feedback control of linear siso systems - … · 3 closed-loop system in study and design of...

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Feedback Control ofLinear SISO systems

Process Dynamics and ControlProcess Dynamics and Control

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Open-Loop ProcessOpen-Loop Process

The study of dynamics was limited to The study of dynamics was limited to open-loop systemsopen-loop systemsObserve process behavior as a result of specific input signals

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Closed-Loop SystemClosed-Loop System

In study and design of control systems, we are concernedwith the dynamic behavior of a controlled or Closed-loopSystems

Feedback Control System

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Feedback ControlFeedback Control

Control is meant to provide regulation of process outputs about areference, , despite inherent disturbances

The deviation of the plant output, ,from its intendedreference is used to make appropriate adjustments in the plant input,


SystemController

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Feedback ControlFeedback Control

Process is a combination of sensors and actuators

Controller is a computer (or operator) that performs the requiredmanipulations

e.g. Classical one degree-of-freedom feedback control loop

Computer Actuator

Process

Sensor

-+++

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Closed-Loop Transfer FunctionClosed-Loop Transfer Function

Block Diagram of Closed-Loop Process

- Open-Loop Process Transfer Function - Controller Transfer Function - Sensor Transfer Function - Actuator Transfer Function

Computer Actuator

Process

Sensor

-+++

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Closed-Loop Transfer FunctionClosed-Loop Transfer Function

For analysis, we assume that the impact of actuator andFor analysis, we assume that the impact of actuator andsensor dynamics are negligiblesensor dynamics are negligible

Closed-loop process reduces to the block diagram:Closed-loop process reduces to the block diagram:


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Closed-loop Transfer FunctionsClosed-loop Transfer Functions

The closed-loop process hasThe closed-loop process has Two inputsTwo inputs

The reference signalThe reference signal The disturbance signal The disturbance signal

Two outputsTwo outputs The manipulatedThe manipulated (control) variable signal(control) variable signal The The output (controlled) variable signaloutput (controlled) variable signal

We want to see how the inputs affect the outputsWe want to see how the inputs affect the outputs Transfer functions relatingTransfer functions relating ,, and , and ,

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Closed-loop Transfer functionClosed-loop Transfer function

There are four basic transfer functionsThere are four basic transfer functions

They arise from three so-called sensitivity functionsThey arise from three so-called sensitivity functions

Highlights the dilemma of control system designHighlights the dilemma of control system design Only one degree of freedom to shapeOnly one degree of freedom to shape the three sensitivitythe three sensitivity

functionsfunctions

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Sensitivity functions:Sensitivity functions: The sensitivity function:The sensitivity function:

The complementary sensitivity function: The complementary sensitivity function:

The control sensitivity function:The control sensitivity function:

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Overall transfer function for the output:Overall transfer function for the output:

Servo response is the response of the output to Servo response is the response of the output to setpoint setpoint changechange Regulatory response is the response of theRegulatory response is the response of the output to disturbanceoutput to disturbance

changeschanges

REGULATORYRESPONSE

SERVORESPONSE

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Servo mechanism requires that:Servo mechanism requires that:

Regulatory response requires that:Regulatory response requires that:

SinceSince

The two objectives are complementaryThe two objectives are complementary

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Note thatNote that

oror

requires that the controllerrequires that the controller is largeis large

This leads to large control sensitivityThis leads to large control sensitivity

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PID ControllerPID Controller

Most widespread choice for the controller is the PID controller

The acronym PID stands for:P - ProportionalI - IntegralD - Derivative

PID Controllers:greater than 90% of all control implementationsdates back to the 1930svery well studied and understoodoptimal structure for first and second order processes (given

some assumptions)always first choice when designing a control system

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PID ControlPID Control

PID Control Equation

PID Controller ParametersKc Proportional gain

Integral Time ConstantDerivative Time ConstantController Bias

Proportional Action

IntegralAction

DerivativeAction

ControllerBias

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PID Controller Transfer Function

or:

