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
Try to find a quadratic energy function for the system? What is the system? and what is the ODE like?
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
PolyTech Forum website is not affiliated with any of the manufacturers or service providers discussed here. All logos and trade names are the property of their respective owners.