Spring 2010
Advanced Topics (EENG 4010-003)
Control Systems Design (EENG 5310-001)
What is a Control System?

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System- a combination of components that act
together and perform a certain objective
Control System- a system in which the
objective is to control a process or a device or
environment
Process- a progressively continuing
operations/development marked by a series of
gradual changes that succeed one another in a
relatively fixed way and lead towards
a particular result or end.
Control Theory

Branch of systems theory (study of interactions
and behavior of a complex assemblage)
Manipulated
Variable(s)
Manipulated
Variable(s)
Control System
Control System
Feedback
function
Control
Variable(s)
Control
Variable(s)
Open Loop
Control
System
Closed Loop
Control
System
Classification of Systems
Classes of Systems
Lumped Parameter
Distributed Parameter (Partial
Differential Equations,
Transmission line example)
Deterministic
Stochastic
Continuous Time
Linear
Time Varying
Discrete Time
Nonlinear
Constant Coefficient
Non-homogeneous Homogeneous (No External
Input; system behavior depends
on initial conditions)
Example Control Systems

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
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Mechanical and Electo-mechanical (e.g. Turntable) Control Systems
Thermal (e.g. Temperature) Control System
Pneumatic Control System
Fluid (Hydraulic) Control Systems
Complex Control Systems
Industrial Controllers
– On-off Controllers
– Proportional Controllers
– Integral Controllers
– Proportional-plus-Integral Controllers
– Proportional-plus-Derivative Controllers
– Proportional-plus-Integral-plus-Derivative Controllers
Mathematical Background

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
Why needed? (A system with differentials, integrals
etc.)
Complex variables (Cauchy-Reimann Conditions,
Euler Theorem)
Laplace Transformation
– Definition
– Standard Transforms
– Inverse Laplace Transforms
Z-Transforms
Matrix algebra
Laplace Transform

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
Definition L[ f (t )]  F (s)  0 f (t )est dt
Condition for Existence   0  Limit e | f (t ) | 0
Laplace Transforms of exponential, step, ramp,
sinusoidal, pulse, and impulse functions
t
Translation of f (t ) and multiplication by e
Effect of Change of time scale
Real and complex differentiations, initial and final
value theorems, real integration, product theorem
Inverse Laplace Transform
t
suchthat
t 
Inverse Laplace Transform

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

c  j
Definition
 F (s)e ds
Formula is seldom or never used; instead,
Heaviside partial fraction expansion is used.
2
d
Illustration with a problem: 2y  4 dy  3 y  2r (t )
dt
dt
Initial conditions: y(0) = 1, y’(0) = 0, and
r(t) = 1, t >= 0. Find the steady state response
s  2s  3
F
(
s
)

Multiple pole case with
( s  1)



L
 ( s  a)    and
Use thes ideas
to
find



a
1
L [ F ( s)]  f (t ) 
2j
1
st
c i
2
3
1
2
L1 
2
2
 ( s  a)   
2
Applications

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Spring-mass-damper- Coulomb and viscous
damper cases
RLC circuit, and concept of analogous variables
Solution of spring-mass-damper (viscous case)
DC motor- Field current and armature current
controlled cases
Block diagrams of the above DC-motor problems
Feedback System Transfer
functions and Signal flow graphs
Block Diagram Reduction
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Combining blocks in a cascade
Moving a summing point ahead of a block
Moving summing point behind a block
Moving splitting point ahead of a block
Moving splitting point behind a block
Elimination of a feedback loop
H2
-
R(s)
+
G1 +
-
G2
+
+
H3
Y(s)
G3
H1
G4
Signal Flow Graphs

a11
Mason’s Gain Formula
r1
a11x1  a12 x2  r1  x1
x1
a21
a21x1  a22 x2  r2  x2
r2
a12
x2
a22
Solve these two equations and generalize to
F 
get Mason’s Gain Formula G 
ijk
k
ij
G1
H2
H3
G2
G3
R(s)
G5
G6
H8
G7
H7
G4
G8
Y(s)
Find Y(s)/R(s)
using the formula

ijk
Another Signal Flow Graph
Problem
G7
1
R(s)
G1
G2
G3
G4
-H4
-H2
-H3
G8
G5
G6
C(s)
-H1
Homework Problem
H1(s)
R1(s)
-
G1(s)
+
X1(s)
+
+
G2(s)
G3(s)
G4(s)
R2(s)
+
+
G5(s)
H2(s)
+
+
X2(s)
G6(s)
Control System Stability: RouthHurwitz Criterion

