EEE 317: Control System I
SYLLABUS
Introduction to control systems. Linear system models: transfer function, block diagram and
signal flow graph (SFG), State variables: SFG to state variables, transfer function to state
variables and state variable to transfer function. Feedback control System: closed-loop systems,
parameter sensitivity, transient characteristics of control systems, effect of third pole and zero on
the system response and system types and steady-state error. Routh stability criterion, Analysis
of feedback control system: root-locus method and frequency response method. Design of
feedback control system: controllability and observability, root locus, frequency response and
state-variable methods. Digital control systems: introduction, sampled data systems, stability
analysis in z-domain.
REFERENCE BOOKS
1. Control systems engineering – I. J. Nagrath
2. Modern control engineering – Katsuhiko Ogata
3. Control systems engineering – Norman S. Nise
4. Analog and digital control system – Chen
5. Advanced control engineering – Roland S. Burns
TOPICS COVERED
 Introduction to control systems
 Mathematical Models of Physical systems
 Feedback characteristics of control systems
 Time-domain analysis of control systems
 Concept of stability
 The root-locus technique
 Frequency response analysis
 Control system design by root-locus method
 Control system design by frequency response
 Sampled-data control systems
OBJECTIVE OF THE COURSE
The objective of this course is to highlight the following:
1. Representing a system using ordinary differential equations and Laplace transforms.
2. Analysis of single input/ single output systems and their components considering their
input/ output relationship is linear.
3. Design controllers for single input/ single output systems that meet design requirements.
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Introduction to Control Systems
The goal of control engineering is to design and build real physical systems to perform given
tasks.
An engineer is asked to design an install a heat exchanger to control the temperature and
humidity of a large building. He determines the required capacity of the exchanger and then
proceed to install the system. After installation the exchanger is found to be insufficiently
powerful to control the buildings environment. He has to replace the unit and place a powerful
one. This is an example of empirical method.
Consider again the task of sending astronauts in moon and bringing them back safely. It cannot
be carried out by empirical method. In this case analytical method is indispensable. This method
consists of the following steps:
- modeling
- setting up mathematical equations
- analysis and design
Empirical methods may be expensive and dangerous. Analytical methods are simulated in
computers to see the result. If the design is satisfactory, the system is implemented using
physical devices.
A control system is an interconnection of components or devices so that the output of the overall
system will follow as closely as possible a desired signal. The reasons of designing control
systems include:
- Automatic control (e.g., control of room temperature)
- Remote control (e.g., antenna position control)
- Power amplification(e.g., control system will generate sufficient power to turn
the heavy antennas)
Control system components can be mechanical, electrical, hydraulic, pneumatic, thermal, or it
may be a computer program. It plays an important role in the development of modern
civilization. We use heating and air-conditioning in domestic domain for comfortable living. It
has found application in quality control of manufacturing products, machine-tool control,
weapon systems, power systems, robotics and many other places.
Classification of Control Systems
Control systems are basically classified as –
 Open-loop control system
 Closed-loop control system
In open-loop system the control action is independent of output. In closed-loop system control
action is somehow dependent on output. Each system has at least two things in common, a
controller and an actuator (final control element). The input to the controller is called reference
input. This signal represents the desired system output.
Open-loop control system is used for very simple applications where inputs are known ahed of
time and there is no disturbance. Here the output is sensitive to the changes in disturbance
inputs. Disturbance inputs are undesirable inputs that tend to deflect the plant outputs from their
