I Need Guidance on Stability Analysis for Nonlinear State-Space Models

bakon72915

Hey guys........

I am currently working on a project where I am modeling a nonlinear dynamic system using a state-space approach, but I'm encountering some challenges when it comes to stability analysis. The system I am working with exhibits some nonlinearity in both its state equations and output equations, making it difficult for me to apply the traditional methods I'm familiar with.

I have tried linearizing the system around different equilibrium points, but the results haven't been very conclusive. I was wondering if anyone here could share some insights or techniques they’ve found helpful for performing stability analysis on nonlinear state-space models? Are there any specific tools or theorems that you would recommend using for such a system, especially for non-smooth or discontinuous nonlinearities?

Additionally, I am also looking for advice on how to approach the Lyapunov stability criterion in this context. I've read up on Lyapunov functions, but I’m struggling to construct a suitable one for my system due to the nonlinearities involved.

I also check this: https://state-space.ieeecss.org/d/4219-postdoc-in-dynamic-regulation-of-photosynthesis-in-light-acclimated-organisms salesforce-dev But I have not found any solution. Could anyone guide me about this? Any references to papers, books, or software that could help would be greatly appreciated. I’d also love to hear about any personal experiences or approaches that worked for you in similar situations.

Thanks in advance!

mebbaid

I have tried linearizing the system around different equilibrium points, but the results haven't been very conclusive

I assume that by the above you meant, given an equilibrium point you are interested in, you computed the Linear Tangent Model and checked the spectrum of the system Matrix and found that all eigenvalues either have negative real part or zero real part (because in case they positive real part then that equilibrium point is an unstable equilibrium for the dynamics by the "Principle of stability in the first approximation" ). If that is the case, then around the same equilibrium point you need to employ the Center Manifold Theorem (See for example A. Isidori Nonlinear Control Systems - Appendix B.1). Given the above situation, a center manifold exists, and the study of stability of your system can be reduced to the study of stability of a reduced dynamics evolving on that manifold.

I hope the above reference and brief explanation helps