9
Usually, capital letters (e.g., A, B, F) will denote matrices,
while lower case letters (e.g., x, y, z) shall denote vectors. Both
standard lower case letters and Greek lower case letters shall indicate
a scalar. Any deviations made from the notation for matrices, vectors,
and scalars will be clear from context.
Main Results for the Nonlinear Servomechanism Problem
Consider the nonlinear system
N: x(t) = f(x(t), u(t), w(t))
y(t) = Hx(t)
e(t) = r(t) y(t) (2-2)
where x(t) e Rn is the state of the system, u(t) e RP is the control
input to the system, w(t) e Rd is a disturbance signal, y(t) e RP is the
output of the system, and e(t) e RP is the error which occurs when
tracking the reference signal r(t) e RP. Note that the output is
assumed to be a linear function of the state and the dimension of the
input is the same as that of the output.
It is our objective to design a closed-loop controller which will
asymptotically regulate against disturbances and also asymptotically
track a reference signal. In particular, we desire e(t) = r(t) y(t)
+ 0 as t + where r(t) is a specified output chosen from a given
class of functions.
The primary concern here will be tracking and disturbance rejection
when the disturbance w(t) and the reference r(t) are comprised of