1. Control Systems EE 4314 Lecture 7 February 4, 2014
Spring 2014
Woo Ho Lee

2. Announcement Lab#2: Identification of DC motor transfer function
Location: NH250
Feb. 4, Tuesday
101A (3:30-5:20PM)
102A (5:30-7:20PM)
Feb. 5, Wednesday
103A (2:00-3:50PM)
104 (4:00-5:50PM)
Class website:
Homework #1: Due by Feb. 6.
Lab #1 report is due by Feb. 13.
Lab #2 handout is posted.

3. TAs Update TAs: Sajeeb Rayhan: Home work grading and office hours
Office hours: Tue/Thu 10AM-12PM, Mon 4PM-6PM at NH250
Corina Bogdan: Lab preparation & homework and report grading
Office: NH250
Joe Sanford: Lab lecture

4. Labs Schedule Four Sessions (Total: 42 students)
Session 101: Tue: 3:30PM-5:20PM (12 students)
101A (6)
101B (6)
Session 102: Tue: 5:30PM-7:20PM (11 students)
102A (6)
102B (5)
Session 103: Wed: 2:00PM-3:50PM (12 students)
103A (6)
103B (6)
Session 104: Wed: 4:00PM-5:50PM (7 students)

5. Labs #2 Schedule Lab #2: NH250 101A and 102A: Feb. 4 (Tue)
103A and 104: Feb. 5 (Wed)
101B and 102B: Feb. 11 (Tue)
103B: Feb. 12 (Wed)
Tuesday
Wednesday
101 (3:30-5:20)
103 (2-3:50)
102 (5:30-7:20)
104 (4-5:50)

6. Session (12)101A & 101B 101A 101B Saad Akhtar X Sanjeeb Banjara
Asrat Beshah
Blake Farmer
Hawariya Gebremedhien
Nadim Giotis
Daniel Goodman
Leighlan Jensen
Kevin Oseguera
Prabesh Poudel
Eric Reiser
Caroline Storm x

7. Session (11) 102A & 102B 102A 102B Laury Arcos Matthew Barboza X
Monica Beltran
Victoria Brandenburg
Israel Fierro
John Fierro
Haile Fintie
Samuel Luce
Blen Mamo
Nisha Shrestha
Christopher Williams x

8. Session (12) 103A & 103B Joshua Berry Pasquier Biyo Aaron DyresonX
Pasquier Biyo
Aaron Dyreson
Pursottam Giri
Prem Kattel
Gregory Martin x
Bardia Mojra
Vihang Parmar
Abison Ranjit
Thyag Ravi
Sharad Timilsina
Hannah Vuppula

9. Electromechanical Systems
Physics
Law of motors: 𝐹=𝐵𝑙𝑖
Convert electric energy (i) to mechanical work (F)
Law of generator: 𝑒 𝑡 =𝐵𝑙𝑣
Mechanical motion  electric voltage
Where 𝐵: strength of magnetic field
𝑙: length of a coil
𝑣: velocity of the conductor
𝐹: Force acting on the conductor
𝑒(𝑡): voltage across the conductor

10. Magnetic Force on Current Carrying Wire
Force F = 𝑙 × B I
I: current
B: strength of magnetic field
𝑙: length of a wire that carries current I through a magnetic field 𝑙

11. Torque in Magnetic Field
Force F=B𝑙I
Torque T=2F𝑎=2B𝑙aI=KtI
Torque constant
Kt=2B𝑙a
a=radius of wire loop B F I 𝑙

12. Torque in Permanent Magnet DC Motor
Torque T=2nB𝑙aI=KtI
Torque constant
Kt=2nB𝑙a
n = number of loops
n=5

13. DC Motor
Find dynamic equations
Find transfer function
𝑚 𝑣𝑎
= 𝑑𝜃 𝑑𝑡

14. DC Motor

15. DC Motor Block Diagram

16. Loudspeaker Force acting on moving mass 𝐹=𝐵𝑙𝑖 l=2an
n: number of turns
a: radius of core 𝐹

17. Magnetic Levitation Model
Applying KVL
Applying Newton’s law

18. Heat Flow Heat flow 𝑞= 1 𝑅 ( 𝑇 1 − 𝑇 2 )
q: heat energy flow (J/sec)
R: thermal resistance
T: temperature
Relation between temperature of the substance and heat flow
𝑇 = 1 𝐶 𝑞
C: thermal capacity

19. Heat Flow
Find the differential equations that determine the temperature in the room 𝑇 1 (four sides are thermally insulated)
𝑇 1

20. Heat Flow
Find the differential equations that determine the temperature in the room 𝑇 1 (four sides are thermally insulated)
𝑇1 = 1 𝐶1 (−𝑞1−𝑞2)= 1 𝐶1 1 𝑅1 𝑇 0 − 𝑇 𝐶1 1 𝑅2 𝑇 0 − 𝑇 1
= 1 𝐶1 ( 1 𝑅1 + 1 𝑅2 ) 𝑇 0 − 𝑇 1
𝑇 1