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Why poles need to be in Left Hand Plane
Necessary condition involving Characteristic
Equation (Polynomial) Coefficients
Proof that the above condition is not sufficient
Ex: s3+s2+2s+8.
Routh-Hurwitz Criterion- Necessary & Sufficient
Routh-Hurwitz Criterion: Some
Typical Problems

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2nd and 3rd order systems
q(s)=s5+2s4+2s3+4s2+11s+10 (first element of a
row 0; other elements are not)
q(s)=s4+s3+s2+s+K (Similar to above case)
q(s)=s3+2s2+4s+K (for k = 8, first element of a
row 0; so are other elements of the row)
q(s)= s5+4s4+8s3+8s2+7s+4 (Use auxiliary eqn.)
q(s)= s5+s4+2s3+2s2+s+1: Repeated roots on
imaginary axis; Marginally stable case
Root-Locus Method: What and Why?


Plotting the trajectories of the poles of a closed
loop control system with free parameter variations
Useful in the design for stability with out
Y(s)
sacrificing much on performance R(s)
G(s)
Closed Loop Transfer Function
+
-

Y ( s)
G( s)

R( s ) 1  G ( s ) H ( s )


H(s)
m
( s  zi )

K ( s  z1 )( s  z2 )...( s  zm )
G ( s) H ( s) 
 K . in1
( s  p1 )( s  p2 )...( s  pn )
 (s  p j )
Let Open Loop Gain
Roots of the closed loop characteristic
equation 1  G ( s) H ( s)  0 depend on K.
j 1
Relationship between closed loop
poles and open loop gain



When K=0, closed loop poles match open loop
poles
When K  , closed loop poles match open
loop zeros.
Hence we can say, the closed loop poles start
at open loop poles and approach closed loop
zeros as K increases and thus form trajectories.
Mathematical Preliminaries of
Root Locus Method


Complex numbers can be
expressed as (absolute value,
angle) pairs.
Now, 1  G(s) H (s)  0  G(s) H (s)  1
|s|

q

s=|s|.ejq
|s| q
| G(s) H (s) | 1 & G(s) H (s)  1800 (2k  1)

s=+j

|s+s1|
The loci of closed loop poles can
be determined using the above
f
-s1=-1j1
constraints (particularly, the
angle constraint) on G(s)H(s).
s=+j

s+s1=|s+s1|ejf
Root Locus Method- Step1 thru 3
of a 7-Step Procedure
Step-1: Locate poles and zeros of G(s)H(s).
Step 2: Determine Root Locus on the real-axis using angle
constraint. Value of K at any particular test point s can be
calculated using the magnitude constraint.
Step 3: Find asymptotes by using angle constraint in
G( s) H ( S ) . Find asymptote centroid
Limit
s0
j 3
(

p
)

(

z
)



. This formula may be
 
n
j 1
A
m
j
i 1
i
A
nm
m
obtained by setting 1  (s  K )
nm
 1  G( s) H ( s)  1 
K  ( s  zi )
A
i 1
n
 (s  p )
j
Illustrative Problem:
G( s) 
K
; H ( s)  1
s( s  1)( s  2)
j 1
-2
-1
0
j 3
Root Locus Method- Step 4
Step 4: Determine breakaway points (points where two
or more loci coincide giving multiple roots and then
deviate). Now, from the characteristic equation
Z ( s)
0
P( s )
f ( s)  P( s)  K .Z ( s)  0
df
get, at a multiple pole s1,
|s s1  0 , because
where 1  G(s) H (s)  1  K .
, we
at s1,
ds
f ( s )  ( s  s1 ) ( s  s2 )...( s  sn ) . Thus we get at s1,
r
P' ( s )
P' ( s )
; f ( s )  P( s ) 
Z ( s)  0
 P( s ).Z ' ( s )  P' ( s ).Z ( s)  0
Z ' ( s)
Z ' ( s)
P( s) dK
P( s) Z ' ( s)  P' ( s).Z ( s)
K 
,

 0 at s = s1. Thus, we
Z ( s) ds
Z 2 ( s)
K 
Since
break points by setting dK/ds=0. In the
example, we get s = -0.4226 or -1.5774 (invalid).
get
Root Locus Method- Step 5
Step 5: Determine the points (if any) where the root loci
cross the imaginary axis using Roth-Hurwitz Stability
Criterion.
Illustration with the Example Problem
Characteristic equation for the problem:s3+3s2+2s+K
S3 1
2
From the array, we know that the system
S2 3
K
is marginally stable at K=6. Now, we can 1
S (6-K)/3
get the value of  (imaginary axis
S0 K
crossing) either by solving the second row
3s2+6 =0 or the original equation with s=j.
Root Locus Method- Step 6 and 7
Step 6: Determine angles of departure at complex poles and
arrival at complex zeros using angle criterion.
Step 7: Choose a test point in the broad neighborhood of
imaginary axis and origin and check whether sum of the
angles is an odd multiple of +180 or -180. If it does not
satisfy, select another one. Continue the process till
sufficient number of test points satisfying angle condition
are located. Draw the root loci using information from
steps 1-5.
Root Locus approach to Control
System Design

