3 fundamental feedback controller

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FEEDBACK CONTROLLERProportional, Integral,

Derivative (PID)

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FEEDBACK CONTROLLERS

•Controller – brain of the control loop – performs the decision (D) operation in the

control system

•Operation 1. Compares the process signal it receives the controlled

variable with the set point.2. Sends an appropriate signal to the control valve (or

any other control element) in order to maintain the controlled variable at its set point.

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(Load disturbances )(Uncontrolled variables)

Heat ExchangerFluid in

Steam out

Ti T desiredFluid out

TT

TCSteam in(Manipulated variables)

SP

• reverse action– When an increase in signal to the controller

requires a decrease in controller output

• direct action– When an increase in signal to the controller

requires an increase in controller output

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Action of Controllers

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Action of Controllers

• Signal from the temperature transmitter increase, indicating the outlet temperature has increased above the set point.

• To return the temperature to the set point, the controller must close the steam valve by some amount. The controller must reduce its output signal to the valve.

• When an increase in signal to the controller requires a decrease in controller output, the controller must be set reverse action.

Case 1: heat exchanger control loop

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• Signal from the level transmitter increase, indicating that the level in the tank has increased above the set point.

• To return the level to the set point, the controller must open the steam valve by some amount. The controller must increase its output signal to the valve.

• When an increase in signal to the controller requires an increase in controller output, the controller must be set direct action.

Case 2: level control loop

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Type of Feedback controllersThe feedback controllers make a decision is by solving an equation based on the difference between the controlled variables and the set point.

In feedback control, the objective is to reduce the error signal to zero where

(8-1)sp me t y t y t

and

error signal

set point

measured value of the controlled variable

(or equivalent signal from the sensor/transmitter)

sp

m

e t

y t

y t

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Conventional block diagram representation for the controller

• P(s) - controller output• Gc(s)- transfer function that describes how the

controller acts on an error

Ysp(s), %TO

Ym(s), %TO

E(s), %TO P(s), %COGc(s)

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For proportional control, the controller output is proportional to the error signal,

(8-2)cp t p K e t where:

controller output

bias (steady-state) valuecontroller gain (usually dimensionless)c

p t

pK

Proportional Control

is controller output when the error is zero

-The proportionality is given by the controller gain, Kc-The controller gain determines how much the output from the controller changes for a given change in error

p

(usually set at mid-scale 50%CO)

Y= c + mx

The controller output will saturate (level out) at pmax=15 psig or 20 mA at the upper end and at pmin=3 psig or 4 mA at the lower end of the output. The ideal transfer function does not predict this saturation phenomenon.

0

1 * * (8-8)τ

tc

Ip t p K e t e t dt

If controller output, p(t) @ MV saturates = exceeds p(t) max, the controller cannot bring CV to SP.

The controller saturation occurs when the load disturbance or SP change is so large that it is beyond the range of MV.

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The key concepts behind proportional control are the following:

1. The controller gain can be adjusted to cause a quick change to the control valve upon a load disturbance or a change at the set point.

2. the sign of Kc can be choosed to make the controller output increase (or decrease) as the error signal increases.

3. If the process response is within the quality limits, then the controller setting is considred acceptable.

Reverse or Direct Action• The controller gain can be made either negative or positive.• For proportional control, when Kc > 0, the controller output p(t)

increases as its input signal ym(t) decreases

(8-22)c sp mp t p K y t y t • This controller is an example of a reverse-acting controller.

• When Kc < 0, the controller is said to be direct acting because the controller output increases as the input increases.

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A high proportional gain results in a large change in the controller output for a given change in the error.

If the proportional gain is too high, the system can become unstable.

In contrast, a small gain results in a small output response to a large input error, and a less responsive (or sensitive) controller.

If the proportional gain is too low, the control action may be too small when responding to system disturbances.

http://upload.wikimedia.org/wikipedia/commons/2/2b/Change_with_Kp.png

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AdvantageOne tuning parameter, Kc

DisadvantagePure proportional control will not settle at its target value, but will

retain a steady state error (offset) that is a function of the proportional gain and the process gain

The controlled variables operates with an offset (steady state deviation of the controlled variables from set point)

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transfer function for an ideal proportional controller

(8-6)cP s

KE s

Some controllers have a proportional band setting instead of a controller gain. The proportional band PB (%) is defined as

100% (8-3)c

PBK

Proportional band, PB (%) and Controller gain, Kc (dimensionless)

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The controller is a proportional only controller with the gain of 2.5, its bias value is set at mid-scale. The set point of the controlled variable is 20%TO. Calculate the controller output if the input signal to the controller is 30%TO.

Question

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Integral ControlFor integral control action, the controller output depends on the integral of the error signal over time,

0

1 * * (8-7)τ

t

Ip t p e t dt

where , an adjustable parameter referred to as the integral time or reset time, has units of time.

τI

Integral control action is widely used because it provides an important practical advantage, the elimination of offset (by pure proportional control).

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• Integral control action is normally used in conjunction with proportional control as the proportional-integral (PI) controller :

• When the error is introduce at t=0, the controller output changes immediately by an amount equal to Kc. This is the response due to the proportional mode

• Under PI controller as long as the error is present, the controller keeps changing its output (integrating the error). Once the error disappears (goes to zero), the controller do not change the output anymore (integrates a function with a value of zero).

0

1 * * (8-8)τ

tc

Ip t p K e t e t dt

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• The corresponding transfer function for the PI controller in is given by

τ 111 (8-9)τ τ

Ic c

I I

P s sK KE s s s

• The PI controller has two parameters, Kc andτI

• Integral time, I (time)• To drive the process toward the set point.• Decrease I makes controller action faster (aggressive)

SP1

PV1

PV2

SP2

Time

Idealresponse

Decreasing II is not strong enough to drive the process toward SP2

I is too strong. Process becomes oscillatory.

