to
components which are either constant or sinusoidal in nature. To
simplify the development, we shall consider a particular disturbance
signal, say w (t), and a particular reference signal, say r (t), to be
represenative signals from a given class of functions. Once the
controller is derived with respect to these signals, the results can be
generalized to cover a class of functions for which r*(t) and w*(t) are
assumed to belong.
We now make the following assumptions:
(A.l) For some chosen reference signal r (t) and a particular
"Jc "k
disturbance w (t) there exists an open-loop control u (t) and an
initial state x (0) = x such that
o
x*(t) = f(x*(t), u*(t), w*(t))
y*(t) = Hx*(t) = r*(t) (2-3)
e(t) = r*(t) y*(t) = 0 for all t > 0
-k k
(A.2) The elements of both x (t) and u (t) satisfy the scalar, linear
differential equation
()(r) + ar-l(*)(r_1) + + ai(*)U) + a0(,) = 0 (2"4)
where the characteristic roots of (2-4) are all in the closed right
half-plane.
The first assumption is merely a way of stating that it is possible
to provide output tracking. A typical example where (A.l) would not
hold is for a system having more outputs than inputs. This particular