Model-based control was introduced as a method of achieving a setpoint response that emulated open-loop response, thereby eliminating the overshoot and cycling commonly experienced under PID control. It is also ideal for dead-time-dominant processes such as paper machines. If there is no exponential lag in the path of the load disturbance, it also provides satisfactory load regulation.
Its limitation is that the lag-dominant processes commonly encountered in fluid processing have the same dominant exponential lag in the path of the load disturbance as in the path of the controller output. This results in a slow exponential recovery following a step load change. The limitation imposed by these "disturbance dynamics" was previously explained by the author in some detail in Control (May 2011, "Meditating on Disturbance Dynamics").
The reason for the slow exponential recovery is the use of the model to estimate the load change and match it with an identical change in controller output. The appearance of open-loop control predominates in load response, just as it does in response to setpoint changes. Much faster load response is achievable if the manipulated variable is made to overshoot the load change. The amount of overshoot that produces the best response varies with the ratio of dead time to the exponential lag.
Here we define the best load response (regulation) for a first-order-plus-dead-time process, and examine the possibility of approaching it. While real fluid processes are more complex, the behavior of this fundamental model is very easy to visualize and forms a basis for understanding what is possible using real controllers on real processes.
This pure dead-time process is easy enough to understand. A step in load produces a step in the controlled variable one dead time later. A model-based controller will respond by stepping its output an amount equal to the load step as estimated through the model. If the two changes are indeed matched, the controlled variable will return to setpoint following the next dead time. Figure 1 shows the best response to a step load change for the pure dead-time process having a steady-state gain of 2.0 (in blue)—a load step of 10% produces a deviation of 20%, which the controller converts to an output step of 10% to eliminate the deviation one dead time later.
The limitation encountered here is model mismatch, which has two dimensions: gain and time. If the model gain is too low, the loop gain is < 1, and return to setpoint will follow a series of decaying exponential steps; if too high, recovery will follow a cycle having a period of two dead times, damped if the loop gain is < 2. A dynamic mismatch is more serious, producing harmonic vibrations. To increase robustness and dampen any harmonics, filtering or sampling are generally applied, both of which reduce performance, as they effectively add to the loop dead time.
Pure dead-time fluid processes are uncommon, but static mixers can be dead-time-dominant. Their lags filter out potential harmonics, but also make control more difficult. Of more consequence here is the pursuit of best regulation for lag-dominant processes.
Non-Self-Regulating (NSR) Processes
Integrating processes such as liquid level do not self-regulate. Level is the integral of the difference between vessel inflow and outflow. A level controller left in manual will eventually result in overflow or emptying of the vessel. Flow in and out must be perfectly matched to hold level constant, and changing level will not affect either inflow or outflow, absent controller action. Model-based controllers are not suitable here, because there is no steady-state relationship between the level and the manipulated flow. If the level changes, the model can't predict a flow change that will restore it to setpoint.
Following a step change in load (inflow for example), the level will ramp during the following dead time, as shown in the red trace of Figure 1. Control action can't begin until the level starts to change, and it can have no effect until another dead time elapses. The best regulator for a non-self-regulating (NSR) process would step its output as soon as the controlled variable deviates from setpoint. But, stepping it an amount equal to the estimated load step would leave the level offset by the amount it changed during the dead-time interval between the two steps. To eliminate any offset, the step in controller output must overshoot the load step. Figure 1 shows the best response, attained by the controller output doubling the load step, followed by a matching of the load at the end of the next dead time.