- Algorithms for deterministic balanced subspace identification
A new algorithm for identification of a balanced state space representation is proposed. It is based on a procedure for estimation of the impulse response and sequential zero input responses directly from data. The proposed algorithm is more efficient than the existing alternatives that compute the whole Hankel matrix of Markov parameters. It is shown that the computations can be performed on Hankel matrices of the input-output data of various dimensions. By choosing wider matrices, we need persistency of excitation of smaller order. Moreover, this leads to computational savings and improved statistical accuracy when the data is noisy. Using a finite amount of input-output data, the existing algorithms compute finite time balanced representation and the identified models have a lower bound on the distance to an exact balanced representation. The proposed algorithm can approximate arbitrarily closely an exact balanced representation. Moreover, the finite time balancing parameter can be selected automatically by monitoring the decay of the impulse response. [Paper]
Co-ordinator(s): Bart De Moor, Jan Willems, Researcher(s): Ivan Markovsky
- Hammerstein system identification using LS-SVM's
Least-Squares Support Vector Machines can be used for nonlinear static function regression. LS-SVM's have also been used for blackbox modelling of nonlinear dynamic systems of the form y(t) = f(y(t-1,...),u(t-1,.....)) in the same manner of static function regression. As fully nonlinear blackbox models are often overly broad and difficult to tune we propose the use of LS-SVM's for the identification of Hammerstein models, which can be identified as additive nonlinear models with a collinearity constraint.
More information in Goethals I., Pelckmans K., Suykens J.A.K., De Moor B., "NARX identification of Hammerstein Models using least squares support vector machines", in Proc. of the 6th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2004), Stuttgart, Germany, Sep. 2004, pp. 507-512.
Co-ordinator: Johan Suykens, Researcher: Kristiaan Pelckmans
- Application of structured total least squares for system identification and model reduction
We consider the following generic problem: minimize the L2 norm of the difference between a given time series and an estimated one under the constraint that the estimated time series is a trajectory of a linear time invariant system with a fixed order. The generic problem can be viewed as maximum likelihood system identification in the errors-in-variables setup under suitable assumptions for the measurement noises. Multiple time series and latent variables can be considered in the identification problem as well as measurement errors. Special cases of the generic problem are output only identification, noisy realization, and finite time optimal L2 model reduction. We relate the generic problem to the structured total least squares problem. In contrast to previous results using the structured total least squares approach, we do not restrict to single input single output systems. An efficient software package that implements the theory in practice is used. The performance of the proposed method and software is tested on examples from the database for the identification of systems DAISY. [Paper]
Co-ordinator(s): Bart De Moor, Jan Willems, Sabine Van Huffel, Researcher(s): Ivan Markovsky
- New techniques for the blind identification of linear systems
We have derived new techniques for the blind identification of multiple-input multiple-output (MIMO) finite impulse response (FIR) filters with temporally independent identically distributed and spatially independent non-Gaussian inputs. First we have followed a time-domain approach, and we limited ourselves to the case of 2 outputs and 2 inputs. After a classical prewhitening, the remaining problem is the blind identification of a paraunitary filter. For this task, we have derived a multilinear algebraic algorithm. This procedure is a generalization of the well-known algorithm for independent component analysis (ICA) (i.e., blind identification of a scalar mixture) proposed by Comon. Next, in collaboration with B. Chen and A. Petropulu of Drexel University, Philadelphia, we have proposed a novel frequency domain approach for MIMO systems with at least as many outputs as inputs. The system frequency response is obtained via a singular value decomposition (SVD) of a matrix constructed from the power-spectrum and slices of the polyspectrum of the system output. Since several slices could be used, the SVD can be replaced by a joint diagonalization of a set of matrices. The flexibility to select the polyspectrum slices allows us to bypass the frequency dependent permutation and phase ambiguity problems, which are usually associated with a frequency domain SVD.
Researcher: Lieven De Lathauwer
- System Identification with LS-SVM
In the framework of nonlinear system identification, LS-SVM is a powerful technique for black-box modelling. This research is aimed to the development of methodologies and algorithms for the application of LS-SVM within the context of data-driven (prediction error) system identification, dealing with issues of large-scale implementation, model selection, model structure, etc. In particular, the research is oriented to develop and exploit model structures that may contain linear components, in such a way that the complexity of the final model can be reduced substantially.
Researcher: Marcelo Espinoza
- Distances and angles between linear models
Quantifying the distance between signals and/or their models is important in many applications, e.g. clustering, classification, fault detection, ... We have studied several distance measures for linear models. Furthermore, we have defined a new notion of subspace angles between models. These angles also indicate how far apart two models are and they can be used to deduce several distance measures.
The subspace angles between two linear time-invariant models are defined as the principal angles between certain subspaces derived from the models. We have established a relation between the subspace angles between two models and a recently defined cepstral metric.
Co-ordinator: Bart De Moor, Researcher: Katrien De Cock, Jeroen Boets
- Recursive subspace identification
Subspace identification is a well known and robust technique to identify linear models. It is less known that efficient recursive implementations exist for subspace algorithms. Over the last few years, several recursive input-output subspace algorithms have been proposed which mainly follow the following two steps:
* Replace the QR decomposition which is usually the first step of a subspace implementation by a recursive QR decomposition. Note that only R is needed in classical subspace algorithms. The same remains true for the recursive algorithms which is a big asset.
