8.7.1 Experiment 1—The Inverted Pendulum
Part 1. The linearized equations of the inverted pendulum, obtained by assuming
that the pendulum mass is concentrated at its center of gravity [15, 16] are given by
"!$#
&%
'
'()
!$*+,
(8.148)
where
is the angle of the pendulum from the vertical position,
-
is the position of
the cart,
*.
is the force applied to the cart,
%
is the mass of the cart,
is the mass of
the pendulum,
is the gravitational constant, and
is the moment of inertia about the
center of mass. Assuming that normalized values are given by
!0/
,
!1/
,
!324 5/
,
%6!7/
, and
!8#4-/
, derive the state space form
9
:
;!3<
:
=
*+
where
:
!1>
9
?
9
@-A
and
<BDCEB
and
=
BFCHG
are the corresponding matrices.
Part 2. Using MATLAB, determine the following:
(a) The eigenvalues, eigenvectors, and characteristic polynomial of matrix
<
.
(b) The state transition matrix at the time instant
I!J/
.
(c) The unit impulse response (take
and
as the output variables) for
#)KLMKN/
with
O
I!P#E4-/
. Plot the system output response.
(d) The unit step response for
#LKQKR/
and
O
S!T#4U/
. Draw the system output
response.
(e) The unit ramp response for
#VKP(KR/
and
O
W!J#4U/
. Draw the system output
response. Compare the response diagrams obtained in (c), (d), and (e).
(f) The system state response resulting from the initial condition
:
&#X!1>YI/ /Z/Z/H@ A
and the input
*+[!]\^-_`
for
#RKa0K6b
and
O
!c#E4-/
.
(g) The inverse of the state transition matrix
&dfehg&Xi
G
for
j!0b
.
(h) The state
:
at time
M!$b
assuming that
:
f/D#F