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

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ME 779 Control Systems

Practical systems

Topic # 4

Reference textbook:

Control Systems, Dhanesh N. Manik, Cengage Publishing, 2012

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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

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ELECTRIC CIRCUITS

RC circuit (voltage source)

( ) ( )

R

e t i t R  1 ( ) ( )

C

e t i t dt C 



i R C

e e e  

Voltage across resistance Voltage across capacitance Total voltage drop

Control Systems: Practical Systems

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ELECTRIC CIRCUITS

1 ( ) ( )

i

E s I s R Cs        

RC circuit (voltage source) Laplace transform

( ) 1 1 ( ) 1 1

C i

E s E s RCs s     

System transfer function between voltage drop across the capacitance and input voltage RC= is the time-constant



Static sensitivity K=1

Control Systems: Practical Systems

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ELECTRIC CIRCUITS

RL circuit (current source)

( )

R

e t i R 

( )

L

di e t L dt 

a R L

i i i  

Voltage across the resistance Voltage across the inductance Total current

Control Systems: Practical Systems

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ELECTRIC CIRCUITS

RL circuit (current source)

1 1 ( ) ( )

a

I s E s R Ls        

Laplace transform of the current source

( )

L

E I s Ls 

Laplace transform of the current through the inductance

( ) 1 1 ( ) 1 1

L a

I s L I s s s R     

Transfer function between the inductance current to the source current

L/R= is the time-constant

K=1

Control Systems: Practical Systems



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ELECTRIC CIRCUITS

RLC circuit (voltage source)

( ) ( )

R

e t i t R 

( ) ( )

L

di t e t L dt 

1 ( ) ( )

C

e t i t dt C 



Voltage across inductance Voltage across capacitance

Control Systems: Practical Systems

Voltage across resistance

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ELECTRIC CIRCUITS

RLC circuit (voltage source)

1 ( ) ( ) E s I s R Ls Cs         

Laplace transform of voltage and current

 

2 2 2

( ) 1 1 ( ) 2

C n n n

E s E s Cs R Ls Cs s s               

Transfer function between capacitance voltage and source voltage

2 n

K m  

1

n

LC  

2 R L C  

Control Systems: Practical Systems

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ELECTRIC CIRCUITS

RLC circuit (current source)

( ) ( )

R

e t i t R 

1 ( ) ( )

L

i t e t dt L  

( ) ( )

C

de t i t C dt 

Current through resistor Current through inductance Current through capacitance

( ) ( ) ( ) ( )

R L C

i t i t i t i t   

Total current

Control Systems: Practical Systems

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ELECTRIC CIRCUITS

( ) 1 1 1 ( ) E s I s Cs R Ls   

RLC circuit (current source) Laplace transform of voltage and current

2 2 2 2

( ) 1 1 ( ) ( 2 )

n L n n

I s s I s s LC s RC LC               

System transfer function between inductance current and source current

1

n

LC  

1 2 L C R  

2 n

K m  

Control Systems: Practical Systems

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FILLING SYSTEMS

Incompressible fluids

( ) ( ) ( )

i F

p t p t q t R  

q(t): flow rate pi(t): inlet pressure p(t): pressure in the tank RF: flow resistance

Control Systems: Practical Systems

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FILLING SYSTEMS

Incompressible fluids

( ) ( )

F

dh t A dp dp q t A C dt g dt dt     g A CF  

equivalent fluid capacitance

( ) ( )

F F i

dp p t R C p t dt  

Governing differential equation System transfer function

Control Systems: Practical Systems

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FILLING SYSTEMS

Incompressible fluids Pressure-voltage analogy Tank-filling system Electrical Pressure, p(t) Voltage, e(t) flow rate, q(t) Current, i(t) Fluid resistance, RF Electrical resistance, R Fluid capacitance, CF Electrical capacitance, C

( ) 1 ( ) 1

i F F

P s P s R C s  

Control Systems: Practical Systems

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FILLING SYSTEMS

Incompressible fluids Pressure-current analogy Fluid Electrical Pressure, p Current, I flow rate, Q Voltage, E Fluid resistance, RF Electrical compliance, 1/R Fluid capacitance, CF Electrical inductance, L

