How can I create state space model of this system ? ?

It doesn't seem to be like general ss model. Because, normally system has to have just one derivative term at the left side in the each line. Although, it's different equation which is given above. Question is how can I create ss model of this system, then find the equilibrium point( u=0, u=-sin(x2) ) and are they isolated ?

ALL INITIAL CONDITIONS ARE ZERO

Thanks for your help

Answer 1

[Aykut Satici]

This is a simple exercise in linear algebra. What multiplies the acceleration variables (x3, x4)

M = [1, cos(x1); cos(x1), 2],

is called the mass matrix in the dynamics literature. If you want to express these equations in the form

x_dot = f(x,u),

all you need to do is to solve that linear equation for the states x3_dot and x4_dot. Since the mass matrix is nonsingular (why?), you can find the equilibrium points of this dynamical system with, say, (u = 0), by equating the right-hand side of this system of equations to zero. That gives

x3 = x4 = 0,
sin(x2) = 0 ==> x2 = k2*\pi, k2 in integers
sin(x1+x2) = 0 ==> sin(x1)cos(x2) = 0 ==> x1 = k1*pi, k1 in integers.

Note that, around each of these equilibrium points, there exists an open neighborhood which contains no other equilibrium points. Therefore, the equilibrium points are isolated. You can also make this deduction by showing that the equilibrium points are regular values of the drift vector field (right-hand side of these equations); in other words, by computing the Jacobian matrix of the drift vector field and showing that it is nonsingular at the equilibrium points.

You should go through the same calculations for u = -sin(x2).