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International audience; This paper considers a particular parameter estimator for switched systems and analyzes its properties. The estimator in question is defined as the map from the data set to the solution set of an optimization problem where the to-be-optimized cost function is a sum of pointwise infima over a finite set of sub-functions. This is a hard nonconvex problem. The paper studies some fundamental properties of this problem such as uniqueness of the solution or boundedness of the estimation error regardless of computational considerations. The interest of the analysis is to lay out the main influential properties of the data on the performance of this (ideal) estimator.

This paper concerns the incremental L2-gain stability of piecewise-affine (PWA) systems. We propose sufficient conditions derived from dissipativity theory to compute an upper bound on the incremental L2-gain. This is achieved by constructing piecewise-polynomial storage functions through the use of sum of squares (SOS) relaxations. The constraints are expressed as linear matrix inequalities (LMIs), which can be solved numerically in an efficient way. The proposed conditions are verified to be less conservative than previous results found in the literature by means of a numerical example.

International audience; For Prediction Error Identication, there are two main ingredients to get a consistent estimate: one of them is the data informativity with respect to (w.r.t.) the considered model structure. One common criterion used for the informativity is the positive deniteness of the input density spectral power (DSP) matrix at all frequencies. This criterion is not appropriate for multisine excitation but can be used for ltered white noise excitation for many identication problems. However, this criterion is not necessary and its application for some identication problems might not be possible. In this paper, we propose a necessary and sucient condition for the data informativity in the case of multiple-inputs single-output (MISO) nite impulse response (FIR) model structure in open-loop.

In Prediction Error Identification, the consistency of the identified parameter vector is only guaranteed if the data are informative enough i.e. if the excitation signal is sufficiently rich. For single-input single-output systems, one can verify whether a given excitation is sufficiently rich for a system based on the number of frequencies at which its power spectrum is nonzero. The extension of this criterion to multivariate systems is not straightforward. In the literature, one has proposed criteria based on the number of frequencies at which the power spectrum matrix of the excitation signal is strictly positive definite. However, this criterion is too restrictive as it does not cover the case of multisine excitations, while it is well known that such excitation signals can lead to consistent estimates. This paper proposes less restrictive conditions for the consistency of the identified parameter vector when FIR multiple-inputs single-output systems have to be identified with multisine signals in the open loop configuration.

This paper discusses an interval-valued state estimator for linear dynamic systems. In particular, we derive an expression of the tightest possible interval-valued estimator in the sense that it is the intersection of all interval-valued estimators. This estimator appears, in a general setting, to be an infinite dimensional dynamic system. Therefore, practical implementation requires some over-approximations which would yield a good trade-off between computational complexity and tightness., 7 pages, 4 figures

International audience; Lur’e-type nonlinear systems are virtually ubiquitous in applied control theory, which explains the great interest they have attracted throughout the years. The purpose of this paper is to propose conditions to assess incremental asymptotic stability of Lur’e systems that are less conservative than those obtained with the incremental circle criterion. The method is based on the approximation of the nonlinearity by a piecewise-affine function. The Lur’e system can then be rewritten as a so-called piecewise-affine Lur’e system, for which sufficient conditions for asymptotic incremental stability are provided. These conditions are expressed as linear matrix inequalities (LMIs) allowing the construction of a continuous piecewise-quadratic incremental Lyapunov function, which can be efficiently solved numerically. The results are illustrated with numerical examples.

We consider in this paper the problem of estimating a parameter matrix from observations which are affected by two types of noise components: (i) a sparse noise sequence which, whenever nonzero can have arbitrarily large amplitude (ii) and a dense and bounded noise sequence of "moderate" amount. This is termed a robust regression problem. To tackle it, a quite general optimization-based framework is proposed and analyzed. When only the sparse noise is present, a sufficient bound is derived on the number of nonzero elements in the sparse noise sequence that can be accommodated by the estimator while still returning the true parameter matrix. While almost all the restricted isometry-based bounds from the literature are not verifiable, our bound can be easily computed through solving a convex optimization problem. Moreover, empirical evidence tends to suggest that it is generally tight. If in addition to the sparse noise sequence, the training data are affected by a bounded dense noise, we derive an upper bound on the estimation error., Comment: To appear in IEEE Transactions on Automatic Control

