This gives
1 ,* Atr 1 *
K2[n (t) - n (t)] = K2e [no - no] = 0 (2-34)
1 2
and the vector [n - n] is not observable which is a contradiction.
0 0
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.1) 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 K1 be arbitrary. Under these
conditions, there exist initial states x(O) = x0 and n(O) = n0 such that in the closed-loop system NC, e(t) = Er*(t) - y(t)] = 0 for all 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 [xo, no] denote this initial state and let [x*(t), n*(t)] be the
00
corresponding state trajectory. The following relationship must then hold for the system NC
x (t) = f(x*(t), u*(t), w*(t))
, (t) = An*(t) (2-35)
u (t) = -Klx*(t) - K2n*(t)
e(t) = r*(t) - Hx*(t) = 0
Henceforth, the initial state of the combined plant and controller will be grouped in a pair as [x n]. The state trajectory which results from this initial state will e grouped as [x(t), n(t)].