The controlled object with multiple poles and self-regulating characteristics has the following transfer function, including two identical large time constants.
The step response test for this object is as follows:
From the response curve, it can be observed that this is a self-regulating object. Naturally, we can follow the method shown in the figure above, using Lambda tuning to plot and obtain the characteristics of the controlled object. Alternatively, the bisection method proposed by Professor Sigurd can be used to obtain the model parameter product. The model parameters obtained from these two methods are not entirely identical but do not affect the results.
The above equation represents the PI control parameters obtained using the Lambda tuning method. The results of the closed-loop test with these parameters are shown in the figure below:
For an object without pure delay, due to its complex characteristics, the Lambda tuning method can only achieve a slow response. Is there a better approach? Of course, there is: incorporating derivative action by using a PID controller. The PD controller employs incomplete differentiation, with parameters as follows:
For this controlled object, if PI control is used directly, the proportional action cannot be too large. When a PID controller is employed, the PD component in the PID can cancel one pole, significantly improving the characteristics of the controlled object. Therefore, there is some truth to the statement by some instructors that PI control can only achieve a reasonably acceptable closed-loop performance.
Everything comes with trade-offs. By using a PID controller, we gain a lot, but at what cost? With PI control, the controller output peaks at 1.8. With PID control, the controller output peaks at 20,000. This is the necessary price to pay to counteract the large time constant and multiple poles.
When the delay is not a true pure delay caused by multiple capacities, derivative action is ineffective. Many high-order controllers that rely on strict zero-pole cancellation, when considering factors such as model uncertainty and controller output, struggle to guarantee theoretical performance in practical applications.
In simple terms, when PI control has been tuned to its performance limit and still cannot meet requirements, adding derivative action is a method worth trying. Eliminating disturbance sources and enhancing variable structure are commonly used engineering approaches. Of course, if necessary, derivative action, lead-lag correction, or high-order controllers can also be employed.
