Hybrid System Identification

Abstract

This chapter introduces the main topic of the book, that is, hybrid system identification. It details the various models of hybrid systems before posing the hybrid system identification problem and discussing the inherent trade-off between the number of modes and the error of the model. This leads to different formulations of the problem, depending on the assumptions and on which part of the trade-off is focused on. Thus, for both arbitrarily switched and piecewise-defined models, two cases are detailed: for a fixed number of submodels and for a given upper bound on the error. The chapter ends with a brief discussion on other problems studied in other fields that are related to hybrid system identification.

Notes

PWA systems are introduced in [1] and arbitrarily switched systems in [2]. The universal approximation capability of PWA models was proved in [3, 4]. The approximation of a NARX model by a PWARX one as described in this chapter is taken from [5]. The various types of partitions of the regression domain for PWA systems have been evaluated in terms of computational complexity for digital implementation in [6]. System identification with nonlinear partitions was considered in [7].

Switching regression was introduced by [8], and early algorithms include the one of [9] and the expectation-maximization methods of [10] for the two-mode case, further generalized by [11, 12].

Regression trees [13], and subsequent improvements [14, 15], are well-known early examples of piecewise regression models, together with the mixtures of experts [16], which however usually consider smooth switchings.

In the machine learning community, the mixture-of-experts framework and expectation-maximization algorithm were extended to model hybrid dynamical systems with hidden Markov models in [17]. A number of works based on such a probabilistic view can be found in [18, 19] or in [20, 21], where so-called jump Markov linear systems are considered.

More recently, most of the work in this field was produced by the control community for hybrid system identification. A first review of the methods operating in that context appeared in 1982 in [22], while the revival of the field was described in the tutorial of [2] with a review of some of the new major techniques at the time that are also experimentally compared in [23]. An updated survey followed in 2012 in [24].

The bounded-error approach in which the submodels are estimated one by one until a predefined error is achieved was first considered in this context in [25] and then followed in [5, 26] with different methods to estimate the jth submodel at each step.

The Minimum-of-Errors formulation in (4.21) was considered in [27], where its direct optimization with the multilevel coordinate search solver of [28] was proposed (see Sect. 6.1.2).

Several works aiming at directly modeling PWA systems with nonlinear models can be found in [29], and particularly multilayer neural networks with PWA activation functions in [30, 31].

The connection between subspace clustering and hybrid system identification was first discussed in [32] and resulted in the so-called algebraic methods. For more details on subspace clustering and related methods, see [33, 34].