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DYNAMIC ERROR COEFFICIENTS:
A characteristic of the definition of static error coefficients is that only one of the
coefficients assumed a finite value for a given system. The other coefficients are either
zero or infinity. The steady state error obtained through the static error coefficients is
either zero, a finite nonzero value, or infinity. Thus, the variation of the error with time
can not be obtained through the use of such coefficients The dynamic error coefficients
presented below provide some information about how the error varies with time, namely,
whether or not the steady state error of the system with a given input increases in
proportion to t, t 2 , etc.
SYSTEMS WITH DIFFERENT DNAMIC ERROR BUT INDENTICAL STATIC
ERROR COEFFICIENTS:
We shall first demonstrate two systems having different dynamic errors can have
identical static error coefficients. Consider the following two systems:
G1( s) 
10
s( s  1)
,
G 2( s) 
10
s(5s  1)
The static error coefficients are given by:
K p1  
,
K p2  
K v1  10
,
K v 2  10
K a1  0
,
K a2  0
Thus, the two systems have the same steady state error for the same step input. Similar
comments apply to the steady state errors for ramp and parabolic inputs. This analysis
indicates that it is impossible to estimate the system dynamic error from the static error
coefficients.
DYNAMIC ERROR COEFICIENTS:
We shall now introduce dynamic error coefficients to describe dynamic error. We shall limit our
systems to unity feed back ones. By dividing the numerator polynomial of E(s)/R(s) by its
denominator polynomial, E(s)/R(s) can be expanded into a series in ascending power of as
follows:
E ( s)
1
1
1
1 2



s
s  ........
R( s) 1  G ( s) K1 K 2
K3
The coefficients K1 , K 2 , K 3 ,…. of the power series are defined to the dynamic error
coefficients.
Namely,
K1  dynamic position error coefficient
K 2  dynamic velocity error coefficient
K 3  dynamic acceleration error coefficient
In a given system, the dynamic error coefficients are related to the static error coefficients.
Consider the following type 0 system with unity feed back:
G( s) 
K
Ts  1
The static position error, static velocity error and static acceleration error coefficients are,
respectively,
Kp  K
Kv  0
Ka  0
Since E(s)/R(s) can be expanded as:
E ( s)
1  Ts
1
TK



s  .....
R( s ) 1  K  Ts 1  K (1  k ) 2
The dynamic coefficients are given in terms of the static error coefficients as follows:
The dynamic position error coefficient is:
K1  1  K  1  K p
The dynamic velocity error coefficient is:
(1  K ) 2
K2 
TK
As another example, consider the unity feed back control system with the following feed back
transfer function:
G( s) 
 n2
s 2  2n s
the static error coefficients are given by
K p  lim G ( s)  
s0
K v  lim sG ( s ) 
n
2
s0
K a  lim s 2 G ( s )  0
s0
Since E(s)/R(s) can be expanded as:
s 2  2n s
E ( s)

R( s) s 2  2n s   n2
2
n

1

s
2
n
1

2
n
s
s2
s2
 n2
1  4 2  2
s
 s  ....
2
n
 n 
2
The dynamic velocity error coefficient is equal to the static velocity error coefficient; namely
K2 
n
 Kv
2
The dynamic acceleration error coefficient is given by
K3 
 n2
1  4 2
If a similar analysis is made for higher order systems, we can show that for a type N system the
dynamic error coefficients are given by:
K n1  
for
K n 1  lim S N G ( s )
for
nN
nN
Where n=0, 1, 2, ….. the values of K n 1 for n>N are determined by the results of the
expansion of E(s)/R(s) near the origin.
ADVANTAGE OF DYNAMIC ERROR COEFFICIENTS:
An advantage of the dynamic error coefficients becomes clear when E(s) is written in the
following form:
E (s) 
1
1
1 2
R( s) 
sR ( s ) 
s R( s )  ....
K1
K2
K3
The region of convergence of this series is the neighborhood of s=0. This corresponds to t  
in the time domain. The corresponding time solution or the steady state error is given, assuming
all initial conditions are zero and neglecting impulses at t=0 , as follows:
1

1
1
lim e(t )  lim  r (t ) 
r ' (t ) 
r" (t )
K2
K3
 K1

t  t 
The steady state error due to the input function and its derivatives can thus be given in terms of
the dynamic error coefficients. This is an advantage of the dynamic error coefficients.
From the foregoing analysis, it can be seen that if E(s)/R(s) is expanded around the origin into
a power series, successive coefficients of the series indicate the dynamic error of the system
when it is subjected to a slowly varying input. The dynamic error coefficients provide a simple
way estimating the error signal to arbitrary inputs and the steady state error, without requiring
us actually to solve the system differential equation.
A REMARK ON DNAMIC VELOCITY ERROR COEFFICIENTS:
It is important to point out that the dynamic velocity error coefficient K 2 can be estimated from
the time constant of the first order system which approximates the given closed-loop transfer
function in the neighborhood of s=0 .
Consider a unity feedback control system with the following closed-loop transfer function :
C ( s) 1  Ta s  Tb s 2  ......

R( s) 1  T1 s  T2 s 2  .....
In the neighborhood of s=0:
lim
C ( s)
1

R( s ) 1  (T1  Ta ) s  ....
The dynamic velocity error coefficient K 2 is given by:
K2 
1
T1  Ta
This can be verified by expanding the function E(s)/R(s) about the origin as follows:
E (s)
C (s)
1
 1
 1
 1  1  (T1  Ta ) s  ....  (T1  Ta ) s  ....
R( s)
R( s)
1  (T1  Ta )
The dynamic velocity error coefficient K 2 
1
T1  Ta
From the preceding analysis, we may conclude that if C(s)/R(s) is approximated by:
C ( s)
1

R(s) 1  Teq s
for
s 1
Then the dynamic velocity error coefficient K 2 is:
K2 
1
Teq
Note that the dynamic velocity error coefficient K 2 thus obtained is the same as the static
velocity error coefficient. Since C(s)/R(s) can be written:
1
C ( s)
1


R( s ) 1  Teq s
Teq s
G (s)

1
1  G ( s)
1
Teq s
The static velocity error coefficient K v is:
K v  lim sG ( s)  lim s
s0
1
Teq s
s0
Back

1
Teq