Note:

numerator of PID transfer function cancels second order dynamics denominator provides integration to remove possibility of steady-state

errors

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Controller Transfer Function:

or,

Note:

Many variations of this controller existEasily implemented in MATLAB/SIMULINKeach mode (or action) of controller is better studied individually

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Proportional FeedbackProportional Feedback

Form:

Transfer function:

or,

Closed-loop form:

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Example:Given first order process:

for P-only feedback closed-loop dynamics:

Closed-LoopTime Constant

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Final response:

Note: for “zero offset response” we require

Possible to eliminate offset with P-only feedback (requires infinitecontroller gain)

Need different control action to eliminate offset (integral)

Tracking Error Disturbance rejection

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Proportional Feedback

Servo dynamics of a first order process under proportional feedback

increasing controller gain eliminates off-set

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Proportional Feedback

High-order processe.g. second order underdamped process

increasing controller gain reduces offset, speeds response and increasesoscillation

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Important points: proportional feedback does not change the order of the system

started with a first order processclosed-loop process also first orderorder of characteristic polynomial is invariant under proportional

feedback

speed of response of closed-loop process is directly affected by controllergain increasing controller gain reduces the closed-loop time constant

in general, proportional feedback reduces (does not eliminate) offset speeds up response for oscillatory processes, makes closed-loop process more oscillatory

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Integral ControlIntegral Control

Integrator is included to eliminate offset

provides reset action usually added to a proportional controller to produce a PI

controllerPID controller with derivative action turned offPI is the most widely used controller in industryoptimal structure for first order processes

PI controller form

Transfer function model

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PI FeedbackPI Feedback

Closed-loop response

more complex expression degree of denominator is increased by one

Assuming the closed-loop system is stable, we get

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ExamplePI control of a first order process

Closed-loop transfer function

Note:offset is removedclosed-loop is second order

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Example (contd)effect of integral time constant and controller gain on closed-loop

dynamics

(time constant) natural period of oscillation

damping coefficient

integral time constant and controller gain can induce oscillationand change the period of oscillation

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PI Feedback

Effect of integral time constant on servo dynamics

Small integral time constant induces oscillatory (underdamped)closed-loop response

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PI Feedback

Effect of controller gain on servo dynamics

affects speed of response increasing gain eliminates offset quicker

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PI Feedback

Effect of integral action of regulatory response

reducing integral time constant removes effect of disturbancesmakes behavior more oscillatory

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Important points:

integral action increases order of the system in closed-loop

PI controller has two tuning parameters that can independently affect speed of response final response (offset)

integral action eliminates offset

integral action should be small compared to proportional action tuned to slowly eliminate offsetcan increase or cause oscillationcan be de-stabilizing

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Derivative ActionDerivative Action

Derivative of error signalUsed to compensate for trends in output

measure of speed of error signal changeprovides predictive or anticipatory action

P and I modes only response to past and current errorsDerivative mode has the form

if error is increasing, decrease control actionif error is decreasing, decrease control action

Usually implemented in PID form

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PID FeedbackPID Feedback

Transfer Function

Closed-loop Transfer Function

Slightly more complicated than PI form

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PID FeedbackPID Feedback

Example:PID Control of a first order process


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PID Feedback

Effect of derivative action on servo dynamics

Increasing derivative action leads to a more sluggish servo responseIncreasing derivative action leads to a more sluggish servo response

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PID Feedback

Effect of derivative action on regulatory response

increasing derivative action reduces impact of disturbances on controlledvariable

slows down servo response and affects oscillation of process

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PD FeedbackPD Feedback

PD ControllerPD Controller

ProportionalProportional Derivative Control is common in mechanicalDerivative Control is common in mechanicalsystemssystems Arise in application for systems with an integratingArise in application for systems with an integrating behaviourbehaviour

Example : System in series with an integratorExample : System in series with an integrator

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Transfer Function

Closed-loop Transfer Function

Slightly more complicated than PI form

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DCDC Motor example:Motor example: In terms of angular velocityIn terms of angular velocity (velocity control)(velocity control)

In terms of the angle (position control)In terms of the angle (position control)

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Closed-loop transfer functionClosed-loop transfer function

SimplifyingSimplifying

Notice thatNotice that

Same effect as a PID controller.Same effect as a PID controller.

