Ind. Eng. Chem. Res. 1989,28, 1481-1489 1481 Ulrich, G. D.; Milnes, B. A.; Subramanian, N. S. Particle Growth in Flames: 11. Experimental Resulta for Silica Particles. Combust. Sci. Technol. 1976, 14, 243. Wang, C. S.; Friedlander, S. K. The Self-preserving Particle Size Distribution for Coagulation by Brownian Motion. J. Colloid Interface Sci. 1967, 24, 170. Wu, J. J.; Flagan, R. C. Onset of Runaway Nucleation in Aerosol Reactors. J. Appl. Phys. 1987, 61, 1365. Wu, J. J.; Flagan, R. C. A Discrete-SectionalSolution to the Aerosol Dynamic Equation. J. Colloid Interface Sci. 1988, 123, 339. Received for review October 14, 1988 Revised manuscript received May 8, 1989 Accepted June 15, 1989 PROCESS ENGINEERING AND DESIGN Control of a Multivariable Open-Loop Unstable Process Apostolos Georgiou,? Christos Georgakis, and William L. Luyben* Chemical Process Modeling and Control Research Center, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015 A systematic approach is presented for the design of a control system for a multivariable open-loop unstable process. The closed-loop stability, dynamic performance, and robustness of the multiloop single-input-single-output controllers are established by an effective damping coefficient and its corresponding effective closed-loop time constant. Controller tuning is reduced to a four-step optimization problem. The procedure is illustrated on a complex system of two reactors in series, with a separator and recycle, where an exothermic second-order reaction occurs. The system is highly nonlinear and has three open-loop poles in the right half of the s plane. The effects of variation of reactor size and kinetic parameters are used to explore the robustness of the control system. Performance is evaluated by simulation of the nonlinear mathematical model. A number of processes in the chemical industry are multivariable and open-loop unstable (e.g., exothermic reactors). There have been many theoretical studies of multivariable controller design techniques for these pro- cesses: (1) pole placement methods (see Wonham (1967), Power (1975), Edgar (1976),Friedland (1986)); (2) internal model control (IMC) (see Zafiriou and Morari (1987));and (3) model predictive control for unstable systems (Cheng and Brosilow (1987)). Concerning (1) above, it has been pointed by many au- thors (see above cited references) that while these methods have found application in other disciplines, for example in the aerospace industry, they have not found wide ap- plication in the chemical/petroleum industry. Some of the reasons are (a) complexity, excessive engineering man- power requirements, operator nonacceptance; (b) re- quirement of measurements (or estimates) of the states (concentration, etc.) that are not available because of physical or economical constraints; (c) difficulty in han- dling integral action which is very important in chemical processes; (d) no guidance to the engineer for tuning these methods on-line (chemical processes are highly nonlinear and usually require on-line tuning); and (e) lack of ro- bustness (sensitivity to model/plant mismatch). Concerning (2) and (3), the recent IMC work (Zafiriou and Morari (1987)) and model predictive control work (Cheng and Brosilow (1987)) are still in the initial stages of design. * Author to whom correspondence should be addressed. Present address: Chemical Engineering Department, Princeton University, Princeton, NJ 08544. Therefore, most multivariable industrial processes are still controlled satisfactorily by simple three-mode (pro- portional-integral-derivative (PID)) single-input-single- output (SISO) controllers. Pure optimization techniques based on minimizing a performance index such as the integral square error (ISE) or integral absolute error (IAE) of the outputs or inputs have been proposed, but these methods have several problems: (a) they do not provide any guidance for how to weight the outputs or the inputs in the performance index; (b) they do not provide any guidance for incorporating important specifications of a controller design (overshoot, settling time, robustness, etc.) in the performance index; (c) many times, these pure op- timization procedures find a local optimum and not a global optimum. We used the IAE optimization procedure (Georgiou et al. (1986)) to tune the SISO controllers, but the perform- ance was unsatisfactory compared to that obtained by the procedure proposed in this paper. We are not aware of any practical systematic procedure for designing multiloop SISO conventionalPID controllers for multivariable open-loop unstable processes. This paper proposes such a procedure. It should be emphasized that multivariable controllers are not considered in this paper. The objective of this paper is to illustrate a controller design for a multivariable open-loop unstable process. The modeling of the multivariable open-loop unstable system is discussed first. This system consists of two jacketed CSTR’s with recycle and separator where an exothermic second-order reaction occurs. The distinctive features of the dynamics of the system are the strong nonlinearity of temperature involved in the kinetics and 0888-5885/89/2628-1481$01.50/0 0 1989 American Chemical Society