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CHAPTER – 6
TIME-DOMAIN ANALYSIS OF CONTROL SYSTEMS
6.1. INTRODUCTION:
Since time is used as an independent variable in most control systems, it is usually of interest to evaluate
the state and output responses with respected to time, or simply the time response. In the analysis problem, a
reference input signal is applied to a system, and the performance of the system is evaluated by studying the
system response in the time domain. For instance, if the objective of the control system is to have the output
variable track the input signal, starting at some initial time and initial condition, it is necessary to compare the
input signal, starting at some initial time and initial condition, it is necessary to compare the input and output
responses as functions of time. Therefore, in most control system problems, the final evaluation of performance of
the system is based on the time responses.
The time response of a system is the output of the system as a function of time, when subjected to a known
input.
The time response of a control system is usually divided into two parts: the transient response and the
steady-state response. Let y (t ) denote the time response of a continuous data system; then, in general, it can be
written as
y (t ) = y t (t ) + y ss (t ) ………………………………...…………………… (6.1)
Where y t (t ) denotes the transient response and y ss (t ) denotes the steady-state response.
In control system, transient response is defined as the part of the time response that goes to zero as time goes
to infinity. Thus, y t (t ) has the property:
t
lim ∞ y t (t ) = 0 ………………………………………………...………... (6.2)
The steady-state response is simply the part of the total response that remains after the transient has died
out. Thus, the steady-state response can still vary in a fixed pattern, such as a sine wave, or a ramp function that
increases with time.
All real, stable control systems exhibit transient phenomena to some extent before the steady state is
reached. Since inertia, mass, and inductance are unavoidable in physical systems, the response of a typical control
system cannot follow sudden changes in the input instantaneously, and transients are usually observed. Therefore,
the control of the transient response is necessarily important, because it is a significant part of the dynamic
behavior of the system; and the deviation between the output response and the input or the desired response,
before the steady state is reached, must be closely controlled.
The steady-state response of a control system is also very important, since it indicates where the system
output ends up at when time becomes large. For a position control system, the steady-state response when
compared with the desired reference position gives an indication of the final accuracy of the system. In general, if
the steady-state response of the output does not agree with the desired reference exactly, the system is said to have
a steady-state error.
Steady state error
output
0 Transient response Steady State
Output Response t
The study of a control system in the time domain essentially involves the evaluation of the transient and
the steady-state responses of the system. In the design problem, specifications are usually given in terms of the
transient and the steady-state performances, and controllers are designed so that the specifications are all met by
the designed system.
6.2. TYPICAL TEST SIGNALS FOR THE TIME RESPONSE OF CONTROL SYSTEMS:
For the analysis of time response of a control system, the following input signals are used.
, - 94 -
¾ Step Signal: The step is a signal whose value changes from one level (usually zero) to another level ‘A’
in zero time. The step signal is applied to the system to study the behavior of the system for a sudden
change in input. The mathematical representation of the step function is
r (t ) = A u (t )
Where, u (t ) = 1; t ≥ 0.
= 0; t < 0.
In the Laplace transform form,
A
R(s) =
S
¾ Unit step signal: If the magnitude ‘A’ of the step signal is unity, then the step signal is known as unit
step signal is denoted by u(t). That is
u (t ) = 1; t ≥ 0.
= 0; t < 0.
The Laplace transform of the unit step signal u(t) is
1
L[u(t)] =
S
The graphical representation of a step signal is shown in fig. 6.2(a).
r(t) r(t)
A 1
0 t 0 t
Fig. 6.2(a) step signal. Unit step signal.
¾ Ramp Signal: The ramp is a signal which starts at a value of zero and increases linearly with time.
Mathematically, r (t ) = A t ; for t > 0.
= 0; for t < 0.
Where , ‘A’ represents the slope of the line. The Laplace transform of the ramp signal is
A
R(s) =
s2
The graphical representation of a ramp signal is shown in fig. 6.2(b), it is shown that a ramp signal is integral
of a step signal. (If the slope A is unity, then the ramp signal is known as unit ramp signal.)
r(t)
0 t
Fig.6.2(b) ramp signal.
¾ Parabolic Signal: The instantaneous value of a parabolic signal varies as square of the time from an
initial value of zero at t = 0 as shown in fig.6.2(c).
Mathematical representation of the parabolic signal is
At 2
r (t ) = ; t ≥ 0.
