Time Response of discrete time systems

 Page 1


Digital Control Module 4 Lecture 1
Module 4: Time Response of discrete time systems
Lecture Note 1
1 Time Response of discrete time systems
Absolute stability is a basic requirement of all control systems. Apart from that, good relative
stability and steady state accuracy are also required in any control system, whether continuous
time or discrete time. Transient response corresponds to the system closed loop poles and
steady state response corresponds to the excitation poles or poles of the input function.
1.1 Transient response speci?cations
Inmanypracticalcontrolsystems, thedesiredperformancecharacteristicsarespeci?edinterms
of time domain quantities. Unit step input is most commonly used in analysis of a system since
it is easy to generate and represent a su?ciently drastic change thus providing useful informa-
tion on both transient and steady state responses.
The transient response of a system depends on the initial conditions. It is a common prac-
tice to consider the system initially at rest.
Consider the digital control system shown in Figure1.
r(t)
e(t)
T
T
+
Plant Hold
Digital
Controller
c(t) c(kT)
-
Figure 1: Block Diagram of a closed loop digital system
Similar to the continuous time case, transient response of a digital control system can also be
characterized by the following.
1. Rise time (t
r
): Time required for the unit step response to rise from 0% to 100% of its
?nal value in case of underdamped system or 10% to 90% of its ?nal value in case of
overdamped system.
2. Delay time (t
d
): Time required for the the unit step response to reach 50% of its ?nal
value.
I. Kar 1
Page 2


Digital Control Module 4 Lecture 1
Module 4: Time Response of discrete time systems
Lecture Note 1
1 Time Response of discrete time systems
Absolute stability is a basic requirement of all control systems. Apart from that, good relative
stability and steady state accuracy are also required in any control system, whether continuous
time or discrete time. Transient response corresponds to the system closed loop poles and
steady state response corresponds to the excitation poles or poles of the input function.
1.1 Transient response speci?cations
Inmanypracticalcontrolsystems, thedesiredperformancecharacteristicsarespeci?edinterms
of time domain quantities. Unit step input is most commonly used in analysis of a system since
it is easy to generate and represent a su?ciently drastic change thus providing useful informa-
tion on both transient and steady state responses.
The transient response of a system depends on the initial conditions. It is a common prac-
tice to consider the system initially at rest.
Consider the digital control system shown in Figure1.
r(t)
e(t)
T
T
+
Plant Hold
Digital
Controller
c(t) c(kT)
-
Figure 1: Block Diagram of a closed loop digital system
Similar to the continuous time case, transient response of a digital control system can also be
characterized by the following.
1. Rise time (t
r
): Time required for the unit step response to rise from 0% to 100% of its
?nal value in case of underdamped system or 10% to 90% of its ?nal value in case of
overdamped system.
2. Delay time (t
d
): Time required for the the unit step response to reach 50% of its ?nal
value.
I. Kar 1
Digital Control Module 4 Lecture 1
3. Peak time (t
p
): Time at which maximum peak occurs.
4. Peak overshoot (M
p
): The di?erence between the maximum peak and the steady state
value of the unit step response.
5. Settling time (t
s
): Time required for the unit step response to reach and stay within 2%
or 5% of its steady state value.
However since the output response is discrete the calculated performance measures may be
slightly di?erent from the actual values. Figure 2 illustrates this. The output has a maximum
value c
max
whereas the maximum value of the discrete output is c
*
max
which is always less than
or equal to c
max
. If the sampling period is small enough compared to the oscillations of the
response then this di?erence will be small otherwise c
*
max
may be completely erroneous.
t
1.0
cmax
c
*
max
c(t)
Figure 2: Unit step response of a discrete time system
1.2 Steady state error
The steady state performance of a stable control system is measured by the steady error due
to step, ramp or parabolic inputs depending on the system type. Consider the discrete time
system as shown in Figure 3.
T
+
_
c(t) r(t)
e(t)
e
*
(t)
Gp(s)
1-e
-Ts
s
H(s)
Figure 3: Block Diagram 2
From Figure 2, we can write
E(s) =R(s)-H(s)C(s)
I. Kar 2
Page 3


