Design For Accessibility: Getting it right from the start
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04 Transfer Function.pptx
1. AE 313
AE Systems & Control
04 SYSTEM MODELLING
Dr. Syed Saad Azhar Ali
75-110
syed.ali@kfupm.edu.sa
2. Catching upโฆ
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โข Trying to learn about control systems
โ An interconnection of components in a configuration that helps us achieve our desired output
โ The interconnecting configuration can be open loop or closed loop
โข In this interconnection, there is
โ The process/system/plant to be controlled
โ The actuator
โ The sensor
โ Feedback
โ Comparator
โ Controller
โข Once the overall scheme (control and controlled variables, reference value, specifications,
objectives) is finalized, the most important step is to know the system
โข Need physical laws and respective mathematical representation - > differential equations
โข solving the differential equation will take time and effort
โ Time domain is easier to comprehend for us but not easy to solve
โ Convert to Frequency Domain -> Laplace Transform
5. Mechanical System
Spring-Mass Damper
โข Displacement of the mass M (y)
โข The mass is suspended with
spring (spring constant k)
โข moves inside the walls with
friction (coefficient b)
โข r(t) is the applied force that
โ Moves -> Ma
โ Takes care of spring -> ky
โ Overcomes the friction -> bv
Ma bv ky
๐ ๐ = ๐ด๐ + ๐๐ + ๐๐
๐ ๐ = ๐ด
๐ ๐๐
๐ ๐๐
+ ๐
๐ ๐
๐ ๐
+ ๐๐ This is the differential equation
for the spring-mass damper
๐ ๐ = ๐ด๐ + ๐๐ + ๐๐
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7. Mechanical System
Spring-Mass Damper
Ma bv ky
Using the Laplace of derivative of a function
๐ ๐ ๐ = ๐ด๐
๐ ๐
๐
๐ ๐๐
+ ๐๐
๐ ๐
๐ ๐
+ ๐๐ ๐
๐
๐โ๐
โ
๐น ๐ = ๐ด๐๐
๐ ๐ โ ๐๐
๐ ๐ โ ๐โฒ ๐ +๐๐๐ ๐ โ ๐ ๐ +๐๐ ๐
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8. Mechanical System
Spring-Mass Damper
Ma bv ky
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๐น ๐ = ๐ด๐๐๐ ๐ โ ๐๐ ๐ โ ๐โฒ ๐ + ๐๐๐ ๐ โ ๐ ๐ + ๐๐ ๐
โข Now assuming the system is initially at
rest or relaxed
โข i.e. no displacement, velocity or
acceleration
โข This means ZERO INITIAL CONDITIONS
โข Zero displacement ๐ ๐ = ๐
โข Zero velocity ๐โฒ ๐ = ๐
= ๐
= ๐
= ๐
๐น ๐ = ๐ด๐๐
๐ ๐ + ๐๐๐ ๐ + ๐๐ ๐
9. Mechanical System
Spring-Mass Damper
Ma bv ky
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๐น ๐ = ๐ด๐๐๐ ๐ + ๐๐๐ ๐ + ๐๐ ๐
๐น ๐ = (๐ด๐๐
+ ๐๐ + ๐)๐ ๐
๐(๐)
๐น(๐)
=
๐
๐ด๐๐ + ๐๐ + ๐
This expression is called the
Transfer Function
10. TRANSFER FUNCTION
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โข
๐(๐)
๐น(๐)
=
๐
๐ด๐๐+๐๐+๐
โข So what is a transfer function
โข Output over input (ratio of output and input)
โข Is this a transfer function - >
๐(๐)
๐(๐)
โข Output over input in frequency domain
๐(๐)
๐น(๐)
,
๐(๐)
๐น(๐)
,
๐(๐)
๐น(๐)
โข Anything missingโฆ
ยป Lets review โฆ
NOโฆ!!
11. TRANSFER FUNCTION
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โข
๐(๐)
๐น(๐)
=
๐
๐ด๐๐+๐๐+๐
โข So what is a transfer function
โข Output over input in frequency domain
๐(๐)
๐น(๐)
,
๐(๐)
๐น(๐)
,
๐(๐)
๐น(๐)
โข With zero initial conditions
12. Electrical System
Parallel RLC Circuit
โข The total current is divided in all
the parallel branches
โข r(t) is divided in
โ resistance -> iR(t) = v(t)/R
โ inductance -> iL(t) =
1
๐ฟ
v(t)๐๐ก
โ Capacitance-> iC(t) = C
๐v(t)
๐๐ก
This is the called the integro-
differential equation
๐ ๐ = ๐๐น + ๐๐ณ + ๐๐ช
๐ ๐(๐)
๐ ๐
= C
๐ ๐
v(t)
๐๐ก2 +
1
๐
๐ v(t)
๐ ๐
+
1
๐ฟ
v(t)
๐ ๐ =
v(t)
๐น
+
1
๐ฟ
v(t)๐๐ก + C
๐v(t)
๐๐ก
Differentiating the equation
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13. Electrical System
Parallel RLC Circuit ๐ ๐(๐)
๐ ๐
= C
๐ ๐v(t)
๐๐ก2 +
1
๐
๐ v(t)
๐ ๐
+
1
๐ฟ
v(t)
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determine the transfer function
Donโt forget to set the initial
conditions ZERO
๐ฝ(๐)
๐น(๐)
=
๐
๐ช๐๐ +
1
๐
๐ +
1
๐ฟ
14. 15
Project 1 (Competition)
โข Group Project
โข Competition (I will try to get some prize as well)
โข Design the glider
โข Get theoretical results
โข Competition day - we will get real results
โข 11 March 2023 โ KFUPM Beach
โข 5%
8/23/2023 Dr Syed Saad Azhar Ali