Recursive and State-Space Identification of Hybrid Systems

Abstract

This chapter focuses on hybrid system identification issues with a more control-oriented perspective. As such, it first considers recursive methods that apply on-line to switched and piecewise linear input–output systems. An approach that alternates between assigning the current point to the closest mode and updating the corresponding model is detailed. Other approaches are based on the recursive identification of a single model to track the output signal, with two flavors: one that follows the algebraic approach and directly yields the hybrid model and another one based on the a posteriori detection of abrupt changes in the model parameters. Then, the chapter introduces the various batch approaches proposed for the identification of linear hybrid systems in state-space form.

Notes

Input–Output Models

In the framework of parallel identifiers, the recursive approach alternating mode detection and parameter update is due to [1]. As an extension, the submodel recognition including the cluster covariance matrices is inspired from [2, 3], where, for MIMO PWA systems, parameter update is performed using an inverse QR factorization approach. The Bayesian approach for hybrid system identification is due to [4], where the pdfs are approximated by particle filters (see [5] for a more detailed overview of these approaches).

The recursive algebraic procedure to identify SARX models has been proposed in [6,7,8] extending the condition of persistence of excitation for recursive ARX model identification of [9].

Reference [10] proposed a sparse estimation approach for the recursive identification of PWARX models.

Regarding the identification of SOE models, [11] adapted the batch clustering-based technique of [12] by replacing the least squares estimation of the ARX submodels by the estimation of OE submodels, using instrumental variables. Several works deal with recursive approaches. An extension of [1] cited above, which detects the mode mismatches and resets the variance matrices of the corresponding parameter vectors, is applied to OE submodels in [13]. In [14], a recursive identification algorithm for SOE systems with bounded noise is presented with convergence properties.

Already in the review of [15], different aspects of model adaption and signal tracking are discussed and the importance of prior assumptions about the parameter variations (random walk, jump changes, Markov chain) is highlighted to yield efficient algorithms based on a trade-off between tracking ability and noise rejection. For detection of abrupt changes, one can refer to [16]. One can see also [17], for a more complete treatment, and many applications. The Adaptive Forgetting through Multiple Models (AFMM) algorithm was proposed by [18].

State-Space Models

Various works deal with off-line identification of hybrid SS systems. The equivalence between switched affine ARX (SARX) (I/O) models and switched affine SS models is explored in [19].

In the segmentation of the signals according to the submodel changes, the change detection technique for stable systems is due to [20]; the one based on checking the dimension of the observability subspaces as well as the common basis recovering can be found in [21,22,23].

The procedure based on small sets division is detailed in [24], with the similarity transformation borrowed from [25].

For the coordinate descent algorithm, see [26]. An approach bounding the total number of switchings for segmentation is presented in [27] for SS models and [28] for I/O models.

Few recursive approaches have been developed. One can refer to [29, 30].