desired values. They must be calibrated and adjusted at regular intervals to ensure proper
operation.
Closed-loop systems are also called feedback control systems. Feedback is the property of the
closed-loop systems which permits the output to be compared with the input of the system so
2
that appropriate control action may be formed as a function of inputs and outputs. Feedback
systems has the following features:
- reduced effect of nonlinearities and distortion
- Increased accuracy
- Increased bandwidth
- Less sensitivity to variation of system parameters
- Tendency towards oscillations
- Reduced effects of external disturbances
The general block diagram of a control system is shown below.
Figure: Closed-loop control system
Some Definitions
Reference input – It is the actual signal input to the control system.
Output (Controlled variable) – It is the actual response obtained from a control system.
Actuating error signal – It is the difference between the reference input and feedback signal.
Controller – It is a component required to generate control signal to drive the actuator.
Control signal – The signal obtained at the output of a controller is called control signal.
Actuator – It is a power device that produces input to the plant according to the control signal,
so that output signal approaches the reference input signal.
Plant – The combination of object to be controlled and the actuator is called the plant.
Feedback Element – It is the element that provides a mean for feeding back the output quantity
in order to compare it with the reference input.
Servomechanism – It is a feedback control system in which the output is mechanical position,
velocity, or acceleration.
Example of Control Systems
Toilet tank filling system:
Figure: Toilet tank filling system
3
Position control system: [antenna]
Figure: Position control system
Velocity control system: [audio/ video recorder]
Figure: Velocity control system
Clothes Dryer:
Figure: Automatic dryer
4
Temperature control system: [oven, refrigerator, house]
Figure: Temperature control system
Computer numerically controlled (CNC) machine tool:
(a)
(b)
Figure: CNC machine tool control system
5
Control System Design
In order to design and implement a control system, we need knowledge about the following
things:
 Knowledge of desired value, (performance specification)
 Knowledge of the output value, (feedback sensor, its resolution and dynamic
response)
 Knowledge of controlling device,
 Knowledge of actuating device,
 Knowledge of the plant.
With all of this knowledge and information available for the control system designer, he can
start the design steps shown below in the flow diagram.
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Mathematical Models of Physical Systems
We use mathematical models of physical systems to design and analyze control systems.
Mathematical models are described by ordinary differential equations. If the coefficients of the
describing differential equations are function of time, then the mathematical model is linear
time-varying. On the other hand, if the coefficients describing differential equations are
constants, the model is linear time-invariant.
The differential equations describing a LTI system can be reshaped into different forms
for the convenience of analysis. For transient response or frequency response analysis of singleinput-single-output linear systems, the transfer function representation is convenient. On the
other hand, when the system has multiple inputs and outputs, the vector-matrix notation may be
more convenient.
Powerful mathematical tools like Fourier and Laplace transforms are available for linear
systems. Unfortunately no physical system in nature is perfectly linear. Certain assumptions
must always be made to get a linear model. In the presence of strong nonlinearity or in presence
of distributive effects it is not possible to obtain linear models.
A commonly adopted approach is to build a simplified linear model by ignoring certain
nonlinearities and other physical properties that may be present in a system and thereby get an
approximate idea of the dynamic response of the system. A more complete model is then built
for more complete analysis.
Transfer function
The transfer function of an LTI system is the ratio of Laplace transform of the output variable to
the Laplace transform of the input variable assuming zero initial conditions. Following are some
examples of how transfer functions can be determined for some dynamic system elements.
Eo ( s )
1

