I have the differential equation: $$M(q)\ddot{q}+C(q,\dot{q})\dot{q} = J(q)[K_p(t)e+K_d(t)\dot{e}]$$ that represents Euler-Lagrange dynamic of a closed-loop system including the control input law, where $q$ is the state and $e$ the error signal. How to find the equilibrium point of the equation?
UPDATE1: The equation in the state space form $\dot{x}=f(x,u)$ where $x_1=q , x_2=\dot{q}$ is: $$\dot{x_1}=x_2$$ $$\dot{x_2}=M(x_1)^{-1}[-C(x_1,x_2)+J(x_1)u(t)]$$ where $u(t)$ the input law is $$u(t)=K_p(t)e+K_d(t)\dot{e}$$ UPDATE2: e is defined as $e=[e_x\:e_y]^T$ where $e_x=x-x_{ref}$ and so $e_y=y-y_{ref}$ where $x,y,x_{ref},y_{ref}$ are the outputs of the kinematic model and $q=[q_1\:q_2]^T$ is the generalized coordinates of the dynamic model.
To summerized this is equation of a 2DOF robotic arms where q represents the angles of joints and x,y the cartesian coordinate of the end-effector and it is controlled by Transpose Jacobian control law:
$$\tau=J(q)[K_p(t)e+K_d(t)\dot{e}]$$
In an article the author said that equilibrium point of the aforementioned closed-loop equation is the solution of: $$K_p(t)e+K_d(t)\dot{e}=0$$
But I didn't catch why.