23
Definition: If the state trajectory for the closed-loop system NC
k
converges to [x (t), n (t)] for a set of initial states in the
neighborhood of [xQ, nQ] then we say there is local tracking of r (t)
fc
with disturbance w (t). If this convergence occurs for all initial
states then we say there is global tracking of r*(t) with disturbance
w*(t).
To give conditions under which global or local tracking will occur,
we first define a new set of state vectors as follows
x(t) = x(t) x*(t)
*n(t) = n(t) n*(t)
(2-37)
where [x(t), n(t)] is the state trajectory of NC resulting from an
"k "k
arbitrary initial state and [x (t), n (t)] is the trajectory which
"k 'k
gives e(t) = 0, t > 0 and results from the initial state [x n ].
o o
Since it is our goal to have the trajectory [x(t), n(t)] converge,
* k
eventually, to the trajectory [x (t), n (t)] we may think of
Cx(t), (t)] as the transient trajectory. Using (2-29) and (2-35), it
is then possible to write a dynamic equation modeling the transient
response of the closed-loop system. This will be referred to as the
closed-loop transient system NCT. The system NCT is given by
NCT:
x(t) = f(x*(t)+x(t), u*(t)+u(t), w*(t)) f(x*(t), u*(t), w*(t))
^{t) = Ari(t) BHx(t)
u(t) = -Kxx(t) K2n(t)
(2-38)