## Process response depends on integration degree, delays & lags

Process response depends on process variable’s self-regulation

An indication of the ease with which a process may be controlled can be obtained by plotting the process reaction curve. This curve is constructed after having first stabilized the process temperature under manual control and then making a nominal change in heat input to the process, such as 10%. A temperature recorder then can be used to plot the temperature versus time curve of this change.[1]

Process response of self-regulating process variable vs. time:
vessel temperature after heating increase

Process response of integrating process variable vs. time:
vessel level after inflow increase [2]

Another phenomenon associated with a process or system is identified as the steady-state transfer-function characteristic. Since many processes are nonlinear, equal increments of heat input do not necessarily produce equal increments in temperature rise. The characteristic transfer-function curve for a process is generated by plotting temperature against heat input under constant heat input conditions. Each point on the curve represents the temperature under stabilized conditions, as opposed to the reaction curve, which represents the temperature under dynamic conditions. For most processes this will not be a straight-line, or linear, function.

Temperature gain vs. heat input of endothermic process:
heating a liquid till it boils and absorbs more heat

As the temperature increases, the slope of the tangent line to the curve has a tendency to decrease. This usually occurs because of increased losses through convection and radiation as the temperature increases. This process gain at any temperature is the slope of the transfer function at that temperature. A steep slope (high ΔT/ΔH) is a high gain; a low slope (low ΔT/ΔH) is a low gain.

Temperature gain vs. heat input of exothermic process:
heating a mixture of reactants till their reaction gives off reaction heat;
melting a plastic till its flow gives off frictional heat

This curve follows the endothermic curve up to the temperature level D. At this point the process has the ability to begin generating some heat of its own. The slope of the curve from this point on increases rapidly and may even reverse if the process has the ability to generate more heat than it loses. This is a negative gain since the slope ΔT/ΔH is negative. This situation would actually require a negative heat input, or cooling action.

Process response depends on process variable’s dead time delays & capacity lags

Temperature response vs. time of single capacity:
vessel-temperature lag

Temperature response vs. time of single capacity with dead time:
vessel-temperature lag with hot-water piping delay

Temperature response vs. time of two capacities:
vessel-temperature lag and vessel-wall lag

Temperature response vs. time of three capacities:
vessel-temperature lag, vessel-wall lag, and thermowell-wall lag

Two characteristics of these curves affect the process controllability, (1) the time interval before the temperature reaches the maximum rate of change, A, and (2) the slope of the maximum rate of change of the temperature after the change in heat input has occurred, B. The process controllability decreases as the product of A and B increases. Such increases in the product AB appear as an increasingly pronounced S-shaped curve on the graph.

The time interval A is caused by dead time, which is defined as the time between changes in heat input and the measurement of a perceptible temperature increase. The dead time includes two components, (1) propagation delay (material flow velocity delay) and (2) exponential lag (process thermal time constants).

Temperature responses vs. time of temperatures I, II, III, and IV

The maximum rate of temperature rise is shown by the dashed lines which are tangent to the curves.

The tangents become progressively steeper from I to IV. The time interval before the temperature reaches the maximum rate of rise also becomes progressively greater from I to IV.

As the S curve becomes steeper, the controllability of the process becomes increasingly more difficult. As the product of the two values of time interval A and maximum rate B increases, the process controllability goes from easy (I) to very difficult (IV). Response curve IV, the most difficult process to control, has the most pronounced S shape.[1]

1. Stevenson, John. “Control Principles.” Process/industrial instruments and controls handbook, 5th ed., edited by Gregory K. McMillan, McGraw-Hill, 1999, pp. 2.4-2.30.
2. Shinskey, F. Greg. “Fundamentals of Process Dynamics and Control.” Perry’s chemical engineers’ handbook, 8th ed., edited by Don W. Green, McGraw-Hill, 2008, pp. 8-5 – 8-19.

## PID controller responds to error, to error footprint, and to projected change

The top graph shows the measured process variable (the process’s output); the bottom graph shows the controller output (the process’s input). The setpoint is changed at t=0, and the external process load is changed at t=10. The PID control’s D-action is on the process variable only, not on the setpoint. The PID control action is fast and accurate; the PI-only actions keep the valve movements smaller.[1]

PID controller responds to diverse needs

…proportional-integral-derivative (PID) is by far the dominant feedback control algorithm.[2]

PID controllers are found in large numbers in all industries. The PID controller is a key part of systems for motor control. They are found in systems as diverse as CD and DVD players, cruise control for cars, and atomic force microscopes. The PID controller is an important ingredient of distributed systems for process control.[3]

There are approximately three million regulatory controllers in the continuous process industries…

Based on a survey of… controllers in the refining, chemicals and pulp and paper industries… 97% of regulatory controllers utilize a PID feedback control algorithm.[2]

Many sophisticated control strategies, such as model predictive control, are also organized hierarchically. PID control is used at the lowest level; the multivariable controller gives the set points to the controllers at the lower level.

