Topic # 4 Practical systems Reference textbook : Control Systems, - PowerPoint PPT Presentation

ME 779 Control Systems Topic # 4 Practical systems Reference textbook : Control Systems, Dhanesh N. Manik, Cengage Publishing, 2012 1

Control Systems: Practical Systems Learning Objectives • Electric circuits: RC, RL, RLC -Voltage and current sources • Filling systems: incompressible and compressible -Pressure-voltage and pressure-current analogies • Thermal systems - Temperature-voltage and temperature-current analogies • Mechanical systems: spring-mass-damper system -Force-voltage and force-current analogies 2

Control Systems: Practical Systems ELECTRIC CIRCUITS RC circuit (voltage source)  Voltage across e t ( ) i t R ( ) R resistance Voltage across 1   e ( ) t i t dt ( ) capacitance C C   e e e Total voltage drop i R C 3

Control Systems: Practical Systems ELECTRIC CIRCUITS RC circuit (voltage source)   1 Laplace transform     E s I s R ( ) ( ) i   Cs System transfer function between E ( ) s 1 1   C voltage drop across the capacitance    E s ( ) RCs 1 s 1 and input voltage i  Static sensitivity K=1 RC= is the time-constant 4

Control Systems: Practical Systems ELECTRIC CIRCUITS RL circuit (current source)  e t i R ( ) Voltage across the R resistance di Voltage across the  L e t ( ) L dt inductance   i i i Total current a R L 5

Control Systems: Practical Systems ELECTRIC CIRCUITS RL circuit (current source)   1 1 Laplace transform of the current source     I ( ) s E s ( ) a   R Ls E  I ( ) s Laplace transform of the current through the inductance L Ls I ( ) s 1 1 Transfer function between the inductance current   L   to the source current L I ( ) s s 1  s 1 a R  L/R= is the time-constant K=1 6

Control Systems: Practical Systems ELECTRIC CIRCUITS RLC circuit (voltage source)  e t ( ) i t R ( ) Voltage across R resistance di t ( ) Voltage across  e t ( ) L dt inductance L 1  Voltage across  e ( ) t i t dt ( ) C capacitance C 7

Control Systems: Practical Systems ELECTRIC CIRCUITS RLC circuit (voltage source)   Laplace transform of voltage and 1      E s ( ) I s ( ) R Ls current   Cs E ( ) s 1  C Transfer function between   1 E s ( )   capacitance voltage and source   Cs R Ls   Cs voltage  2  n       2 2 s 2 s n n K 1 R       2 n n L LC m 2 C 8

Control Systems: Practical Systems ELECTRIC CIRCUITS RLC circuit (current source) e t ( )  i ( ) t Current through R R resistor   1 Current through i t ( ) e t dt ( ) L inductance L Current through de t ( )  capacitance i ( ) t C C dt    i t ( ) i t ( ) i t ( ) i t ( ) Total current R L C 9

Control Systems: Practical Systems ELECTRIC CIRCUITS RLC circuit (current source) E s ( ) 1 Laplace transform of voltage and current  1 1 I s ( )   Cs R Ls  2 System transfer function I ( ) s 1   L n between inductance       2 2 s 1 I s ( ) ( s 2 )   2 current and source   LC s n n   RC LC current K L 1     2 1 n C n   LC m 2 R 10

Control Systems: Practical Systems Incompressible fluids FILLING SYSTEMS  p t ( ) p t ( )  i q t ( ) R F q(t): flow rate p i (t): inlet pressure p(t): pressure in the tank R F : flow resistance 11

Control Systems: Practical Systems Incompressible fluids FILLING SYSTEMS A dh t A dp dp ( )  equivalent fluid    C F q t ( ) A C   g F capacitance dt g dt dt dp Governing differential equation   p t ( ) R C p t ( ) F F i dt System transfer function 12

Control Systems: Practical Systems Pressure-voltage analogy FILLING SYSTEMS Incompressible fluids Tank-filling Electrical system Pressure, p(t) Voltage, e(t) flow rate, q(t) Current, i(t) Fluid Electrical resistance, R resistance, R F Fluid Electrical capacitance, capacitance, C P s ( ) 1 C F   P s ( ) 1 R C s i F F 13

