Hardware Reference

In-Depth Information

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
.

(2.39)

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

⎨

p

2aαU
m
|e| −
U
k
2
),

sgn(e)(

|e| >e
l
,

f
PTOS
=

,

(2.40)

⎩

k
k
2
e,

|e| ≤ e
l

and

½

k
2
f
PTOS
−
x
2

U
m

¾

u = U
m
sat

.

(2.41)

The saturation function sat(w)isdefined as

½

sgn(w),

|w| ≥ 1,

sat(w)=

(2.42)

w,

|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
.

r

2k
1

aα

k
2
=

(2.43)

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