2
= 0; t < 0.
The Laplace transform of the parabolic signal is given by
A
R(s ) =
s3
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¾ Unit parabolic signal: If the magnitude of the parabolic signal is unity then it is called unit parabolic
signal.
r(t)
r(t)
2/T
1/T
0 t t
0
T/2 T
Fig.6.2(c) Parabolic signal. Fig.6.2(d) Impulse signal.
¾ Impulse Signal: A unit impulse is defined as a signal which has zero value every where except at t = 0,
where its magnitude is infinite. It is generally called the δ function and has the following property:
δ (t ) = 0 ; t ≠ 0.
+ε
∫ δ (t)dt = 1.
−ε
Where ε tends to zero.
Since a perfect impulse cannot be achieved in practice, it is usually approximated by a pulse of small
width but unit area as shown in fig.6.2(d).
Mathematically, an impulse function is the derivative of a step function i.e.
δ (t ) = u (t )
The Laplace transform of a unit impulse is
Lδ (t ) = 1 = R(s)
C ( s)
The impulse response of a system with transfer function = G ( s ) , is given by
R( s)
C(s) = G(s)R(s)
= G(s)
Or, c(t ) = L−1G ( s) = g (t )
Thus the impulse response of a system, indicated by g(t), is the inverse Laplace transform of its transfer
function. This is sometimes referred to as weighting function of the system. Weighting function of a system can
be used to find the system’s response to any input r (t ) (even though its Laplace transform cannot be found) by
means of the convolution integral. Thus
t
c(t ) = ∫ g (t − τ )r (τ )dτ
0
6.3. TIME RESPONSE OF FIRST-ORDER SYSTEMS:
Let us consider the first order system of fig.6.3 with unity feedback.
E(s) 1 C(s) 1
R(s) ≡
+ sT R(s) 1 + sT C(s)
-
Figure 6.3 Block diagram of a first order system.
The transfer functions of the system from fig.6.3. is
C ( s) 1
= ……………………………………………. (6.3)
R( s) 1 + sT
In the following sections we shall analyze the system response to unit step and unit ramp inputs assuming zero
initial conditions.
CHAPTER – 6
TIME-DOMAIN ANALYSIS OF CONTROL SYSTEMS
6.1. INTRODUCTION:
Since time is used as an independent variable in most control systems, it is usually of interest to evaluate
the state and output responses with respected to time, or simply the time response. In the analysis problem, a
reference input signal is applied to a system, and the performance of the system is evaluated by studying the
system response in the time domain. For instance, if the objective of the control system is to have the output
variable track the input signal, starting at some initial time and initial condition, it is necessary to compare the
input signal, starting at some initial time and initial condition, it is necessary to compare the input and output
responses as functions of time. Therefore, in most control system problems, the final evaluation of performance of
the system is based on the time responses.
The time response of a system is the output of the system as a function of time, when subjected to a known
input.
The time response of a control system is usually divided into two parts: the transient response and the
steady-state response. Let y (t ) denote the time response of a continuous data system; then, in general, it can be
written as
y (t ) = y t (t ) + y ss (t ) ………………………………...…………………… (6.1)
Where y t (t ) denotes the transient response and y ss (t ) denotes the steady-state response.
In control system, transient response is defined as the part of the time response that goes to zero as time goes
to infinity. Thus, y t (t ) has the property:
t
lim ∞ y t (t ) = 0 ………………………………………………...………... (6.2)
The steady-state response is simply the part of the total response that remains after the transient has died
out. Thus, the steady-state response can still vary in a fixed pattern, such as a sine wave, or a ramp function that
increases with time.
All real, stable control systems exhibit transient phenomena to some extent before the steady state is
reached. Since inertia, mass, and inductance are unavoidable in physical systems, the response of a typical control
system cannot follow sudden changes in the input instantaneously, and transients are usually observed. Therefore,
the control of the transient response is necessarily important, because it is a significant part of the dynamic
behavior of the system; and the deviation between the output response and the input or the desired response,
before the steady state is reached, must be closely controlled.
The steady-state response of a control system is also very important, since it indicates where the system
output ends up at when time becomes large. For a position control system, the steady-state response when
compared with the desired reference position gives an indication of the final accuracy of the system. In general, if
the steady-state response of the output does not agree with the desired reference exactly, the system is said to have
a steady-state error.