Effect of Addition of Poles to Open Loop Function: Pulls
the root locus right; lowers system’s stability and slows
down the settling of response.
j
x

j
j
x

x
x x x


Effect of Addition of Zeros to Open Loop Function: Pulls
the Root Locus to Left; improves system stability and
speeds up the settling of response
j
o x
x x
j

x o x
j
x

x
x o x

Performance Criteria Used In
Design
We consider 2nd order systems here, because higher
order systems with 2 dominant poles can be
approximated to 2nd order systems e.g.
1
T ( s)  2
when | a | 10 | n |
2
( s  2n s  n )( s  a)

For
2nd
order system
n 2
Y (s)  2
R( s )
2
s  2n s  n
1
1
e  t sin( n 1   2 t  q )
For unit step input R( s)  ; y (t )  1 
s
1  2
1
Where q  cos  .
 Two types of performance criteria (Transient and
Steady State)
 Stability is a validity criterion (Non-negotiable).
n
Transient Performance Criteria
overshoot
ess
y (t )  1 
1
1  2
e  nt sin( n 1   2 t  q )
•Rise Time TR= Time to reach Value 1.0
y(t)
1.0
TR
TP
t
TS
nTr1
•Rise Time Tr1= Time from 0.1 to 0.9
2.16  0.6
2.0
T

Empirical Formula is r1
n
for 0.3    0.8
0.6
• Settling time (Time to settle to within 98% of 1.0)=4/n

T

• Peak Time
 1 
P
2
n
Percentage Overshoot = 100e
 / 1 2

Series Compensators for
Improved Design



RC OP-Amp Circuit for phase lead (or lag)
compensator
Lead Compensator for Improved Transient
Response; Example: G(s)  4 Required to
s( s  2)
reduce rise time to half keeping  = 0.5.
Lag Compensator for Improved steady-state
performance. Example: G(s)  1.06
s( s  1)( s  2)

C ( s)
1.06
1.06


R( s) s( s  1)( s  2)  1.06 ( s  0.3307  j 0.5864)( s  0.3307  j 0.5864)( s  2.3386)
Frequency Response Analysis
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
Response to x(t) = X sin(t)
G(s) = K/(Ts+1) and G(s)=(s+1/T1)/(s+1/T2) cases
Frequency response graphs- Bode, and Nyquist
plots of K , ( j ) , (1  jT ) ,[1  2 ( j /  )  ( j /  ) ]
Resonant frequency and peak value
Nichols Chart
Nyquist Stability Criterion
1
1
2
n



n
1
Control System Design Using
Frequency Response Analysis
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Lead Compensation
Lag Compensation
Lag-Lead Compensation
State Space Analysis

State-Space Representation of a Generic
Transfer Function in Canonical Forms:
–
–
–
–

Controllable Canonical Form
Observable Canonical Form
Diagonal Canonical Form
Jordan Canonical Form
Eigenvalue Analysis
Solution of State Equations






  Ax
Solution of Homogeneous Equations x
At
e
Interpretation of
Show that the state transition matrix
is
given by (t )  e At  L1 (sI  A) 1 
Properties of (t )
Solution of Nonhomogeneous Equations
Cayley-Hamilton Theorem
Controllability and Observability



Definitions of Controllable and Observable
Systems
Controllabililty and Obervability Conditions
Principle of Duality
Control System Design in State
Space



Necessary and Sufficient Condition for
Arbitrary Pole Placement
Determination of Feedback Gain Matrix by
Ackerman’s formula
Design of Servo Systems
Introduction to Sampled Data
Control Systems



Z-transform and Inverse Z-transform
Properties of Z-Transform and Comparison
with the Corresponding Laplace Transform
Properties
Transfer Functions of Discrete Data Systems
Analysis of Sampled Data Systems



Input and Output Response of Sampled Data
Systems
Differences in the Transient Characteristics of
Continuous Data Systems and Corresponding
Discrete (Sampled) Data Systems
Root Locus Analysis of Sampled Data Systems