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Lecture #9
Control Engineering
REVIEW SLIDES
Reference: Textbook by Phillips and Habor
Mathematical Modeling
Models of Electrical Systems
R-L-C
series circuit, impulse voltage source:

Model of an RLC parallel circuit:

Kirchhoff’ s voltage law:
The algebraic sum of voltages around any closed loop in
an electrical circuit is zero.

Kirchhoff’ s current law:
The algebraic sum of currents into any junction in an
electrical circuit is zero.
Models of Mechanical Systems
Mechanical translational systems.

Newton’s second law:

Device with friction (shock absorber):

B is damping coefficient.
Translational system to be defined is a spring (Hooke’s
law):
K is spring coefficient

Model of a mass-spring-damper system:

Note that linear physical systems are modeled by linear
differential equations for which linear components can be
added together. See example of a mass-spring-damper
system.

Simplified automobile suspension system:
Mechanical rotational systems.

Moment of inertia:

Viscous friction:

Torsion:

Model of a torsional pendulum (pendulum in clocks inside
glass dome);
Moment of inertia of pendulum bob denoted by J
Friction between the bob and air by B
Elastance of the brass suspension strip by K

Differential equations as mathematical models of physical
systems: similarity between mathematical models of
electrical circuits and models of simple mechanical
systems (see model of an RCL circuit and model of the
mass-spring-damper system).
Laplace Transform
Name
Unit Impulse
Unit Step
Unit ramp
nth-Order ramp
Exponential
Time function f(t)
(t)
u(t)
t
tn
e-at
nth-Order exponential t n e-at
Sine
sin(bt)
Cosine
cos(bt)
Damped sine
e-at sin(bt)
Damped cosine
e-at cos(bt)
Diverging sine
t sin(bt)
Diverging cosine
t cos(bt)
Laplace Transform
1
1/s
1/s2
n!/sn+1
1/(s+a)
n!/(s+a)n+1
b/(s2+b2)
s/(s2+b2)
b/((s+a)2+b2)
(s+a)/((s+a)2+b2)
2bs/(s2+b2)2
(s2-b2) /(s2+b2)2
Find the inverse Laplace transform of
F(s)=5/(s2+3s+2).
Solution:
Find inverse Laplace Transform of
Find the inverse Laplace transform of
F(s)=(2s+3)/(s3+2s2+s).

Solution:
Laplace Transform Theorems
Transfer Function
Transfer Function

After Laplace transform we have
X(s)=G(s)F(s)

We call G(s) the transfer function.

System interconnections

Series interconnection
Y(s)=H(s)U(s) where H(s)=H1(s)H2(s).
 Parallel interconnection
Y(s)=H(s)U(s) where H(s)=H1(s)+H2(s).

Feedback interconnection
Transfer function of a servo motor:
Mason’s Gain Formula
This gives a procedure that allows us to find
the transfer function, by inspection of either
a block diagram or a signal flow graph.
 Source Node: signals flow away from the
node.
 Sink node: signals flow only toward the
node.
 Path: continuous connection of branches
from one node to another with all arrows in
the same direction.

Loop: a closed path in which no node is
encountered more than once. Source node
cannot be part of a loop.
 Path gain: product of the transfer functions
of all branches that form the loop.
 Loop gain: products of the transfer
functions of all branches that form the loop.
 Nontouching: two loops are non-touching
if these loops have no nodes in common.

An Example
Loop 1 (-G2H1) and loop 2 (-G4H2) are not
touching.
 Two forward paths:

State Variable System:

Solutions of state equations:
Responses
System Responses (Time Domain)

First order systems:
Transient response
Steady state response
Step response
Ramp response
Impulse response

Second order systems
Transient response
Steady state response
Step response
Ramp response
Impulse response
Time Responses of first order systems
The T.F. for first order system:
b0
Y ( s)
K
G ( s) 


R( s ) s  a0 s  1
a0  1 / 
b0  K /  themeaningof  and K will be clear below.
In general, with initialcondition y0 :
y0
(K / )
Y ( s) 

R( s)
s  (1 /  ) s  (1 /  )
Unit step response: R(s)  1/ s
(K / )
K
K
Y ( s) 
 
s[ s  (1 /  )] s s  (1 /  )
y (t )  K (1  e t /  ),
t 0
y (t )  K (1  e t /  )u (t )
First term steady - stateresponse(from thepole of input R(s)).
Second term natural(or transient)response(originatefrom
thepole of theT .F.)


 is called the time constant
Ex. Position control of the pen of a plotter for a
digital computer:   1sec is too slow,   0.1sec
is faster.
lim t  y (t )  lim s 0 sY ( s )  K
 K  steady - state response
and   time constant
for general first order T.F.
System DC Gain

In general:
limt  y (t )  lim s 0 sY ( s )  lim s 0 sG ( s ) R( s )
For unit step input : R(s)  1 / s
limt  y (t )  lim s 0 G ( s )  G (0).
T hereforeG (0) is thesteady - stategain for constant
(unit step)input.T hisis true,independent of theorder
of thesystem.Hence G(0) is called theDC gain.