A novel subspace identification approach with enforced causal models

Abstract

Subspace identication methods (SIMs) for estimating state-space models have been proven to be very useful and numerically ecient. They exist in several variants, but have one feature in common: as a rst step, a collection of high-order ARX models are estimated from vectorized inputoutput data. In order not to obtain biased estimates, this step must include future outputs. However, all but one of the submodels include non-causal input terms. The coecients of them will be correctly estimated to zero as more data become available. They still include extra model parameters which give unnecessarily high variance, and also cause bias for closed loop data. In this paper, a new model formulation is suggested that circumvents the problem. Within the framework, the system matrices (A,B,C,D) and the Markov parameters can be estimated separately. It is demonstrated through analysis that the new methods generally give smaller variance in the estimate of the observability matrix and it is supported by simulation studies that this gives lower variance also of the system invariants, such as poles. AbstractSubspace identification methods (SIMs) for estimating state-space models have been proven to be very useful and numerically efficient. They exist in several variants, but all have one feature in common: as a first step, a collection of high-order ARX models are estimated from vectorized input-output data. In order not to obtain biased estimates, this step must include future outputs. However, all but one of the submodels include non-causal input terms. The coefficients of them will be correctly estimated to zero as more data become available. They still include extra model parameters which give unnecessarily high variance, and also cause bias for closed-loop data. In this paper, a new model formulation is suggested that circumvents the problem. Within the framework, the system matrices (A, B, C, D) and Markov parameters can be estimated separately. It is demonstrated through analysis that the new methods generally give smaller variance in the estimate of the observability matrix and it is supported by simulation studies that this gives lower variance also of the system invariants such as the poles. ᭧

A Bootstrap Subspace Identification Method: Comparing Methods for Closed Loop Subspace Identification by Monte Carlo Simulations

“…The PARSIM-E (Qin and Ljung (2003), Qin et al (2005) and DSR (Di Ruscio (1996), DSR e (Di Ruscio (2008, 2009) methods may be viewed as subspace methods of this class which are based on matrix equations where the states are eliminated from the problem.…”

On consistency of closed-loop subspace identification with innovation estimation

“…The observability matrix, Γ f , can be estimated similarly to the order determination procedure in [9].…”

“…Further, very few SIMs are applicable to closedloop identification, even though the data satisfy identifiability conditions for traditional methods such as PEMs. In [9], we give the reasons why subspace identification approaches Contrary to the open loop SIMs, the traditional SIMs (e.g., CVA, N4SID and MOESP) are biased under closedloop condition. Verhaegen [10] proposed a closed-loop SIM via the identification of an overall open-loop state space model followed by a model reduction step to obtain state space representations of the plant and controller.…”

Subspace identification of Hammerstein systems with nonparametric input backlash and switch nonlinearities

“…(2) Subspace identification methods are used for the estimation. The use of such methods is mainly motivated by a set of interesting properties: the simplicity, the intrinsic numerical robustness and their straightforward application to MIMO systems ( [26], [29], [30], [10], [18], [23], [24], [9]). Some subspace methods adapted to Hammerstein systems have been introduced ( [15], [14], [3], [33], [28], [17], [21]) but, to the best of our knowledge, these methods aren't adapted to the case of backlash or switch nonlinearity.…”