control system

1. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Control Systems
Lecture 4

2. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Modeling different systems: Mechanical
Translational system
• Mechanical Translational system
(mass-spring-dashpot)
• Mass: The Mass is an inertial element
– Force (F) Acceleration Reaction force

3. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
• Dashpot (damper)
– The reaction damping force F is approximated by
the product of damping f and relative velocity if
any.
– F(t) = f (v1-v2) = fv
Modeling different systems: Mechanical
Translational system

4. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
• Spring
– Restoring force Reaction Force on each
end is same
– The spring element force equation, in accordance
with Hooke’s Law is given by:
– Fk = K (xc-xd)
Modeling different systems: Mechanical
Translational system

5. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Modeling different systems: Mechanical
Rotational system
Rotational Systems are similar to Translational Systems
except for the difference that the torque equations are written
Extending the Newton’s Law of motion
𝑇𝑜𝑟𝑞𝑢𝑒𝑠 = 𝐽𝛼
Elements of the Rotational System are;
1.Inertia element
Tj = Jdω/dt =J.d2ϴ/dt2
2.Torsional spring element
Tk = K (ϴ1 - ϴ2) = Kϴ
3.Damper element
TB = B (ω1 – ω2) = Bω

6. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Example 1

7. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
1. Decide input and the output
2. Draw the free body diagram of the masses (optional)
3. Convert to the frequency-domain representations
4. Create transfer function
Modeling Steps:
Identify Modeling steps

8. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Step 1: Decide Input and Output
)(tT
Any angular position, velocity
or acceleration of the
rotational masses
Output variable:Input variable:

9. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Step 2: The free body diagram J1 (optional)
Step 3: The frequency response representation
1
11J
1J
11D
1K
2k
TKKDJ  211111  T
Terms related to θ1
Step 2: Free Body Diagram

10. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
2
22J
2J
22D
2K
32D
1K
013222222   KDKDJ 
Terms related to θ2
Step 2: The free body diagram J2 (Optional)
Step 3: The frequency response representation
Step 2: Free Body Diagram

11. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
3
33J
3J
32D
33D
22D
022333233    DDDJ
Terms related to θ3
Step 2: The free body diagram J3 (Optional)
Step 3: The frequency response representation
Step 2: Free Body Diagram

12. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Step 4: Transfer Function


































0
0
)(0
0
3
2
1
323
2
2
222
2
11
2
T
DDsJssD
sDKsDJsK
KKsDJs



Inertial 1:
Inertial 2:
Inertial 3:
TsKsKssDsJs  )()()()( 211111
2

0)()()()()( 13222222
2
 sKssDsKssDsJs 
0)()()()( 22333233
2
 ssDssDssDsJs 
Put the equations in matrix form

13. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Step 4: Transfer Function
If we are interested to control the position of the mass J3, then solve for θ3.
Cramer’s rule:
)(0
0
00
0
323
2
2
222
2
11
2
2
22
2
11
2
3
DDsJssD
sDKsDJsK
KKsDJs
sD
KsDJsK
TKKsDJs









































0
0
)(0
0
3
2
1
323
2
2
222
2
11
2
T
DDsJssD
sDKsDJsK
KKsDJs




14. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Example 2
Find the transfer function between )(and)( 2 ttT 

15. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Example 2