Suppose a system is described by a set of non-linear differential equations. For implementation of some control techniques, the nonlinear functions need to be smooth. Say, the output is the first state variable and the control input appears in the last state equation (say, 8th state).
If some discontinuous functions are present in the 1st state equation, Is there any way to approximate or transform this discontinuous functions to smooth nonlinear functions?
For example,
Let the 1st state equation be:
dx1/dt = k*x2 - x1*x3 + F1 -F2 -F3 +D(t)
where, x1,x2,x3:state variables; F1,F2,F3:dis-continous functions; D(t):Disturbance
The dis-continous functions are defined below:
(1) if F1>=0 ; F1= 14.07(1-x3);
otherwise, F1=0
(2) F2 = 12.98*x1/(x1+1)
(3) if x1>=7.75, F3 = 1.15(x1-7.75)
otherwise, F3 = 0
Anirudh,
Unfortunately, there are many systems with discontinuities in state and input matrix where Paul's suggestion won't work !
There are methods, but in many cases they are system specific. I would suggest you study the nature of discontinuity in your system very carefully, and then try to find a way around the problem.
-Sanjay
Dear Anirudh,
I just saw your question and maybe it is late to answer.
It is not very clear why and where you feel the discontinuous function is bothering you. Yes, standard mathematical stuff requires continuity even for the mere existence of solutions, others for the proofs of stabilty.
At the time, I was bothered by these assumablke necessary continuity conditions. My first thought was: if in order to move my robot from A to B and then back to A, I am using a discontinuous pulse command, does it break the robot or the solution suddenly disappears?
Going back to those mathematical claims, they looked like: "We prove that on a suny day with clear sky you can walk." Very true and correct. Should we then conclude that just because of some clouds or even without seeing the sun we canot walk? NO.
OK, mathematically this required more than just an opinion. You don't seem to assume that discontinuity does not allow solutions. Then, do you worry about guarnteeing stabilty of your system? Here, in case you are interested, there are new developments that eliminate the need for continuity.
Thanks, Itzhak
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