Page 366 - Modern Control Systems

340 Chapter 5 The Performance of Feedback Control Systems

A more sophisticated approach attempts to match the frequency response of
the reduced-order transfer function with the original transfer function frequency
response as closely as possible. Although frequency response methods are covered
in Chapter 8, the associated approximation method strictly relies on algebraic ma-
nipulation and is presented here. We will let the high-order system be described by
the transfer function

n l
a ms m + a m- iS' ~ + ••• + 0 ^ + 1
GH(S) = K"; n n , 7 ,,_! , r^T^i (5.51)
1
b ns + 6,,-is"" + ••• + b xs + 1
in which the poles are in the left-hand s-plane and m ^ n. The lower-order approx-
imate transfer function is

p
C nS + ••• + C]S + 1
G L(s) = K^—- (5.52)

where p < g < n. Notice that the gain constant, K, is the same for the original
and approximate system; this ensures the same steady-state response. The method
outlined in Example 5.9 is based on selecting C/ and d L in such a way that G L{s) has
a frequency response (see Chapter 8) very close to that of G H{s). This is equiva-
lent to stating that G H(j<o)/G L(j<o) is required to deviate the least amount from
unity for various frequencies. The c and d coefficients are obtained by using the
equations

d k
(k
M \s) =-^ M(s) (5.53)
dsr
and
(
A * t o ^ A ( j ) , (5.54)
ds k
where M(s) and A ( J ) are the numerator and denominator polynomials of
G H(s)/G L(s), respectively. We also define

2, ( - l f # > ( Q ) M ^ - ^ ( Q )
M = = 2 5 55
* S *,<2*-*)! ' * °^ • * * ( - >

and an analogous equation for A 2f/. The solutions for the c and d coefficients are
obtained by equating

= (5.56)
M lq A 2/?
for q = 1,2,... up to the number required to solve for the unknown coefficients.
Let us consider an example to clarify the use of these equations.