I am having trouble modeling a state-space model where the states and inputs are multiplied together.
Consider this example:
Xdot = X*sin(u)
This is a simplified example but it covers my problem. Here X represents the state and u the input. They are multiplied together.
How can I separate them into
Xdot= Ax+Bu
My approach was to linearize the model by taking partial derivative with respect to U. But this way the equation will be a constant multiplied by U.
Xdot = Bu
But this way it will not depend on X which I find intuitively not correct.
Can someone give me help?
Your system: Xdot = f(X,u) is a nonlinear system. If you want in the form Xdot = A X + B u form which represents a set of linear first order differential equations the linearization is correct. In linearized equations you cannot get (X.u) terms.
Hi Asser Roshdy
I'm unsure if the system is linearizable at the equilibrium point, x = 0. However, your example x' = x*sin(u) is a stabilizable open-loop system. By choosing 2πn − π < u < 2πn, where n ∈ ℤ, the eigenvalue of the system will have a strictly negative real part.
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