Digital Control Module 4 Lecture 1
Module 4: Time Response of discrete time systems
Lecture Note 1
1 Time Response of discrete time systems
Absolute stability is a basic requirement of all control systems. Apart from that, good relative
stability and steady state accuracy are also required in any control system, whether continuous
time or discrete time. Transient response corresponds to the system closed loop poles and
steady state response corresponds to the excitation poles or poles of the input function.
1.1 Transient response speci?cations
Inmanypracticalcontrolsystems, thedesiredperformancecharacteristicsarespeci?edinterms
of time domain quantities. Unit step input is most commonly used in analysis of a system since
it is easy to generate and represent a su?ciently drastic change thus providing useful informa-
tion on both transient and steady state responses.
The transient response of a system depends on the initial conditions. It is a common prac-
tice to consider the system initially at rest.
Consider the digital control system shown in Figure1.
r(t)
e(t)
T
T
+
Plant Hold
Digital
Controller
c(t) c(kT)
-
Figure 1: Block Diagram of a closed loop digital system
Similar to the continuous time case, transient response of a digital control system can also be
characterized by the following.
1. Rise time (t
r
): Time required for the unit step response to rise from 0% to 100% of its
?nal value in case of underdamped system or 10% to 90% of its ?nal value in case of
overdamped system.
2. Delay time (t
d
): Time required for the the unit step response to reach 50% of its ?nal
value.
I. Kar 1
Digital Control Module 4 Lecture 1
3. Peak time (t
p
): Time at which maximum peak occurs.
4. Peak overshoot (M
p
): The di?erence between the maximum peak and the steady state
value of the unit step response.
5. Settling time (t
s
): Time required for the unit step response to reach and stay within 2%
or 5% of its steady state value.
However since the output response is discrete the calculated performance measures may be
slightly di?erent from the actual values. Figure 2 illustrates this. The output has a maximum
value c
max
whereas the maximum value of the discrete output is c
*
max
which is always less than
or equal to c
max
. If the sampling period is small enough compared to the oscillations of the
response then this di?erence will be small otherwise c
*
max
may be completely erroneous.
t
1.0
cmax
c
*
max
c(t)
Figure 2: Unit step response of a discrete time system
1.2 Steady state error
The steady state performance of a stable control system is measured by the steady error due
to step, ramp or parabolic inputs depending on the system type. Consider the discrete time
system as shown in Figure 3.
T
+
_
c(t) r(t)
e(t)
e
*
(t)
Gp(s)
1-e
-Ts
s
H(s)
Figure 3: Block Diagram 2
From Figure 2, we can write
E(s) =R(s)-H(s)C(s)
I. Kar 2
Digital Control Module 4 Lecture 1
We will consider the steady state error at the sampling instants.
From ?nal value theorem
lim
k?8
e(kT) = lim
z?1

(1-z
-1
)E(z)

G(z) = (1-z
-1
)Z

G
p
(s)
s

GH(z) = (1-z
-1
)Z

G
p
(s)H(s)
s

C(z)
R(z)
=
G(z)
1+GH(z)
Again,E(z) = R(z)-GH(z)E(z)
or,E(z) =
1
1+GH(z)
R(z)
?e
ss
= lim
z?1

(1-z
-1
)
1
1+GH(z)
R(z)

The steady state error of a system with feedback thus depends on the input signal R(z) and
the loop transfer function GH(z).
1.2.1 Type-0 system and position error constant
Systems having a ?nite nonzero steady state error with a zero order polynomial input (step
input) are called Type-0 systems. The position error constant for a system is de?ned for a step
input.
r(t) = u
s
(t) unit step input
R(z) =
1
1-z
-1
e
ss
= lim
z?1
1
1+GH(z)
=
1
1+K
p
where K
p
= lim
z?1
GH(z) is known as the position error constant.
1.2.2 Type-1 system and velocity error constant
Systems having a ?nite nonzero steady state error with a ?rst order polynomial input (ramp
input) are called Type-1 systems. The velocity error constant for a system is de?ned for a
ramp input.
r(t) = u
r
(t) unit ramp
R(z) =
Tz
(z-1)
2
=
TZ
-1
(1-Z
-1
)
2
e
ss
= lim
z?1
T
(z-1)GH(z)
=
1
K
v
where K
v
=
1
T
lim
z?1
[(z-1)GH(z)] is known as the velocity error constant.
I. Kar 3
Page 4