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Derivative ActionDerivative Action

Important Points:

Characteristic polynomial is similar to PI derivative action does not increase the order of the system adding derivative action affects the period of oscillation of the process

good for disturbance rejectionpoor for tracking

the PID controller has three tuning parameters and can independentlyaffect, speed of response final response (offset) servo and regulatory response

derivative action should be small compared to integral actionhas a stabilizing influencedifficult to use for noisy signalsusually modified in practical implementation

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Closed-loop Stability

Every control problem involves a consideration of closed-loop stability

General concepts:

Bounded Input Bounded Output (BIBO) Stability:

An (unconstrained) linear system is said to be stable if the outputresponse is bounded for all bounded inputs. Otherwise it is unstable.

Comments:Stability is much easier to prove than instabilityThis is just one type of stability

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Closed-loop dynamics

Let

then,

The closed-loop transfer functions have a commondenominator

called the characteristic polynomial

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Closed-loop stability

General Stability criterion:

“ A closed-loop feedback control system is stable if and only if allroots of the characteristic polynomial are negative or have negativereal parts. Otherwise, the system is unstable.”

Unstable region is the right half plane of the complex plane.

Valid for any linear systems.

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Problem reduces to finding roots of a polynomial (forpolynomial systems, without delay)

Easy (1990s) way : MATLAB function ROOTS (or POLE)

Traditional:1. Routh array:

Test for positivity of roots of a polynomial2. Direct substitution

Complex axis separates stable and unstable regions Find controller gain that yields purely complex roots

3. Root locus diagram Vary location of poles as controller gain is varied Of limited use

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Closed-loop stability

Routh array for a polynomial equation

is

where

Elements of left column must be positive to have roots with negative realparts

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Example: Routh Array

Characteristic polynomial

Polynomial Coefficients

Routh Array

Closed-loop system is unstable

2 36 149 058 121 0 42 0 78 05 4 3 2. . . . . .s s s s s+ ! + + + =

a a aa a ab b bc cd de

5 3 1

4 2 0

1 2 3

1 2

1 2

1

2 36 058 0 42149 121 0 782 50 082 00 72 0 78189 00 78

( . ) ( . ) ( . )( . ) ( . ) ( . )( . ) ( . ) ( )( . ) ( . )( . ) ( )( . )

!

! !

a a a a a a5 4 3 2 1 02 36 149 058 121 0 42 0 78= = = ! = = =. , . , . , . , . , .

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Direct Substitution

Technique to find gain value that de-stabilizes the system.

Observation: Process becomes unstable when poles appear on right half plane

Find value of that yields purely complex poles

Strategy: Start with characteristic polynomial

Write characteristic equation:

Substitute for complex pole

Solve for and

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Example: Direct Substitution

Characteristic equation

Substitution for

Real Part Complex Part

System is unstable if

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Root Locus Diagram

Old method that consists in plotting poles of characteristic polynomial ascontroller gain is changed

e.g.

Characteristic polynomial

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Stability and Performance

Given plant model, we assume a stable closed-loop system can bedesigned

Once stability is achieved - need to consider performance of closed-loop process - stability is not enough

All poles of closed-loop transfer function have negative real parts - canwe place these poles to get a “good” performance

S: Stabilizing Controllers for a given plantP: Controllers that meet performance

S

PC

Space of all Controllers

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Controller Tuning

Can be achieved by Direct synthesis : Specify servo transfer function required and calculate

required controller - assume plant = model

Internal Model Control: Morari et al. (86) Similar to direct synthesisexcept that plant and plant model are concerned

