Page 498 - Mechanical Engineers' Handbook (Volume 2)

7 Steady-State Performance and System Type 489

Figure 33 Tracking system: (a) open-loop system; (b) closed-loop system.

The CLCE is

G (s) 1
1 1 0 (95)
s 2 s 2
3
2
That is, s 2s G (s) 0.
1
Now for closed-loop stability, G (s) must be of the form
1
2
G (s) s s 3
1
1
2
yielding the CLCE
3
s ( 2)s s 0 (96)
2
1
2
3
From the Routh–Hurwitz stability criterion it can be shown that for asymptotic stability
2, 0, and ( 2) 0. So pick 4, 1, 1, yielding the
1 3 1 2 3 1 2 3
controller
2
4s s 1
G (s) (97)
c
s 2
In an actual design situation the choice of , , and will be dictated by the transient
1 2 3
design specifications.
7.4 Indices of Performance
Sometimes the performance of control systems is given in terms of a performance index. A
performance index is a number that indicates the ‘‘goodness’’ of system performance. Such
performance indices are usually employed to get an optimal design in the sense of mini-
mizing or maximizing the index of performance. The optimal parameter values depend di-
2
rectly upon the performance index chosen. Typical performance indices are 0 e (t) dt, 0
te (t) dt, 0 e(t) dt, 0 t e(t) dt. In control system design the task is to select the controller
2
parameters to minimize the chosen index.
7.5 Integral-Square-Error (ISE) Criterion
According to the ISE criterion, the quality of system performance is evaluated by minimizing
the integral