robotic manipulator is discussed in Chapter Six.) The true reference
signal shall be denoted r*(t). Also, it is likely that the actual
disturbance, denoted w*(t), will not be the same as the nominal disturbance w (t). When r*(t) and w*(t) are applied to the system then,
if assumption (A.1) holds, tracking will occur provided that a certain state and input trajectory are present. Denoting this state and input as x *(t) and u*(t) respectively, the following definitions can be made
r (t) r*(t) -*(t)
x (t) x *(t) - *(t) (5-3)
U t u* (t)
where r*(t), *(t), k*(t), and u* (t) denote the deviations of the true
signals from the nominal signals. Note that because the system under consideration is nonlinear, Mx(t) and U (t) are not necessarily the state and input trajectories which give tracking of r (t) with the disturbance W (t).
Now as before, the objective of the control will be to cause the actual state trajectory x(t) to asymptotically approach the state trajectory x*(t). In addition, this will be accomplished with an input u(t) which asymptotically approaches u*(t).
Consider the control scheme given in Figure 5-1. In relation to this scheme we have used the following definitions
(t) x(t) - *(t)
(5-4)
^U(t) UM ut -u u(t)