Representability of stochastic systems (Q2781441)

Finite-dimensional linear systems have several equivalent representations, namely the State Space Representation (SSR), the Matrix Fraction Description (MFD), and the AutoRegressive Moving Average (ARMA) representation. The representability theory of the stationary stochastic systems deals with the representability, i.e. the description of the representations of all possible input/output and state space of linear stationary stochastic systems, and with their representability, i.e. reconstruction of the entire statistics of the output process, knowing only the vector white noise processes related to the given representations. The book starts with spectral representation and Wold decomposition of stationary processes. The next part is devoted to purely nondeterministic stationary processes with rational spectral density including weak and strong state-space representations. The chapter on system invariants and canonical forms in SSR, ARMA and MFD representations contains observability invariants, constructibility invariants, reachability invariants, and controllability invariants. In the chapter called ``Geometric'' theory of state-space realizations the word ``geometric'' expresses the fact that instead of the coordinates of the actual processes the analysis is based on the subspaces generated by these processes. The results can be used for constructing acausal realizations, when the pole structure is arbitrary. A chapter is devoted to an analysis of inner functions leading finally to the factorization theorem of all-pass functions. Then the generalized Wiener-Hopf factorization of all-pass functions is discussed without minimality conditions and finally, the authors deal with the reconstruction of the output process from input white noises. Especially, it is shown that if the all-pass function is minimal, then the only factorization is determined by the forward and backward innovations of an output process. A list of some canonical cases of the reconstruction of stationary processes from given forward and backward input white noise is introduced in the last chapter. The canonical cases are obtained by means of the extremal possibilities of the input white noises. The book is written for researchers in signal processing, stochastic modeling and system identification, or in control system design.