This best response curve is identified by two characteristics: its peak deviation, eb (reached after one dead time), and Integrated Error, IEb. Both have economic consequences: Peak deviation determines how closely the setpoint can be positioned to operating constraints such as trip points, and IE is a measure of excess energy use or product giveaway. Both should be minimized, and they are related. For the NSR process:
eb = ∆qτd/τ (1)
IEb = ebτd = ∆qτd2/τ (2)
where ∆q is the size of the load change; τd is the loop dead time; and τ is the integrating time constant of the process. (In Figure 1, the controlled variable changed twice as much as the load during the dead time, so the integrating time of this process is one-half the dead time.) Note that IEb varies with the square of the dead time. With a real controller, peak deviation will be larger and occur later, with a resulting much higher IE.
Manipulated-variable overshoot is essential in achieving a prompt and complete return to setpoint. However, the approach to this best load regulation is hindered in two ways. The output step must be initiated at the earliest detection of a deviation to achieve earliest recovery, and information obtained at that time is least accurate in estimating the required size of the output step. Based on the ramp rate of the level, the step size can be more accurately estimated with time. Then, when the direction of the level finally changes, the overshoot must be promptly removed, with a similar demand on accuracy.
No controller has the accuracy required to do this. Secondly, the derivative action needed to determine ramp rate can't be applied to level measurements due to its sensitivity to noise. So we're limited to PI control of liquid level (with feed-forward as necessary with boiler-drum applications). But the tuning of the PI controller should reflect the output motion that resembles the best, 100% output overshoot being an essential feature.
Most processes are self-regulating, having a proportional relationship between the controlled and manipulated variables. Model-based controllers relate the two to predict the result of output moves, but typically provide no output overshoot for prompt recovery from load changes. Only pure dead-time processes require no output overshoot for best response. For all others, the size of the output step in relation to the load step follows the equation below (derived in Reference 2 in my book, Process Control Systems, 4th ed., McGraw-Hill, New York, 1996, p. 35):
∆m/∆q = 1+ετd/τ1 (3)
where ∆m and ∆q are the changes in manipulated variable and load respectively; τd is dead time in the loop; τ1 is the time lag in the load path; and ε is 2.718, the base of natural logarithms. At the end of the following dead time, m and q would be set equal.
Figure 2 shows best load responses for two lag-dominant processes: a self-regulating process in blue, and an unstable process in red, to be described later. Their required output overshoots both follow the above equation.
Peak deviation for the stable (self-regulating) process is
εb = ∆qKp(1-e-τd/τ1) (4)
where Kp is the process steady-state gain, having a value of 2.0 in this example; and τ1 is its exponential lag time. The departure and recovery curves are complementary, so that IEb = ebτd as before. In Figure 2, τ1 is twice τd, requiring a peak output/load ratio of 1.607, or 60.7% overshoot. Most lag-dominant processes feature a τd/τ1 ratio of around 1/7, requiring an output/load ratio of 1.867 or 86.7% overshoot, approaching the 100% for NSR processes.
An unstable process is one that runs away from setpoint, given the slightest disturbance. Exothermic reactors are typical of this class. Any increase in temperature will cause the heat produced by the reaction to increase more than the heat transferred to the cooling medium, which accelerates the departure from equilibrium. Given enough heat-transfer surface, an exothermic reactor can be stable, but they are known to become unstable when that surface becomes sufficiently fouled (See my article, "Exothermic Reactors: The Stable, the Unstable, the Uncontrollable," Chem. Eng., March 2004).
An unstable process is simulated using a negative time constant and gain. Figure 2 shows the best step-load response for an unstable process having a time constant twice its dead time and a gain of -2.0. (This is a reasonable simulation of an unstable exothermic reactor because as dead time approaches its time constant, the process becomes uncontrollable.) The increasing heat load requires a lower coolant temperature, representing the variable manipulated by the controller output. Observe how the controlled variable follows a divergent trajectory, in contrast to the self-regulating process, where, absent control, it would approach a new steady state.