67
derived assuming u(t) is the only external input to the system and no
consideration is made for uncontrollable inputs such as a disturbance or
reference signal.
In the linear servomechanism problem, the dynamics of the plant and
the controller can be modeled by the following equation
LC:
x(t)
n(t)
F 0
-BH A
x(t)
n(t)
u(t) +
E
w(t) +
0
0
r(t)
B
u(t) = -Kjxit) K2n(t)
y(t) = Hx(t)
(4-44)
where x(t) e Rn is the state of the plant, n(t) e RPr is the state of
the internal model system, u(t) e Rm is the input, y(t) e RP is the
output, w(t) e R^ is a disturbance, and r(t) e RP is the reference. The
system LC is essentially the linear version of the system NC given in
equation (2-29) for the nonlinear servomechanism problem. Note that for
the linear system, it is not necessary for the dimension of the input
and the dimension of the output to be the same.
As already discussed, if the internal model system is chosen
appropriately and the feedback gives asymptotic stability to the system
LC without the exogenous inputs w(t) and r(t); then tracking and
disturbance rejection will occur when w(t) and r(t) are applied. The
precise conditions as to when it is possible to obtain a stabilizing
feedback are conditions (B.l) and (B.2).
Suppose that the conditions for a solution to the linear servo
mechanism problem have been satisfied and the closed-loop system LC has