# Fixed point stability of piecewise linear system

I have an autonomous system of nonlinear equations of the form:

$$Mx'' + C(\omega)x' + K(\omega)x + F_{nl}(x) = 0$$

where $M$ is the mass, $C$ the damping and $K$ the stiffness matrix. The nonlinear force term $F_{nl}$ depends on the solution and is of a piecewise linear shape, where the nonsmoothness occurs at $x=0$

$$F_{nl}(x) = \begin{cases} 0 & \text{if }x < 0 \\ 10x & \text{if }x \ge 0 \end{cases}$$

The system results from two rings rotating relative to each other at $\omega$. For certain rotational speeds, energy can be transferred from the rotating ring to the stationary ring, resulting in a divergence of the zero solution.

I am trying to study the stability of the zeros solution fixed point ($x=0$). Floquet multipliers won't work as the linearization at the fixed point will only take into account the left or right sided derivative.

Except for doing time simulation with a small perturbation around $x=0$, I am lost as to how to study stability in a more academically sound way. Any help is appreciated!

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## 1 Answer

It seems there is a proof that this kind of system is stable using Lyapunov theory, see:

Shorten, R.N. and Narendra, K.S.: "On the stability and existence of common Lyapunov functions for stable linear switching systems", Proceedings of the 37th IEEE Conference on Decision and Control, http://dx.doi.org/10.1109/CDC.1998.761788.

In essence, the paper states that if the system matrices for each of the modes (i.e., the cases x<0 and x >= 0) are stable and simultaneously triangularizable (which is the case here), the resulting hybrid system is stable as well.

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Thanks for the answer, but I still have some troubles. I end up with a system of the form $\dot{x}(t) = A_i(t)x(t)$ where the $A_i$ are my linear, time dependent and periodic coefficient matrices. Checking for the existence of quadratic Lyapunov matrices, seems to entail matrices with constant coefficents describing the different systems $i$. Is this idea still applicable to time dependent/periodic coefficient matrices? –  Markus Mar 6 '12 at 17:22
I did not notice that your system had time-dependent matrices. In that case, I do not think the result I linked to is applicable. Nevertheless, you may be interested in Lyapunov functions for stability analysis, since they provide an approach that can deal with non-linearities. –  Johannes Kloos Mar 9 '12 at 12:54