Kernel-based Impulse Response Identification with Side-Information on Steady-State Gain

Abstract

In this paper, we consider the problem of system identification when side-information is available on the steadystate (or DC) gain of the system. We formulate a general nonparametric identification method as an infinite-dimensional constrained convex program over the reproducing kernel Hilbert space (RKHS) of stable impulse responses. The objective function of this optimization problem is the empirical loss regularized with the norm of RKHS, and the constraint is considered for enforcing the integration of the steady-state gain side-information. The proposed formulation addresses both the discrete-time and continuous-time cases. We show that this program has a unique solution obtained by solving an equivalent finite-dimensional convex optimization. This solution has a closed-form when the empirical loss and regularization functions are quadratic and exact side-information is considered. We perform extensive numerical comparisons to verify the efficiency of the proposed identification methodology.

Kernel-based identification with frequency domain side-information

“…More precisely, for a choice of hyperparameters, the first 100 points of D are used for training the model and following this, the cost function to be optimized by Bayesian optimization is defined as the validation error calculated using the remaining points of data. For more details, see Appendix I in Khosravi and Smith (2021d). Starting from P 0 = ