An &amp;#x2113;&lt;inf&gt;0&lt;/inf&gt;&amp;#x2212;&amp;#x2113;&lt;inf&gt;1&lt;/inf&gt; norm based optimization procedure for the identification of switched nonlinear systems

Bako, Laurent; Boukharouba, K.; Lecœuche, S.

doi:10.1109/cdc.2010.5717199

Cited by 12 publications

(28 citation statements)

References 24 publications

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“…1). In particular, these crossings potentially generate undesired switches between the submodels and violate the assumption required by the method in [8].…”

Section: A Illustrative Examplementioning

confidence: 99%

“…3) Estimate the modeλ i for each data point bŷ (8) and classify the data into n subsets accordingly. 4) Re-estimate the submodels with a nonlinear estimator applied independently to each data subset.…”

Section: Reduced-size Kernel Models For Large-scale Problemsmentioning

confidence: 99%

“…Note that the crucial issue in this approach in order to deal with large data sets is to fix the submodel size and thus to limit the number of optimization variables. The only other reference directly dealing with a similar problem is [8], where a sparse optimization based method is proposed to iteratively estimate the submodels one by one. However, it relies on the assumption that the difference between the outputs of the nonlinear subsystems is larger than a bound on the noise for all input, which is unrealistic as soon as the subsystems are defined by intersecting functions f j .…”

Section: Introductionmentioning

confidence: 99%

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Reduced-Size Kernel Models for Nonlinear Hybrid System Identification

Bloch

Lauer

2011

IEEE Trans. Neural Netw.

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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.

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“…1). In particular, these crossings potentially generate undesired switches between the submodels and violate the assumption required by the method in [8].…”

Section: A Illustrative Examplementioning

confidence: 99%

Section: Reduced-size Kernel Models For Large-scale Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Reduced-Size Kernel Models for Nonlinear Hybrid System Identification

Bloch

Lauer

2011

IEEE Trans. Neural Netw.

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“…The quality of the estimates thus obtained rely on the capabilities of global optimization solvers, and thus cannot be guaranteed for models with many parameters. On the other hand, the approach of [2], which extends the method of [1], relies on a sequence of convex optimizations to estimate the submodels one by one, and thus does not suffer from local minima issues. More precisely, the method relies on the formulation of the identification as a sparse optimization problem and its convex relaxation.…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, the method relies on the formulation of the identification as a sparse optimization problem and its convex relaxation. However, the analysis of the method provided in [2] is only valid for noiseless data. This is all the more unsatisfactory in the nonlinear setting, since the uncertainty on the model structure can be interpreted as a form of noise.…”

Section: Introductionmentioning

confidence: 99%

Learning nonlinear hybrid systems

Lauer

Bako³

et al. 2013

Proceedings of the 16th International Conference on Hybrid Systems: Computation and Control

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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.

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Model structure selection for switched NARX system identification: A randomized approach

et al. 2021

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The identification of switched systems is a challenging problem, which entails both combinatorial (sample-mode assignment) and continuous (parameter estimation) features. A general framework for this problem has been recently developed, which alternates between parameter estimation and sample-mode assignment, solving both tasks to global optimality under mild conditions. This article extends this framework to the nonlinear case, which further aggravates the combinatorial complexity of the identification problem, since a model structure selection task has to be addressed for each mode of the system. To solve this issue, we reformulate the learning problem in terms of the optimization of a probability distribution over the space of all possible model structures. Then, a randomized approach is employed to tune this distribution. The performance of the proposed approach on some benchmark examples is analyzed in detail.

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An ℓ<inf>0</inf>−ℓ<inf>1</inf> norm based optimization procedure for the identification of switched nonlinear systems

Cited by 12 publications

References 24 publications

Reduced-Size Kernel Models for Nonlinear Hybrid System Identification

Reduced-Size Kernel Models for Nonlinear Hybrid System Identification

Learning nonlinear hybrid systems

Model structure selection for switched NARX system identification: A randomized approach

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