To apply Jacobian linearization procedure, we should find equilibrium points of the system. But, in some nonlinear systems, it is impossible to find equilibrium points set which make system's differential equations equal to zero. So, we have to handle the problem in another way.
I have this kind of nonlinear system and cannot find equilibrium points. What do you suggest me to linearize this nonlinear system with Jacobian matrices?
I tried to solve this problem with optimization methods in MATLAB via fmincon, but I think it is not a clear way.
My reference source is: http://www.sosmath.com/diffeq/system/nonlinear/linearization/linearization.html
Thanks a lot!
For linearization, it is not necessarily that the linearization point is an equilibrium. For $x$ being the state and $u$ being the input signal, you have $$\dot{x} = f(x,u) = f(x_0,u_0)+A(x-x_0)+B(u-u_0)+H.O.T.$$ Here $A$ and $B$ are the linearization (Jacobian) matrices, $$A = \frac{\partial}{\partial x}f(x,u)|_{x=x_0,\, u=u_0}$$ and $$B = \frac{\partial}{\partial u}f(x,u)|_{x=x_0,\, u=u_0}.$$
If $x_0$, $u_0$ are an equilibrium, then you have $f(x_0,u_0) = 0$; otherwise, the constant term will remain.
For $x_0=0$ and $u_0=0$ we have the standard linearization: $\dot{x} \approx Ax + Bu$. However, if $x_0$, $u_0$ is not the origin, then you should work with the deviations $\tilde{x}=x-x_0$ and $\tilde{u}=u-u_0$.
Finally, you obtain $$\dot{\tilde x} = A \tilde x + B \tilde u + C,$$ where $C = f(x_0,u_0)$ is the constant term that you must take into account for control design.