In this paper we formulate a solution of the robust linear regression problem in a general framework of correntropy maximization. Our formulation yields a unified class of estimators which includes the Gaussian and Laplacian kernel-based correntropy estimators as special cases. An analysis of the robustness properties is then provided. The analysis includes a quantitative characterization of the informativity degree of the regression which is appropriate for studying the stability of the estimator. Using this tool, a sufficient condition is expressed under which the parametric estimation error is shown to be bounded. Explicit expression of the bound is given and discussion on its numerical computation is supplied. For illustration purpose, two special cases are numerically studied., Comment: 10 pages, 5 figures, To appear in Automatica

International audience; We consider the problem of recovering a finite number of linear subspaces from a collection of unlabeled data points that lie in the union of the subspaces. The data are such that it is not known which data point originates from which subspace. To address this challenge, we show that the clustering problem is amenable to a sparse optimization problem. Considering a candidate subspace and the distances of the data points to that subspace, the foundation of the proposed method lies in the maximization of the number of zero distances. This can be relaxed into a convex optimization. Efficiency of the relaxation can be significantly increased by solving a sequence of reweighted convex optimization problems.

In this paper we are interested in the incremental stability of piecewise-affine (PWA) systems. We present conditions to compute an upper bound on the incremental ℒ2-gain based on dissipativity analysis. These conditions are expressed as linear matrix inequalities (LMI) allowing the construction of a continuous piecewise quadratic storage function. It is also shown that these conditions imply incremental asymptotic stability of the system. The result is illustrated with numerical examples.

International audience; In this note, we investigate the convergence of a robust recursive identifier for linear models subject to impulsive disturbances. Under the assumption that the disturbance is unknown and can be of arbitrarily large magnitude, the analyzed algorithm attempts to minimize online the sum of absolute errors so as to achieve a sparse prediction error sequence. It is proved that the identifier converges exponentially fast into an euclidean ball whose size is determined by the richness properties of the estimation data, the frequency of occurrence of impulsive errors and the parameters of the algorithm.

The paper addresses the problem of minimality and identifiability of Switched AutoRegressive eXogenous (abbreviated by SARX) models. We propose a notion of identifiability and minimality for SARX models which depends only on the parameters of the model, not on data. We formulate conditions for minimality and identifiability of SARX systems. In particular, we show that SARX systems are generically identifiable.

In this work, a recursive procedure is derived for the identification of switched linear models from input–output data. Starting from some initial values of the parameter vectors that represent the different submodels, the proposed algorithm alternates between data assignment to submodels and parameter update. At each time instant, the discrete state is determined as the index of the submodel that, in terms of the prediction error (or the posterior error), appears to have most likely generated the regressor vector observed at that instant. Given the estimated discrete state, the associated parameter vector is updated based on recursive least squares or any fast adaptive linear identifier. Convergence of the whole procedure although not theoretically proved, seems to be easily achieved when enough rich data are available. It has been also observed that by appropriately choosing the data assignment criterion, the proposed on-line method can be extended to deal also with the identification of piecewise affine models. Finally, performance is tested through some computer simulations and the modeling of an open channel system.

International audience; The design of a nonlinear Luenberger observer fora parametrized linear system is studied. From an observabilityassumption of the system, the existence of such an observer isconcluded. In a second step, a constructive novel algorithm forthe identification of multi-input multi-output linear systems issuggested and is implemented on a second order system

International audience; Identifying switched linear models directly from input-output measurements only is known to be a non-trivial identification problem. When switched state-space models are considered in a general setting and both the continuous state and the discrete mode are unmeasured, the problem proves to be a much harder realization problem. The present paper describes a method for identifying discrete-time switched linear state-space models from input-state-output measurements. While the discrete mode is unknown, we assume here that the continuous state is measured along with the input and output signals. Given a finite collection of such measured data, we propose a sparsity-inducing optimization approach for estimating the matrices associated with each submodel.

International audience; 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 epsilon-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.

