Thank you Group! for your input.
What I mean by a single input is that we have one actuator connected to
the cart. While all of the following can be sensed for feedback:
position, velocity of cart; and angle and angular velocity of the inverted
pendulum.
I approached the problem following way:
1) Linearized the nonlinear equations of motion. Assumed cart position
and angle of the pendulum to be available for feedback.
2) Got transfer function x/f and a/f where, f is force, x is cart displ
and a is angle.
3) Because of linearization we have got polezero cancellation by the
system. Below are the TFs:
1.6 s^2  40
x/f=  zeros at 5 and 5
s^4 + 0.16 s^3  40 s^2  4 s
4 s^2
a/f=  zeros at 0, 0
s^4 + 0.16 s^3  40 s^2  4 s
poles at 0 0.1 6.3 6.3
4) Please observe that the system has nonminium phase zeros and has
unstable poles.
5) I am using cascaded controller with inner loop for pendulum and outer
loop for cart position.
6) I can stabilize the inner loop (i.e. pendulum angle) using lead and lag
compensator with angle as feedback.
7) I am also able to get stable open loop (for complete system) with
feedback from both, angle and position. However, position feedback is
making inner loop unstable (Note: inner loop is initially designed with
feedback from angle only)
8) I am using rltool/SISO tool in matlab for design purpose.
I am trying hard to get solution to the problem. This is my first exposure
to control. Any input for any of the steps above would be of great help.
I checked controllability and observability in matlab. The system is
controllable and observable (matlab returns rank=4 for both). I looked
into the IEEE paper, most part of it is Greek and Latin for me.
Regards,
Shashikant
(sarada at tamu.edu)
On Fri, 14 Nov 2003, dave y. wrote:
Tell you what, why don't you assume just a tiny bit of noise in the
system and then tell me you can do this. In the meantime, just to
experiment with the concept of open loop a bit, the next time you're
driving down the xway, why don't you point the car straight down
the road, then take your hands off the wheel and close your eyes.
You can use the time remaining to contemplate the merits of your
position (at least in theory).


wrote:

>
>> Are you sure you've got the assignment described correctly? You have
>> to measure both cart position and pendulum angle to make this work,
>> and that represents a two input system. If, for example, you only
>> measure cart position and don't know what the pendulum angle is, you
>> don't have a chance of keeping it balanced.
>
>It is definitely possible (at least in theory...) to control both cart
>position and stabilize upright position of the pendulumm using force
>on cart as (single) input. This is done in numerous published
>articles. One that springs to mind, is
>
> AUTHOR = {Atassi, Ahmad N. and Khalil, Hassan K.},
> TITLE = {A separation principle for the stabilization of a class of
> nonlinear systems},
> JOURNAL = {IEEE Trans. Aut. Control},
> FJOURNAL = {Institute of Electrical and Electronics Engineers.
> Transactions on Automatic Control},
> VOLUME = ,
> YEAR = ,
> NUMBER = ,
> PAGES = ,
> ISSN = ,
>}
>
>> On Mon, 10 Nov 2003 23:53:47 0600, Shashikant N Sarada
>>
>> >Hi
>> >I am working on a project on inverted pendulum. I have to control pendulum
>> >rotation as well as cart position within given specifications. I am using
>> >Matlab/Simulink and am designing as a single inputmultiple output system.
>> >I am able to control pendulum angle but not the cart position. Has
>> >anybody worked on similar problem?
>> >
>> >Any input will be of great help!
>> >
>> >Regards,
>> >
>> >Shashikant


