A Behavioral Approach to the Pole Structure of One-Dimensional and Multidimensional Linear Systems

We use the tools of behavioral theory and commutative algebra to produce a new definition of a (finite) pole of a linear system. This definition agrees with the classical one and allows a direct dynamical interpretation. It also generalizes immediately to the case of a multidimensional (nD) system. We make a natural division of the poles into controllable and uncontrollable poles. When the behavior in question has latent variables, we make a further division into observable and unobservable poles. In the case of a one-dimensional (1D) state-space model, the uncontrollable and unobservable poles correspond, respectively, to the input and output decoupling zeros, whereas the observable controllable poles are the transmission poles. Most of these definitions can be interpreted dynamically in both the 1D and nD cases, and some can be connected to properties of kernel representations. We also examine the connections between poles, transfer matrices, and their left and right matrix fraction descriptions (MFDs). We find behavioral results which correspond to the concepts that a controllable system is precisely one with no input decoupling zeros and an observable system is precisely one with no output decoupling zeros. We produce a decomposition of a behavior as the sum of subbehaviors associated with various poles. This is related to the integral representation theorem, which describes every system trajectory as a sum of integrals of polynomial exponential trajectories.