1,721,056 research outputs found
Title: A module theoretic interpretation of multiplicity and rank of a stationary random process
No full textThis paper is in a sense a celebration of Fuhrmann’s book Linear Systems and Operators in Hilbert Space. Inspired by one of Fuhrmann’s unifying ideas, in this note we shall discuss module theory applied to stationary stochastic processes. In particular, we shall reframe the notion of multiplicity and rank of a second order stochastic process in a module theoretic framework. This will (hopefully) clarify some issues which are still occasion of some confusion in the literature
Consistency Analysis of some Closed-Loop Subspace Identification Methods
Full text linkWe study statistical consistency of two recently proposed subspace identification algorithms for closed-loop systems. These algorithms
may be seen as implementations of an abstract state-space construction procedure described by the authors in previous work on stochastic
realization of closed-loop systems.A detailed error analysis is undertaken which shows that both algorithms are biased due to an unavoidable
mishandling of initial conditions which occurs in closed-loop identification. Instability of the open loop system may also be a cause of
trouble
Realization of Systems with Exogenous Inputs and Subspace Identification Methods
No full textThis paper solves the stochastic realization problem for a discrete-time stationary process with an exogenous input. The oblique projection of the future outputs on the space of the past observations along the space of the future inputs is factorized as a product of the extended observability matrix and the state vector. The state vector is chosen by using the canonical correlation analysis (CCA) of past and future conditioned on the future inputs. We then derive the state equations of the optimal predictor of the future outputs in terms of the state vector and the future inputs. These equations lead to a forward innovation model for the output process in the presence of exogenous inputs. The basic step of the realization procedure is a factorization of the conditional covariance matrix of future outputs and past data given future inputs. This factorization is based on CCA and can be easily adapted to finite input–output data. We derive four stochastic subspace identification algorithms which adapt the realization procedure to finite input–output data. Numerical results are also included
Subspace identification by data orthogonalization and model decoupling
No full textIt has been observed that identification of state-space models with inputs may lead to unreliable results in certain experimental conditions even when the input signal excites well within the bandwidth of the system. This may be due to ill-conditioning of the identification problem, which occurs when the state space and the future input space are nearly parallel.
We have in particular shown in the companion papers (Automatica 40(4) (2004) 575; Automatica 40(4) (2004) 677) that, under these circumstances, subspace methods operating on input–output data may be ill-conditioned, quite independently of the particular algorithm which is used. In this paper, we indicate that the cause of ill-conditioning can sometimes be cured by using orthogonalized data and by recasting the model into a certain natural block-decoupled form consisting of a “deterministic” and a “stochastic” subsystem. The natural subspace algorithm for the identification of the deterministic subsystem is then a weighted version of the PI-MOESP method of Verhaegen and Dewilde (Int. J. Control 56 (1993) 1187–1211). The analysis shows that, under certain conditions, methods based on the block-decoupled parametrization and orthogonal decomposition of the input–output data, perform better than traditional joint-model-based methods in the circumstance of nearly parallel regressors
Acausal Models and Balanced realizations of stationary processes
No full textSPECIAL ISSUE ON SYSTEMS THEOR
Irreducible representations of L/sup 2/-signals[1991] Proceedings of the 30th IEEE Conference on Decision and Control
No full textOn the Ill-conditioning of subspace identification with inputs
No full textThere is experimental evidence that the performance of standard subspace algorithms from the literature (e.g. the N4SID method) may be surprisingly poor in certain experimental conditions. This happens typically when the past signals (past inputs and outputs) and future input spaces are nearly parallel. In this paper we argue that the poor behavior may be attributed to a form of ill-conditioning of the underlying multiple regression problem, which may occur for nearly parallel regressors. An elementary error analysis of the subspace identification problem, shows that there are two main possible causes of ill-conditioning. The first has to do with near collinearity of the state and future input subspaces. The second has to do with the dynamical structure of the input signal and may roughly be attributed to “lack of excitation”. Stochastic realization theory constitutes a natural setting for analyzing subspace identification methods. In this setting, we undertake a comparative study of three widely used subspace methods (N4SID, Robust N4SID and PO-MOESP). The last two methods are proven to be essentially equivalent and the relative accuracy, regarding the estimation of the (A,C) parameters, is shown to be the same
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