I'm trying to linearize a non-linear model and write it in state-space form (derive state-space matrices) for control and optimization purposes. The nonlinear model is: x(k+1)=f(x(k),u(k),d(k)), and it contains the minimum of three terms:
f(x(k),u(k),d(k))=x(k)+min(a.u(k),b.x(k),c.d(k))+...
a,b and c are constants and the three terms inside the min() function are all linear. How do I go about linearizing this model? Note that I want to use the linearized model as a prediction model for MPC.
Thank you in advance.
Linearization essentially considers the small perturbation dynamics about some system state. The linearized system description will depend upon the values of a.x, b.u and c.d. Thus first you need to decide at which point in the state space you want to linearize about. Often the chosen point is the so-called steady state solution where u(k), d(k) are defined constants and x(k+1)=x(k)=constant is the steady state solution. If the min function is the only non-linear term in you equations, then once you have calculated the steady state value of the state the rest is trivial.
Related to your query I suggest you to follow http://www.control.lth.se/fileadmin/control/Education/EngineeringProgram/FRTF05_China/slides_Lec2_20191029.pdf
https://ethz.ch/content/dam/ethz/special-interest/mavt/dynamic-systems-n-control/idsc-dam/Lectures/Control-Systems-1/Lectures/L1802-Modeling-JT.pdf
Linearization essentially considers the small perturbation dynamics about some system state. The linearized system description will depend upon the values of a.x, b.u and c.d. Thus first you need to decide at which point in the state space you want to linearize about. Often the chosen point is the so-called steady state solution where u(k), d(k) are defined constants and x(k+1)=x(k)=constant is the steady state solution. If the min function is the only non-linear term in you equations, then once you have calculated the steady state value of the state the rest is trivial.
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