Characteristic of I in PI & PID mode

Unit 3: Characteristics of PB, I & D

© Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)

(Steady state errors are eliminated more quickly)

• Integral time, I (time)• To drive the process toward the set point.• Decrease I makes controller action faster (aggressive)

PV1

PV2

Idealresponse

Decreasing II is not strong enough to drive the process toward SP2

I is too low. Process becomes oscillatory.

Observe! The steepnest of response curve increases with decreasing I. Be alert that too low I’s may result in instability.

Decreasing too much I can cause the present value to overshoot the setpoint value

Characteristic of I in PI mode & PID mode

Unit 3: Characteristics of PB, I & D


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• The PI controller has two parameters, Kc and • The smaller value of the faster the controller integrates.• Reset time:

τI

τI

1

1

RI

Disadvantage

An inherent disadvantage of integral control action is phenomenon known as reset windup.

• When a sustained error occurs, the integral term becomes quite large and the controller output eventually saturates.

• Further buildup of the integral term while the controller is saturated is referred to as reset windup or integral windup.

• Inlet temp drops will reduce the outlet temp.• The PI will ask the steam valve to open.• The signal from the controller will increase

until the outlet temp equal to the set point.• The controller integrates up to 100% because

the inlet pressure drop is too large.• So the valve is fully open but the outlet temp

is not at the set point.• The controller will try to correct by further

increasing the output (integrating the error) even though the valve will not open more after 100%.

• Further buildup of the integral term while the controller is saturated is referred to as reset windup or integral windup

• When inlet temp is increasing, the outlet temp will start to increase.

• The valve remains wide open and not closing because the controller must integrate from 107% to 100% before is starts closing. 26

P(t),%

Closing &Opening valvegraph

CV

MV

3 psig =

15 psig =

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Derivative Control

• The function of derivative control gives the controller the capability to anticipate where the process is heading by calculating the derivative error (slope of the error curve)

• Thus, for ideal derivative action,

τ (8-10)D

de tp t p

dt

where , the derivative time, has units of time.• Derivative control is used to reduce the magnitude of the

overshoot produced by the integral component and improve the combined controller-process stability

• Recommended to be used in slow processes (processes with long time constant)

• The derivative term slows the rate of change of the controller output

τD

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• The outlet temp start to decrease when the inlet temp decreases.

• At time ta, the amount of the error is positive and small. The control correction provide by P and I is small.

• However the derivative of this error ( slope of error curve) is large and positive. Control correction provide by D is large.

• By looking at the derivative of the error, the controlled variable is heading away from set point and use this fact to help controlling

• At tb, the error is still large and positive.• However, the derivative of this error is

negative and the error is decreasing. The variable has started to come back to set point

PID control for heat exchanger

• Derivative time, D (time)• To accelerate the process.• Increase D makes oscillation period shorter.

SP1

PV1

PV2

SP2

Time

Idealresponse Increasing D

Characteristic of D in PI & PID mode

Unit 3: Characteristics of P, I & D


• Derivative time, D (time)• To accelerate the process.• Increase D makes oscillation period shorter

PV1

PV2

Idealresponse

With increasing D, oscillation period becomes shorter,

decrease overshoot, but slow down transient response (the

rate of change of the controller output)

For noisy process, avoid using D (or use lower value).

Good for slow process.


Increasing D



PV1

PV2

Idealresponse

Increasing D

Be alert. Too much D make controller action faster and may result in instability.

• Good for slow processes such as temperature control.




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Proportional-Integral-Derivative (PID) Control

Now we consider the combination of the proportional, integral, and derivative control modes as a PID controller.

Form of PID Control

The form of the PID control algorithm is given by

0

1 * * τ (8-13)τ

tc D

I

de tp t p K e t e t dt

dt

0

* * (8-16)t

c I Dde t

p t p K e t K e t dt Kdt

Expanded Form of PID Control

The corresponding transfer function is:

11 τ (8-14)τc D

I

P sK s

E s s

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For example, an ideal PD controller has the transfer function:

1 τ (8-11)c D

P sK s

E s

• By providing anticipatory control action, the derivative mode tends to stabilize the controlled process.

PD controller

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PID - Most complicated to tune (Kc, I, D) .- Better performance than PI- No offset

PI - More complicated to tune (Kc, I) .- Better performance than P- No offset- Reset windup

P - Simplest controller to tune (Kc).- Offset with sustained disturbance or setpoint change.

Controller Comparison

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y

Typical Response of Feedback Control SystemsConsider response of a controlled system after a sustained disturbance occurs (e.g., step change in the disturbance variable)

Typical process responses with feedback control.

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ProportionalLarger values typically mean faster response since the larger the error, the larger the proportional term compensation. An excessively large proportional gain will lead to process instability and oscillation.

Integralsmall values imply steady state errors are eliminated more quickly. Excessively small integral term cause the present value to overshoot the setpoint value.

DerivativeLarger values decrease overshoot, but slow down transient response and may lead to instability.

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Question

1. Consider the temperature control system. The controller is a proportional only controller with the gain of 2.5. Develop the block diagram with the transfer function of this controller.

2. Now, Assume the controller is proportional- integral (PI) and a gain of 3 %CO/%TO, integral time is 5 min. Develop the block diagram with the transfer function of this controller.

3. Now, Assume the controller is proportional- integral-derivative (PID) with a gain of 3 %CO/%TO, integral time is 5 min and derivative time is 0.2 min. Develop the block diagram with the transfer function of this controller.

3 fundamental feedback controller

Engineering

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