Instability in the Simplest Class of Continuous Switched Linear Systems with Stable Components

For discontinuous switched linear systems, even when they are built by composing stable systems, examples of unstable systems are known. Here, threedimensional homogeneous continuous piecewise linear systems composed by two linear systems sharing a boundary plane are considered. If the two linearization matrices are non-singular, then the only equilibrium point is at the origin, which is in the separation plane of the linear regions. For the important case where both matrices have complex eigenvalues and their whole spectra are in the open-left half of the complex plane, the possible counter-intuitive instability of the origin is analytically proved. Thus, in this paper, examples of unstable continuous switched linear system with just two stable components are shown. Copyright c ©2005 IFAC.