33
The above restriction is actually somewhat misleading.
Specifically, consider a system having more inputs than outputs (i.e.,
m > p). It is quite possible that when only p out of the m inputs are
used, all conditions for solving the servomechanism problem will be
satisfied. In this case, we can define a new system NP which is nothing
more than the original system N operating with only p inputs. This is
shown by the following equations:
NP:
x(t) = f(x(t), u(t), w(t)) = fp(x(t), up(t), w(t))
y(t) = Hx(t) (2-56)
u(t) = Mup(t)
where M e RmxP is of full rank. If there exists an M such that the
system Np with input Up(t) meets all conditions given in the previous
sections, then the problem can be solved. Working with the input Up(t),
let the feedback law which stabilizes the closed-loop transient system
be
uP(t)
x(t) K ?n(t)
K 9C
(2-57)
In terms of the original system N, the feedback law will then be
where
u(t) = -K-^xit) K2n(t)
K1 "
MK
P1
K9 = MK 9
2 p,2
(2-58)
(2-59)