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|a We use the tools of behavioural 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 system. We make a natural division of poles into controllable and uncontrollable poles. When the behaviour in question has latent variables, we make a further division into observable and unobservable poles. In the case of a 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 MFDs. We find behavioural 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 behaviour as the sum of sub-behaviours 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.
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