2
Ei ( s ) LCs  RCs  1
[See chapter 3 of Ogata for details]
Eo ( s )
1

2
Ei ( s ) R1C1 R2C2 s  ( R1C1  R2C2  R1C2 )s  1
The insertion of an isolation amplifier between the two RC-circuits will produce no loading
effect.
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A. Transfer Function of Armature Controlled DC Motor
  K f i f ; TM  K1K f i f ia  KT ia
eb  K b
di
d
; La a  Ra ia  eb  e
dt
dt
J
d 2
d
 f0
 TM  KT ia
2
dt
dt

KT
s ( Ra  sLa )( Js  f 0 )  KT Kb 
In Laplace domain,
 Eb ( s )  K b s ( s )

( La s  Ra ) I a ( s )  E ( s )  Eb ( s ) ;
 2
( Js  f 0 s ) ( s )  KT I a ( s )
Neglecting La , G ( s) 
 G( s) 
 ( s)
E ( s)
KT / Ra
K /R
Km
 T a 
;
Js  s ( f 0  KT K b / Ra ) s ( Js  f ) s ( s m  1)
2
where, f  f 0  KT Kb / Ra and Km  KT / Ra f ;  m  J / f .
K m and  m are called the motor gain and time constant respectively. These two parameters are
usually supplied by the manufacturer.
The block diagram model is,
B. Transfer Function of a Field-controlled DC Motor

'
TM  K1 ia  K1 K f i f ia  K T i f

 di f
 Rf if  e
;
Lf
dt

 d 2
d
 TM  KT i f
J 2  f
dt
 dt
We obtain,
G (s) 

( L f s  R f ) I f ( s)  E ( s)
  2


( Js  fs) ( s)  TM ( s)  KT I f ( s)
 (s)
E (s)

KT 
s ( L f s  R f )( Js  f )
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Mechanical System
Translational system and rotational system
du
d 2x
m 2
dt
dt
ud  k1 (u2  u1 ); us  k2 ( x2  x1 )
um  ma  m
d 2
d
J
;
2
dt
dt
Td  k1 (1  2 ); Ts  k2 (1   2 )
TJ  J
d 2 y (t )
dy (t )
m
 k1
 k2 y (t )  u
2
dt
dt
Mechanical Acclerometer
For a constant input acceleration y becomes constant.
Then, x  k y . Taking Laplace transform of the previous equation, X (s)  ( s 2  cs  k )Y (s) ;
X (s)


1
Y (s)


s  cs  k 
2
Rotary Potentiometer
  (t ) 
 Vmax
Vo (t )  
V

 max 
  max 
  max
or, Vo (t )  k p (t )

  (t )

9
Example: Draw the block diagram of the following system.
Example: Control of flaps in airplane
10
Example:
Figure below shows a reduction gearbox being driven by a motor that develops a torque Tm (t ) . It
has a gear reduction ratio of ‘n = b / a’. Find a differential equation relating the motor
torque Tm (t ) and the output angular position  o (t ) .
Ta (t )  Tm (t )  I m
T (t ) a 1 
d 2 m
d m
d 2o
d
b
;

C
T
(
t
)

I
 C0 o ; a   ; m   n
m
b
o
2
2
Tb (t ) b n  o a
dt
dt
dt
dt

d 2o
do 
d 2o
do
From above, n Tm (t )  nI m 2  nCm
;
  I 0 2  Co
dt
dt 
dt
dt

N.B.  d m
2
d 2o dm
d 
n o
 2 n 2 ;
dt
dt
dt
dt 

Gearbox parameters:
,
From above,  ( I o  n I m ) o  (Co  n 2Cm ) o  nTm (t )
,
2
Inputting parameters,
Example:
0.0225 o  0.3 o  50Tm (t )
Gear train and its equivalence
[Ans]
11
V ( s )  k p  r ( s )   o ( s ) 
k p  Vbatt /  max
Tmotor  J1eq1  f1eq1
Tachometers
V ( s)  kt ( s)  kt s ( s)
Error detector
V ( s )  k p  r ( s )   o ( s ) 
k p  Vbatt /  max
Error detector using op amp
e = r - vw
Example
Obtain the transfer function X(s)/E(s) for the
electromechanical system shown left assuming
dx
that the coil has a back emf eb  K1 and the
dt
coil current i2 produces a force Fc  k2i2 on the
mass M.
K2
X ( s)

