The approach to dynamic system problem can be
listed as follow:
1. Define the system and its components
2. Formulate the mathematical model and list the
necessary assumptions.
3. Write the differential equations describing the
model.
4. Solve the equation
Introduction
to Control Systems
11 INTRODUCTION
Control theories commonly used today are classical control theory (also called con
ventional control theory), modern control theory, and robust control theory. This book
presents comprehensive treatments of
Sensitivity of control systems
Suppose system G(s) changes to G(s)+G(s) due to
the environment, ageing
Y(s) changes to Y(s)+Y(s)
Y (G ( s ) + G ( s ) = Y (G ( s ) +
For Open Loop
For Closed Loop
dY ( s )
G ( s )
dG ( s )
Y ( s) = G ( s) R( s)
Y ( s ) =
CHAPTER 9
9.0 TRANSIENT
RESPONSE
AND
STEADY
STATE
ERROR
ANALYSIS
9.1
Introduction:
In most practical systems, the input signal is not known ahead
of time and can be random in nature.
It is only in some
special cases where the input can be known in advance
CHAPTER
5
5.0 INTRODUCTION TO CONTROL SYSTEMS
5.1
Introduction
Automatic Control has played a vital role in the advancement
of engineering and Science. For example automatic control is
essential in such industrial processes such as controlling
pressure, t
Applying input to cause system variables to
conform to desired values called the
reference.
 Cruisecontrol car: f_engine(t)=? speed=90
kph
Ecommerce server: Resource allocation?
T_response=5 sec
Embedded networks: Flow rate? Delay = 1 sec
Computer
10Jan15
The time response of a control system consists of two
parts:
1. Transient response
2. Steadystate response
 from initial state to the final
 the manner in which the
state purpose of control
system output behaves as t
systems is to provide a d
OPENLOOP CONTROL
Output quantity does not influence the
input quantity hence has no effect on
control action
Output is neither measured nor
feedback for comparison with input
Each input has corresponding output
condition
With disturbance system will not
Goal: Determining whether the system is stable or
unstable from a characteristic equation in
polynomial form without actually solving for the
roots.
Rouths stability criterion is useful for determining
the ranges of coefficients of polynomials for
stabi
CHAPTER 10
10.0 STABILITY ANALYSIS
10.1 Concept of stability
The stability of control system whether continuous or
discrete is determined by its impulse response to inputs
or disturbances. A stable system is the one that remains
at rest unless exited by a
Signal Flow Graphs
SIGNAL FLOW GRAPH
SignalFlow Graph Models
Block diagram reduction procedures are
sometimes difficult to implement.
Another method (Developed by Mason) for
determining the relationship between system
variables can be used.
This metho
Laplace Transform Inversion
Inverse
Laplace
Transformation
(i) Find the partial fraction expansion of F(s)
(ii) Obtain the inverse transform f(t) using the
transform tables.
Case I: Simple Real Roots
Case II: Complex Conjugate and Simple
Real Roots
Cas
THE RATIONALE FOR
MATHEMATICAL MODELING
DYNAMIC VERSUS STEADYSTATE MODEL
Dynamic model
Describes time behavior of a process
Changes in input, disturbance, parameters, initial
condition, etc.
Described by a set of differential equations: ordinary
(ODE),
Any physical control suffers from steady
state errors in response due to certain inputs
Errors can be attributed to:
Transient period due to changes in input
Components imperfections
Static friction, backlash and amplifier drift
Aging or deterioration
I
5
Transient and SteadyState
Response Analyses
51 INTRODUCTION
In early chapters it was stated that the first step in analyzing a control system was to derive a mathematical model of the system. Once such a model is obtained, various methods are available
6
ControlSystem Characteristics
6.1
INTRODUCTION
Open and closedloop transfer functions have certain basic characteristics
that permit transient and steadystate analyses of the feedbackcontrolled
system. Five factors of prime importance in feedbackc
CS 321: Introduction to Control
Systems Engineering
COURSE LECTURERS:
LOGISTICS
WEEKLY
2 HRS LECTURE
1 HR TUTORIAL
2 HRS LABORATORY WORK
Mr. N.H. MVUNGI
Ms. V. Msoka
Quizzes will be any day during any of the sessions for the
1st 10 minutes of the session.
Linear approximations of
physical systems
Nonlinearity is essential and pervasive
A system is defined as linear in terms of the
system excitation and response.
A linear system satisfies the properties of
superposition and homogeneity
Introduction
maj
It is intended you to be able to:
Aim 3.1: Apply Newtons Laws to form the
equations of motion for lumped parameter
mechanical systems (Translational and
rotational) comprising masses, springs, and
dampers.
Aim 3.2: Apply Kirchhoffs voltage law to form
the
yd
The most important characteristic of the
dynamic response is absolute stability, that is,
whether the system is stable or unstable.
A system is:
stable if its transient response decays
&
unstable if it does not decay.
e
+

Controller: C(s) e.g. PID
CHAPTER 7
7.0 MODELLING OF CONTROL SYSTEMS
7.1
Introduction
To analyse any system, a mathematical model of that system must
be obtained. A mathematical model of a dynamic system is defined
as a set of equations that represents the dynamics of the system
a
Basic components of a control system
Basic concepts of a control system
1.Plant: a physical object to be controlled
such as a mechanical device, a heating furnace,
a chemical reactor or a spacecraft.
plant
Plant/Process
Controlled Variable
Expected Value
The time response solution is obtained by:
Obtain the D.E.
Obtain the Laplace Transf. of the D.E.
Solve the resulting algebraic transform of the
variable of interest.
The Laplace transform exists for linear
D.E. for which the transformation integral
CHAPTER
6
6.0 LAPLACE TRANSFORMATION
6.1
Introduction to Laplace Transformation
The Laplace transform method is an operational method that can be
used for solving linear differential Equations.
By Laplace
transform one can convert many common functions su
27 LINEARIZATION OF NONLINEAR MATHEMATICAL MODELS
Nonlinear Systems. A system is nonlinear if the principle of superposition does
not apply. Thus, for a nonlinear system the response to two inputs cannot be calculated
by treating one input at a time and a
Two of the earliest examples
Water clock (270 BC)
Selfleveling wine vessel (100BC)
The idea is still
used today, i.e.
flush toilet
Watts
Steam Engine
Newcomens steam engine
(1712)
had limited success
Beginning of systems
engineering
Watts systems