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Control 1: Feedback Control Systems

1

Feedback Control systems

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2

Introduction 3

Feedback 5

Proportional Control 7

First Order Process 7

Second Order Process 10

Proportional Gain and Proportional Band 12

Steady State Errors (Offset) 13

Stability Consideration 16

Stability from Frequency Response 17

Proportional plus Integral Control Action 17

First Order Process 18

Step Response of P + I Controller 20

Proportional plus Derivative Control Action 21

Step Response of a P + D controller 23

Ramp Response of a P + D controller 23

Proportional plus Integral plus Derivative Controller (P + I + D) 24

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Introduction

A closed loop control system consists of a process, a measurement system and a controller.

The controller automatically adjusts one of the inputs to the process in response to a

measurement signal. Feedback, as a method of control, may be adopted for a number of reasons:

• To minimise the effect of disturbances. All process systems will have disturbances. In a

majority of cases, the role of the control system is to minimise the effect of the

disturbances. That is, hold the system at some steady state operating condition. In certain

circumstances, it may be necessary to move the system operating conditions. For

example, in a batch plant, in this case, the control system adjusts the process to follow the

desired value or set point.

• To alter the transient of response of a system. Typically to influence the overshoot and

the settling time.

• To reduce the effect of parameter variations. The parameters of a system may change

with age, environment, (e.g., temperature, humidity, atmospheric pressure). Feedback can

be used to reduce the sensitivity of the system to these variations. Consider the following

example:

Kθi θo

The gain of the system, K, is nominally 10. Unfortunately, the gain can vary between 9

and 11, i.e., ± 10% error. It is decided to stabilise the overall gain of the system using

feedback.

Kθ i θ o

K1000

0.1

-

+


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The closed loop gain can be found from:

( )

( )

( )

θ θ

θ θ

θ θθ θ

θθ

o K i o

o K i K

o K o Ko K

o

i

K

K

= −

= −

+ =+ =

=+

1000 01

1000 100

100 10001 100 1000

1000

1 100

. θ

θ

θ

o

iK i

K Closed Loop Gain % Error

9 9.989 -0.11

10 9.99 -0.1

11 9.991 -0.09

Feedback will only compensate for relatively small variations. Larger variations will

require an adaptive or self-tuning control system.

• To linearise a non-linear system. For example a control valve positioner uses the

feedback of valve position to reduce any inherent hysterisis in the valve steady state

characteristic.

Consider the following non-linear system. The system consists of an amplifier whose gainis not constant across its operating region. It is assumed that the input to the system is

sinusoidal.

0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

θ i

θ o

θi θo

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Input

Output


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The following feedback system can be used to improve linearity.

θ i100

-

+

θo

0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Input

Output

A feedback control system may have been designed for only one of the purposes outlined

above. However other advantages will generally be present, e.g., valve positioner may be

used to minimise non-linearities, but the system response time will probably also be

improved

Feedback

The general form of a feedback system is shown:

Gp(s)θ i θ o

Kc Gc(s)

H(s)

-

+

θ e


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where KcGc(s) is the controller transfer function

Gp(s) is the process transfer function

H(s) is the transfer function in feedback path, e.g., measurement transducer

s is the Laplace variable

Consider the signal passing around the loop:

( ) ( ) ( )θ θo e K G s G sc c p= 1

and ( ) ( )θ θ θe i o H s= − 2

substitute equation (2) into equation (1) for θe

( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( )

( ) ( ) ( )( )

θ θ θ

θ θ θ

θ θ θ

θ θ

θθ

o i o H s K G s G s

o i K G s G s o H s K G s G s

o o H s K G s G s i K G s G s

o H s K G s G s i K G s G s

o

i

K G s G s

H s K G s G s

c c p

c c p c c p

c c p c c p

c c p c c p

c c p

c c p

= −

= −

+ =

+ =

=+

1

1

( ) ( )K G s G sc c p is the forward path transfer function

( )H s is the feedback path transfer function

( ) ( ) ( )H s K G s G sc c pis the open loop transfer function

( ) ( )

( ) ( ) ( )( )K G s G s

H s K G s G s

c c p

c c p1 +is the closed loop transfer function

The closed loop transfer function is also given by:

( )θθo

i

Forward Path

Forward Path x Feedback Parth=

+1

The performance of the control system may be judged by both dynamic and steady state

responses to various inputs and disturbances. In general the process transfer function will be

fixed and only the controller transfer function is open to choice. Note, the control engineer

should always be mindful of possible process changes, i.e., changes to Gp(s) which may

improve the overall system response.


