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342 Chapter 5 The Performance of Feedback Control Systems

Equation (5.56) with q = 1 requires that M 2 = A 2; therefore,
49
+ d 2 = — (5.64)
-2d 2
Completing the process for M 4 = A 4, we obtain

2
d 2 = ^ . (5.65)
Solving Equations (5.64) and (5.65) yields d x = 1.615 and d 2 = 0.624. (The other
sets of solutions are rejected because they lead to unstable poles.) The lower-order
system transfer function is

G L(s) = - = 3^5 ( 5 66)
1 + 1.615.V + 0.624^ 2 s 2 + 2.5905 + 1.60 v '

It is interesting to see that the poles of G H(s) are s = —I, 2 , 3 , whereas the poles
-
-
of G L(s) are 5 = —1.024 and -1.565. Because the lower-order model has two poles,
we estimate that we would obtain a slightly overdamped step response with a set-
tling time to within 2% of the final value in approximately 3 seconds. •
It is sometimes desirable to retain the dominant poles of the original system,
G H(s), in the low-order model. This can be accomplished by specifying the denomi-
nator of G L(s) to be the dominant poles of G H(s) and allowing the numerator of
G L(s) to be subject to approximation.
Another novel and useful method for reducing the order is the Routh approxi-
mation method based on the idea of truncating the Routh table used to determine
stability. The Routh approximants can be computed by a finite recursive algorithm
that is suited for programming on a digital computer [19].
A robot named Domo was developed to investigate robot manipulation in unstruc-
tured environments [22-23].The robot shown in Figure 5.33 has 29 degrees of freedom,
making it a very complex system. Domo employs two six-degree-of-freedom arms and
hands with compliant and force-sensitive actuators coupled with a behavior-based sys-
tem architecture to achieve robotic manipulation tasks in human environments. Design-
ing a controller to control the motion of the arm and hands would require significant
model reduction and approximation before the methods of design discussed in the sub-
sequent chapters (e.g., root locus design methods) could be successfully applied.

5.9 DESIGN EXAMPLES

In this section we present two illustrative examples. The first example is a simplified
view of the Hubble space telescope pointing control problem. The Hubble space tele-
scope problem highlights the process of computing controller gains to achieve de-
sired percent overshoot specifications, as well as meeting steady-state error
specifications. The second example considers the control of the bank angle of an air-
plane. The airplane attitude motion control example represents a more in-depth look