* Replace the SVD decomposition by at least a recursive version, but preferably by a subspace tracking algorithm, which is orders of magnitudes faster. Efficient implementations have been obtained, e.g. by using an Instrumental Variable - Projection Approximation (IV-PAST).
In our research, we deal with recursive output-only algorithms and recursive stochastic realization on the one hand and input-output recursive algorithms applied to data from airplanes on the other hand.
More information in Goethals I., Mevel I., Benveniste A., De Moor B., "Recursive output-only subspace identification for in-flight flutter monitoring", in Proc. of the 22nd International Modal Analysis Conference (IMAC-XXII), Dearborn, Michigan, Jan. 2004 and De Cock K.,
Mercere G., De Moor B., Recursive
subspace identification for in-flight modal analysis of
airplanes, in Proc. of the International Conference on
Noise and Vibration Engineering (ISMA 2006), Leuven, Belgium,
Sept. 2006, pp. 1563-1577.
Researcher: Katrien De Cock
- Errors-in-variables filtering
We consider estimation problems for linear system with a state disturbance and additive errors on the input and the output. The problem formulation and the estimation principle are deterministic. The derived filter is identical to the stochastic Kalman filter. The problem formulation with additive error on both the input and the output, however, is more symmetric then the classical Kalman filter one and allows interpretation in terms of misfit and latent variables. [Paper]
Co-ordinator(s): Bart De Moor, Jan Willems, Researcher(s): Ivan Markovsky
- Spurious mode rejection in modal analysis
Together with industrial partners LMS
(Belgium), Dassault and EADS-Airbus and Flemish and French
universities, a EUREKA project was started in 2001 to tackle the
problem of flutter detection in in-flight analysis and the appearance
of spurious modes in vibration data. Due to changes in speed and air
pressure, the modal characteristics of an airplane change
continuously during in-flight measurements. If the damping of a
certain mode suddenly decreases the aircraft may experience heavy
vibrations, putting a lot of stress on the body of the craftm which
may ultimately lead to the destruction of the aircraft. When these
strong vibrations occur, the aircraft is said to go into flutter. In
order to detect flutter before the aircraft is destroyed, a
continuous monitoring of the plane during in-flight measurements is
necessary. Fast, preferrably recursive techniques can then be used to
update a dynamical model describing the structural vibrations.
Vibration modes and their dampings can easily be derived from these
models. A problem however lies in the fact that together with the
identified vibration modes of the plane, all identification
procedures that are commonly used today also return some to many
modes that have no physical relevance. This results from the fact
that in modal analysis, the modelling order is usually chosen much
higher that the true system order to reduce the bias on the
estimates. It is important to separate these spurious modes from the
true ones, a cumbersome procedure which heavily relies on user
In light of the Flite project, several methods were investigated to automatize the detection of spurious modes. Within SCD we studied the relation between subspace identification and balanced model reduction. It was found that subspace identification methods are fast, reliable, and therefore ideally suited for use in online analysis. Detection of spurious modes proved much harder, as a strong theoretical theory about spurious modes is lacking. Several heuristic techniques were proposed, involving non-balanced model reduction, modal energy analysis and pole/zero cancellations.
Researcher(s): Katrien De Cock, Ivan Markovsky, Jeroen Boets, Bart Vanluyten
- Realization and filtering for hidden Markov models
In this research we study stochastic dynamical systems, in particular
systems of which the output space is finite. Such systems are used to
model speech, images, economical and biological phenomena. A popular
way to represent these systems is by means of discrete time hidden
Markov models (HMM). This kind of models is very analogous to the well-
known discrete time linear time-invariant stochastic models (LTISM).
However, for linear stochastic systems the theoretic framework has been
investigated thoroughly, while for hidden Markov models lots of
theoretical questions remain open. For instance for linear stochastic
models, it is well known what equivalence of systems means. For hidden
Markov models, we will investigate if it is possible to define
equivalence classes in the same way. Next, the (approximate) quasi-
realization problem, the (approximate) realization problem and the
filtering problem will be studied. Concerning the realization problem,
we will check whether it can be solved using the recently introduced
nonnegative matrix factorization or variants of this factorization.
This decomposition then plays the role for hidden Markov models that
the singular value decomposition plays for linear stochastic systems.
Co-ordinator(s): Bart De Moor, Jan Willems, Researcher(s): Bart Vanluyten, Katrien De Cock
- Optimal filtering in the presence of unknown inputs
Faults, disturbances, systematic measurement error,... can all be represented as unknown inputs. The optimal filtering problem for linear discrete-time systems in the presence of unknown inputs has therefore received a lot of attention during the last decades. A popular method is to augment the state vector with the input vector and then design a Kalman filter for the augmented model. However, this method is limited to the case where a dynamical model for the unknown input is available. In this research, we develop optimal filters/estimators which are based on the assumption that no prior information about the unknown input is available. We consider the state estimation problem as well as the input estimation problem. Furthermore, the filtering problem as well as the smoothing problem are addressed. Finally, the relation to system inversion is explored.[Paper1][Paper2]
Co-ordinator(s): Bart De Moor, Researcher(s): Steven Gillijns