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INTRODUCTION & THEORY
STATIC ERROR COEFFICIENTS:
The transient-response characteristics are important performance features of control
systems. Another important feature is concerned with system error. Errors in a control
system can be attributed to many factors. Changes in the reference input will cause
unavoidable errors during transient periods and may also cause steady-state errors.
Imperfections in the system components, such as static friction, backlash, and amplifier
drift, as well as aging or deterioration, will cause errors at steady state.
The steady state error is a measure is a measure of the accuracy of a control system. The
steady state performance of a stable control system is generally judged by the steady state
error due to step, ramp, or acceleration inputs. We shall investigate such performance.
Namely, we shall investigate a type of steady state error which is caused by the
incapability of a system to follow particular types of inputs.
Any physical control system inherently suffers steady state error in response to certain
types of inputs. A system may have no steady state error to a step input., but the same
system may exhibit nonzero steady state error to a ramp input. ( the only way we may be
able to eliminate this error is to modify the system structure). Whether or not a given
system will exhibit steady state error for a given type of input depends upon the type of
open-loop transfer function of the system.
CALSSIFICATION OF CONTROL SYSTEMS:
Control systems may be classified according to their ability to follow step inputs, ramp
inputs, parabolic inputs, etc. this is a reasonable classification scheme because actual
inputs may frequently be considered combinations of such inputs. The magnitudes of the
steady state errors due to those individual inputs are indicative of the “goodness” of the
system. Consider the following open-loop transfer function G(s)H(s):
G( s) H ( s) 
K (Ta s  1)(Tb s  1)....(Tm s  1)
s N (T1 s  1)(T2 s  1)....(T2 s  1)
It involves the term S N in the denominator, representing a pole of multiplicity N at the
origin. The present classification scheme is based on the number of integration indicated
by the open-loop transfer function. A system is called type 0, type 1, type 2, …if N=0,
N=1, N=2,…. Respectively. Note that this classification is different from that of the order
of a system. As the type number is increased, accuracy is improved; however, increasing
the type number aggravates stability problem. A compromise between steady state
accuracy and relative stability is always necessary. In practice, it is rather exceptional to
have type 3 or higher systems because we find it generally difficult to design stable
systems having more than two integration in the feed forward path.
We shall see later if G(s)H(s) is written so that each term in the numerator and
denominator, except the term S N , approaches unity as S approaches zero, then openloop gain K is directly related to the steady state error.
STEADY STATE ERRORS:
Consider the system shown below, The closed loop transfer function is:
R(s)
+
E(s)
G(s)
C(s)
H(s)
C ( s)
G( s)

R( s ) 1  G ( s) H ( s )
The transfer function between the actuating error signal e(t) and the input signal r(t) is:
E ( s)
G( s) H ( s)
1
 1

R( s)
R( s )
1  G( s) H ( s)
Where the actuating error e(t) is the difference between the input signal and the feedback
signal.
The final value theorem provides a convenient way to find the steady state performance
of a stable system. Since E(s) is:
E ( s) 
1
R( s)
1  G( s) H ( s)
The steady state actuating error is:
sR ( s )
1  G ( s) H ( s)
s0
ess  lim e(t )  lim
t 
The static error coefficients defined in the following are figures of merit of control
systems. The higher the coefficients, the smaller the steady state error. In a given system,
the output may be the position, pressure, temperature, etc. The physical form of the
output, however, is immaterial to the present analysis. Therefore, in what follows, we
shall call the output “position”, the rate of charge of the output “velocity”, etc. This
means that in a temperature control system “position” represents the output temperature,
“velocity” represent the rate of charge of the output temperature, etc.
STATIC POSITION ERROR COEEFFICIENT Kp
The steady state actuating error of the system for a unit step input is:
sR ( s )
1
1

1  G ( s ) H ( s ) s 1  G (0) H (0)
s0
ess  lim e(t )  lim
The static position error coefficient K p is defined by:
K p  lim G ( s) H ( s)  G (0) H (0)
s0
Thus, the steady state actuating error in terms of the static position error coefficient K p is
given by:
ess 
1
1 K p
For a type 0 system,
K p  lim
K (Ta s  1)(Tb s  1)....
K
(T1 s  1)(T2 s  1)....
s0
For a type 1 or higher system,
K p  lim
s0
K (Ta s  1)(Tb s  1)....
 ........( N  1)
(T1 s  1)(T2 s  1)....
Hence, for a type 0 system, the static position error coefficient K p is finite, while for a
type 1 or higher system K p is infinite.
For a unit step input, the steady state actuating error e ss may be summarized as follows:
e ss 
1
1 K
e ss =0
for type 0 system
for type 1 or higher system
From the foregoing analysis, it is seen that the response of a unity-feedback control
system to a step input involves a steady state error if there is no integration in the
feedback path. (If small errors for step inputs can be tolerated, then a type 0 system may
be permissible, provided that the gain K is sufficiently large. If the gain K is too large,
however, it is difficult to obtain reasonable relative stability). If zero steady state error for
a step input is desired, the type of the system must be one or higher.
STATIC VELOCITY ERROR COEFFICIENT Kv
The steady state actuating error of the system with a unit step input (unit velocity input) is
given by:
s
1
1
 lim
2
1  G(s) H (s) s
1  sG ( s ) H ( s )
s0
s0
ess  lim e(t )  lim
The static velocity error coefficient K v is defined by:
K v  lim sG ( s) H ( s)
s0
Thus, the steady state actuating error in terms of the static velocity error coefficient K v is
given by:
e ss 
1
Kv
The term velocity error is used here to express the steady state error for a ramp input. The
dimension of the velocity error is the same as the system error. That is, velocity error is
not an error in velocity, but is an error in position due to a ramp input.
For a type zero system:
K v  lim
sK (Ta s  1)(Tb s  1)....
0
(T1 s  1)(T2 s  1)....
s0
For a type 1 system:
sK (Ta s  1)(Tb s  1)....
K
(T1 s  1)(T2 s  1)....
K v  lim
s0
For a type 2 system:
K p  lim
sK (Ta s  1)(Tb s  1)....
 ........( N  2)
s N (T1 s  1)(T2 s  1)....
s0
The steady state actuating error e ss for the unit ramp input can be summarized as follows:
ess 
1