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The following sections are concerned with the performance of systems using different

controller transfer functions. In process systems the controller generally takes one of a small

number of possible forms, namely proportional, proportional plus integral, proportional plus

integral plus derivative.

Proportional Control

First Order Process

The controller is a simple gain term such that the controller output is proportional to the loop

error. Only changes are being considered therefore the bias is neglected. Consider a system

containing a single lag as shown:

θ i θ o

( )Kp

sτ + 1 Kc

Kl

-

+

θ e +

+

θ l

θi is the set-point

θl is the load disturbance

For a change of set-point, i.e., with θl = 0.

( )( )

( )( )

θθ

ττ

τ τ

o

i

KcKp

sKcKp

s

KcKp

s KcKp

KcKp

KcKp sKcKp

=

+ ++

⎛

⎝ ⎜

⎞

⎠⎟

=+ +

=

++

+⎛

⎝ ⎜

⎞

⎠⎟1 1

1

11

11

This results is in the standard form for a first order system.

( )( )

( )

( ) ( )

( ) ( )

θθ τ τ

ττ τ

τ

o

i

K

s

KcKp

KcKp sKcKp

KKcKp

KcKp

KcKp

KcKp

andKcKp KcKp

closed loop

closed loop

closed loop

closed loop

=+

=

++

+⎛

⎝ ⎜

⎞

⎠⎟

∴ =+ +

<

=+ +

<

11

11

1 11

1 1

,

,


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8

)

Therefore closing the loop around a first order system results in a first order system. Thus,

the response of the closed loop system to a step change of set-point will be an exponential.

This response is faster than in open loop since the time constant is reduced by a

factor ( . The step response for a number of values of gain is shown in the following

graph. The set-point change equals,

1 + KcKp

θiA

s= .

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

KcK =1

KcK =2

KcK =3

KcK =5

KcK =10

( )t

Secondsτ

θA

o

Offset

The response settles out at a level less than the demanded input level, θi = A, this is offset.

The steady value of the output can be found from the final value theorem:

( )( )

( )( )

( ) ( )

( )( )

( )

θ θ τ τ

θ θτ

o i KcKp

KcKp sKcKp

As

KcKp

KcKp sKcKp

o tt

Lims o s LimsA

s

KcKp

KcKp sKcKp

AKcKp

KcKps s

=+

++

⎛

⎝ ⎜

⎞

⎠⎟

=+

++

⎛

⎝ ⎜

⎞

⎠⎟

→∞= =

++

+⎛

⎝ ⎜

⎞

⎠⎟

→+→ →

11

1 11

1

11

110 0

. .

Therefore as KcKp is increased the offset is reduced.

Consider the response of the closed loop system to a load change θl, i.e., with θi = 0.

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( )( )

( )( )

θθ

ττ

τ τ

o

l

KlKp

sKcKp

s

KlKp

s KcKp

KlKp

KcKp sKcKp

=

+ ++

⎛

⎝ ⎜

⎞

⎠⎟

=+ +

=

++

+⎛

⎝ ⎜

⎞

⎠⎟1 1

1

11

11

The effective time constant is the same for both set-point and load change. The following

graph shows typical response for θlA

s Kl= .

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

( )t

Secondsτ

θo

AKlKcKp=1

KcKp=2

KcKp=5

KcKp=10

KcKp=0, i.e.,

no control

Offset

Whilst the disturbance exists there is an offset or steady state error whose value can be

reduced by increasing the loop gain KcKp.

The responses show that, for both set-point and load changes, increasing the gain reduces the

time required to reach a new steady state value and in addition decreases the steady state

error, (offset). It would appear that the maximum possible controller gain should be used. A

real process will have other lags or delays in the control loop, all of which will effect the

response at high gain and lead to unstable operation. If the largest time constant is two or

three orders of magnitude greater than any of the other lags the maximum gain will be high.

