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practical situations, a near time-optimal solution is implemented by combining
time-optimal control for large errors, where it make sense, with linear control
for small errors, and ensuring smooth transition between the two modes. One
such solution, Proximate Time Optimal Servomechanism,wasfirst proposed
in 1987 [209]. Realization of the PTOS for continuous-time and discrete-time
systems were presented in [210] and [211], respectively.
2.6.2 Proximate Time Optimal Servomechanism
The proximate time-optimal servomechanism (PTOS) provides a smooth, bump-
free merger of linear state feedback control for small errors with nonlinear
time-optimal control for large errors. Linear state feedback controller is used
for position errors bounded by an upper limit, i.e.,
|e| ≤ e l ,
x 1
x 2
u l = Kx=[k 1
k 2 ]
= k 1 x 1 + k 2 x 2 .
The state variables are the position error, x 1 = e = y r − y, and the error
velocity x 2 = e = −y. For error larger than e l , an approximation of time
optimal solution is applied. To implement this, the velocity profile and control
law are modified according to
2aαU m |e| − U k 2 ),
|e| >e l ,
f PTOS =
k k 2 e,
|e| ≤ e l
k 2 f PTOS x 2
U m
u = U m sat
The saturation function sat(w)isdefined as
|w| ≥ 1,
|w| < 1.
The state feedback gains are selected such that the conditions given in equa-
tion 2.43 are satisfied. These conditions ensure continuity of the velocity profile
as well as its first derivative with respect to e at |e| = e l .
2k 1
k 2 =
U m
k 1
e l =
The acceleration discount factor α (α ≤ 1) in equation 2.40 allows us to ac-
commodate uncertainty in the acceleration factor a of the plant. Using α < 1
however makes the response slightly slower. With α = 1, the nonlinear part of
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