Control Systems EE 4314 Lecture 7 February 4, 2014 Spring 2014 Woo Ho Lee whlee@uta.edu

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: www.uta.edu/ee/ngs/ee4314_control Homework #1: Due by Feb. 6. Lab #1 report is due by Feb. 13. Lab #2 handout is posted.

TAs Update TAs: Sajeeb Rayhan: Home work grading and office hours mdsajeeb.rayhan@mavs.uta.edu Office hours: Tue/Thu 10AM-12PM, Mon 4PM-6PM at NH250 Corina Bogdan: Lab preparation & homework and report grading Email: ioanacorina.bogdan@mavs.uta.edu Office: NH250 Joe Sanford: Lab lecture Email: joe.sanford@MAVS.UTA.EDU

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)

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)

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

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

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

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

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

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

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

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

DC Motor

DC Motor Block Diagram

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

Magnetic Levitation Model Applying KVL Applying Newton’s law

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

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

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 1 + 1 𝐶1 1 𝑅2 𝑇 0 − 𝑇 1 = 1 𝐶1 ( 1 𝑅1 + 1 𝑅2 ) 𝑇 0 − 𝑇 1 𝑇 1

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

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)

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

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

Hydraulic Actuator Flow goes inside of piston 𝑄 1 = 1 𝜌 𝑅 1 𝑝 𝑠 − 𝑝 1 𝑥 Flow come out of piston 𝑄 2 = 1 𝜌 𝑅 2 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

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

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 +

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

Block Diagram: Simplification Rules = =

Block Diagram: Reduction Rules = =

Block Diagram Simplification Example: Simplify the block diagram

Example