The PID controller can thus be said to be the “bread and butter” of control engineering.[3]

PID controller responds to error with proportional action

PID controllers are defined by the control algorithm, which generates an output based on the difference between setpoint and process variable (PV). That difference is called the error…

…the most basic controller would be a proportional controller. The error is multiplied by a proportional gain and that result is the new output.

When the error does not change, there is no change in output. This results in an offset for any load beyond the original load for which the controller was tuned. A home heating system might be set to control the temperature at 68˚F. During a cold night, the output when the error is zero might be 70%. But during a sunny afternoon that is not as cold, the output would still be 70% at zero error. But since not as much heating is required, the temperature would rise above 68˚F. This results in a permanent off-set.

PID controller responds to error footprint with integral action

Integral action overcomes the off-set by calculating the integral of error or persistence of the error.

This action drives the controller error to zero by continuing to adjust the controller output after the proportional action is complete. (In reality, these two actions are working in tandem.)

PID controller responds to projected change with derivative action

And finally, there is a derivative term that considers the rate of change of the error. It provides a “kick” to a process where the error is changing quickly…

Derivative action is sensitive to noise in the error, which magnifies the rate of change, even when the error isn’t really changing. For that reason, derivative action is rarely used on noisy processes and if it is needed, then filtering of the PV is recommended.

Since a setpoint change can look to the controller like an infinite rate of change and processes usually change more slowly, many controllers have an option to disable derivative action on setpoint changes and instead of multiplying the rate of change of the error, the rate of change of the PV is multiplied by the derivative term.

Derivative is not often required, but can be helpful in processes that can be modelled as multiple capacities or second order.[4]

PID controller responds simply and intuitively

The PID controller is a simple implementation of feedback.

It has the ability to eliminate steady-state offsets through integral action, and it can anticipate the future through derivative action.[3]

1. Skogestad, Sigurd, and Chriss Grimholt. “The SIMC method for smooth PID controller tuning.” PID Control in the Third Millennium. Springer London, 2012. 147-175.
2. Desborough, Lane, and Randy Miller. “Increasing customer value of industrial control performance monitoring-Honeywell’s experience.” AIChE symposium series 326 (2002): 172-192.
3. Åström, Karl Johan, and Tore Hägglund. Advanced PID control. ISA-The Instrumentation, Systems and Automation Society, 2006, p. 1.
4. Heavner, Lou. “Control Engineering for Chemical Engineers.Chemical Engineering 124.3 (Mar. 2017): 42-50.

## Control difficulty comes from less capacity and more dead time

Figure 3.9. Temperature measurement device located downstream.

Control difficulty is increased by dead time

The dead time can be computed as the length divided by the speed.

…in Figure 3.9. As the mixing of steam and cold water to produce hot water occurs in the tank, the temperature of the water in the tank is the true indicator of the state of the process. When the hot water temperature transmitter is located downstream, the temperature controller does not know the current value of the temperature of the water in the tank. Instead, the controller only knows what the temperature was at some time in the past. In essence, the controller is taking actions based on old information, which is never a good idea.

You do not know where you are; you only know where you have been. Your driving performance is seriously impaired. Furthermore, if you insist on driving this way, take some advice: slow down!

Actually, we have to give this same advice to the temperature controller for the hot water process. We do this by reducing the controller gain.

When the dead time is large, the gain must be reduced substantially, resulting in a very slow loop. The usual complaint in large dead time processes is that the controller is so slow that it is useless.

As the dead time increases, more overshoot is generally observed, and… oscillations will become pronounced.

The larger the dead time,… the slower the response.[1]

Control difficulty is reduced by capacity

Capacity is where a process stores variable amounts of mass or energy.[2]

The time constant… is always the ratio of holdup to throughput. For material balances, the holdup is material and the throughput is material flow. For energy balances, the holdup is energy and the throughput is energy flow.

Control difficulty can be quantified

…controls… can cope with time constants… far better than with dead time.[1]

The parameter

…has the property 0 ≤ control difficulty ≤ 1…

…processes with small control difficulty are easy to control, and the difficulty in controlling the system increases as control difficulty increases. Systems with control difficulty = 1 correspond to processes with pure time delay, which are difficult to control well.[3]

…dead time is the “difficult element to control.”[1]

1. Smith, Cecil L. Practical process control: tuning and troubleshooting. John Wiley & Sons, 2009, pp. 72, 83, 85, 87-90.
2. Shinskey, F. Greg. Process control systems: application, design and tuning. 4th ed., McGraw-Hill, Inc., 1996, p. 22.
3. Åström, Karl Johan, and Tore Hägglund. Advanced PID control. ISA-The Instrumentation, Systems and Automation Society, 2006, p. 26.