Identification of piecewise affine systems via mixed-integer programming

Roll, Jacob; Bemporad, Alberto; Ljung, Lennart

doi:10.1016/j.automatica.2003.08.006

Cited by 351 publications

(256 citation statements)

References 40 publications

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“…These times show that the complexity of the proposed algorithms scales linearly with the number of data. As a result, the method can be applied to very large data sets, which cannot be handled by previous approaches such as the ones described in [1,3,6,12]. Note that the programs, including the MCS optimization algorithm, are fully implemented in non-compiled Matlab code and that the variability over the 100 runs comes from the random sampling of the true parameters generating the data, not from the optimizer.…”

Section: Large-scale Experimentsmentioning

confidence: 99%

“…However, this approach is sensitive to initialization. Note that problem (2) can also be solved directly by using mixed integer programming techniques, as proposed in [12] for hinging hyperplane models. These latter optimization techniques can guarantee to find the global minimum, but, due to their high complexity, they can only be used in practice for small data sets.…”

Section: General Formulation Of the Problemmentioning

confidence: 99%

“…This involves solving a combinatorial optimization problem, which can be prohibitively time consuming. The mixed integer programming (MIP) approach [12] and the bounded-error approach [1] fall into this category.…”

Section: Introductionmentioning

confidence: 99%

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A continuous optimization framework for hybrid system identification

2011

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We propose a new framework for hybrid system identification, which relies on continuous optimization. This framework is based on the minimization of a cost function that can be chosen as either the minimum or the product of loss functions. The former is inspired by traditional estimation methods, while the latter is inspired by recent algebraic and support vector regression approaches to hybrid system identification. In both cases, the identification problem is recast as a continuous optimization program involving only the real parameters of the model as variables, thus avoiding the use of discrete optimization. This program can be solved efficiently by using standard optimization methods even for very large data sets. In addition, the proposed framework easily incorporates robustness to different kinds of outliers through the choice of the loss function.

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Section: Large-scale Experimentsmentioning

confidence: 99%

Section: General Formulation Of the Problemmentioning

confidence: 99%

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A continuous optimization framework for hybrid system identification

2011

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“…Unfortunately it is known that, in the presence of unknownbut-bounded noise, this scenario leads to an NP-hard problem [21,8]. Several approaches have been proposed to address this difficulty [14,3,23,12]. While these are successful when dealing with relatively small noise levels or moderate size problems, performance deteriorates as the noise level or problem size increases.…”

Section: Introductionmentioning

confidence: 99%

Set membership identification of switched linear systems with known number of subsystems

2015

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This paper addresses the problem of robust identification of a class of discrete-time linear hybrid systems, switched linear models, in a set membership framework. Given a finite collection of noisy input/output data the objective is twofold: (i) establish whether this data was generated by a system that switches amongst an a-priori known number of subsystems, and (ii) in that case identify a suitable set of linear models along with a switching sequence that can explain the available experimental information. Our main result shows that these problems are equivalent to minimizing the rank of a matrix whose entries are affine in the optimization variables, subject to a convex constraint imposing that these variables are the moments of an (unknown) Borel measure with finite support. The use of well known (tight) convex relaxations of rank allows for further reducing the problem to a semidefinite optimization that can be efficiently solved. In the second part of the paper we extend these results to handle sensor failures that result in corrupted input/output measurements. Assuming that these failures are infrequent, we show that the problem can be recast into an optimization form where the objective is to simultaneously minimize the rank of a matrix and the number of nonzero rows of a second one. In both cases, appealing to well known convex relaxations of rank and sparsity leads to overall semidefinite optimization problems that can be efficiently solved. These results are illustrated with multiple examples showing substantially improved identification performance in the presence of noise and sensor faults.

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“…Moreover, piecewise affine systems can be used to identify or approximate generic nonlinear systems via multiple linearizations at different operating points [29], [11], [27]. Although hybrid systems (and in particular PWA systems) are a special class of nonlinear systems, most of the nonlinear system and control theory does not apply because it requires certain smoothness assumptions.…”

Section: A Linear Hybrid Systemsmentioning

confidence: 99%

Efficient Evaluation of Piecewise Control Laws Defined Over a Large Number of Polyhedra

Optimal Control of Constrained Piecewise Affine Systems

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Abstract-We consider the class of piecewise state feedback control laws applied to discrete-time systems, motivated by recent work on the computation of closed-form MPC controllers. The on-line evaluation of such a control law requires the determination of the state space region in which the measured state lies, in order to decide which 'piece' of the piecewise control law to apply. This procedure is called the point location problem, and the rate at which it can be solved determines the minimal sampling time of the system. In this paper we present a novel and computationally efficient search tree algorithm utilizing the concept of bounding boxes and interval trees that significantly improves this point-location search for piecewise control laws defined over a large number of (possibly overlapping) polyhedra. Furthermore, the required off-line preprocessing is low and so the approach can be applied to very complex controllers. The algorithm is compared with existing methods in the literature and its effectiveness is demonstrated for large examples.

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Identification of piecewise affine systems via mixed-integer programming

Cited by 351 publications

References 40 publications

A continuous optimization framework for hybrid system identification

A continuous optimization framework for hybrid system identification

Set membership identification of switched linear systems with known number of subsystems

Efficient Evaluation of Piecewise Control Laws Defined Over a Large Number of Polyhedra

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