Digital Control Module 4 Lecture 1
Module 4: Time Response of discrete time systems
Lecture Note 1
1 Time Response of discrete time systems
Absolute stability is a basic requirement of all control systems. Apart from that, good relative
stability and steady state accuracy are also required in any control system, whether continuous
time or discrete time. Transient response corresponds to the system closed loop poles and
steady state response corresponds to the excitation poles or poles of the input function.
1.1 Transient response speci?cations
Inmanypracticalcontrolsystems, thedesiredperformancecharacteristicsarespeci?edinterms
of time domain quantities. Unit step input is most commonly used in analysis of a system since
it is easy to generate and represent a su?ciently drastic change thus providing useful informa-
tion on both transient and steady state responses.
The transient response of a system depends on the initial conditions. It is a common prac-
tice to consider the system initially at rest.
Consider the digital control system shown in Figure1.
r(t)
e(t)
T
T
+
Plant Hold
Digital
Controller
c(t) c(kT)
-
Figure 1: Block Diagram of a closed loop digital system
Similar to the continuous time case, transient response of a digital control system can also be
characterized by the following.
1. Rise time (t
r
): Time required for the unit step response to rise from 0% to 100% of its
?nal value in case of underdamped system or 10% to 90% of its ?nal value in case of
overdamped system.
2. Delay time (t
d
): Time required for the the unit step response to reach 50% of its ?nal
value.
I. Kar 1
Digital Control Module 4 Lecture 1
3. Peak time (t
p
): Time at which maximum peak occurs.
4. Peak overshoot (M
p
): The di?erence between the maximum peak and the steady state
value of the unit step response.
5. Settling time (t
s
): Time required for the unit step response to reach and stay within 2%
or 5% of its steady state value.
However since the output response is discrete the calculated performance measures may be
slightly di?erent from the actual values. Figure 2 illustrates this. The output has a maximum
value c
max
whereas the maximum value of the discrete output is c
*
max
which is always less than
or equal to c
max
. If the sampling period is small enough compared to the oscillations of the
response then this di?erence will be small otherwise c
*
max
may be completely erroneous.
t
1.0
cmax
c
*
max
c(t)
Figure 2: Unit step response of a discrete time system
1.2 Steady state error
The steady state performance of a stable control system is measured by the steady error due
to step, ramp or parabolic inputs depending on the system type. Consider the discrete time
system as shown in Figure 3.
T
+
_
c(t) r(t)
e(t)
e
*
(t)
Gp(s)
1-e
-Ts
s
H(s)
Figure 3: Block Diagram 2
From Figure 2, we can write
E(s) =R(s)-H(s)C(s)
I. Kar 2
Digital Control Module 4 Lecture 1
We will consider the steady state error at the sampling instants.
From ?nal value theorem
lim
k?8
e(kT) = lim
z?1

(1-z
-1
)E(z)

G(z) = (1-z
-1
)Z

G
p
(s)
s

GH(z) = (1-z
-1
)Z

G
p
(s)H(s)
s

C(z)
R(z)
=
G(z)
1+GH(z)
Again,E(z) = R(z)-GH(z)E(z)
or,E(z) =
1
1+GH(z)
R(z)
?e
ss
= lim
z?1

(1-z
-1
)
1
1+GH(z)
R(z)