Pole placement

Tuning relations:Cohen-Coon - 1/4 decay ratiodesigns based on ISE, IAE and ITAE

Frequency response techniquesBode criterionNyquist criterion

Field tuning and re-tuning

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Direct Synthesis

From closed-loop transfer function

Isolate

For a desired trajectory and plant model , controller isgiven by

not necessarily PID form inverse of process model to yield pole-zero cancellation (often inexact

because of process approximation) used with care with unstable process or processes with RHP zeroes

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Direct Synthesis

1. Perfect Control

cannot be achieved, requires infinite gain

2. Closed-loop process with finite settling time

For 1st order open-loop process, , it leads to PI control For 2nd order open-loop process, , get PID control

3. Processes with delay

requires again, 1st order leads to PI control 2nd order leads to PID control

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IMC Controller Tuning


In terms of implemented controller, Gc

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IMC Controller Tuning

1. Process model factored into two parts

where contains dead-time and RHP zeros, steady-state gain scaledto 1.

2. Controller

where is the IMC filter

The constant is chosen such the IMC controller is proper based on pole-zero cancellation

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Example

PID Design using IMC and Direct synthesis for the process

Process parameters:

1. Direct Synthesis: (Taylor Series) (Padé)

Servo Transfer function

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ExampleExample

1.1. IMC Tuning:IMC Tuning:

a)a) Taylor Series:Taylor Series:•• FilterFilter

•• Controller (PI)Controller (PI)

b)b) PadPadé é approximation:approximation:•• FilterFilter

•• Controller (Commercial PID)Controller (Commercial PID)

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ExampleExample

Servo ResponseServo Response

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ExampleExample

Regulatory responseRegulatory response

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IMC TuningIMC Tuning

For unstable processes,For unstable processes,

Must modify IMC filter such that the value of atMust modify IMC filter such that the value of at is 1is 1

Usual modificationUsual modification

Strategy is to specifyStrategy is to specify and solve forand solve for such thatsuch that

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ExampleExample

Consider the processConsider the process

Consider the filterConsider the filter

Let then solve forLet then solve for

Yields aYields a PI controllerPI controller

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ExampleExample

Servo responseServo response

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Pole placementPole placement

Given a process modelGiven a process model

a controller of the form,a controller of the form,

and an arbitrary polynomialand an arbitrary polynomial

Under what condition does there exist a unique controllerUnder what condition does there exist a unique controllerpair and such thatpair and such that

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Pole placementPole placement

We say thatWe say that andand are are primeprime if they do not have if they do not haveany common factorsany common factors

Result:Result:

Assume that Assume that and and are (co) prime. Let are (co) prime. Let be anbe anarbitraty arbitraty polynomial of degreepolynomial of degree . Then there exist. Then there existpolynomialspolynomials and of degreeand of degree suchsuchthatthat

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Pole PlacementPole Placement

ExampleExample

This is a second order systemThis is a second order system The polynomialsThe polynomials andand are primeare prime The requiredThe required degree of the characteristic polynomial isdegree of the characteristic polynomial is

The The degree of the controller polynomialdegree of the controller polynomial andand areare

Controller is given byController is given by

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Performance objective:Performance objective: 3rd order polynomial3rd order polynomial

Characteristic polynomialCharacteristic polynomial is given byis given by

Solving forSolving for and and by equating polynomialby equating polynomialcoefficients on both sidescoefficients on both sides

Obtain a system of 4 equations in 4 unknownsObtain a system of 4 equations in 4 unknowns

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System of equationsSystem of equations

Solution isSolution is

CorrespondingCorresponding controller is a PI controllercontroller is a PI controller

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Tuning Relations

Process reaction curve method: based on approximation of process using first order plus delay model

1. Step in U is introduced2. Observe behavior3. Fit a first order plus dead time model

Manual Control

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Tuning Relations

Process response

4. Obtain tuning from tuning correlationsZiegler-NicholsCohen-Coon ISE, IAE or ITAE optimal tuning relations

0 1 2 3 4 5 6 7 8-0.2

0

0.2

0.4

0.6

0.8

1

1.2

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Ziegler-Nichols Tunings

Controller

PI-only

- Note presence of inverse of process gain in controllergain- Introduction of integral action requires reduction incontroller gain- Increase gain when derivation action is introduced

