system dynamics time response 2021. 1. 28. · block diagrams •a subsystem is represented as a...
TRANSCRIPT
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System DynamicsBlock diagram and feedback control
Mohamed Abdou Mahran Kasem, Ph.D.
Aerospace Engineering Department
Cairo University
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Block diagrams – Closed loop systems
• Multiple subsystems are represented in two ways:as block diagrams and as signal-flow graphs.
• We will concentrate on block diagram, sinceblock diagrams are usually used for frequency-domain analysis and design.
• We will develop techniques to reduce eachrepresentation to a single transfer function.
• Block diagram algebra will be used to reduce block diagrams.
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Block diagrams
• A subsystem is represented as a block with aninput, an output, and a transfer function.
• Many systems are composed of multiplesubsystems.
• When multiple subsystems areinterconnected, a few more schematicelements must be added to the block diagram.
• These new elements are summing junctionsand pickoff points.
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Techniques for reducing complicated systems to one Block diagram – Cascade form
For cascaded subsystems, eachsignal is derived from theproduct of the input times thetransfer function.
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Techniques for reducing complicated systems to one Block diagram – Parallel form
Parallel subsystems have acommon input and an outputformed by the algebraic sumof the outputs from all of thesubsystems.
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Techniques for reducing complicated systems to one Block diagram – Feedback form
• The feedback system forms thebasis for our study of controlsystems engineering.
• It represents a closed loopsystem.
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Techniques for reducing complicated systems to one Block diagram – Feedback form
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Techniques for reducing complicated systems to one Block diagram – Moving Blocks
• The familiar forms (cascade, parallel, and feedback) are not alwaysapparent in a block diagram.
• In this section, we discuss basic block moves that can be made inorder to establish familiar forms when they almost exist.
• It will explain how to move blocks left and right past summingjunctions and pickoff points.
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Techniques for reducing complicated systems to one Block diagram – Moving Blocks
Equivalent block diagrams formedwhen transfer functions are movedleft or right past a summingjunction.
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Techniques for reducing complicated systems to one Block diagram – Moving Blocks
Equivalent block diagrams formedwhen transfer functions are movedleft or right past a pickoff point.
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Techniques for reducing complicated systems to one Block diagram – Example
Reduce the block diagram shown in the shown Figure to a single transfer function.
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Techniques for reducing complicated systems to one Block diagram – Example
Step 1:the three summing junctions can be collapsed into a single summing junction.
Solution:
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Techniques for reducing complicated systems to one Block diagram – Example
Step 2: recognize that the three feedback functions, H1(s), H2(s), and H3(s),are connected in parallel.
They are fed from a common signal source, and their outputs are summed.
Solution:
Step 3:Also recognize that G2(s) and G3(s)
are connected in cascade.
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Techniques for reducing complicated systems to one Block diagram – Example
Thus, the equivalent transfer function is the product
Solution:
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Techniques for reducing complicated systems to one Block diagram – Example
Reduce the system shown in Figure to a single transfer function
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Techniques for reducing complicated systems to one Block diagram – Example
Step 1: First, move G2(s) to the left past the pickoff point to create parallel subsystems,
and reduce the feedback system consisting of G3(s) and H3(s).
Solution:
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Techniques for reducing complicated systems to one Block diagram – Example
Step 2: reduce the parallel pairconsisting of 1/G2(s) andunity, and push G1(s) to theright past the summingjunction, creating parallelsubsystems in the feedback.
Solution:
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Techniques for reducing complicated systems to one Block diagram – Example
Step 3: collapse the summing junctions,
add the two feedback elements together, and combine the last two cascaded blocks.
Solution:
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Techniques for reducing complicated systems to one Block diagram – Example
Step 4: use the feedback formula.
Solution:
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Techniques for reducing complicated systems to one Block diagram – Example
Step 5: Finally, multiply the two cascaded blocks and obtain the final result
Solution:
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Feedback Control
• Consider the system shown in Figure, which can model a control system such as the antenna azimuth position control system.
• Where K models the amplifier gain, that is, the ratio of the output voltage to the input voltage.
• As K varies, the poles move through the three ranges of operation of a second-order system: overdamped, critically damped, and underdamped.
The closed loop T.F.
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Feedback Control
For K between 0 and 𝑎2
4, the poles of the system
are real and are located at
The closed loop T.F.
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Feedback Control
• As K increases, the poles move along the real axis, and
the system remains overdamped until K =𝑎2
4.
• At that gain, or amplification, both poles are real and equal, and the system is critically damped.
The closed loop T.F.
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Feedback Control
For gains above𝑎2
4, the system is underdamped, with
complex poles located at
The closed loop T.F.
as K increases, the real part remains constant and the imaginary
part increases. Thus, the peak time decreases and the percent
overshoot increases, while the settling time remains constant.
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Feedback Control – Example
For the system shown in Figure, find the peak time, percent overshoot, and settling time
Solution:
The closed-loop transfer function takes the form,
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Feedback Control – Example
For the system shown in Figure, find the peak time, percent overshoot, and settling time
Solution:
The closed-loop transfer function takes the form,
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Feedback Control – System design Example
Design the value of gain K, for the feedback control system of the Figure, so that the system will respond with a 10% overshoot.
Solution:
The closed-loop transfer function takes the form,
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Feedback Control – System design Example
Design the value of gain K, for the feedback control system of the Figure, so that the system will respond with a 10% overshoot.
Solution:
The closed-loop transfer function takes the form,
⇒