The steady state error of a system with feedback thus depends on the input signal R(z) and
the loop transfer function GH(z).
1.2.1 Type-0 system and position error constant
Systems having a ?nite nonzero steady state error with a zero order polynomial input (step
input) are called Type-0 systems. The position error constant for a system is de?ned for a step
input.
r(t) = u
s
(t) unit step input
R(z) =
1
1-z
-1
e
ss
= lim
z?1
1
1+GH(z)
=
1
1+K
p
where K
p
= lim
z?1
GH(z) is known as the position error constant.
1.2.2 Type-1 system and velocity error constant
Systems having a ?nite nonzero steady state error with a ?rst order polynomial input (ramp
input) are called Type-1 systems. The velocity error constant for a system is de?ned for a
ramp input.
r(t) = u
r
(t) unit ramp
R(z) =
Tz
(z-1)
2
=
TZ
-1
(1-Z
-1
)
2
e
ss
= lim
z?1
T
(z-1)GH(z)
=
1
K
v
where K
v
=
1
T
lim
z?1
[(z-1)GH(z)] is known as the velocity error constant.
I. Kar 3
Digital Control Module 4 Lecture 1
1.2.3 Type-2 system and acceleration error constant
Systems having a ?nite nonzero steady state error with a second order polynomial input
(parabolic input) are called Type-2 systems. The acceleration error constant for a system
is de?ned for a parabolic input.
R(z) =
T
2
z(z+1)
2(z-1)
3
=
T
2
(1+z
-1
)z
-1
2(1-z
-1
)
3
e
ss
=
T
2
2
lim
z?1
(z+1)
(z-1)
2
[1+GH(z)]
=
1
lim
z?1
(z-1)
2
T
2
GH(z)
=
1
K
a
where K
a
= lim
z?1
(z-1)
2
T
2
GH(z) is known as the acceleration error constant.
Table 1 shows the steady state errors for di?erent types of systems for di?erent inputs.
Table 1: Steady state errors
System Step input Ramp input Parabolic input
Type-0
1
1+K
p
8 8
Type-1 0
1
K
v
8
Type-2 0 0
1
K
a
Example 1: Calculatethesteadystateerrorsforunitstep,unitrampandunitparabolicinputs
for the system shown in Figure 4.
+
-
C(s)
+
-
ZOH
E(s)
R(s)
E
*
(s) 1
10s
1
s
1000
500
Figure 4: Block Diagram for Example 1
Solution: The open loop transfer function is:
G(s) =
C(s)
E
*
(s)
=G
ho
(s)G
p
(s)
=
1-e
-Ts
s
1000/10
s(s+500/10)
I. Kar 4
Page 5


Digital Control Module 4 Lecture 1
Module 4: Time Response of discrete time systems
Lecture Note 1
1 Time Response of discrete time systems
Absolute stability is a basic requirement of all control systems. Apart from that, good relative
stability and steady state accuracy are also required in any control system, whether continuous
time or discrete time. Transient response corresponds to the system closed loop poles and
steady state response corresponds to the excitation poles or poles of the input function.
1.1 Transient response speci?cations
Inmanypracticalcontrolsystems, thedesiredperformancecharacteristicsarespeci?edinterms
of time domain quantities. Unit step input is most commonly used in analysis of a system since
it is easy to generate and represent a su?ciently drastic change thus providing useful informa-
tion on both transient and steady state responses.
The transient response of a system depends on the initial conditions. It is a common prac-
tice to consider the system initially at rest.
Consider the digital control system shown in Figure1.
r(t)
e(t)
T
T
+
Plant Hold
Digital
Controller
c(t) c(kT)
-
Figure 1: Block Diagram of a closed loop digital system
Similar to the continuous time case, transient response of a digital control system can also be
characterized by the following.
1. Rise time (t
r
): Time required for the unit step response to rise from 0% to 100% of its
?nal value in case of underdamped system or 10% to 90% of its ?nal value in case of
overdamped system.
2. Delay time (t
d
): Time required for the the unit step response to reach 50% of its ?nal
value.
I. Kar 1
Digital Control Module 4 Lecture 1
3. Peak time (t
p
): Time at which maximum peak occurs.
4. Peak overshoot (M
p
): The di?erence between the maximum peak and the steady state
value of the unit step response.
5. Settling time (t
s
): Time required for the unit step response to reach and stay within 2%
or 5% of its steady state value.
However since the output response is discrete the calculated performance measures may be
slightly di?erent from the actual values. Figure 2 illustrates this. The output has a maximum
value c
max
whereas the maximum value of the discrete output is c
*
max
which is always less than
or equal to c
max
. If the sampling period is small enough compared to the oscillations of the
response then this di?erence will be small otherwise c
*
max
may be completely erroneous.
t
1.0
cmax
c
*
max
c(t)
Figure 2: Unit step response of a discrete time system
1.2 Steady state error
The steady state performance of a stable control system is measured by the steady error due
to step, ramp or parabolic inputs depending on the system type. Consider the discrete time
system as shown in Figure 3.
T
+
_
c(t) r(t)
e(t)
e
*
(t)
Gp(s)
1-e
-Ts
s
H(s)
Figure 3: Block Diagram 2
From Figure 2, we can write
E(s) =R(s)-H(s)C(s)
I. Kar 2
Digital Control Module 4 Lecture 1
We will consider the steady state error at the sampling instants.
From ?nal value theorem
lim
k?8
e(kT) = lim
z?1

