# Lyapunov stability ??? How ?

Hi all,
I need some help. I have nonlinear system described by ODE, and I have to investigate its stability using Lyapunov approach. The problem is that I
don't know how to find the most appropriate Lyapunov "candidate" functions. Maybe some of you have more experience in this kind of stability checking method. I will be appreciate any help. Greg
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Try to find a quadratic energy function for the system? What is the system? and what is the ODE like?
niclas

functions.
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It is a simplification of the Maglev system (Magnetic Levitation Vehicle). There are 3 quations for: verical displacement (levitation - Z variable ), latteral displacement (guidance - X variable), current (I - varaiable) also addidtional varaible V voltage:
Z'' = k*(I^2)* [-(d-X)/(Z^2) - (4*X)/(4*Z^2 + pi*Z*X) ]/M + g - (f*Z')/M X'' = k*(I^2)* [-1/Z + 4/(4*Z + pi*X) ]/M - (f*X')/M I'=[Z*(V-R*I)]/p
where: p, k, d, f, M, g, pi, R are constants
Input to the system is a voltage, output is a gap Z, also X latteral displacemnet is the output but at this state of my work the most important is to keep only constant gap Z_ref=0.01 m. I did some work linearizng above equations around operation point (I0,Z0,X0), and I applied PID controller to control the gap - works pretty good. Unfortunatelly its not enough my professor wants more !!! :). There are many sources which tells you about Lyapunov stability but it is really hard to find such to explain exactly how to assume candidate functions. According to your message I guess that current is my quadratic term of energy k*I^2/M, but what about X and Z ??? and what about the expresion for relation between current and voltage ??
Thank yor for your intrest Greg

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