PID Control of Unstable Processes: Temperature Control of an Unstable CSTR
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Being able to control temperature in exothermic reactors is essential in order to obtain the wanted production rate, stay in the region of acceptable conversion and of course to ensure safety during operation. This work investigated the possibility of creating a simple and more effective method for tuning PID controllers for unstable systems, with a special focus on an unstable CSTR at the Perstorp plant in Warrington, England. The tuning rule should be a function of measureable process parameters only, and be useable for a range of different operating points. A linearized, dynamic model was developed from mass and energy balance equations for an unstable CSTR with temperature control and implemented in MATLAB and Simulink for simulation. The optimal PID tuning parameters were obtained by using the FMINCON function in MATLAB to minimize the integral absolute error term at a given robustness, for setpoint and disturbance changes. An optimal weighted controller was then obtained using the method of Grimholt and Skogestad .Different IMC tuning procedures were tested and compared with the optimal PID tuning parameters at the most likely operating point inside the temperature range of operation. It was possible to reduce the resulting second order unstable transfer function to a first order unstable transfer function, which made implementation of some of the IMC tuning rules simpler. The IMC procedures derived by Lee et al  and Cho et al  were the most promising ones, so these were chosen for further investigation. The Lee IMC procedure yielded close to optimal results for both setpoint and disturbance changes.The model was linearized around new temperature operating points in order to envelop the entire temperature range of interest, spanning from 40% to 90% conversion at temperatures from 311 K to 344 K. Using the reduced transfer function for each of these temperature operating points was still possible, but the approximation deteriorated slightly with increasing temperature. This made the quality of the IMC methods based on first order models slightly less desirable at higher temperatures. The Lee IMC method was still relatively close to optimal for all operating points and it was used as a basis for developing a simple, temperature dependent approximation of the PID tuning parameters.Plotting the calculated Lee IMC PID tuning parameters as a function of temperature and using curve fitting in MATLAB, resulted in an approximation of the controller gain, Kc, as an exponential function. Integral time, τI, and derivative time τD, were close to constant over the temperature span. This approximation gave almost identical response as the originally calculated Lee IMC, and was close to optimal in the temperature region of interest. A suggestion for a simple tuning equation based on these temperature dependent parameters was made, together with the suggestion of implementing the Lee IMC tuning procedure directly. The advantage of the proposed method is that it only uses measurable process variables for tuning. Estimating each of the components of the Kc equation as functions of temperature however, resulted in an unreliable, non-robust controller.