Multivariable Optimal Pid Control of a Heat Exchanger with Bypasses

Nowadays, there are a lot of different control methodologies that could be used within industrial processes. Some of these methodologies have a complex design and also demands an extra engineering effort to design the controller with a superior performance. Some other controllers may not lead to a desirable performance although they are too easy to design. The most common controller used on chemical plants is the PID that presents a simple design (such as the heuristic tuning methods) and easy implementation. If the chemical process presents a lot of inputs/outputs, the PID design will demand an supplementary engineering attempt to tune and design the controller. In this situation, the DMC (Dynamic Matrix Controller) controller, which is the most suitable control strategy to be used within industrial processes that involve a lot of inputs/outputs, complex dynamics, dead-time or inverse output response. The drawback of the DMC is that it needs too much engineering contribution to be designed, which is normally true for all Predictive Controllers. Midway between the simplest and the most complex controller designs, there is an intermediate solution based on optimal control theory. This work intends to present this intermediate solution, i.e., a relative simplicity in design combined with a superior performance, applied to a heat exchanger with bypasses. An optimal control strategy, called Linear Quadratic Regulator (LQR), was developed successfully for the heat exchanger and the LQR controller law is implemented in conjunction with a state observer, since the states are not all accessible on the heat exchanger model. Simulations results obtained show that even being so straightforward to design, it presents a reasonable performance, i.e., the variables become almost totally decoupled and the associated settling-time can be adjusted by means of a simple tuning procedure. The main contribution of this work is to demonstrate how a PID controller may be optimally designed by using the LQR controller design as its support. The LQR control law is mapped to an equivalent control law in the form of a PID controller, using the A, B and C constant matrices from the state-space form of the heat exchanger model, in order to obtain the P, I and D gains of the controller. To illustrate, the performance obtained with the optimal PID controller is compared with the LQR controller.