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Classical Control Theory

Proportional - Integral - Derivative

Dr. Matt StablesDr. James Taylor

23/02/2010

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Introduction

History of ControlBeen used since antiquity for control of Mechanical systems.

Babylonian and Greek development of Water Clock

Feedback Control;Such as a float valve, controlling temperature, speed or fluid levels

With Modern computer systemsSensor output can be compared with desired output, input adjusted accordingly.

Example :Temperature control can be achieved through use of a heater or a fan

Challenge comes when deciding level of control input for desired results !

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Introduction

Proportional-Integral-Derivative (PID) control accounts for more than 90% of

the controls and automation applications today.

Primarily because;Effective and simple to implement.

Originally intended for linear, time-invariant systems

The PID algorithm has evolved, to control systems with more complex dynamics.

In this lecture;

The need for feedback control

The influences of Proportional, Integral and Derivative aspects

Basic parameter tuning methods

Concentrate on Continuous Time control ( s – operator )

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A common actuator is a DC motor

Provides

• Direct rotary motion or, via

drums and cables, translational

motion

•The Stationary magnetic field

provides a force on a current

carrying conductor

•For control purposes, there is

Input - voltage - u

Output - rotational velocity - y

Introduction

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Voltage

RPM

Time

RPM

Characteristics

Introduction

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Proportional Control

System

Output YMotor Speed

Input UVoltage

Controller

Desired Output DMotor Speed

Open Loop Control

User determines desired response

Controller is an electronic amplifier, determines input voltage

Amplifier voltage U = K . D where K is a Gain

Actual motor speed depends on motor dynamics and load disturbances

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System

Output YInput U

Controller

Desired Output D


Sensor

Closed Loop Control

Output is measured with a suitable sensor

Controller compares desired output with actual output

An electronic amplifier produces a voltage

proportional to the error U = K . (D – Y)

Reduces Sensitivity to disturbance

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Roots of Characteristic equation determine stability

Closed Loop Transfer Function


Output Y+_Desired Output D

G1 (s)

G2 (s)

Y = x D G1 (s)

1 + G1 (s)G2 (s)

G2 (s)G1 (s)Output YDesired Output D

Y = G1 (s) G1 (s) D

G2 (s)

G1 (s) Output YDesired Output D

Y = G1 (s) + G1 (s) D

UnstableStable

Real

Imaginary

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UnstableStable

Real

Complex

Output YMotor Speed

Input UVoltage

K = 2+_

Desired Output DMotor Speed

6.5

12s + 1

Y =

K 6.5

12s + 1

K 6.5

12s + 11 +

x D = K 6.5

12s + K 6.5 +1x D

Roots of Characteristic equation determine stability

Root = -1.167 STABLE

Closed Loop Stability


= 13

12s + 14x D

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Example: Ventilation Control

Output YVentilation Rate

Input UVoltage

K +_

Desired Output DVentilation rate

1.5

10s + 1

K 1.5

10s + K 1.5 +1

x D Y = Closed Loop Transfer Function

Steady state Gain = K 1.5

K 1.5 +1

Always results in a steady state error !!

Time

m3/s

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•Gain is applied to integral of error

Proportional to both magnitude &

duration

•Summing error over time gives an

accumulated offset previously uncorrected

•Results in Zero Steady State Error

•Can cause overshoot of setpoint

•Greater complexity in closed loop Transfer

Function – may become unstable

Output YInput U

K +_

Desired Output D

1

SG (s)

Y = x D

G (s)

1 +

K 1

S

K 1

SG (s)

K G (s)

s + K G (s)Y = x D


Integral Control

The Integral Gain

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Integral Control

Example: Ventilation Control

Output YVentilation Rate

Input UVoltage

K +_

Desired Output DVentilation rate

1.5

10s + 1

K 1.5

10s2 + s +K 1.5

x D Y = Closed Loop Transfer Function

Steady state Gain = K 1.5

K 1.5

Zero steady state error !!

