Abstract-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.

This paper proposes a learning framework and a set of algorithms for nonsmooth regression, i.e., for learning piecewise smooth target functions with discontinuities in the function itself or the derivatives at unknown locations. In the proposed approach, the model belongs to a class of smooth functions. Though constrained to be globally smooth, the trained model can have very large derivatives at particular locations to approximate the nonsmoothness of the target function. This is obtained through the definition of new regularization terms which penalize the derivatives in a location-dependent manner and training algorithms in the form of convex optimization problems. Examples of application to hybrid dynamical system identification and image reconstruction are provided.

Abstract-The paper focuses on the identification of nonlinear hybrid dynamical systems, i.e., systems switching between multiple nonlinear dynamical behaviors. Thus the aim is to learn an ensemble of submodels from a single set of inputoutput data in a regression setting with no prior knowledge on the grouping of the data points into similar behaviors. To be able to approximate arbitrary nonlinearities, kernel submodels are considered. However, in order to maintain efficiency when applying the method to large data sets, a preprocessing step is required in order to fix the submodel sizes and limit the number of optimization variables. This paper proposes four approaches, respectively inspired by the fixed-size least-squares support vector machines, the feature vector selection method, the kernel principal component regression and a modification of the latter, in order to deal with this issue and build sparse kernel submodels. These are compared in numerical experiments, which show that the proposed approach achieves the simultaneous classification of data points and approximation of the nonlinear behaviors in an efficient and accurate manner.

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 1-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.

Abstract-This paper deals with the identification of linear hybrid systems switching between multiple linear subsystems. We propose a new approach based on the geometric properties of hybrid systems in parameter space. More precisely, the data are mapped in that space such that each submodel is represented by a hypersphere. Then, we show how these hyperspheres can be easily separated by Principal Component Analysis (PCA) and derive a condition under which this separation is optimal for systems with two modes. Finally, classical (robust) regression is applied to estimate the system parameters from the classified data set. A simple procedure is also proposed to extend the method to the identification of switched systems with multiple modes. Experiments show that the final algorithm can accurately estimate both the parameters and the number of modes while being simple to apply and far more robust to noise than other methods.

Motivated by recent approaches to switched linear system identification based on sparse optimization, the paper deals with the recovery of sparse solutions of underdetermined systems of linear equations. More precisely, we focus on the associated convex relaxation where the 1 -norm of the vector of variables is minimized and propose a new iteratively reweighted scheme in order to improve the conditions under which this relaxation provides the sparsest solution. We prove the convergence of the new scheme and derive sufficient conditions for the convergence towards the sparsest solution. Experiments show that the new scheme significantly improves upon the previous approaches for compressive sensing. Then, these results are applied to switched system identification.

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