Example:PI:PID:

PID

P

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Example

Ziegler-Nichols Tunings: Servo response

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Example

Regulatory Response

Z-N tuning Oscillatory with considerable overshoot Tends to be conservative

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Cohen-Coon Tuning Relations

Designed to achieve 1/4 decay ratio fast decrease in amplitude of oscillation

Example:

PI: Kc=10.27 τI=18.54PID: Kc=15.64 τI=19.75 τd=3.10

Controller Kc Ti TdP-only ]3/1)[/)(/1( !""! +pKPI

]12/9.0)[/)(/1( !""! +pK)/(209)]/(330[

!"!""

++

PID

]12163

)[/)(/1(!

!""! +pK )/(813

)]/(632[!"!""

++

)/(2114

!""

+

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Tuning relations

Cohen-Coon: Servo

More aggressive/ Higher controller gains Undesirable response for most cases

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Tuning Relations

Cohen-Coon: Regulatory

Highly oscillatory Very aggressive

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Integral Error Relations

1. Integral of absolute error (IAE)

2. Integral of squared error (ISE)

penalizes large errors3. Integral of time-weighted absolute error (ITAE)

penalizes errors that persist

ITAE is most conservative ITAE is preferred

ISE e t dt= !"( )2

0

ITAE t e t dt= !"( )

0

IAE e t dt= !"( )

0

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ITAE Relations

Choose Kc, τI and τd that minimize the ITAE:

For a first order plus dead time model, solve for:

Design for Load and Setpoint changes yield different ITAE optimum

!!

!!"

!!"

ITAEK

ITAE ITAEc I d

= = =0 0 0, ,

Type ofInput

Type ofController

Mode A B

Load PI P 0.859 -0.977I 0.674 -0.680

Load PID P 1.357 -0.947I 0.842 -0.738D 0.381 0.995

Set point PI P 0.586 -0.916I 1.03 -0.165

Set point PID P 0.965 -0.85I 0.796 -0.1465D 0.308 0.929

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ITAE Relations

From table, we getLoad Settings:

Setpoint Settings:

Example

( ) ( )Y A KK A BB

c Id= = = = +!

"""

!"

"" ,

( )Y A KKB

c Id= = = =!

"""

""

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ITAE Relations

Example (contd)Setpoint Settings

Load Settings:

( )KK

K K

c

c

= =

= = =

!1357 4 2437

4 2437 4 24370 3 1415

930

0 947. .

. .. .

.( )

( )

!!

! !

!!

! !

I

I

d

d

= =

= = =

= =

= =

"0842 2 0474

2 0474302 0474 14 65

0 381 930 01150

01150 34497

930

0 738

0 995

. .

. . .

. .

. .

.

.

( )KK

K K

c

c

= =

= = =

!0 965 2 6852

2 6852 2 68520 3 8 95

930

085. .

. .. .

.( )

( )

!!

! !

!!

! !

I

I

d

d

= " =

= = =

= =

= =

0 796 01465 0 7520

0 7520300 7520 39 89

0 308 930 01006

01006 30194

930

0 929

. . .

. . .

. .

. .

.

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ITAE Relations

Servo Response

design for load changes yields large overshoots for set-point changes

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ITAE Relations

Regulatory response

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Tuning Relations

In all correlations, controller gain should be inversely proportional toprocess gain

Controller gain is reduced when derivative action is introduced

Controller gain is reduced as increases

Integral time constant and derivative constant should increase asincreases

In general,

Ziegler-Nichols and Cohen-Coon tuning relations yield aggressivecontrol with oscillatory response (requires detuning)

ITAE provides conservative performance (not aggressive)

!"

!"

!!

dI= 0 25.

feedback control of linear siso systems - … · 3 closed-loop system in study and design of...

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