Note, an on-off controller may be considered to be a proportional controller with infinite

gain, on-off controllers can be used where one lag dominates e.g. heating system.


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Second Order Process

Consider a system consisting of two first order lags in series, e.g., jacketed reactor where the

mixture temperature is controlled by adjusting the flow of coolant to the jacket.

θ i θ o

( )K

s

2

2 1τ +Kc

Kl

-

+

θ e +

+

θl

( )K

s

1

1 1τ +

For a change in set-point.

( )

( )( )( )

( )( )

( )( )

( ) ( ) ( )

θ

θτ τ

τ τ

τ τ

τ τ τ ττ τ

τ τ

τ τ τ τ

o s

i s

K K K

s sK K K

s s

K K K

s s K K

K K K

s s K K K

K K K

s sK K K

c

c

c

c

c

c

c

c

=

+ + ++ +

⎡

⎣⎢

⎤

⎦⎥

=+ + +

=+ + + +

=

++

++⎛

⎝

⎜⎞

⎠

⎟

1 2

1 2

1 2

1 2

1 2

1 2 1

1 2

21 2 1 2 1 2

1 2

1 22

1 2

1 2

1 2

1 2

1 1 11 1

1 1

1 1

K2

Comparing this to the standard form of a second order transfer function gives:

( )

( ) ( ) ( ) ( )

( )( )

( )

( )

θ

θ

ω

ζ ω ωτ τ

τ τ

τ τ τ τ

ζτ τ

τ τ ω τ τ

o s

i s

K

s s

K K K

s sK K K

KK K K

K K K K K K

K K K

n

n n

c

c

c

c cn

c

=+ +

=

++

++⎛

⎝ ⎜

⎞

⎠⎟

∴

= + =+

+ =+

2

2 2

1 2

1 22

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2 1 2

1 2

1 2

2 1

1 2 1

1, ,

Increasing the open loop gain Kc K1 K2 results in:

• ωn increasing, i.e., the system responds faster.

• ζ decreases, i.e., the system is more oscillatory.

• K tends to 1, i.e., the steady state gain tends to 1, therefore offset is reduced.

The following responses show the effect of increasing the controller gain, given that τ1 = τ2 =

1.


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0 2 4 6 8 100

0.5

1

1.5

KcK1K2=20

KcK1K2=5

KcK1K2=2

KcK1K2=1

KcK1K2=0.5

θo

Time (Seconds)

Consider the response of the system to a load change:

( )

( )( )

( )( )( )

( )

( )

( )( )

( )

( )

( )

( ) ( )

θ

θτ

τ ττ

τ

τ

τ τ

τ

τ τ τ τ

τ

τ ττ τ

τ τ τ τ

o s

i s

Kl K

sK K K

s s

Kl K

sK K K

s

Kl K s

s s K K

Kl K s

s s K K K

Kl K s

s sK K K

c c c

c c

=

+ ++ +

⎡

⎣⎢

⎤

⎦⎥

=+ +

+

=+

+ + +

=+

+ + + +=

+

++

++⎛

⎝ ⎜

⎞

⎠⎟

2

21 2

1 2

2

2

1 2

1

2 1

2 1 1

2 1

21 2 1 2 1 2

2 1

1 22

1 2

1 2

1 2

1 2

1 11 1

11

1

1 1

1

1

1

1

K2

The denominators of both transfer functions are the same. Therefore load and set-pointchanges give a similar transient response.

Typical load responses are shown below, given that τ1 = τ2 = 1.


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0 2 4 6 8 10-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

θo

Time (Seconds)

Kc K1 K2 = 1

Kc K1 K2 = 0.5

Kc K1 K2 = 2

Kc K1 K2 = 5

Kc K1 K2 = 20

In summary the effect of increasing the gain of a proportional control system:

• will cause the system to respond faster.

• will reduce the offset but will cause a reduction in the damping factor, i.e., a reduction in

stability.

Steady state error and stability will be given further consideration in the following sections

before the effect of integral and derivative action is considered.

