I am working on a nonlinear model of two wheeled robot system which has 4 states [position, velocity, angle, angular velocity]. I have check the observability of my system with Lie Algebra and see that my system has rank less than n=4. I am assuming that only the angular velocity is measurable. I have little knowledge that I can see it using Null space of the Observability matrix but I am unable to find any details in Applied Nonlinear control by Slotine and Nonlinear Control system by Hassan K. Khalil. Please help me
Hi Muhammad,
Your problem seems very close to the ones related to these following papers:
https://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwipsqGmle_RAhVW_WMKHeLUCwsQFggvMAE&url=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F220104213_Lie_algebra_application_to_mobile_robot_control_A_tutorial&usg=AFQjCNE0RkP9HvO6OZ3CsJXwqx1qTLkcvQ
http://calhoun.nps.edu/bitstream/handle/10945/44185/Yun_Stability.pdf?sequence=1
Best regards
The model equations are attached in the form of a report in which I have checked the observability and controllability of the system for a single initial condition. Please give your comments.
Have a look to http://sebastiannilsson.com/wp-content/uploads/2013/05/Self-balancing-two-wheeled-robot-report.pdf
In particular, Section V.C.
Thankyou Stephane. It is indeed the same point where I reached. Would you please give a look to my recent question? Thanks
Keep on by investigating minimal state-space realization adapted to nonlinear systems. Here are some introducing links:
http://ac.els-cdn.com/S0377042700003411/1-s2.0-S0377042700003411-main.pdf?_tid=1ec6cd10-e939-11e6-bc0c-00000aacb362&acdnat=1486034402_9fa520a2b3ed54117cd4c93edfaa5992
https://pure.strath.ac.uk/portal/files/32751923/paper.pdf
http://www.bg.ic.ac.uk/research/g.stan/IEEE_TCNS_2015.pdf
Aoa Dear Zeeshan
I won't be able to address your question exactly, but, I have one suggestion which may work: If you can linearize your sytem, and then find the similary transformation matrix, and eventually you can come up with a modified output matrix. The transformed output matrix can tell you which states are observable and which are not
Obviously x1:=x is unobservable from x4:=\dot\theta. Observability means that you can reconstruct the whole state from the knwoledge of the (theoretically) known signals, i.e. y,u, and their derivatives (this is precisely what the nonlinear Kalman rank condition says). In your case, when you differentiate x4 twice, you will get 3 relations allowing you to express x4,x3,x2 in function of x4,\dot x_4, \ddot x_4,u,\dot u (the 3 relations should be generically full rank with respect to x4,x3,x2); differentiating further brings no new information, in particular x1 will never show up.
By the way, you mention in your post "I have checked the observability and controllability of the system for a single initial condition", which is somewhat weird. What you are interested in is the generic rank around a generic point, or often around an equilibrium point (in which case the nonlinear observability at this equilibrium amounts to the linear observability of the tangent system around that point). Usually, you are happy with the linear observability, since it implies nonlinear observability.
For a nonlinear system, we are concerned with the local observability at some operating point as already pointed by Philipe Martin. You can see an example of a nonlinear system in the appendix of the paper
Nonlinear state observers and extended Kalman filters for battery systems,
and how the observability is calculated.
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