76
robotic manipulator is discussed in Chapter Six.) The true reference
signal shall be denoted r*(t). Also, it is likely that the actual
ft
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.l) holds, tracking will occur provided that a certain
state and input trajectory are present. Denoting this state and input
ic 'if
as x (t) and u (t) respectively, the following definitions can be made
?*(t)
:= r*(t)
- r (t)
tf(t)
w*(t)
- w*(t)
X (t)
x*(t)
- X (t)
(5-3)
/N*, x
U (t)
:= u*(t)
- U (t)
where r*(t), w^t), x^t), and 1j*(t) denote the deviations of the true
signals from the nominal signals. Note that because the system under
consideration is nonlinear, x^t) and "ij*(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
'if
trajectory x (t). In addition, this will be accomplished with an input
if
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
x{t) := x(t) x*(t)
tiit) := u(t) u*(t)
(5-4)