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) F+o
= + u(t) + [ E]ww(t) + r(t)
BH A n(t)0B
u(t) = -K1x(t) - K2n(t) (4-44)
y(t) = Hx(t)
where x(t) e Rn is the state of the plant, n(t) . 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 Rd 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.1) and (B.2).
Suppose that the conditions for a solution to the linear servomechanism problem have been satisfied and the closed-loop system LC has