where τf is the time constant of the first-order filter in the controller, and e is the exponential constant 2.718. With this evidence of poor load regulation on lag-dominant processes, one wonders why proponents continue to push model-based controls.
Identifying Load Components
The load in a process may have multiple components. Consider, for example, a steam-heated exchanger raising the temperature of a process fluid flowing at rate F and having a specific heat C from inlet temperature T1 to the desired (set) exit temperature T2. The manipulated steam flow W required to supply the required heat transfer rate would be
where H represents the latent heat of the steam. Flow F and inlet temperature T1 are the components of the heat load, and both are subject to change.
Product composition from some distillation columns is controlled by manipulating the external material balance, where the flow of a product such as distillate is used to control its composition. The flow of distillate has to vary in direct proportion to the flow of lights entering the column (the load), which is the product of feed rate and the concentration of lights in the feed, a mass balance much like the heat-balance equation above. If instead, reflux flow is manipulated to control composition, it does not need to change much with feed composition, although it must change in proportion to feed rate, and also with heat load factors such as boil-up, feed enthalpy and reflux temperature.
In both of these examples, the point at which the load components enter the process is not the same as where the manipulated variable enters. Steam flow typically enters the shell of a heat exchanger, whereas liquid flows through the tubes. The shell has more heat capacity than the tubes, and therefore, exit temperature will respond slower to a change in steam flow than to a change in liquid flow or inlet temperature. Another important consideration in a heat exchanger is that its dynamic parameters vary inversely with flow through the tube bundle because its heat capacity is fixed, while flow rate varies.
In a distillation column, dead time will usually be longer for the effect of feed changes on product composition than for corresponding changes in boil-up or reflux flow. However, this does depend on whether the feed is vapor or liquid, and where the product composition is measured, as temperature on a tray or in a product analyzer.
In any process where load and manipulated flows do not enter at the same point, differences in dynamic responses are to be expected. Especially complex is a boiler, where load changes can come from variations in steam demand at one end and fuel quality at the other, with heat capacities distributed through both sides of the heat-transfer surface. Applying feed-forward control to these processes involves selection and tuning of dynamic compensators, attempting to balance manipulated flows against measured loads dynamically, in an effort to minimize transients in critical controlled variables during and following load swings.
Effects on Controller Tuning
The earliest publication of effective PID tuning rules came from Ziegler and Nichols in the early 1940s. (Ziegler, J. G. and N. B. Nichols, “Optimum Settings for Automatic Controllers,” Trans. ASME, November 1942, pp. 750-768.) Their tests were conducted by inserting a step change in the path of the controller output and observing the response of the controlled variable in recovering from it. This was equivalent to a load step with identical dynamics in the load path and controller output. Many others have published tuning rules in the meantime, some taking strong exception to the Ziegler and Nichols rules. The principal source of the disparity has been the use of the setpoint as the disturbing variable instead of the load, without there being dynamic elements in the setpoint path.
Typical of setpoint response methods is one described in a 1995 Control article, called “lambda” tuning, where lambda is specified as the closed-loop time constant for setpoint response, equivalent to the IMC filter τf. (Thomasson, F. Y., “Five Steps to Better PID Control,” Control, April 1995, pp. 65-67) There is no mention of load response in the Thomasson article, although the title implies a general applicability: “Five Steps to Better PID Control.” The load response of lambda-tuned loops is similar to that produced by IMC control—long exponential recovery curves having excessive integrated error.
Perhaps the reason for introducing a setpoint disturbance to test or tune a loop is that the load is not often accessible. Ziegler and Nichols simulated their load changes by introducing a bias into the controller output signal. This feature is actually available in some controllers and is useful for simulating a load change. Where it is not, a load change can be simulated by transferring the controller to manual while in a steady-state with no deviation, stepping the controller output, and then immediately transferring back to automatic before any deviation develops. This is effective for all controllers having bumpless transfer between manual and automatic operation.
However, this type of test is only accurate for processes where the load dynamics are essentially identical to those in the controller-output path. Figure 5 shows what can happen when those dynamics are not the same. A process was simulated from 20 interacting time lags, representative of distributed processes such as heat exchangers, stirred tanks, and distillation columns. (Here, Στ is the 63.2%, open-loop response time of a multi-lag processes, also the residence time of a stirred tank.) With the total lag Στ equal in the two paths, a PID controller was tuned to produce a minimum integrated absolute error (IAE) in response to a step change in load—this is the central curve (red), where Στq/Στm = 1. A minimum-IAE curve has a symmetrical peak (similar to a Gaussian distribution curve), followed by minimal overshoot and good damping.