I have a nonlinear MIMO system
$\dot{x}=f(x)+g(x)u$, also the sliding surface for my outputs happens to be a vector S=0. Now I can write $\dot{S}=F(x)+G(x)U$. Can I not extract U from this relationship by enforcing attraction condition. i.e $\dot{V}< -\eta*|s|*-k*s$?
or is it necessary to convert the system into normal form first?
You can do it, the only condition is that the surface converge and you can prove it using lyapunov theory
Off couse you can using Lyapunov theory as Mr. Medrano commented. In addition you must analyze what the stability of the system \dot{x}=f(x)+g(x)u once its trajectories reach S=0 and remains there \dot{S}=0. For this last analysis, canonical forms as the regular form are very useful.
@Jesus
Yeah that is why I used the attraction condition, that will ensure "attractiveness" of sliding surface and hence vector S=0 becomes the invariant set. Actually the problem was, my system was constantly blowing up when I was trying to integrate it using solvers like dopri5, Lsoda etc.
Also since $\dot{s}=F+GU, control vector becomes 1/G(-F-eta*sign(s)-k*s), my system blows up as soon as G vanishes! and I guess that is the reason why taking G inverse in this case isnt always the best idea and thats why I think I need to convert the system into normal form? Whats your take on this?
@Juan!
Hi! Actually, there are various ways of arriving at control vector U, in brute force approach, as I have been doing all along, I find the sliding surface for each output variables depending on the relative degree of that particular output and then enforce the attraction condition to make the S=0 the invariant set. But using this traditional approach, I wasnt able to get desired answer.
So I decided to look for alternate ways of formulating control vector U in MIMO case, I found out that one of my seniors had first converted the system into normal form using lie derivatives. He then used this transformed state($\zeta$) in the expression of error while defining sliding sirface.
That is the reason why I got confused.
This is the paper of my senior- I couldnt figure out as to why did he bother converting his system into normal form via lie derivatives etc.
https://arc.aiaa.org/doi/abs/10.2514/1.56361
Do you have relative degrees higher than one in some of the outputs?
Actually no! Both of my outputs have relative degree of 1. Shall I post my exact problem?
@Juan
Yes, I figured out that I had two outputs and one of them had relative degree of 2 and one had RD=1.
You can use zero-dynamic design method which is also very popular method in nonlinear control design. In this method it is not required to convert the system into normal form.
Yes is true, but on some systems is not trivial to prove the stability. I recommend transforming the system to regular form and then determinate the control, this is the easier way.
To transform your system use the following paper:
"A generalized regular form for multivariable sliding mode control"
W. Perruquetti, J.P. Richard, and P.Borne
The paper is based on the work of Lukyanov and Utkin. You can use this paper to solve your problem.
Thanks debashish, I've read of zero dynamics design method, however I've never tried it for my system. I have solved my problem now , the problem I faced was due to un invertibility of G matrix. So I used dynamic extension technique, wherein I considered one of the controls to be state. Like u1dot=w
@Jesus, also, how would you ensure stability of "left-over dynamics"? I mean in a simple made up example like the one below, the left over dynamics actually blows up.
x1dot=x2+u1
x2dot=x1+u1+u2
x3dot=x1*x3+u1
y=(x1,x2)
in this case, the left over dynamics i.e x3dot just blows up if I choose reference signal to be a simple step.
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