Laurent Bako

Review: Propagator-based methods for recursive subspace model identification

The problem of the online identification of multi-input multi-output (MIMO) state-space models in the framework of discrete-time subspace methods is considered in this paper. Several algorithms, based on a recursive formulation of the MIMO Output Error State-Space (MOESP) identification class, are developed. The main goals of the proposed methods are to circumvent the huge complexity of eigenvalues or singular values decomposition techniques used by the offline algorithm and to provide consistent state-space matrices estimates in a noisy framework. The underlying principle consists in using the relationship between array signal processing and subspace identification to adjust the propagator method (originally developed in array signal processing) to track the subspace spanned by the observability matrix. The problem of the (coloured) disturbances acting on the system is solved by introducing an instrumental variable in the minimized cost functions. A particular attention is paid to the algorithmic development and to the computational cost. The benefits of these algorithms in comparison with existing methods are emphasized with a simulation study in time-invariant and time-varying scenarios.

Parameterization and identification of multivariable state-space systems: A canonical approach

In this paper, the problem of determining a canonical state-space representation for multivariable systems is revisited. A method is derived to build a canonical state-space representation directly from data generated by a linear time-invariant system. Contrary to the classic construction methods of canonical parameterizations, the technique developed in this paper does not assume the availability of any observability or controllability indices. However, it requires the A-matrix of any minimal realization of the system to be non-derogatory. A subspace-based identification algorithm is also introduced to estimate such a canonical state-space parameterization directly from input-output data.

On-line structured subspace identification with application to switched linear systems

This article is concerned with the identification of switched linear multiple-inputs–multiple-outputs state-space systems in a recursive way. First, a structured subspace identification scheme for linear systems is presented which turns out to have many attractive features. More precisely, it does not require any singular value decomposition but is derived using orthogonal projection techniques; it allows a computationally appealing implementation and it is closely related to input–output models identification. Second, it is shown that this method can be implemented on-line to track both the range space of the extended observability matrix and its dimension and thereby, the system matrices. Third, by making use of an on-line switching times detection strategy, this method is applied to blindly identify switched systems and to label the obtained submodels. Simulation results on noisy data illustrate the abilities and the benefits of the proposed approach.

A sparse optimization approach to state observer design for switched linear systems

Abstract A new continuous state observer is derived for discrete-time linear switched systems under the assumptions that neither the continuous state nor the discrete state are known. A specificity of the proposed observer is that, in contrast to the state of the art, it does not require an explicit prior estimation of the discrete state. The key idea of the method consists in minimizing a non-smooth l 2 -norm-based weighted cost functional, constructed from the matrices of all the subsystems regardless of when each of them is active. In the light of recent development in the literature of compressed sensing, the minimized cost functional has the ability to promote sparsity in a way that makes the knowledge of the discrete mode sequence unnecessary.

Identification of piecewise affine systems based on Dempster-Shafer Theory

Abstract In this paper, we present a novel approach for the identification of Piecewise ARX systems from input-output data. The proposed method is able to solve simultaneously the problem of the data assignment, the parameter estimation and the submodels’ number thanks to an evidential procedure. Our approach exploits the fact that, in PWARX systems, the data are locally linear. Thats why our strategy aims at simultaneously minimizing the error between the measured output and each submodels output and the distance between data belonging to the same submodel. A soft multi-category Support Vector Classifier is used to find a complete polyhedral partition of the regressor set by discriminating all the classes simultaneously. Finally, the performance and the feasibility of the proposed approach are proved via a numerical example and an experimental data collected from an electronic component placement process in a pick-and-place machine.

On the notion of persistence of excitation for linear switched systems

The paper formulates the concept of persistence of excitation for discrete-time linear switched systems. In addition, the paper provides sufficient conditions for an input signal to be persistently exciting. Persistence of excitation is formulated as a property of the input signal, and it is not tied to any specific identification algorithm. The results of the paper rely on realization theory and on the notion of Markov-parameters for linear switched systems.

Identification of switched linear state space models without minimum dwell time

We consider the problem of identifying switched linear state space models from a finite set of input-output data. This is a challenging problem, which requires inferring both the discrete state and the parameter matrices associated with each discrete state. An important contribution of our work is that we do not make the restrictive assumption of minimum dwell time between the switches, as it is customary in methods that deal with such models. We first propose a technique for eliminating the unknown continuous state from the model equations under an appropriate assumption of observability. On a time horizon, this gives us a new switched input-output relation that involves structured intermediary matrices, which depend on the state space representation matrices. To estimate the intermediary matrices, we present a randomly initialized algorithm that alternates between data classification and parameter update via recursive least squares. Given these matrices, the parameters associated to the different discrete states can be computed after a correct estimation of the discrete state.

A Recursive Sparse Learning Method: Application to Jump Markov Linear Systems

Abstract This paper addresses the problem of identifying linear multi-variable models from the input-output data which is corrupted by an unknown, non-centered, and sparse vector error sequence. This problem is sometimes referred to as error correcting problem in coding theory and robust estimation problem in statistics. By taking advantage of some recent developments in sparse optimization theory, we present here a recursive approach. We then show that the proposed identification method can be adapted to estimate parameter matrices of Jump Markov Linear Systems (JMLS), that is, switched linear systems in which the discrete state sequence is a stationary Markov chain. Some numerical simulation results illustrate the potential of the new method.

Learning nonlinear hybrid systems: from sparse optimization to support vector regression

This paper deals with the identification of hybrid systems switching between nonlinear subsystems of unknown structure and focuses on the connections with a family of machine learning algorithms known as support vector machines. In particular, we consider a recent approach to nonlinear hybrid system identification based on a convex relaxation of a sparse optimization problem. In this approach, the submodels are iteratively estimated one by one by maximizing the sparsity of the corresponding error vector. We extend this approach in several ways. First, we relax the sparsity condition by introducing robust sparsity, which can be optimized through the minimization of a modified l1-norm or, equivalently, of the ε-insensitive loss function. Then, we show that, depending on the choice of regularizer, the method is equivalent to different forms of support vector regression. More precisely, the submodels can be estimated by iteratively solving a classical support vector regression problem, in which the sparsity of support vectors relates to the sparsity of the error vector in the considered hybrid system identification framework. This allows us to extend theoretical results as well as efficient optimization algorithms from the field of machine learning to the hybrid system framework.

Incremental and Decremental Multi-category Classification by Support Vector Machines

In this paper we propose an online multi-category support vector classifier dedicated to non-stationary environment. Our algorithm recursively discriminates between datasets of three or more classes, one sample at a time. With its incremental and decremental procedures, it can achieve an efficient update of the decision function after the incorporation/elimination of the incoming/oldest data. The key idea is to keep the KKT conditions of one single optimization problem satisfied, while adding or eliminating data. Compared to the QP approach, our classifier is able to provide accurate results. The performance of the proposed algorithm is shown on synthetic and experimental data.