Proportional Gain and Proportional Band

θ θc Kc e=

Controller gain Kc is the change in output divided by the change in input. Some process

controllers have the proportional gain term calibrated as proportional band.

Proportional Band is the percentage change in measured variable required to produce 100

percent change in output, expressed as a percentage of the instrument full scale reading, i.e.,

gain KcMeasured Variablechange togive change inoutput i e oportional Band

oportional BandKc

,, . . , Pr

Pr

=

∴ =

100%

100%

100%


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Steady State Errors (Offset)

An important consideration is the steady state error that can exist between the system input,

i.e., set-point, and the system output as time becomes large.

For example, suppose a process consists of heating a continuous flow of water via a steam

heater. The flow of steam being adjusted by a control valve which is connected to a

proportional controller. Suppose, at some condition of flow, the controller has been adjusted

so that zero steady state error exists. This is achieved by adjusting the bias in the controller.

Let the water rate be increased by 10 %, then 10 % more heat is required. The steam valve

must open further to supply this heat. This it can only do if the water temperature falls below

the desired value by an amount sufficient to give the required valve opening. The water

temperature will continue to fall below the desired value (θi) until at a given error (θe)the

valve is open wide enough to maintain the water temperature at (θi-θe)oC. This error θe is

the 'offset' and it will persist as long as conditions remain unchanged.

Steady state errors may be examined more formally by use of the system transfer function

and the final value theorem.

Consider a closed loop system with unity feedback.

θ i θ oKcKpG(s)

-

+

θ e

( )( ) ( )

( )( )

( ) ( ) ( ) ( )( ) ( )

( )( )

( ) ( ) ( )( )

θθ

θ θ θ θθ

θθ

o si s K K G s

K K G s

now error e s i s o s i si s K K G s

K K G s

e si s

K K G s

c p

c p

c p

c p

c p

=+

= − = −+

= +

1

1

1

,

Steady state error can be found from the final value theorem:

( ) ( )( )

( )( )θ θ

θe t Lim s e s Lim s

i s

K K G sts s c p→∞ → →

= =+0 0 1

. . . .


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Kc Kp G(s) is the open loop transfer function and is of the general form:

( )( )( ) ( )

( )( ) ( )K K G s

K K s s s

s s s sc p

c p v

nw

=+ + +

+ + +

τ τ τ

τ τ τ

1 2

1 2

1 1

1 1 1

........

........

1

Systems may be classified according to the value of n1, i.e., the number of integrators in the

loop. Using this classification a system of Type O has no integrators, Type 1 a single

integrator, Type 2 a double integrator and so on.

Consider a step input, ( )θi sA

s= , then

( )( )( ) ( )( )

θe t Lim sA

s K K G s

A

Lim K K G sts c p

sc p→∞ →

→

=+

=+0

0

1

1 1. . .

If the system is Type O, i.e., n = 0 then ( )Lim K K G s K Ks

c p c p→

=0

and the steady state error

( )( )

θe tA

K Kt c p→∞

=+1

Therefore with n = 0 offset will exits, the magnitude of which can be reduced by increasing

Kc.

If the system is Type 1 or greater, i.e., n > 0 then ( )Lim K K G ssc p

→ = ∞0 and the steady state

error

( )θe tt→∞

= 0

Therefore the presence of an integrator in the loop will cause the steady state error to be

reduced to zero.

The same general formula may be used to determine the steady state error for any type of

input. In general with process systems the only other form of set-point change, other than astep, will be a ramp. This can occur when the system is being moved from one operating

condition to another. For a ramp input the following can be derived:

• Type O, i.e., n = 0 the system response lags further and further behind the set-pointchange.

• Type 1, i.e., n = 1 the system response settles down to follow the input whilst laggingbehind it.

• Type 2 and greater, i.e., n ≥ 2 the system response will succeed in following the inputexactly once the initial transient has decayed away.


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Consideration must also be given to steady state errors due to load disturbances. Consider

the following system.

θ i θ o

-

+

θe +

+

θ l

KcK1G1(s) K2G2(s)

( ) ( )θ θe s o s= − , since ( )θi s = 0 , i.e., no change in set-point.

( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( )( ) ( )

( ) ( )( )

θ θ θ

θ θ θ

θ θ

θ θθ

o s K G s l s o s K K G s

o s l s K G s o s K K K G s G s

o s K K K G s G s l s K G s

e s o sl s K G s

K K K G s G s

c

c

c

c

= −

= −

+ =

= − =−

+

2 2 1 1

2 2 1 2 1 2

1 2 1 2 2 2

2 2

1 2 1 2

1

1

Steady state error can be found from the final value theorem:

( ) ( ) ( ) ( )( ) ( )( )

θ θ θe t Lim s e s Lim s l s K G s

K K K G s G sts s c→∞ → →

= = −+0 0

2 2

1 2 1 21. . . .

Consider a step change in load, ( )θl sA

s= , then

( )( ) ( )

( ) ( )( )θ

θe t Lim s

A

s

l s K G s

K K K G s G sts c→∞ →

=−

+0

2 2

1 2 1 21. . .

• If G1(s) and G2(s) contain no integrators, then

( )( )

θe t LimAK

K K Kts c→∞ →

=−

+0

2

1 21. , i.e., offset.

• If G1(s) contains no integrators and G2(s) contains integrators, then

( )θe t LimA

K Kts c→∞ →

=−

0 1

. , i.e., offset.

• If G1(s) contains integrators, then

( )θe tt→∞

= 0 , i.e., no offset.


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Thus, an integrator is required between the error and the entry point of the disturbance. In

practice this may lead to the requirement of an integrator even though the system transfer

function contains an integrator. This can lead to stability problems.

Stability Consideration

Consider the following closed loop system.

θ i θ o

-

+

θe

Kc ( )

2

3 1s + ( )1

2 1s +1

s

The closed loop transfer function is given by:

( )

( ) ( )( )

θ

θ

o s

i s

K

s s s K

c

c

=+ + +

2

3 1 2 1 2

let Kc = 1.0( )

( ) ( )( )θ

θ

o s

i s s s s s s s=

+ + +=

+ − +

2

6 5 2

2

1 6 23 2 2

With Kc equal to one the closed loop system behaves as an exponential lag in series with a

second order, (quadratic), lag. Comparing the second order lag with the standard second

order transfer function, indicates negative damping, thus any small changes in θi will lead to

sinusoidal oscillations of increasing magnitude. The fact that the closed loop system is

unstable for certain values of Kc is not obvious from the diagram or from the closed loop

transfer function. Therefore tests for stability are an important part of feedback controlsystem design. There are several ways in which stability can be examined. Here we will

restrict ourselves to consideration using frequency response techniques.


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Stability from Frequency Response

A simplistic non-rigorous approach will be used. Consider the general system shown below,

in which the load disturbance has a periodic component ( )B Sin tl ω .

( )θ i s ( )θ o s

-

+( )θe s +

+

( ) ( )θ ωl s B S in tl=

Kc KpGp(s)

Suppose the frequency of the load disturbance is such that the phase lag through the process

is 1800, the output, θo, is thus 180

0behind θl. Although there is no delay in the controller,

proportional action only, its output lags its input by 1800

because of the negative sign in the

comparitor. Thus, the controller output is in phase with the load disturbance and if the gain

round the loop is greater than 1 the amplitude of the disturbance fedback round the loop will

be greater than the original disturbance. If the external disturbance, θl, was removed the

oscillation round the loop would continue to increase in magnitude until some element

saturated, i.e., the system is unstable. If the gain around the loop was less than 1, then the

oscillation would decay away if θl became zero, i.e., the loop would be stable. The point of entry of the disturbance is not important from stability consideration, neither is there a

requirement that it should be a perfect sinusoid. If a system is unstable, there will always be a

component of some disturbance that will set the oscillation up.

Thus, for stability we may examine the system frequency response and determine the loop

gain at a frequency whose phase shift would be -1800. If the loop gain is greater than 1, then

the system will be unstable.

Proportional plus Integral Control Action

It has been shown that the use of an integrator in a control loop will eliminate the offset

inherent with proportional controllers. A controller with pure integral action can be used but

it is more usual to combine integral with proportional action to give a proportional plus

integral (P+I) controller.


