentially noise-free nature of the measurements. The identification method performed satisfactorily, and gave the values of the parameters in the mathematical model which described the process. The digital control system was shown to give better control than the classical scheme, especially in the steady state, where the predictive nature of the optimal controller effectively prevented overshoot, whilst assuring a fast stable response to disturbances.

Application of estimation theory to a real process under digital computer control

This paper demonstrates methods for applying combined optimum control and estimation theory to serial systems with time delay. The case of serial linear systems with time delay is considered in detail; a result analogous to the separation theorem of linear systems is presented. Illustrative examples of serial chemical reactors, rolling mills, and second-order plus dead-time approximations of higher order systems are discussed. Numerical results for a second-order plus dead-time system are presented: these results are compared with a suboptimal feedback controller (modified Smith predictor) and the open-loop response. It is shown that, in this case, the optimum estimation and control gains may be approximated by constants which further simplify the DDC algorithm. In the second-order plus dead-time approximation to higher order overdamped systems, the optimum algorithm can be reduced to recursive estimation with constant gain and linear state variable feed forward control. This algorithm may be used as a direct replacement for digital controllers used in the process industries.

C.H. Wells

Papers

Application of combined optimum control and estimation theory to direct digital control