(1-z
-1
)E(z)

G(z) = (1-z
-1
)Z

G
p
(s)
s

GH(z) = (1-z
-1
)Z

G
p
(s)H(s)
s

C(z)
R(z)
=
G(z)
1+GH(z)
Again,E(z) = R(z)-GH(z)E(z)
or,E(z) =
1
1+GH(z)
R(z)
?e
ss
= lim
z?1

(1-z
-1
)
1
1+GH(z)
R(z)

The steady state error of a system with feedback thus depends on the input signal R(z) and
the loop transfer function GH(z).
1.2.1 Type-0 system and position error constant
Systems having a ?nite nonzero steady state error with a zero order polynomial input (step
input) are called Type-0 systems. The position error constant for a system is de?ned for a step
input.
r(t) = u
s
(t) unit step input
R(z) =
1
1-z
-1
e
ss
= lim
z?1
1
1+GH(z)
=
1
1+K
p
where K
p
= lim
z?1
GH(z) is known as the position error constant.
1.2.2 Type-1 system and velocity error constant
Systems having a ?nite nonzero steady state error with a ?rst order polynomial input (ramp
input) are called Type-1 systems. The velocity error constant for a system is de?ned for a
ramp input.
r(t) = u
r
(t) unit ramp
R(z) =
Tz
(z-1)
2
=
TZ
-1
(1-Z
-1
)
2
e
ss
= lim
z?1
T
(z-1)GH(z)
=
1
K
v
where K
v
=
1
T
lim
z?1
[(z-1)GH(z)] is known as the velocity error constant.
I. Kar 3
Digital Control Module 4 Lecture 1
1.2.3 Type-2 system and acceleration error constant
Systems having a ?nite nonzero steady state error with a second order polynomial input
(parabolic input) are called Type-2 systems. The acceleration error constant for a system
is de?ned for a parabolic input.
R(z) =
T
2
z(z+1)
2(z-1)
3
=
T
2
(1+z
-1
)z
-1
2(1-z
-1
)
3
e
ss
=
T
2
2
lim
z?1
(z+1)
(z-1)
2
[1+GH(z)]
=
1
lim
z?1
(z-1)
2
T
2
GH(z)
=
1
K
a
where K
a
= lim
z?1
(z-1)
2
T
2
GH(z) is known as the acceleration error constant.
Table 1 shows the steady state errors for di?erent types of systems for di?erent inputs.
Table 1: Steady state errors
System Step input Ramp input Parabolic input
Type-0
1
1+K
p
8 8
Type-1 0
1
K
v
8
Type-2 0 0
1
K
a
Example 1: Calculatethesteadystateerrorsforunitstep,unitrampandunitparabolicinputs
for the system shown in Figure 4.
+
-
C(s)
+
-
ZOH
E(s)
R(s)
E
*
(s) 1
10s
1
s
1000
500
Figure 4: Block Diagram for Example 1
Solution: The open loop transfer function is:
G(s) =
C(s)
E
*
(s)
=G
ho
(s)G
p
(s)
=
1-e
-Ts
s
1000/10
s(s+500/10)
I. Kar 4
Digital Control Module 4 Lecture 1
Taking Z-transform
G(z) = 2(1-z
-1
) Z

1
s
2
-
10
500s
+
10
500(s+5000)

= 2(1-z
-1
)

Tz
(z-1)
2
-
10z
500(z-1)
+
10z
500(z-e
-50T
)

=
1
250

(500T-10+10e
-50T
)z-(500T +10)e
-50T
+10
(z-1)(z-e
-50T
)

Steady state error for step input =
1
1+K
p
where K
p
= lim
z?1
G(z) =8.?e
step
ss
= 0.
Steadystateerrorforrampinput=
1
K
v
whereK
v
=
1
T
lim
z?1
[(z-1)G(z)] = 2.?e
ramp
ss
= 0.5.
Steady state error for parabolic input =
1
K
a
where K
a
=
1
T
2
lim
z?1
[(z-1)
2
G(z)] = 0. ?
e
para
ss
=8.
I. Kar 5