Efficient multidimensional regularization for Volterra series estimation

Abstract

This paper presents an efficient nonparametric time domain nonlinear system identification method. It is shown how truncated Volterra series models can be efficiently estimated without the need of long, transient-free measurements. The method is a novel extension of the regularization methods that have been developed for impulse response estimates of linear time invariant systems. To avoid the excessive memory needs in case of long measurements or large number of estimated parameters, a practical gradient-based estimation method is also provided, leading to the same numerical results as the proposed Volterra estimation method. Moreover, the transient effects in the simulated output are removed by a special regularization method based on the novel ideas of transient removal for Linear Time-Varying (LTV) systems. Combining the proposed methodologies, the nonparametric Volterra models of the cascaded water tanks benchmark are presented in this paper. The results for different scenarios varying from a simple Finite Impulse Response (FIR) model to a 3rd degree Volterra series with and without transient removal are compared and studied. It is clear that the obtained models capture the system dynamics when tested on a validation dataset, and their performance is comparable with the white-box (physical) models.

Continuous-time system identification with neural networks: Model structures and fitting criteria

“…on the test dataset, respectively. These results compare favorably with stateof-the-art black-box nonlinear identification methods applied to this benchmark (Birpoutsoukis, Csurcsia, & Schoukens, 2018;Relan, Tiels, Marconato, & Schoukens, 2017;Svensson & Schön, 2017). To the best of our knowledge, the best previously published result was obtained in (Svensson & Schön, 2017) using a state-space model with priors for basis function expansion inspired by Gaussian Processes, and trained using a sequential Monte Carlo method.…”

Continuous-time identification of dynamic state-space models by deep subspace encoding

“…(a) CCT benchmark Method RMS BLA [24] 0.75 Volterra model [5] 0.54 State-space with GP-inspired prior [31] 0.45 SCI [13] 0.40 NL-SS + NLSS2 [24] 0.34 TSEM [13] 0.33 Tensor B-splines [17] 0.30 neural ODE [9] with normalization (∆t/τ = 0.03) 0.18 (0.33) Grey-Box with physical overflow model [26] 0. 1 also contains neural ODE with normalization.…”

Phase-based order separation for Volterra series identification

“…Hammerstein, Wiener or Wiener-Hammerstein structure, where the system is decomposed with only linear filters and static nonlinearities; for specific structures, estimation methods with good results are available (Rébillat, Hennequin, Corteel, & Katz, 2011;Tiels & Schoukens, 2014). Alternatively, regularization methods can be used to improve kernels estimation (Birpoutsoukis, Csurcsia, & Schoukens, 2018;Birpoutsoukis, Marconato, Lataire, & Schoukens, 2017).…”