Control Systems: Practical Systems Pressure-current analogy Incompressible fluids FILLING SYSTEMS Fluid Electrical Pressure, p Current, I flow rate, Q Voltage, E Fluid Electrical compliance, resistance, R F 1/R Fluid Electrical inductance, capacitance, C F L P s ( ) 1   P s ( ) 1 R C s i F F 14

Control Systems: Practical Systems Compressible fluids FILLING SYSTEMS  p t ( ) p t ( )  i m R F m : flow rate p i (t): inlet pressure p(t): pressure in the tank R F : flow resistance C F is the equivalent fluid capacitance 15

Control Systems: Practical Systems Compressible fluids FILLING SYSTEMS V dp dp    Mass flow rate equation from perfect m C F gas equation RT dt dt dp   p t ( ) R C p t ( ) Governing differential equation F F i dt 16

Control Systems: Practical Systems Pressure-voltage analogy Compressible fluids FILLING SYSTEMS , Fluid Electrical Pressure, p Voltage, E Mass flow rate, m  Current, I Fluid resistance, R F Electrical resistance, R Fluid capacitance, C F Electrical capacitance, C 17

Control Systems: Practical Systems Pressure-current analogy FILLING SYSTEMS Compressible fluids Fluid Electrical Pressure, p Current, I  Voltage, E Mass flow rate, m Fluid resistance, R F Electrical compliance, 1/R Fluid capacitance, C F Electrical inductance, L 18

Control Systems: Practical Systems Heat flow due to convective heat transfer THERMAL SYSTEMS  a  Q hA T T ( ) Q=rate of heat flow h=coefficient of convective heat transfer of the body surface A= surface area T a =temperature of the surrounding medium T=temperature of the body 19

Control Systems: Practical Systems THERMAL SYSTEMS R T =1/hA Thermal resistance  Heat flow in terms of thermal resistance T T  a Q R T dT dT   Heat flow in terms Q MC C p T of thermal capacitance dt dt dT t ( )   T t ( ) R C T t ( ) Governing differential equation T T a dt 20

Control Systems: Practical Systems THERMAL SYSTEMS Temperature-voltage analogy Thermal Electrical Temperature, T Voltage, E Heat flow rate, Q Current, I Thermal Electrical resistance, R T resistance, R Thermal Electrical capacitance, C T capacitance, C T s ( ) 1   T s ( ) 1 R C s a T T 21

Control Systems: Practical Systems Temperature-current analogy THERMAL SYSTEMS Thermal Electrical Temperature, T(t) Current, i Heat flow rate, q(t) Voltage, e Thermal resistance, R T Electrical compliance, 1/R Thermal capacitance, C T Electrical inductance, L 22

Control Systems: Practical Systems MECHANICAL SYSTEMS Spring-mass-damper k    f(t) mx cx kx f t ( ) m X s ( ) 1 c    2 F s ( ) ms cs k x(t) 1        2 2 m s 2 s n n 23

Control Systems: Practical Systems MECHANICAL SYSTEMS Force-voltage analogy Spring-mass-damper V s ( ) s 1   Transfer function between   2 k velocity and force F s ( ) ms cs k   ms c s 24

Control Systems: Practical Systems MECHANICAL SYSTEMS Force-voltage analogy Spring-mass-damper Mechanical Electrical Force Voltage Velocity Current Mass Inductance Compliance Capacitance (Reciprocal of stiffness) Damping Resistance 25

Control Systems: Practical Systems MECHANICAL SYSTEMS Force-voltage analogy Spring-mass-damper 26

Control Systems: Practical Systems MECHANICAL SYSTEMS Force-current analogy Spring-mass-damper I ( s ) 1  Transfer function between current and voltage of 1 a LRC circuit E ( s )   Ls R Cs 27

Control Systems: Practical Systems MECHANICAL SYSTEMS Force-current analogy Spring-mass-damper Mechanical Electrical Force Current Velocity Voltage Mass Capacitance Compliance Inductance (Reciprocal of stiffness) Damping Conductance(reciprocal of resistance) 28

Control Systems: Practical Systems MECHANICAL SYSTEMS Force-current analogy Spring-mass-damper 29