# Jacobian linearization and equilibrium points for any inputs

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?

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

• Can you post your model equations? Dec 6 '20 at 10:47
– voo
Dec 6 '20 at 11:03
• My outputs are: y1=x1_dot+x3*x2; y2=x2; y3=x3_dot-x1*x2; y4=x1; y5=x3;
– voo
Dec 6 '20 at 11:07

• Sorry, you are wrong about linearizations. It is not true that any system has an equilibrium, e.g., $\dot{x}=1$ does not. If you consider an autonomous system, then the equilibriums (if any) are in the state space. It can be a point, many isolated points, a set. If you consider a system with input, then the equilibriums also depend on the input. And it is also possible that there is no equilibrium for a particular input signal. Dec 6 '20 at 19:56