4. System Response. This module is concern with the response of LTI system. L.T. is used to investigate the response of first and second order systems. Higher order systems can be considered to be the sum of the response of first and second order system.
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c(t)4. System Response
Review of some LTI properties
We will express system as in figure below, system input is r(t), output is c(t), and impulse response is h(t)
Taking the L.T. of (1) yields
C(s) = R(s)H(s) (2)
Where C(s)is the L.T. of c(t)
R(s) is the L.T. of r(t)
H(s) is the L.T. of h(t)
H(s) is called the transfer function (T.F)
If the input is r’(t) then the output is c’(t) where r’(t) denotes the derivative of r(t)
1. Impulse response
Impulse response, denoted by h(t), is the output of the system when its input is impulse (t). h(t) is called the impulse response of the system or the weighting function
If the input is r(t)dt then the output is c(t)dt
5. Poles and Zero
T.F.isusually rational and therefore can be expressed as N(s)/D(s). Poles is the values of s resulting T.F to be infinite. Zeroes is the values of s resulting T.F to be zero
Output of LTI system is the convolution of its input and its impulse response:
c(0)4.1. Time Response of the First Order Systems.
Here we will investigate the time response of the first order systems.
The transfer function of a general first order system can be written as:
Solving for C(s) yields
The eq. can be shown in the block diagram as shown in the figure bellow.
We can found the differential equation first we write (1) as
The diff. Eq. is the inverse L.T. of (2)
Note that the initial condition as an input has a Laplace transform of c(0), which is constant.
The inverse L.T of a constant is impulse (t). Hence the initial condition appears as the impulse function
Here we can see that the impulse function has a practical meaning, even though it is not a realizable signal
Now we take the L.T of (3) and include the initial condition term to get
(3)4.1. Time Response of the First Order Systems.
Since we usually ignore the initial condition
in block diagram, we use the system block
diagram as shown bellow.
The first term originates in the pole of input R(s) and
is called the forced response or steady state response
The second term originates in the pole of the transfer
function G(s) and is called the natural response
Figure below plot c(t)
Suppose that the initial condition is zero then
Unit step response
For unit step input R(s)=1/s, then
The final value or the steady state value of c(t) is K
lim c(t)= K
c(t) is considered to reach final value after reaching
98% of its final value.
The parameter is called the time constant. The
smaller the time constant the faster the system reaches
the final value.
Taking the inverse L.T of (2) yields
(1)4.1. Time Response of the First Order Systems.
A general procedure to find the steady state
value is using final value theorem
lim c(t) = lim sC(s) = lim sG(s)R(s)
t s0 s0
For Unit step input then the final value is
css(t)= lim G(s)
since R(s) = 1/s
System DC gain is the steady state gain to
a constant input for the case that the output
has a final value.
For the input equal to unit ramp function
r(t) = t and R(s) = 1/s2, C(s) is
(4)4.2. Time Response of Second Order System
The standard form second order system is
Case 3: =1 (real equal poles), c(t) is
This system is said to be critically damped
The poles of the TF is s = n jω(12)
Case 4: =0 (imaginary poles), c(t) is
Where = damping ratio
n = natural frequency, or undamped frequency.
Consider the unit step response of this system
This system is said to be undamped
For this system we have
Time constant = = 1/n ; frequency = n
Case 1: 0<<1 (complex poles), c(t) is
This system is said to be underdamped
Case 2: >1 (real unequal poles), c(t) is
This system is said to be overdamped