Steady state error
output
0 Transient response Steady State
Output Response t
The study of a control system in the time domain essentially involves the evaluation of the transient and
the steady-state responses of the system. In the design problem, specifications are usually given in terms of the
transient and the steady-state performances, and controllers are designed so that the specifications are all met by
the designed system.
6.2. TYPICAL TEST SIGNALS FOR THE TIME RESPONSE OF CONTROL SYSTEMS:
For the analysis of time response of a control system, the following input signals are used.
, - 94 -
¾ Step Signal: The step is a signal whose value changes from one level (usually zero) to another level ‘A’
in zero time. The step signal is applied to the system to study the behavior of the system for a sudden
change in input. The mathematical representation of the step function is
r (t ) = A u (t )
Where, u (t ) = 1; t ≥ 0.
= 0; t < 0.
In the Laplace transform form,
A
R(s) =
S
¾ Unit step signal: If the magnitude ‘A’ of the step signal is unity, then the step signal is known as unit
step signal is denoted by u(t). That is
u (t ) = 1; t ≥ 0.
= 0; t < 0.
The Laplace transform of the unit step signal u(t) is
1
L[u(t)] =
S
The graphical representation of a step signal is shown in fig. 6.2(a).
r(t) r(t)
A 1
0 t 0 t
Fig. 6.2(a) step signal. Unit step signal.
¾ Ramp Signal: The ramp is a signal which starts at a value of zero and increases linearly with time.
Mathematically, r (t ) = A t ; for t > 0.
= 0; for t < 0.
Where , ‘A’ represents the slope of the line. The Laplace transform of the ramp signal is
A
R(s) =
s2
The graphical representation of a ramp signal is shown in fig. 6.2(b), it is shown that a ramp signal is integral
of a step signal. (If the slope A is unity, then the ramp signal is known as unit ramp signal.)
r(t)
0 t
Fig.6.2(b) ramp signal.
¾ Parabolic Signal: The instantaneous value of a parabolic signal varies as square of the time from an
initial value of zero at t = 0 as shown in fig.6.2(c).
Mathematical representation of the parabolic signal is
At 2
r (t ) = ; t ≥ 0.
2
= 0; t < 0.
The Laplace transform of the parabolic signal is given by
A
R(s ) =
s3
, - 95 -
¾ Unit parabolic signal: If the magnitude of the parabolic signal is unity then it is called unit parabolic
signal.
r(t)
r(t)
2/T
1/T
0 t t
0
T/2 T
Fig.6.2(c) Parabolic signal. Fig.6.2(d) Impulse signal.
¾ Impulse Signal: A unit impulse is defined as a signal which has zero value every where except at t = 0,
where its magnitude is infinite. It is generally called the δ function and has the following property:
δ (t ) = 0 ; t ≠ 0.
+ε
∫ δ (t)dt = 1.
−ε
Where ε tends to zero.
Since a perfect impulse cannot be achieved in practice, it is usually approximated by a pulse of small
width but unit area as shown in fig.6.2(d).
Mathematically, an impulse function is the derivative of a step function i.e.
δ (t ) = u (t )
The Laplace transform of a unit impulse is
Lδ (t ) = 1 = R(s)
C ( s)
The impulse response of a system with transfer function = G ( s ) , is given by
R( s)
C(s) = G(s)R(s)
= G(s)
Or, c(t ) = L−1G ( s) = g (t )
Thus the impulse response of a system, indicated by g(t), is the inverse Laplace transform of its transfer
function. This is sometimes referred to as weighting function of the system. Weighting function of a system can
be used to find the system’s response to any input r (t ) (even though its Laplace transform cannot be found) by
means of the convolution integral. Thus
t
c(t ) = ∫ g (t − τ )r (τ )dτ
0
6.3. TIME RESPONSE OF FIRST-ORDER SYSTEMS:
Let us consider the first order system of fig.6.3 with unity feedback.
E(s) 1 C(s) 1
R(s) ≡
+ sT R(s) 1 + sT C(s)
-
Figure 6.3 Block diagram of a first order system.
The transfer functions of the system from fig.6.3. is
C ( s) 1
= ……………………………………………. (6.3)
R( s) 1 + sT
In the following sections we shall analyze the system response to unit step and unit ramp inputs assuming zero
initial conditions.
