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The mechanical energy of the pendulum and its time

derivative are as follows:

3)

4)

1

2

= -mZ2d2+mgZ(1-

case),

V =

mld

COS er

It is seen from equation 4) that V can be increased

or decreased by changing the sign of r in accordance

with that of 8cos8. If sgn(i') = sgn(8cos0) (resp.

s gn( i ) = -sgn(ecose)), then V > 0 (resp.

V < 0).

Since the travel of t he cart is finite, r has to be con-

trolled in consideration

of

the constraint on T . The ba-

sic idea of t he design meth od lies in constructing a servo

system having a.sinusoida1 reference input, which is ob-

tained from ( 0 , 8 ) , and generating i: satisfying the sign

condition in order to control V to a prescribed value.

Let

Td

be th e reference input of the servo system for

T , which will be given later. Using Td, we put

U,

=

(M+msin28){f l ( rd - r )

- f 2 r

+ m g cos e sin e +m1e2 sin e.

(5)

Here f i and f are given by

-

W

W n

CO

f i

=

R2, fi = a

0)

when > 0 (resp. a < 0). In order t o make

V

converge

to the reference energy, say

vd

the parameter

a

is given

bY

12)

osgn(V Vd)

if I V

vd

I> bo

ao(V-

) / b o

if

I

v-vd

I 0 and

bo

>

0

are the design parameters.

We see that

a0

relates t o the amplitude of T . Actually

a0 is chosen smaller than the desired amplitude because

g ( w k ) 2 g ( w n ) . According to (12 ) , the amplitude of a

is decreased as V approaches

vd.

The parameter bo is a

number th at determines the time when the amplitude of

a is decreased. Th e time is delayed more if a smaller bo is

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given. These parameters are determined by performing

simulations.

In case of th e swing-up control ,

v d is

equal to the

potential energy of the pendulum at the upward vertical

position, i.e.,

v d =

2mgl. (13)

Although

8

enters the region of 7r/2