21
This gives
1 k A-t* 1 k
K2Cn (t) n (t)] = K2eax[n0 nQ] = 0 (2-34)
1
and the vector [nQ nQ] is not observable which is a contradiction.
Theorem 2.1 Given the p-input, p-output system N suppose that for a
particular reference r*(t) and a particular disturbance w*(t)
assumptions (A.l) and (A.2) both hold. In addition, suppose that the
pair (A,B) defines an internal model system of r*(t) and w*(t) with
respect to N. Furthermore, assume K2 of the system NC is such that the
pair (A,K2) is observable and let Kj be arbitrary. Under these
if
conditions, there exist initial states x(0) = xQ and n(0) = nQ such
that in the closed-loop system NC, e(t) = [r*(t) y(t)] = 0 for all
if if
t > 0 when the exogenous signals r (t) and w (t) are present.
Proof: To prove Theorem 2.1 it is necessary to show that there exists
an initial state* for the system NC such that perfect tracking occurs.
Let [x0, n*] denote this initial state and let [x*(t), n*(t)] be the
corresponding state trajectory. The following relationship must then
hold for the system NC
x*(t) = f(x*(t), u*(t), w*(t))
n*(t) = An*(t) (2-35)
u*(t) = -K]X*(t) K2n*(t)
e(t) = r*(t) Hx*(t) = 0
Henceforth, the initial state
controller will be grouped in a pair as [x
which results from this initial state will
of the combined plant and
nQ]. The state trajectory
grouped as [x(t), n(t)].
&