Question

# How can I find the controllability conditions or controllability distribution for discrete time nonlinear state-space system?

I need to find the controllability distributin for the following discrete time nonlinear state-space system:

x(t)=f(x(t-1)+B(t)u(t)
y(t)=g(x(t)).

There is some algorithm for continuous time nonlinear system however I couldn't find the discrete time case.

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• Erol Kurt · Gazi University
for any controlibility one needs to have the eigenvalues of the system as a result of linear and ninlinear operators. For this just think of the largest eigenvalud which drives the system. See the Jacobian and related subjects from any chaos ir nonlinear dynamics book.
• AFAIK, for nonlinear systems, the equivalent to contralibilty is usally called flatness. Hope this is a helpful starting point.
• Alberto Lorandi Medina · Universidad Veracruzana
• Adolfo Silva · National Institute for Space Research, Brazil
You can applied the controllability test at each iteration from the system driven derive from Jacobian!

I don't know if it useful for you problem!
• Here is the right link from Alberto:
http://www.ece.rutgers.edu/~gajic/psfiles/chap5.pdf
but it does not answer the question, the theory in this pdf is only about linear system...
• Alberto Lorandi Medina · Universidad Veracruzana
Sorry I did not see that need's the nonlinear case
• Hebertt Sira-Ramirez · Center for Research and Advanced Studies of the National Polytechnic Institute
Linearity or affine character in the control inputs doesn't make much sense in discrete time nonlinear systems. This is lost at the second iteration. The concept of vector fields does not directly apply to discrete time nonlinear systems either, much the less the concept of distribution. What you deal with is control parameterized transition maps from instant to instant. However, differentials, or Khaler differentials, and their "linear subspaces" may be used to chacterize a lot of concepts in this field. You may then take their duals and identify these objects as vector fields. The geometry is of a quite different flavor and non-intuitive, though. A good reference is earlier work by M. Fliess and El-Asmi (see also "Reversible linear and nonlinear discrete-time dynamics" in ieee explore. A beautiful paper by J. Grizzle may be found in http://scholar.google.com.mx/scholar_url?hl=es&q=http://citeseerx.ist.psu.edu/viewdoc/download%3Fdoi%3D10.1.1.34.3800%26rep%3Drep1%26type%3Dpdf&sa=X&scisig=AAGBfm3uEBYrXWT4_tJ0L19KwjfOy9-EvA&oi=scholarr&ei=dwF6T7yVF4bq2AX8pbm1Bg&ved=0CBoQgAMoAjAA). Search also under J.F. Pomet et al, Aranda et al. More recently: see papers by C. Moog, Ulle Kotta etc.
• First for discrete time system... it matters how you find the discretized version. If you are using an Euler method with H.o.T. eliminated- it is equivalent to using Zero Order Hold... most of the properties of linear systems are retained and you can use the controllability tests for linear systems to ascertain the stability. If you are discretizing using Trapezoidal or other methods... you need to linearize around different points and then study controllability. Let me know if this helps you.