Time

m3/s

1

S

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•Gain is applied to rate of change of error

Acts to slowdown change

•Most noticeable near setpoint

•Used to reduce magnitude of overshoot

•Used in combination with Proportional

and/or Integral gains

•Greater complexity in closed loop Transfer

Function – may become unstable due to

sensitivity to noise

Output YInput U

K s +_

Desired Output D

G (s)

K s G (s)

1 + K s G (s)Y = x D


Derivative Control

The Derivative Gain

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Control Requirements

Important considerations; Is steady state error to be eliminated ?Does system have restraints on input or output?How rapidly can change be achieved?How rapidly does change NEED to be achieved?

Proportional GainApplicable to Error between setpoint & outputLarger values faster responseVery large values process instability and oscillation. Results in Steady State Error

Integral GainProportional to integral of Error between setpoint & outputLarger values steady state errors rapidly eliminated. Overshoot may lead to instabilityZero Steady State Error

Derivative GainProportional to derivative of Error between setpoint & outputLarger values decreased overshoot, but slower transient response May lead to instability due to signal noise amplification in the differentiation of the error

Control Gains Summary

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Control Combinations

Proportional Derivative Control

Output YInput U

Kp +Kd s +_

Desired Output D

G (s)

Y = x D


G( Kp +Kd s)

1 +G( Kp +Kd s)

Used; Where a steady state error can be toleratedAvoids; Destabilising nature of Integral action

Reduces overshoot effects However; Susceptible to noise, acts to amplify it

Input U

Kp+_

Desired Output D

G (s)

Kd s

Output Y

_+

Alternative Form;

Proportional Velocity Feedback Control

Time

m3/s

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Proportional Integral Control

Output Y

Kp + KI+_

Desired Output D

G (s)

Y = x D


G( s Kp +KI )

s +G( s Kp +KI)

1

SUsed; Where systems are predominantly 1ST Order

Ensures; Zero Steady State Error (if closed loop transfer .

. function is stable )

Increased setpoint tracking speed

However; Increased complexity introduces a Phase Lag ; reducing

. stability

Input U

Kp

+_

Desired Output D

G (s)KI

Output Y

_+1

S

Alternative Form;

Proportional Integral (Feedback ) Control

Time

m3/s

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Proportional Integral Derivative Control

Output YInput U

Kp + KI + Kd s +_

Desired Output D

G (s)

Y = x D


G(s2 Kd + s Kp + KI)

s +G(s2 Kd+ s Kp + KI)

1

SUsed:

Where system is 2ND or high order

Ensures;

Setpoint tracking with zero steady state error

Allows;

Faster response without oscillatory nature of PI control

Greater influence of error response, degree of overshoot

and oscillation

Time

m3/s

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Control Gain Tuning

Manually :

Set KI & KD to zero, increase KP until response begins to oscillate, then halve it

Increase KI until steady state error is eliminated, (but not too much! )

Increase KD until system responds sufficiently rapidly (but not too much! )

Trial & Error !

Experienced operator required.

Control Type Kp KI Kd

P 0.50Kc - -

PI 0.45Kc 1.2Kp / Pc -

PID 0.60Kc 2Kp / Pc KpPc / 8

Ziegler–Nichols method

Set KI & KD to zero, increase KP until response begins to oscillate;Defined as Critical Gain KC with an Oscillation Period PC

Then set gains as follows:-

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Conclusions

Systems have been controlled mechanistically for millennia!

Development of electronics in early 20th Century paved the way for feedback amplifiers and feedback control

Control response is changeable based on feedback gains used.

Electronic control systems using PID structure used in 90% of industrial controllers

Applicable to LINEAR Systems

Advances in Classical Control:

Linearization by Feedback : Non-linearities are cancelled by feeding back inverse of system dynamics

Model-Based Gain calculation for higher order systems. Requires more control gains Proportional Integral PLUS (PIP)

Gain scheduling for time varying systems: Control gains are updated throughout operation

State Dependant Parameter (SDP) control; Gains are updated based on current state of system

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Classical Control Theory

Proportional - Integral - Derivative

Dr. Matt StablesDr. James Taylor

23/02/2010

Any Questions ?