Hybrid dynamical system Identification: geometry, sparsity, and nonlinearities. (Identification de systèmes dynamiques hybrides : géométrie, parcimonie, et non-linéarités)

The main methods in the literature for hybrid system identification, particularly for linear hybrid systems, are reviewed in this chapter. After the presentation of the different points of view (algebraic, bounded-error, clustering, sparsity and convex relaxation, and time order), the methods are considered for switched systems and for piecewise systems. For each type of systems, the methods are classified with respect to the assumption on the number of modes, which can be fixed or not. Chapter 1 introduced the models and problems associated to the identification of hybrid systems from data. The identification of hybrid systems has attracted researchers’ attention since the last two decades with different approaches. In this chapter, we review various identification methods for linear hybrid systems based on the input-output model (1.3): yk = fqk(xk) + vk = θ > qk xk + vk, (2.1) where yk ∈ R is the output at discrete time k, xk = [yk−1, . . . , yk−na , uk−1, . . . , uk−nb ] > ∈ Rd is the regression vector of dimension d = na +nb with the past outputs yk−i and inputs uk−i and the model orders na and nb, qk ∈ {1, . . . , s} is the discrete state or mode, s is the number of submodels, θj ∈ Rd, j = 1, . . . , s, are the parameter vectors defining each submodel and vk ∈ R is a noise term. In the case of affine subsystems, the regression vector xk is simply replaced by x̃k = [xk , 1] >. In most of the methods, na and nb are supposed to be known and Problem 4 of Chapter 1 is considered: Problem 4. Given a data set D = {(xk, yk)}k=1 generated by a linear switched system, estimate the number of submodels s, the submodel parameter vectors θj, j = 1, . . . , s, and the switching sequence {qk}k=1. This chapter continues with the main points of view for Problem 4 in Sect. 2.1. These points of view have been considered in many studies over the two recent decades. The methods are classified in two groups: for SARX systems in Sect. 2.2 and PWARX systems in Sect. 2.3. We further distinguish between methods that work with a free or fixed number of modes. 2.1 Points of view Problem 4 is an NP-hard problem in general [73]. Indeed, the number of submodels s, the submodel parameter vectors θj , j = 1, . . . , s, and the switching sequence {qk}k=1 mutually