29.9. P, I, AND D RESPONSES GRAPHED

2303

29.8.3Derivative control mode (D)

Derivative – sometimes called rate or pre-act – is a control action causing the output signal to be o set by an amount proportional to the rate at which the input is changing. Derivative controller action responds to how quickly the input changes over time, biasing the output signal commensurate with that rate of input change. If proportional (P) action tells the output how far to move when an error appears, derivative (D) action tells the output how far to move when the input ramps. If proportional (P) action acts on the present and integral (I) action acts on the past, derivative (D) action acts on the future: it e ectively “anticipates” overshoot by tempering the output response according to how fast the process variable is rising or falling. When you think of “derivative action” (D), think discretion: this control action is cautious and prudent, working against change.

Mathematically defined, derivative action is the ratio of output o set to input velocity:

Derivative time constant (minutes) = τd = Output o set

Input velocity

Derivative time constant (minutes) = τd = ΔOutput

de dt

For example, if the PV signal begins to ramp at a rate of 5% per minute on a process controller with a derivative time constant of 4 minutes, the output will immediately become o set by 20% (ΔOutput = Derivative value × dedt ). In most PD and PID controllers, derivative response is also multiplied by proportional gain, so the same conditions applied to a PD controller that happened to also have a gain of 2 would result in an immediate o set of 40% (ΔOutput = Gain value × Derivative value × dedt ). The direction (sign) of this o set in relation to the direction of the input ramping depends on whether the controller is configured for direct or reverse action.

29.9P, I, and D responses graphed

A very helpful method for understanding the operation of proportional, integral, and derivative control terms is to analyze their respective responses to the same input conditions over time. This section is divided into subsections showing P, I, and D responses for several di erent input conditions, in the form of graphs. In each graph, the controller is assumed to be direct-acting (i.e. an increase in process variable results in an increase in output).

It should be noted that these graphic illustrations are all qualitative, not quantitative. There is too little information given in each case to plot exact responses. The illustrations of P, I, and D actions focus only on the shapes of the responses, not their exact numerical values.

In order to quantitatively predict PID controller responses, one would have to know the values of all PID settings, as well as the original starting value of the output before an input change occurred and a time index of when the change(s) occurred.

2304	CHAPTER 29. CLOSED-LOOP CONTROL

29.9.1Responses to a single step-change

100
95
90	Derivative response
85
80
75
70	Integral response
65
60
55
% 50
45	Proportional response
40
35
30	PV
25
20	SP
15
10
5
0
	Time

Proportional action directly mimics the shape of the input change (a step). Integral action ramps at a rate proportional to the magnitude of the input step. Since the input step holds a constant value, the integral action ramps at a constant rate (a constant slope). Derivative action interprets the step as an infinite rate of change, and so generates a “spike11” driving the output to saturation.

When combined into one PID output, the three actions produce this response:

100
95
90
85
80
75
70
65
60	PID response
55

%50

45

40

35

30

25		PV
20
			SP
15
10
5
0

Time

11This is the meaning of the vertical-pointing arrowheads shown on the trend graph: momentary saturation of the output all the way up to 100%.

29.9. P, I, AND D RESPONSES GRAPHED

2305

29.9.2Responses to a momentary step-and-return

	100
	95
	90	Derivative response
	85
	80
	75
	70	Integral response
	65
	60
	55
	% 50
	45	Proportional response
	40
	35
	30	PV
	25
	20
		SP
	15
	10
	5
	0

Time

Proportional action directly mimics the shape of the input change (an up-and-down step). Integral action ramps at a rate proportional to the magnitude of the input step, for as long as the PV is unequal to the SP. Once PV = SP again, integral action stops ramping and simply holds the last value12. Derivative action interprets both steps as infinite rates of change, and so generates “spikes13” at the leading and at the trailing edges of the step. Note how the leading (rising) edge causes derivative action to saturate high, while the trailing (falling) edge causes it to saturate low.

12This is a good example of how integral controller action represents the history of the PV − SP error. The continued o set of integral action from its starting point “remembers” the area accumulated under the rectangular “step” between PV and SP. This o set will go away only if a negative error appears having the same percent-minute product (area) as the positive error step.