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θ D V

Kc

-

+

θ e

+

+

( )KTi

e t dtc t θ0∫

θ c

θ M V

Controller output, ( ) ( ) ( )θ θ θc t e t KK

Tie t dtc

c t= + ∫ 0 .

where Ti is integral action time.

This may be transformed to give the controller transfer function:

( )

( )

( )θ

θ

c s

e sK

sTiK

sTi

sTic c= +⎡⎣⎢ ⎤⎦⎥ =

+1

1 1

First Order Process

Consider a P + I controller used with a first order process.

θ i θ o

( )Kp

sτ + 1

Kl

-

+

θ e +

+

θ l

( )K

sTi

sTic

+ 1

For a change in load disturbance, i.e., θi = 0.

( )

( )( )

( )

( )( )

( ) ( ) ( )

( )

θ

θτ

ττ τ

τ ττ τ

o s

l s

K Kp

sK Kp sTi

sTi s

K Kp

sK Kp sTi

sTi

s K Kp

s sK Kp

TisTi

s K Kp

s s sK KpK Kp

Ti

s K Kp

ss K Kp K Kp

Ti

l

c

l

c

l

c

l

c

c

l

c c

=

+ ++

+

⎡

⎣⎢

⎤

⎦⎥

=

+ ++

=+ + +

=+ + +

=

++

+⎛

⎝ ⎜

⎞

⎠⎟

1 11

11

11 1

12 2


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( )

( ) ( ) ( )

( )

θ

θ

ω

ζ ω ωτ

τ τ

ζτ

ωτ

o s

l s

K

s s

s K Kp

ss K Kp K Kp

Ti

K Kp Ti

K Kp

K Kp

Ti

n

n n

l

c c

c

cn

c

=+ +

=

++

+⎛

⎝

⎜⎞

⎠

⎟

∴

=+

=

2

2 2

22 1

1

2,

The addition of integral action has eliminated offset. However the closed loop transfer

function is more complex than with proportional action. The closed loop transfer function is

now second order and thus with gain greater than a certain value the system response will

oscillate.

The response, to a load change, of a second order system with a P + I controller is shown

below.

0 5 10 15 20-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time (Seconds)

Measured

Variable

No Integral

action

Large Ti

Optimum Ti

Small Ti

Empirical techniques for determining the best values of Kc and Ti were considered in

Introduction to Control module.


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Step Response of P + I Controller

The step response of a P + I controller in isolation is shown below, i.e., the controller output

is not connected to the final control element.

Controller Output

θc

Error Signal

θe = MV-DV

Output due to

proportional

action

Output due

to integral

action

Time

t1 t2

If the offset is constant, i.e., a sustained error, the integral action will eventually cause the

output of the controller to saturate at either its maximum or minimum value.

Integral action time is defined as the time taken for the output change due to proportional to

be repeated by integral action, when the error signal is a step.

Therefore integral action time, Ti, = (t2 - t1).

The Units for Ti are:

(Seconds / Repeat), (Minutes / Repeat), (Hours / Repeat) or just Seconds, Minutes or Hours.

Some controllers use a different definition for integral action, i.e., reset rate. Reset rate is the

number of times integral action has repeated the change due to proportional in unit time,

when the error signal is a step.

Therefore reset rate, RR, =( )

1 1

2 1t t T−=

i.

The Units for RR are:(Repeats / Second), (Repeats / Minute) or (Repeats / Hour).


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Proportional plus Derivative Control Action

The controller output now has a component that is proportional to the rate of change of error.

θ D V

Kc

-

+

θ e +

+

K Tdd e

dtc

θ

θ c

θ M V

Bias

Controller output, ( ) ( )( )

θ θθ

c t e t K K Tdd e t

dtc c= + .

where Td is derivative action time.


( )

( ) [ ]

θ

θ

c s

e s K s Tdc= +1

In steady state K Tdd e

dtc

θ= 0 , therefore there is no derivative action, thus derivative cannot

eliminate offset. Derivative action is never used by itself, but is always added to

proportional or proportional plus integral action.

Consider a P + D controller with a system as shown below.

θ i θ o

( )Kp

s sτ + 1

Kl

-

+

θ e +

+

θ l

( )K sTdc + 1

For a change in load disturbance, i.e., θi = 0.