The paper presents realization theory of discrete-time linear switched systems. A discrete-time linear switched system is a hybrid system, such that the continuous sub-system associated with each discrete state is linear. In this paper we present necessary and sufficient conditions for an input-output map to admit a discrete-time linear switched state-space realization. The conditions are formulated as finite rank conditions of a generalized Hankel-matrix. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observable.Further, we prove that minimal realizations are unique up to isomorphism. We also discuss procedures for converting a linear switched system to a minimal one and we present an algorithm for constructing a state-space representation from input-output data.The paper uses the theory rational formal power series in non-commutative variables. The latter theory was successfully applied to bilinear and state-affine systems in the past., Some typos have been corrected and Algorithm 1 (realization algorithm) has been added, together with a theorem stating the correctness of the algorithm

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.

This paper addresses the problem of driving the state of a linear discrete-time system to zero in minimum time. The inputs are constrained to lie in a bounded and convex set. The solution presented in the paper is based on the observation that the state sequence induced by the minimum-time control sequence is the sparsest possible state sequence over a certain finite horizon. That is, the desired state sequence must contain as many zero vectors as possible, all those zeros corresponding to the highest values of the time index. Hence, by taking advantage of some recent developments in sparse optimization theory, we propose a numerical solution. We show in simulation that the proposed method can effectively solve the minimum-time problem even for multi-inputs linear discrete-time systems.

Piecewise Affine maps (PWA) constitute an attractive modeling abstraction for nonlinear dynamic systems. An important feature of PWA model structures is that they carry the quite natural intuition according to which the operating space of a complex physical system can be decomposed into an appropriate number of pieces on which the system can be viewed as being affine. The paper presents a new method for the recovery of piecewise affine systems from observed samples of data. This problem is usually addressed through data clustering procedures which, unfortunately, are rarely guaranteed to be optimal. The ambition of this work is to overcome the need of resorting to a systematic clustering. To this end, some ideas of sparse optimization are used in combination with the local linearity property of PWA maps.

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, contrary to the state-of-art, it does not require an explicit 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 some recent development in the literature of compressed sensing, the minimized cost functional has the ability to promote sparsity in a way that makes prior knowledge/estimation of the discrete mode sequence unnecessary.

International audience; The work presented in this paper is concerned with the identification of switched linear systems from input-output data. The main challenge with this problem is that the data are available only as a mixture of observations generated by a finite set of different interacting linear subsystems so that one does not know a priori which subsystem has generated which data. To overcome this difficulty, we present here a sparse optimization approach inspired by very recent developments from the community of compressed sensing. We formally pose the problem of identifying each submodel as a combinatorial $\ell_0$ optimization problem. This is indeed an NP-hard problem which can interestingly, as shown by recent literature, be relaxed into a (convex) $\ell_1$-norm minimization problem. We present sufficient conditions for this relaxation to be exact. The whole identification procedure allows us to extract the parameter vectors (associated with the different subsystems) one after another without any prior clustering of the data according to their respective generating-submodels. Some simulation results are included to support the potentialities of the proposed method.

We consider the problem of identifying a switched nonlinear system from a finite collection of input-output data. The constituent subsystems of such a switched system are all nonlinear systems. We model each individual subsystem as a sparse expansion over a dictionary of elementary nonlinear smooth functions shaped by the whole available dataset. Estimating the switched model from data is a doubly challenging problem. First one needs, without any knowledge of the parameters, to decide which subsystem is active at which time instant. Second, the representation of each nonlinear subsystem over the considered basis shall be performed in a high dimensional space. We tackle both tasks simultaneously by sparse optimization. More specifically, we view the switched nonlinear system identification problem as the problem of minimizing the l 0 norm of an error vector. We subsequently relax it into an l 1 convex minimization problem for which powerful numerical tools exist.

In this paper we study the identifiability of linear switched systems (LSSs) in discrete-time.The question of identifiability is central to system identification, as it sets the boundaries of applicability of any system identification method; no system identification algorithm can properly estimate the parameters of a system which is not identifiable. We present necessary and sufficient conditions that guarantee structural identifiability for parametrized LSSs. We also introduce the class of semi-algebraic parametrizations, for which these conditions can be checked effectively.

International audience; In this work, a recursive procedure is derived for the identification of switched affine models from input-output data. Starting from some initial values of the parameter vectors that represent the different submodels, the proposed algorithm alternates between data assignment to submodels and parameter update. At each time instant, the discrete state is determined as the index of the submodel that, in term of the prediction error, appears to have most likely generated the regressor vector observed at that instant. Given the estimated discrete state, the associated parameter vector is updated based on recursive least squares. Convergence of the whole procedure although not theoretically proved, seems to be easily achieved when enough rich data are available. Finally performance is tested through some computer simulations and the modeling of an open channel system.