( ) 1 ( ) 1

i F F

P s P s R C s  

Control Systems: Practical Systems

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FILLING SYSTEMS

Compressible fluids

( ) ( )

i F

p t p t m R  

: flow rate pi(t): inlet pressure p(t): pressure in the tank RF: flow resistance

m

CF is the equivalent fluid capacitance

Control Systems: Practical Systems

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FILLING SYSTEMS

Compressible fluids

dt dp C dt dp RT V m

F

  

Mass flow rate equation from perfect gas equation

( ) ( )

F F i

dp p t R C p t dt  

Governing differential equation

Control Systems: Practical Systems

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FILLING SYSTEMS

Compressible fluids Pressure-voltage analogy

,

Fluid Electrical Pressure, p Voltage, E Mass flow rate, m



Current, I Fluid resistance, RF Electrical resistance, R Fluid capacitance, CF Electrical capacitance, C

Control Systems: Practical Systems

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FILLING SYSTEMS

Compressible fluids Fluid Electrical Pressure, p Current, I Mass flow rate, m



Voltage, E Fluid resistance, RF Electrical compliance, 1/R Fluid capacitance, CF Electrical inductance, L Pressure-current analogy

Control Systems: Practical Systems

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THERMAL SYSTEMS

) ( T T hA Q

a 



Heat flow due to convective heat transfer

Q=rate of heat flow h=coefficient of convective heat transfer of the body surface A= surface area Ta =temperature of the surrounding medium T=temperature of the body

Control Systems: Practical Systems

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RT=1/hA

THERMAL SYSTEMS

Thermal resistance

T a

R T T Q  

p T

dT dT Q MC C dt dt  

Heat flow in terms of thermal resistance Heat flow in terms

• f thermal capacitance

( ) ( ) ( )

T T a

dT t T t R C T t dt  

Governing differential equation

Control Systems: Practical Systems

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THERMAL SYSTEMS

Temperature-voltage analogy

( ) 1 ( ) 1

a T T

T s T s R C s  

Thermal Electrical Temperature, T Voltage, E Heat flow rate, Q Current, I Thermal resistance, RT Electrical resistance, R Thermal capacitance, CT Electrical capacitance, C

Control Systems: Practical Systems

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Temperature-current analogy

THERMAL SYSTEMS

Thermal Electrical Temperature, T(t) Current, i Heat flow rate, q(t) Voltage, e Thermal resistance, RT Electrical compliance, 1/R Thermal capacitance, CT Electrical inductance, L

Control Systems: Practical Systems

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MECHANICAL SYSTEMS

Spring-mass-damper

 

2 2 2

( ) 1 ( ) 1 2

n n

X s F s ms cs k m s s        

Control Systems: Practical Systems

( ) mx cx kx f t   

k c m x(t) f(t)

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MECHANICAL SYSTEMS Force-voltage analogy

2

( ) 1 ( ) V s s k F s ms cs k ms c s      

Transfer function between velocity and force

Control Systems: Practical Systems

Spring-mass-damper

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MECHANICAL SYSTEMS Force-voltage analogy

Mechanical Electrical Force Voltage Velocity Current Mass Inductance Compliance (Reciprocal of stiffness) Capacitance Damping Resistance

Control Systems: Practical Systems

Spring-mass-damper

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MECHANICAL SYSTEMS Force-voltage analogy

Control Systems: Practical Systems

Spring-mass-damper

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MECHANICAL SYSTEMS Force-current analogy

Cs R Ls s E s I 1 1 ) ( ) (   

Transfer function between current and voltage of a LRC circuit

Control Systems: Practical Systems

Spring-mass-damper

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MECHANICAL SYSTEMS

Force-current analogy

Mechanical Electrical Force Current Velocity Voltage Mass Capacitance Compliance (Reciprocal of stiffness) Inductance Damping Conductance(reciprocal

• f resistance)

Control Systems: Practical Systems

Spring-mass-damper

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MECHANICAL SYSTEMS Force-current analogy

Control Systems: Practical Systems

Spring-mass-damper