Dynamic Error Coefficients

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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:

10 10 G1(s)  , G2(s)  s(s 1) s(5s 1)

The static error coefficients are given by:

K p1   , K p2  

K v1  10 , K v2  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:

K G(s)  Ts 1

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

K p  K

K v  0

K a  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 K  2 TK

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

2  n G(s)  2 s  2 n s the static error coefficients are given by

K p  limG(s)  

s  0

 n K v  lim sG(s)  2 s  0

K  lim s 2G(s)  0 a s  0

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

2 E(s) s  2 n s  2 2 R(s) s  2 n s   n 2 1 s  s 2   2  n n 2 s2 1 s  2  n  n

2 2 1 4  2  s   2 s  ....  n   n 

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

 K  n  K 2 2 v

The dynamic acceleration error coefficient is given by

 2 K  n 3 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 n  N N K n1  lim S G(s) for 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:

1 1 1 2 E(s)  R(s)  sR(s)  s R(s)  .... K1 K 2 K 3

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  lime(t)  lim r(t)  r'(t)  r"(t)  K1 K 2 K3  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 :

2 C(s) 1 Ta s  Tb s  ......  2 R(s) 1 T1s  T2 s  .....

In the neighborhood of s=0: C(s) 1 lim  R(s) 1 (T1  Ta )s  ....

The dynamic velocity error coefficient K 2 is given by:

1 K 2  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 )

1 The dynamic velocity error coefficient K 2  T1  Ta

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

C(s) 1  for s  1 R(s) 1 Teq s

1 K  Then the dynamic velocity error coefficient K 2 is: 2 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:

C(s) 1 Teq s G(s)    R(s) 1 T s 1 1 G(s) eq 1 Teq s

The static velocity error coefficient K v is:

1 1 K v  lim sG(s)  lim s  Teq s Teq s  0 s  0