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( )

( )( )

( )

( )

( ) ( )

( )

θ

θτ

τ

τ

ττ

τ τ

o s

l s

K Kp

s sK Kp sTd

s s

K Kp

s s K Kp sTd

K Kp

s s s Td K Kp K Kp

K Kp

ss Td K Kp K Kp

l

c

l

c

l

c c

l

c c

=

+ ++

+

⎡

⎣⎢

⎤

⎦⎥

=+ + +

= + + + =+

++

⎛

⎝ ⎜

⎞

⎠⎟

1 11

1

1 1

12

2


( )

( ) ( ) ( )

( )

θ

θ

ω

ζ ω ωτ

τ τ

ζτ

ωτ

o s

l s

K

s s

K Kp

ss Td K Kp K Kp

Td K Kp

K Kp

K Kp

n

n n

l

c c

c

cn

c

=+ +

=

++

+⎛

⎝ ⎜

⎞

⎠⎟

∴

=+

=

2

2 2

22 1

1

2,

The damping factor ζ has been increased by an amountTd K Kp

K Kp

c

c2 τover the control system

with proportional only action, i.e., Td = 0. Since the damping factor is increased, the

controller gain Kc may be increased, over proportional only, for the same degree of stability,

thus the system will respond faster. Proportional plus derivative action cannot eliminate the

offset with load changes, however because Kc can be increased offset will be less than withproportional only.

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Proportional only,

low gain

P + D, optimum gain,

small TdP + D, optimum gain,

optimum Td

P + D, optimum gain,

large Td

Time

θMV


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Step Response of a P + D controller

Controller Output

θc

Error Signal

θe = MV-DV

Time

Proportional

contribution to

the output

Derivative

contribution to

the output

The step response shows an impulse in the controller output due to derivative action. This is

the response of an idealised, theoretical controller. In practice the impulse will be a fast spike,

the larger Td the larger the spike.

Ramp Response of a P + D controller

Controller Outputθc

Error Signal

θe = MV-DV

Output due to

proportional

action

Output due to

Derivative

action

Timet1 t2


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The ramp response shows a step in controller output due to derivative action, the larger Td

the larger this step. Therefore the final control element will suddenly move to a new position.

This in some way explains the increased speed of response over proportional only. For a

proportional controller to move the final control element by the same amount it would have

to start changing its output before the error occurred. In this sense a P + D controller can be

said to have anticipated the error. Derivative action is sometimes referred to as anticipatoryor pre-act control.

Derivative action time is defined as the time taken for the output change due to derivative to

be repeated by proportional action, when the error signal is a ramp.

Therefore derivative action time, Td, = (t2 - t1).

The Units for Td are Seconds, Minutes or Hours.

Proportional plus Integral plus Derivative Controller (P + I + D)

The controller output now contains all three actions.

θ D V

Kc -

+

θ e +

+

K Tdd e

dtc

θ

θ c

θ M V

K Tdd e

dtc

θ

( )Kc

Tie t dt

t θ0∫

+

Controller output, ( ) ( ) ( )( )

θ θ θθ

c t e t KK

Tie t dt K Td

d e t

dtc

c tc= + ∫ +

0.


( )

( )

[ ]θ

θ

c s

e sK

sTis Td K

s Ti Td s Ti

s Tic c= + +⎡⎣⎢

⎤⎦⎥

=+ +

11 12


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The addition of derivative action to P + I action has the effect of increasing the stability of the

system. Thus allowing a larger gain to be used resulting in a faster response to a disturbance.

Integral action will ensure the elimination of offset due to step load or set-point changes.

θ i θ o

( )( )( )2

10 1 20 1 50 1s s s+ + +

Kl

-

+

θ e

+

+

θ l

[ ]K

s Ti Td s Ti

s Tic

2 1+ +( )

1

20 1s +

The following is the response of the above system to a step change in load using P + I + D

and P + I controllers.

0 100 200 300 400 500 600 700 800-0.2

0

0.2

0.4

0.6

0.8

1

P + I

Kc = 0.86

Ti = 95

P + I + D

Kc = 2.2

Ti = 95

Td = 12.6

Time (Seconds)

θo


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