Kv
ess 
1
1

Kv K
ess 
1
0
Kv
For type 0 system
For type 1 system
For type 2 system
The foregoing analysis indicates that a type 0 system is incapable of following a ramp
input in the steady state. The type 1 system with unity feedback can follow the ramp
input with a finite error. In steady state operation, the output velocity is exactly the same
as the input velocity, but there is a positional error. This error is proportional to the
velocity of the input and is inversely proportional to the gain K. The type 2 or higher
system can follow a ramp input with zero input actuating error at steady state.
STATIC ACCELERATION ERROR COEFFICIENT Ka
The steady state actuating error of the system with a unit parabolic input (acceleration
input) which is defined by:
t2
2
0
r (t ) 
for
t0
for t  0
is given by:
s
1
1

3
2
1  G ( s) H ( s) s
lim s G ( s ) H ( s )
s0
s0
e ss  lim e(t )  lim
The static acceleration error coefficient K a is defined by the equation:
K a  lim s 2 G ( s) H ( s)
s0
The steady state actuating error is then:
ess 
1
Ka
Note that the acceleration error, the steady state error due to a parabolic input is an error
in position.
The values of K a are obtained as follows:
For type 0 system:
s 2 K (Ta s  1)(Tb s  1)....
K a  lim
0
(T1 s  1)(T2 s  1)....
s0
For type 1 system:
s 2 K (Ta s  1)(Tb s  1)....
K a  lim
0
s(T1 s  1)(T2 s  1)....
s0
For type 2 system:
s 2 K (Ta s  1)(Tb s  1)....
K a  lim
K
(T1 s  1)(T2 s  1)....
s0
For type 3 or higher:
K a  lim
s 2 K (Ta s  1)(Tb s  1)....

s N (T1 s  1)(T2 s  1)....
( N  3)
s0
Thus, the steady state actuating error for the unit parabolic input is:
ess   for type 0 and type 1 system
1
ess 
for type 2 system
K
ess  0
for type 3 or higher system
Note that both type 0 and type 1 systems are incapable of following a parabolic input in
the steady state. The type 2 system with unity feedback can follow a parabolic input with
a finite actuating error signal. The type 3 or higher system with unity feedback follows a
parabolic input with zero actuating error at steady state.
SUMMARY:
Table below summarizes the steady state errors for type 0, type 1, type 2 systems when
they are subjected to various inputs. The finite values for steady state errors appear on the
diagonal line. Above the diagonal, the steady state errors are infinity; below the diagonal
they are zero.
As we know, the terms “position error “, “ velocity error”, and “acceleration error” mean
steady state deviations in the output position. A finite velocity error Implies that after
transients have died out, input and the output move at the same velocity but have a finite
position difference.
K a describe the ability of a system to reduce or
The error coefficients K p , K v and
eliminate steady state error. Therefore, they are indicative of the steady state
performance. It is generally describe to increase the error coefficients, while maintaining
the transient response within an acceptable range. If there is only conflict between the
static velocity error coefficient and the static acceleration error coefficient, then the later
may be considered less important than the former. It is noted that to improve the steady
state performance, we can increase the type of the system by adding an integration or
integration to the feed forward path. This, however, introduce an additional stability
problem. The design of satisfactory system with more than two integrations in the feed
forward path is generally difficult.
Type 0 system
Type 1 system
Step Input
Ramp Input
Acceleration Input
r(t)=1
r(t)=t
1
1 K
0

1
r(t)= t 2
2

1
K

Type 2 system
0
0
1
K
CORRELATION BETWEEN INTEGRAL OF ERROR IN STEP
RESPONSE AND STEADY STATE ERROR IN RAMP RESPONSE
We shall next the correlation between the integral of the unit step response of a system
and the steady state error in the unit ramp response of the same system.

In a unity feed back control system, the total area under the error curve  e(t )dt as a
0
result of unit step response gives the steady state error in the response to the unit ramp
input. To prove this, let us define:

 e(t )     st e(t )dt  E ( s)
0
Then:


0
0
lim    st e(t )dt   e(t )dt  lim E ( s)
s0
s0
Note that:
E ( s)
C ( s)
1
 1

R( s)
R( s) 1  G ( s )
Hence:

 R( s ) 
 e(t )dt  lim 1  G(s) 
0
s0
For a unit step input:


1
1
1
1
 e(t )dt  lim 1  G(s) s   lim sG(s)  K
0
s0
 Steady state error in unit ramp
v
s0
Hence in a unity feed back control system, we have

 e(t )dt  e
ssr
0
Where:
e(t)=error in the unit step response
e ssr  steady state error in the unit ramp response
If essr is zero, then e (t) must change its sign at least once. This means that a zero
velocity error system (a system having K v   ) will exhibit at least overshoot, when the
system is subjected to a step input.
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