been constructed in agreement with equation (4-44). Since asymptotic
tracking and disturbance rejection will occur, there must be a steadystate solution to (4-44). Consequently, if both w(t) and r(t) are
applied to the system at the time t = 0 then there exist initial states x(O) = xo and n(O) = no such that no transients appear in the state trajectory and the tracking error is zero. We denote this trajectory by the pair [x ss(t), nss (t)]. In terms of this notation, (4-44) becomes
ss ~ s (tF 0 SSW
= + U SS(t) + w(t) + r(t)
Sss(t)] BH A_]Lnsst) M o B
u ss(t) -Kx ss(t) -K nss(t) (4-45)
Ysst) = r(t) = HxSS(t)
If (4-45) is subtracted from (4-44) the following equation results:
F~t2 VF 0l F~t1 G1
= + I(t) (4-46)
T BH A [ (t0 0
where
X(t) = x(t) - xss(t)
n(t) = n(t) - nss(t) (4-47)
i(t) = u(t) - uss(t) = -Kl(t) - K2 (t)