Definition: If the state trajectory for the closed-loop system NC converges to [x (t), n*(t)] for a set of initial states in the neighborhood of [Xo no] then we say there is local tracking of r (t) 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
(t) = x(t) - x*(t)
=(t) = n(t) - n*(t) (2-37)
where [x(t), n(t)] is the state trajectory of NC resulting from an arbitrary initial state and Ex (t), n (t)] is the trajectory which gives e(t) = 0, t > 0 and results from the initial state [x0, no].
Since it is our goal to have the trajectory [x(t), n(t)] converge, eventually, to the trajectory Ex (t), n*(t)] we may think of [(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)+'(t), u*(t)+u(t), w*(t)) - f(x*(t), u*(t), w*(t)) fl(t) = An(t) - BH'(t) (2-38)
"U(t) = -Kl X(t) - K2"(t)