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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, 2t , etc.

SYSTEMS WITH DIFFERENT DNAMIC ERROR BUT INDENTICAL STATICERROR COEFFICIENTS:

We shall first demonstrate two systems having different dynamic errors can have

identical static error coefficients. Consider the following two systems:

)1(

10)(1

sssG ,

)15(

10)(2

sssG

The static error coefficients are given by:

0,0

10,10

,

21

21

21

aa

vv

pp

KK

KK

KK

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:

........111

)(1

1

)(

)( 2

321

sK

sKKsGsR

sE

The coefficients 321 ,, KKK ,…. of the power series are defined to the dynamic error

coefficients.

Namely,

1K dynamic position error coefficient

2K dynamic velocity error coefficient

3K 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:

1)(

Ts

KsG

The static position error, static velocity error and static acceleration error coefficients are,

respectively,

0

0

a

v

p

K

K

KK

Since E(s)/R(s) can be expanded as:

.....)1(1

1

1

1

)(

)(2

sk

TK

KTsK

Ts

sR

sE

The dynamic coefficients are given in terms of the static error coefficients as follows:

The dynamic position error coefficient is:

pKKK 111

The dynamic velocity error coefficient is:

TK

KK

2

2

)1(

As another example, consider the unity feed back control system with the following feed back

transfer function:

sssG

n

n

2

)(2

2

the static error coefficients are given by

0

)(lim

s

sGK p

0

2)(lim

s

ssGK nv

0

0)(lim 2

s

sGsK a

Since E(s)/R(s) can be expanded as:

22

2

2

2

)(

)(

nn

n

ss

ss

sR

sE

22

22

21

12

nn

nn

ss

ss

....412 2

2

2

ss

nn

The dynamic velocity error coefficient is equal to the static velocity error coefficient; namely

vn KK

22

The dynamic acceleration error coefficient is given by

2

2

3 41

nK

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:

NnforK n 1NnforsGSK N

n )(lim1

Where n=0, 1, 2, ….. the values of 1nK 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:

....)(1

)(1

)(1

)( 2

321

sRsK

ssRK

sRK

sE

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:

tt

trK

trK

trK

te )("1

)('1

)(1

lim)(lim321

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 2K 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 :

.....1

......1

)(

)(2

21

2

sTsT

sTsT

sR

sC ba

In the neighborhood of s=0:

....)(1

1

)(

)(lim

1

sTTsR

sC

a

The dynamic velocity error coefficient 2K is given by:

aTTK

12

1

This can be verified by expanding the function E(s)/R(s) about the origin as follows:

....)(....)(11)(1

11

)(

)(1

)(

)(11

1

sTTsTTTTsR

sC

sR

sEaa

a

The dynamic velocity error coefficientaTT

K

1

2

1

From the preceding analysis, we may conclude that if C(s)/R(s) is approximated by:

11

1

)(

)(

sfor

sTsR

sC

eq

Then the dynamic velocity error coefficient 2K is:eqT

K1

2

Note that the dynamic velocity error coefficient 2K thus obtained is the same as the static

velocity error coefficient. Since C(s)/R(s) can be written:

)(1

)(1

1

1

1

1

)(

)(

sG

sG

sT

sT

sTsR

sC

eq

eq

eq

The static velocity error coefficient vK is:

00

11lim)(lim

ss

TsTsssGK

eqeqv

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