Transfer Functions

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  • Topic: Control theory, LTI system theory, Transfer function
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ECM2105 - Control Engineering

Dr Mustafa M Aziz (2010)

________________________________________________________________________________

TRANSFER FUNCTIONS AND BLOCK DIAGRAMS
1. Introduction
2. Transfer Function of Linear Time-Invariant (LTI) Systems
3. Block Diagrams
4. Multiple Inputs
5. Transfer Functions with MATLAB
6. Time Response Analysis with MATLAB

1. Introduction
An important step in the analysis and design of control systems is the mathematical modelling of the controlled process. There are a number of mathematical representations to describe a controlled process:

Differential equations: You have learned this before.
Transfer function: It is defined as the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable, with all zero initial conditions. Block diagram: It is used to represent all types of systems. It can be used, together with transfer functions, to describe the cause and effect relationships throughout the system. State-space-representation: You will study this in an advanced Control Systems Design course. 1.1. Linear Time-Variant and Linear Time-Invariant Systems

Definition 1: A time-variable differential equation is a differential equation with one or more of its coefficients are functions of time, t. For example, the differential equation: d 2 y( t )
t2
+ y( t ) = u ( t )
dt 2
(where u and y are dependent variables) is time-variable since the term t2d2y/dt2 depends explicitly on t through the coefficient t2.
An example of a time-varying system is a spacecraft system which the mass of spacecraft changes during flight due to fuel consumption.
Definition 2: A time-invariant differential equation is a differential equation in which none of its coefficients depend on the independent time variable, t. For example, the differential equation:

d 2 y( t )
dy( t )
m
+b
+ y( t ) = u ( t )
2
dt
dt
where the coefficients m and b are constants, is time-invariant since the equation depends only implicitly on t through the dependent variables y and u and their derivatives.

1

ECM2105 - Control Engineering

Dr Mustafa M Aziz (2010)

________________________________________________________________________________ Dynamic systems that are described by linear, constant-coefficient, differential equations are called linear time-invariant (LTI) systems.

2. Transfer Function of Linear Time-Invariant (LTI) Systems
The transfer function of a linear, time-invariant system is defined as the ratio of the Laplace

 

(driving function) U(s) =

 

transform of the output (response function), Y(s) =

{y(t)}, to the Laplace transform of the input

{u(t)}, under the assumption that all initial conditions are zero.

u(t)

System differential equation

y(t)

Taking the Laplace transform with zero initial conditions,

U(s)

Transfer function:

System transfer function
G (s) =

Y(s)

Y(s)
U(s)

A dynamic system can be described by the following time-invariant differential equation: an

d n y( t )
d n −1 y( t )
dy( t )
+ a n −1
+ L + a1
+ a 0 y( t )
n
n −1
dt
dt
dt
d m u(t)
d m −1 u ( t )
du ( t )
= bm
+ b m −1
+ L + b1
+ b 0 u(t)
m
m −1
dt
dt
dt

Taking the Laplace transform and considering zero initial conditions we have:

(a

n

)

(

)

s n + a n −1s n −1 + L + a 1s + a 0 Y(s) = b m s m + b m −1s m −1 + L + b1s + b 0 U(s)

The transfer function between u(t) and y(t) is given by:
Y(s) b m s m + b m −1s m −1 + L + b1s + b 0 M (s)
=
=
G (s) =
U(s)
N(s)
a n s n + a n −1s n −1 + L + a 1s + a 0
where G(s) = M(s)/N(s) is the transfer function of the system; the roots of N(s) are called poles of the system and the roots of M(s) are called zeros of the system. By setting the denominator function to zero, we obtain what is referred to as the characteristic equation: ansn + an-1sn-1 + ⋅⋅⋅ + a1s + a0 = 0

We shall see later that the stability of linear, SISO systems is...
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