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Feedback ControlSystems Engineering

Lesson 08: Closed-Loop Control SystemPart 2

Joshua NatividadFEEDCON

29 June 2007

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Objectives

After this lesson, the students will

learn how to build closed-loop block systems (done)

be able to simply and reduce closed-loop block

systems using block system manipulationtheorems (done)

analyze systems with multiple inputs (principle ofsuperposition),

analyze real-world closed-loop systems,

familiarize with controllers for closed-loop systems(PI,PID)

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References

R. S. Burns (2001), Advanced ControlEngineering , USA: Butterworth-Heinemann

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Block DiagramManipulation

Review!

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Block DiagramManipulation

Review!

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Block DiagramManipulation

Review!

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Block DiagramManipulation

Review!

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Example

Find the complete output for the systemshown below.

????

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Solution

First, the block diagram is simplifiedFeedback LoopCascade

We need to work on simplified block diagrams!

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Solution

Simplified block

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Solution

ApplyPrinciple of Superposition: putR2(s) = 0 and replace the summing pointby +1 results to

CI(s) is theresponse toR1(s)actingalone!

What isthe

transferfunction?

CI(s)---------------

R1(s)

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Solution

The transfer function is therefore (withR1(s) acting alone)

or

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Solution

Next, putR1(s) = 0 and the summing pointis replaced with -1, then the response CII(s)to inputR2(s) acting alone is done by

Change to:

-1

Set to zero

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Solution

The resulting diagram is shown below

First, we combine the three blocks that form a cascade,next we simplify the positive feedback loop.

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Solution

The resulting transfer function is

or

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Solution

Using the Principle of Superposition, weadd CI(s) and CII(s)

orJust add CI(s) and CII(s)!!!

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Problem Solving Approach

First, simplify the block diagrams usingthe block diagram reduction theorems.

Second, apply principle of superposition if

the system has multiple inputs.

Solve for the transfer functionof the resulting diagram!

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Examples

Transfer functions of system elements:

DC Servo Motors

Linear Hydraulic Actuators

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DC Servo Motor

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DC Servo Motor

The armature-controlled operation of theDC Servo Motor is shown in the blockdiagram

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DC Servo Motor

The behavior of the inputs with respect tothe circuit is given by the equation

and taking the Laplace transform yields

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DC Servo Motor

Steady- State response

or

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DC Servo Motor

The behavior of a field controlled DC servomotor is given by the equation

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DC Servo Motor

From the model

we take the Laplace transform at zeroinitial conditions

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DC Servo Motor

Stead-state Conditions

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Controllers for Closed LoopSystems

The generalized closed-loop system

The control action u(t) will be such that the controlled output c(t) will be equalto the reference input r1(t) for all values of time, irrespective of the value of thedisturbance input r2(t).

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Proportional Control

The control action, or signal, isproportional to the error e(t)

where K1 is the proportional gain constant

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Proportional Controller

For first order plants,

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Proportional Control

The transfer function of the plant is givenby

which can be combined with theproportional control law

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Proportional Control

Re-arranging the equation yields

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Proportional Control

When r1(t) is a unit step, and r2(t) is set tozero, the final value theorem yields

When r2(t) is a unit step, and r1(t) is set tozero, the final value theorem yields

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Proportional Control PlusIntegral (PI)

Consider the control law

Taking its Laplace

transform yields

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Ti

The time interval in which thepart of the control signal due to

integral action increases by anamount equal to the part of thecontrol signal due to proportional

action when the error isunchanging.

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PI Control

The output of the system is

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