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8/13/2019 FEEDCON - Lesson 08 - Closed-Loop Control Systems Part 2
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Page 1
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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BACKUP