Jacobian linearization and equilibrium points for any inputs

Question

I have 3 nonlinear differential equations and implement Jacobian procedure to create state space model. As you know, Jacobian matrices should be calculated for equilibrium points.

My model contains 3 state variables and 2 inputs. So, i have to obtain totaly 5 equilibrium points for my system. My inputs are like ramp input, so they are not constant. Magnitude of inputs are different in each step of time. At this point, i have confusion:

1. Should i change state space model in every step because of input?
2. If first is true, how can i obtain new equilibrium points in every step?
3. May new equilibrium points make my system unstable locally? If i face this kind of problem, what should i do?

Thanks for your support!

https://www.mathworks.com/help/slcontrol/ug/exact-linearization-algorithm.html

Answer 1

There exist linearization around a point or along a trajectory. When you stud the linearization around a point, you obtain a linear time-invariant system. But it should be linearization around a point, i.e., constant values of the state and the input signal. On the other hand, you can also linearize along a trajectory. Then you have some nominal trajectory, e.g., for a ramp input and the state trajectory driven by this input. You study the dynamics of deviations of your trajectory from the nominal one and obtain a linear time-varying system.

But the key question is: are you interested in the behavior around a point or along a trajectory?