Lecture #9Control EngineeringREVIEW SLIDES

Reference: Textbook by Phillips and Habor

Mathematical Modeling

Models of Electrical Systems

R-L-C series circuit, impulse voltage source:R-L-C series circuit, impulse voltage source:

Model of an RLC parallel circuit:Model of an RLC parallel circuit:

Kirchhoff’ s voltage law:Kirchhoff’ s voltage law:

The algebraic sum of voltages around any closed loop in The algebraic sum of voltages around any closed loop in an electrical circuit is zero.an electrical circuit is zero.

Kirchhoff’ s current law:Kirchhoff’ s current law:

The algebraic sum of currents into any junction in an The algebraic sum of currents into any junction in an electrical circuit is zero.electrical circuit is zero.

Models of Mechanical Systems

Mechanical translational systems.Mechanical translational systems. Newton’s second law:Newton’s second law:

Device with friction (shock absorber):Device with friction (shock absorber): BB is damping coefficient. is damping coefficient. Translational system to be defined is a spring (Hooke’s law):Translational system to be defined is a spring (Hooke’s law): KK is spring coefficient is spring coefficient

Model of a mass-spring-damper system:Model of a mass-spring-damper system:

Note that linear physical systems are modeled by linear Note that linear physical systems are modeled by linear differential equations for which linear components can be differential equations for which linear components can be added together. See example of a mass-spring-damper added together. See example of a mass-spring-damper system.system.

Simplified automobile suspension system:Simplified automobile suspension system:

Mechanical rotational systemsMechanical rotational systems.. Moment of inertia:Moment of inertia:

Viscous friction:Viscous friction:

Torsion:Torsion:

Model of a torsional pendulum (pendulum in clocks insideModel of a torsional pendulum (pendulum in clocks inside

glass dome);glass dome);

Moment of inertia of pendulum bob denoted by Moment of inertia of pendulum bob denoted by JJ

Friction between the bob and air by Friction between the bob and air by BB

Elastance of the brass suspension strip by Elastance of the brass suspension strip by KK

Differential equations as mathematical models of physical Differential equations as mathematical models of physical systems: similarity between mathematical models of systems: similarity between mathematical models of electrical circuits and models of simple mechanical electrical circuits and models of simple mechanical systems (see model of an RCL circuit and model of the systems (see model of an RCL circuit and model of the mass-spring-damper system).mass-spring-damper system).

Laplace Transform

NameName Time function f(t)Time function f(t) Laplace TransformLaplace Transform

Unit ImpulseUnit Impulse (t)(t) 11

Unit StepUnit Step u(t)u(t) 1/s1/s

Unit rampUnit ramp tt 1/s1/s22

nth-Order rampnth-Order ramp t t nn n!/sn!/sn+1n+1

ExponentialExponential ee-at-at 1/(s+a)1/(s+a)nth-Order exponentialnth-Order exponential t t nn e e-at-at n!/(s+a)n!/(s+a)n+1n+1

SineSine sin(sin(bt)bt) b/(sb/(s22+b+b22))

CosineCosine cos(cos(btbt)) s/(ss/(s22+b+b22))

Damped sineDamped sine ee-at -at sin(sin(bt)bt) b/((s+a)b/((s+a)22+b+b22))

Damped cosineDamped cosine ee-at -at cos(cos(bt)bt) (s+a)/((s+a)(s+a)/((s+a)22+b+b22))

Diverging sineDiverging sine tt sin( sin(bt)bt) 2bs/(s2bs/(s22+b+b22))22

Diverging cosineDiverging cosine tt cos( cos(bt)bt) (s(s22-b-b22)) /(s/(s22+b+b22))22

Find the inverse Laplace transform of F(s)=5/(s2+3s+2).

Solution:Solution:

Find inverse Laplace Transform of

Find the inverse Laplace transform of F(s)=(2s+3)/(s3+2s2+s). Solution:Solution:

Laplace Transform Theorems

Transfer Function

Transfer Function

After Laplace transform we have After Laplace transform we have X(s)=G(s)F(s)X(s)=G(s)F(s)

We call We call G(s)G(s) the the transfer functiontransfer function..

System interconnectionsSystem interconnections Series interconnectionSeries interconnection

Y(s)=H(s)U(s)Y(s)=H(s)U(s) where where H(s)=HH(s)=H11(s)H(s)H22(s).(s).

Parallel interconnectionParallel interconnection

Y(s)=H(s)U(s)Y(s)=H(s)U(s) where where H(s)=HH(s)=H11(s)+H(s)+H22(s).(s).

Feedback interconnectionFeedback interconnection

Transfer function of a servo motor:Transfer function of a servo motor:

Mason’s Gain Formula

This gives a procedure that allows us to find This gives a procedure that allows us to find the transfer function, by inspection of either the transfer function, by inspection of either a block diagram or a signal flow graph.a block diagram or a signal flow graph.

Source NodeSource Node: signals flow away from the : signals flow away from the node.node.

Sink nodeSink node: signals flow only toward the : signals flow only toward the node.node.

PathPath: continuous connection of branches : continuous connection of branches from one node to another with all arrows in from one node to another with all arrows in the same direction.the same direction.

LoopLoop: a closed path in which no node is : a closed path in which no node is encountered more than once. Source node encountered more than once. Source node cannot be part of a loop.cannot be part of a loop.

Path gainPath gain: product of the transfer functions : product of the transfer functions of all branches that form the loop.of all branches that form the loop.

Loop gainLoop gain: products of the transfer : products of the transfer functions of all branches that form the loop.functions of all branches that form the loop.

NontouchingNontouching: two loops are non-touching : two loops are non-touching if these loops have no nodes in common.if these loops have no nodes in common.

An Example

Loop 1 Loop 1 (-G(-G22HH11) and loop 2 (-) and loop 2 (-GG44HH22) are not ) are not touching.touching.

Two forward paths:Two forward paths:

State Variable System:

Solutions of state equationsSolutions of state equations::

Responses

System Responses (Time Domain)

First order systems:First order systems:

Transient responseTransient response

Steady state responseSteady state response Step responseStep response Ramp responseRamp response Impulse responseImpulse response

Second order systemsSecond order systems Transient responseTransient response

Steady state responseSteady state response Step responseStep response Ramp responseRamp response Impulse responseImpulse response

Time Responses of first order systemsThe T.F. for first order system:The T.F. for first order system:

)()/1(

)/(

)/1()(

:condition initial with general,In

below.clear be will and of meaning the//1

1)(

)()(

0

0

00

0

0

sRs

K

s

ysY

y

KKba

s

K

as

b

sR

sYsG

0),1()(

)/1()]/1([

)/()(

/

teKty

s

K

s

K

ss

KsY

t

ssR /1)( :response stepUnit

T.F.) theof pole the

from (originate response ent)(or transi natural termSecond

R(s)).input of pole the(from response state-steady First term

)()1()( /

tueKty t

is called the is called the time constanttime constant

Ex. Position control of the pen of a plotter forEx. Position control of the pen of a plotter for a a digital computer: is too slow, digital computer: is too slow, is faster. is faster.

sec1

sec1.0

T.F.order first generalfor

constant time and

response state-steady

)(lim)(lim 0

K

KssYty st

System DC GainSystem DC Gain In general: In general:

gain. DC thecalled is G(0) Hence system. theof

order theoft independen true,is This input. step)(unit

constant for gain state-steady theis )0( Therefore

).0()(lim)(lim

/1R(s) :input stepunit For

)()(lim)(lim)(lim

0

00

G

GsGty

s

sRssGssYty

st

sst