Some relations between ergodicity and minimality properties of invariant sets in quantized control systems

Linear dynamical systems controlled by quantized inputs exhibit phenomena which are typically non-linear, including chaotic behaviours. We consider discrete-time single-input models of the type x(k+1)=Ax(k)+bu(k). The construction of invariant sets for this class of hybrid systems is of utmost importance for the stabilization problem. We first review a technique to construct invariant sets when an arbitrary quantized input set is assigned. We hence study minimality properties for invariant sets when inputs take integer values. There is a relation between a so-called strong minimality property and ergodicity of the closed-loop dynamics, in particular, ergodicity implies strong minimality. A condition ensuring strong minimality is given in terms of the coefficients of the characteristic polynomial of the matrix 'A'. Two examples are presented: the first one shows that strong minimality does not imply ergodicity. The second one shows that our condition for strong minimality is only sufficient: this is done by exhibition of an ergodic dynamics for which our condition is not satisfied.