The investigation of linear periodic systems is a prominent subject of research within the domain of linear systems theory. In this context, the state-space realization of such systems is of particular significance. In fact, after successfully treating the realization problem for time-invariant systems, various researchers have directed their attention toward investigating the case of linear periodic systems. Meanwhile, in the late 1980s, Jan C. Willems proposed an approach that broadened the range of systems studied, now referred to as the behavioral approach. This approach views the behavior of a system as its fundamental element, including all signals that adhere to the system laws (also called system trajectories). More recently, the behavioral framework has also been extended separately to periodic and quaternionic behavioral systems. Our work consists of considering linear periodic input/output quaternionic behavioral systems and using recent developments in order to obtain quasi-minimal and uniform state-space realizations.

The state-space realization of linear systems is of utmost importance in linear systems theory. After the realization problem for the time-invariant case has been solved, particular attention was paid to the case of linear periodic systems (see, e.g. 1,2,3,4,5,6,7,8). Recently, such systems have regained importance, for instance, in the context of coding theory (see9), where periodic convolutional encoders play an important role, 10. The majority of the contributions within this area concern the realization of transfer functions as well as impulse responses, thus excluding the case of input/output linear systems without coprime representations. By the end of the eighties of the last century, Jan C. Willems (see11,12) suggested an approach (nowadays known as the behavioral approach) that considers a wider class of systems and allows to overcome this drawback. According to this approach, the central object in a system is its behavior which consists of all the signals that satisfy the system laws (also called system trajectories). Consequently, the behavior of a system with an input/output representation that is not coprime, contains more trajectories than the set of input/output signals defined by the system transfer function. Our work takes this fact into account. Based on results already obtained in 13,14, we revisit the problem of the realization of linear periodic MIMO behaviors and give further insight into this problem, which allows setting up an algorithm to compute a low-dimensional state-space realization of a periodic behavior. The proposed algorithm is based on a chain decomposition of suitable matrices.