21. Water Tank Example Physics governing fluid flow
Continuity equation: 𝑚 = 𝑤 𝑖𝑛 − 𝑤 𝑜𝑢𝑡
where
m: fluid mass within the system (𝑚=𝜌𝑉= 𝜌𝐴ℎ)
win: mass flow rate into the system
wout: mass flow rate out of the system
Differential equation that governs the height of water
ℎ = 1 𝜌𝐴 (𝑤 𝑖𝑛 − 𝑤 𝑜𝑢𝑡 ) (1)
A: area of the tank
𝜌: density of water
h: height of water
𝑚 = 𝜌𝐴 ℎ

22. Water Tank Example Fluid flow through an orifice 𝑤𝑜𝑢𝑡= 1 𝑅 𝑝1−𝑝𝑎 (2)
𝑤𝑜𝑢𝑡= 1 𝑅 𝑝1−𝑝𝑎 (2)
where
𝑝 1 =𝑔ℎ+ 𝑝 𝑎 : hydrostatic pressure
𝑝 𝑎 : ambient pressure
Substituting (2) into (1) gives
ℎ = 1 𝜌𝐴 (𝑤 𝑖𝑛 − 1 𝑅 𝑝1−𝑝𝑎 ) (3)
Linearization involves selecting the operating point
𝑝 1 = 𝑝 𝑜 +𝑝 (4)

23. Water Tank Example Substituting (2) into (1) gives
𝑤𝑜𝑢𝑡= 1 𝑅 𝑝 𝑜 −𝑝𝑎 = 1 𝑅 𝑝 + (𝑝 𝑜 −𝑝𝑎 )
= 𝑝 𝑜 −𝑝𝑎 𝑅 𝑝 + (𝑝 𝑜 −𝑝𝑎 ) 𝑝 𝑜 −𝑝𝑎 = p o −pa R 1+ 𝑝 (𝑝 𝑜 −𝑝𝑎)
= 𝑝 𝑜 −𝑝𝑎 𝑅 [ 𝑝 (𝑝 𝑜 −𝑝𝑎) ] (5)
Substituting (5) into (3) gives
ℎ = 1 𝜌𝐴 (𝑤 𝑖𝑛 − 𝑝 𝑜 −𝑝𝑎 𝑅 [ 𝑝 (𝑝 𝑜 −𝑝𝑎) ]) (6)
Since 𝑝= 𝑔ℎ
ℎ = −𝑔 2𝐴𝑅 𝑝 𝑜 −𝑝𝑎 ℎ+ 𝑤 𝑖𝑛 𝐴 − 𝑝 𝑜 −𝑝𝑎 𝐴𝑅

24. Hydraulic Actuator with Valve
Find nonlinear differential equations relating the movement  of the control surface to the input displacement x of the valve.
Fluid in
Fluid out
Input
Output

25. Hydraulic Actuator Flow goes inside of piston
𝑄 1 = 1 𝜌 𝑅 𝑝 𝑠 − 𝑝 1 𝑥
Flow come out of piston
𝑄 2 = 1 𝜌 𝑅 𝑝 2 − 𝑝 𝑒 𝑥
Continuity relation
𝐴 𝑦 = 𝑄 1 = 𝑄 2
A: piston area
Force equation
𝐴 𝑝 1 − 𝑝 2 −𝐹=𝑚 𝑦
m: mass of piston and attached rod
Moment equation
𝐼 𝜃 =𝐹𝑙 cos 𝜃 − 𝐹 𝑎 𝑑
I: moment of inertia of the control surface and attachment
Kinematic relationship between  and y
𝑦=𝑙 sin 𝜃

26. Key Equations for Dynamic Models
Mechanical system
Newton’s 2nd law (translation): F=ma
Newton’s 2nd law (rotation): M=I
Hook’s law: F=kx
Electrical system
KCL (Kirchhoff’s current law): 𝐼in= 𝐼out
KVL (Kirchhoff’s voltage law ): V closed loop=0
Ohm’s law
Electromechanical system
Law of motors: 𝐹=𝐵𝑙𝑖
Convert electric energy (i) to mechanical work (F)
Law of generator: 𝑒 𝑡 =𝐵𝑙𝑣
Mechanical motion  electric voltage
Torque developed in a rotor: T=𝐾𝑡𝑖
Back emf: 𝑒= 𝐾 𝑒 𝜃 𝑚

27. Chapter 3: Block Diagrams
Block Diagram Model:
Helps understand flow of information (signals) through a complex system
Helps visualize I/O dependencies
Elements of block diagram:
Lines: Signals
Blocks: Systems
Summing junctions
Pick-off points
Transfer Function Summer/Difference Pick-off point + U2 U1
U1+U2 U +

28. Three Examples of Elementary Block Diagrams
(a) Cascaded system G1(s)G2(s)
(b) Parallel system G1(s)+ G2(s)
𝐺1(𝑠) 1+𝐺1 𝑠 𝐺2(𝑠)
(b) Negative feedback system

29. Block Diagram: Simplification Rules= =

30. Block Diagram: Reduction Rules= =

31. Block Diagram Simplification
Example: Simplify the block diagram

32. Example