4
E ( s) RLCMs  L( M  RCB) s3  RC (2 LK
 k1k2 ) s 2  ( RB  2 LK  k1k2 )s  2RK
Example
Sketch the analogus electrical circuit of the following mechanical system.
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Example
Draw the analogous electric circuit of the system below using f-i analogy.
[Ans]
Synchros
A synchro is an electromagnetic
transducer that is used to convert
angular shaft position into an
electric signal. The basic element
of a synchro is a synchro
transmitter whose construction is
very similar to that of the 3-
alternator.
An ac voltage is applied to the
rotor winding through slip-rings.
The schematic diagram of
synchro synchro transmittercontrol transformer pair is shown
above.
Figure: Synchro error detector
vs1 (t )  KVr sin ct cos(  120 )
vr (t )  Vr sin ct ; vs 2 (t )  KVr sin ct cos 
vs 3 (t )  KVr sin ct cos(  240 )
vs1s 2 (t )  3KVr sin ct sin(  240 )
; vs 2 s 3 (t )  3KVr sin ct sin(  120 )
vs 3s1 (t )  3KVr sin ct sin 
When   0 , vs 3s1 (t )  0 and maximum voltage is induced on S2 coil. This position of the rotor is
defined as the “electrical zero” of the transmitter and used as reference position of the rotor.
The output of the synchro transmitter is applied to the stator winding of a “synchro control
transformer”. Circulating current of the same phase but of different magnitude flows through
the two sets of stator coil. The pair acts as an error detector.
The voltage induced in the control transformer rotor is proportional to the cosine of the angle
between the two rotors and is given by,
e(t )  K Vr sin ct cos  ; where,   90     .
 e(t )  K Vr sin ct cos(90     )  K Vr sin ct sin(   )  K Vr (   ) sin ct …(01)
The equation above holds for small angular displacement.
Thus the synchro transmitter-control transformer pair acts as an error detector which gives a
voltage signal at the rotor terminal of the control transformer proportional to the angular
difference between the shaft positions.
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Equation (01) is represented graphically in figure below for an arbitrary time variation of
(    ).
It is seen that the output of the
synchro error detector is a
modulated signal, where the ac
signal applied to the rotor of
synchro transmitter acts as carrier
and the modulating signal is,
 em (t )  K s (   ); K s  K Vr
A.C. Servomotor
For low power application a.c. motors are preferred, because of their light weight, ruggedness
and no brush contact. Two phase induction motors are mostly used in control system.
The motor has two stator windings
displaced 90 electrical degree apart.
The voltages applied to the
windings are not balanced. Under
normal operating conditions a fixed
voltage from a constant voltage
source is applied to one phase. The
other phase, called control phase, is
energized by a voltage of variable
magnitude which is 90 out of
phase w.r.t. the voltage of fixed
phase.
The torque speed characteristic of the motor is different from conventional motor. X / R ratio is
low and the curve has negative slope for stabilization.
The torque-speed curve is not linear. But we assume it as linear for the derivation of transfer
function. The troque is afunction of both speed and the r.m.s. control voltage, ie., TM  f ( , E ) .
14
Using Tailor series expansion about the normal operating point (TM 0 ,  0 , E0 ) we get,
TM  TM 0 
 TM
E
E  E0
 0
( E  E0 ) 
 TM

E  E0
 0
(  0 )
For position control system, E0  0 ,  0  0 , TM 0
Thus, the above equation may be simplified as,
TM  kE  m  J  f 0 ;
where, k 
 TM
E
E  E0
 0
and m 
 TM

E  E0
 0
.
Performing Laplace transform, kE ( s)  ms ( s)  Js 2 ( s)  f 0 ( s )
Km
 (s)
k
k
J
G( s) 
 2

Or,
; where, K m 
and  m 
.
E ( s ) Js  ( f 0  m) s s( m s  1)
f0  m
f0  m
Since ‘m’ is negative the transient part is decaying as  m is positive. If ‘m’ would positive and
m  f 0 the transient part will increase with time and the system would be unstable. k and m are
the slope of the torque-voltage and torque-speed curve.
A.C. Position Control System
The reference motor phase and the synchro transmitter rotor coil are excited from the same
carrier supply. The carrier voltage driving the control phase is amplified and a phase shift of
90is produced by use of two RC networks. The signal flow graph of the control system is
shown above. The overall transfer function of the system is,
c ( s)
R( s)

Km Ka Ks n
 m s 2  (1  K m K a K t ) s  K m K a K s n
K m =motor gain constant,  m =motor time constant, K s =synchro sensitivity in volts/rad.