13This is the meaning of the vertical-pointing arrowheads shown on the trend graph: momentary saturation of the output all the way up to 100% (or down to 0%).

2306	CHAPTER 29. CLOSED-LOOP CONTROL

When combined into one PID output, the three actions produce this response:

	100
	95
	90
	85
	80	PID response
	75
	70
	65
	60
	55
	% 50
	45
	40
	35
	30	PV
	25
	20
		SP
	15
	10
	5
	0

Time

29.9. P, I, AND D RESPONSES GRAPHED

2307

29.9.3Responses to two momentary steps-and-returns

	100
	95
	90	Derivative response
	85
	80
	75
	70	Integral response
	65
	60
	55
	% 50
	45	Proportional response
	40
	35
	30
	25	PV
	20
	15	SP
	10
	5
	0

Time

Proportional action directly mimics the shape of all input changes. Integral action ramps at a rate proportional to the magnitude of the input step, for as long as the PV is unequal to the SP. Once PV = SP again, integral action stops ramping and simply holds the last value. Derivative action interprets each step as an infinite rate of change, and so generates a “spike” at the leading and at the trailing edges of each step. Note how a leading (rising) edge causes derivative action to saturate high, while a trailing (falling) edge causes it to saturate low.

When combined into one PID output, the three actions produce this response:

100
95
90
85
80
75	PID response
70
65
60
55
% 50
45
40
35
30
25	PV
20
15	SP
10
5
0

Time

2308	CHAPTER 29. CLOSED-LOOP CONTROL

29.9.4Responses to a ramp-and-hold

	100
	95
	90	Derivative response
	85
	80
	75
	70	Integral response
	65
	60
	55
	% 50
	45	Proportional response
	40
	35
	30	PV
	25
	20
		SP
	15
	10
	5
	0

Time

Proportional action directly mimics the ramp-and-hold shape of the input. Integral action ramps slowly at first (when the error is small) but increases ramping rate as error increases. When error stabilizes, integral rate likewise stabilizes. Derivative action o sets the output according to the input’s ramping rate.

When combined into one PID output, the three actions produce this response:

	100
	95
	90
	85
	80
	75
	70
	65
	60	PID response
	55
	% 50
	45
	40
	35
	30	PV
	25
	20
		SP
	15
	10
	5
	0

Time

29.9. P, I, AND D RESPONSES GRAPHED

2309

29.9.5Responses to an up-and-down ramp

	100
	95
	90	Derivative response
	85
	80
	75
	70	Integral response
	65
	60
	55
	% 50
	45	Proportional response
	40
	35
	30	PV
	25
	20
		SP
	15
	10
	5
	0

Time

Proportional action directly mimics the up-and-down ramp shape of the input. Integral action ramps slowly at first (when the error is small) but increases ramping rate as error increases, then ramps slower as error decreases back to zero. Once PV = SP again, integral action stops ramping and simply holds the last value. Derivative action o sets the output according to the input’s ramping rate: first positive then negative.

When combined into one PID output, the three actions produce this response:

	100
	95
	90
	85
	80
	75
	70
	65
	60
	55	PID response
	% 50
	45
	40
	35
	30	PV
	25
	20
		SP
	15
	10
	5
	0

Time

2310	CHAPTER 29. CLOSED-LOOP CONTROL

29.9.6Responses to a multi-slope ramp

	100
	95
	90
	85	Derivative response
	80
	75
	70	Integral response
	65
	60
	55
	% 50
	45	Proportional response
	40
	35
	30
	25	PV
	20
	15	SP
	10
	5
	0

Time

Proportional action directly mimics the ramp shape of the input. Integral action ramps slowly at first (when the error is small) but increases ramping rate as error increases, then accelerates its increase as the PV ramps even steeper. Once PV = SP again, integral action stops ramping and simply holds the last value. Derivative action o sets the output according to the input’s ramping rate: first positive, then more positive, then it spikes negative when the PV suddenly returns to SP.

When combined into one PID output, the three actions produce this response:

100
95
90
85
80
75
70	PID response
65
60
55
% 50
45
40
35
30
25	PV
20
15	SP
10
5
0

Time