  We introduce a real-time identification method for discrete-time
state-dependent switching systems in both the input--output and state-space
domains. In particular, we design a system of adaptive algorithms running in
two timescales; a stochastic approximation algorithm implements an online
deterministic annealing scheme at a slow timescale and estimates the
mode-switching signal, and an recursive identification algorithm runs at a
faster timescale and updates the parameters of the local models based on the
estimate of the switching signal. We first focus on piece-wise affine systems
and discuss identifiability conditions and convergence properties based on the
theory of two-timescale stochastic approximation. In contrast to standard
identification algorithms for switched systems, the proposed approach gradually
estimates the number of modes and is appropriate for real-time system
identification using sequential data acquisition. The progressive nature of the
algorithm improves computational efficiency and provides real-time control over
the performance-complexity trade-off. Finally, we address specific challenges
that arise in the application of the proposed methodology in identification of
more general switching systems. Simulation results validate the efficacy of the
proposed methodology.
