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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

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

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  • $\begingroup$ My purpose is linearize my state space model around a point. Every system should have equilibrium point or points. If we apply an input or inputs to a system, we have to obtain different or same equilibrium points. For example, if the input is step, we can use same equilibrium points which can be calculated according to step response. But if we use ramp or sinusoidal input, equilibrium points should be different every time step. These are my opinion about linearization. $\endgroup$
    – voo
    Dec 6 '20 at 19:00
  • $\begingroup$ 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. $\endgroup$
    – Arastas
    Dec 6 '20 at 19:56
  • $\begingroup$ You are right Arastas, i didn't think that there might be no equilibrium points. But even so your answer clarify my view and prove my opinion about effects of inputs on equilibrium. I hope that when i increase number of diff. eqs. of my system, i will see the situations of equilibrium points. Thanks a lot for your support! $\endgroup$
    – voo
    Dec 6 '20 at 20:11

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