The paper introduces a method to estimate a linear state-space model from its input-output data. Its starting point is the conversion of the state-space model into an input-output ARMAX-like model. In order to conveniently estimate the parameters of this ARMAX model, it is necessary, given the structural characteristics of the initial state-space model, to assume that the dynamic state matrix A is nonderogatory. Under this condition, it is shown that the input-output model and the state-space model from which it is derived have the same order. Based on this important result in linear system realization theory, a canonical state-space representation is finally obtained through a simple construction technique. The developed method can be regarded as an alternative to the standard subspace methods particularly in situations where the state coordinates basis need to be maintained constant with respect to the input-output realization used for the estimation.

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. That's why our strategy aims at simultaneously minimizing the error between the measured output and each submodel's 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.

International audience; 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.

International audience; This paper is concerned with the identification of switched linear MIMO state-space systems in a recursive way. Firstly, 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 (SVD) but is derived using orthogonal projection techniques; it allows a computationally appealing implementation and it is closely related to input-output models identification. Secondly, 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. Thirdly, 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 recursive scheme for the identification of SIMO Hammerstein models is presented. In the proposed scheme, first the Markov parameters of the system are determined, by a least squares support vector machines regression through an over-parameterisation technique. Then, a state-space realisation of the system is retrieved using a recursive subspace identification method. Simulation results are provided to demonstrate the effectiveness of the algorithm.

International audience; In this paper, we propose a new method for the identification of linear Multiple Inputs-Multiple Outputs (MIMO) systems. By introducing a particular user-defined matrix that does not change the rank of the extended observability matrix when multiplying this latter matrix on the left, the subspace identification problem is recasted into a simple least squares problem with all regressors available. Therefore, the Singular Value Decomposition algorithm which is a customary tool in subspace identification can be avoided, thus making our method appealing for recursive implementation. The technique is such that the state coordinates basis of the estimated matrices is completely determined by the aforementioned user-defined matrix, that is, given such a matrix, the state basis of the identified matrices does not change with respect to the realization of input-output data.

We consider the problem of identifying the parameters of a multiple-input multiple-output switched ARX model with unknown number of submodels of unknown and possibly different orders. This is a very challenging problem because of the strong coupling between the unknown discrete state and the unknown model parameters. We address this challenge by algebraically eliminating the discrete state from the switched system equations. This algebraic procedure leads to a set of hybrid decoupling polynomials on the input-output data, whose coefficients can be identified using linear techniques. The parameters of each subsystem can then be identified from the derivatives of these polynomials. This exact analytical solution, however, comes with an important price in complexity: The number of coefficients to be identified grows exponentially with the number of outputs and the number of submodels. We address this issue with an alternative scheme in which the input-output data is first projected onto a low-dimensional linear subspace. The projected data is then fit with a single hybrid decoupling polynomial, from which the classification of the data according to the generating submodels can be obtained. The parameters of each submodel are then identified from the input-output data associated with each submodel.

International audience; 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 minimised 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 emphasised with a simulation study in time-invariant and time-varying scenarios. matrices.

The modeling of nonlinear real-life systems subject to parameters variation is a major research problem. In this paper, a weighted combination of multiple local models is proposed to represent the dynamics of an open channel system. The multimodelling strategy followed consists in the determination of a finite number of models along with appropriate weights functions. The families of Gaussian and Boolean weights functions are particularly studied and compared in various aspects on an application example. For the purpose of supervision, the approach of composite linear local models appears to be an interesting alternative to the use of Saint Venant partial differential equations. The efficiency of the multimodelling methods is shown by simulation within the framework of a dam-gallery system.

International audience; This paper is concerned with the identification of piecewise linear MIMO state space systems in a recursive way. The proposed method summons up benefits of recursive parameters estimation, on-line switching times detection and on-line order estimation. A structured identification scheme which applied on-line, allows to track both the extended observability matrix range space and its dimension. This method is used on-line to blindly identify switching systems and to label the different submodels. Since subspace identification methods rely on batch data block matrices, a minimum dwell time in each discrete state is necessary to achieve good performances. Simulation results comfort this point and illustrate the abilities and the benefits of the proposed approach.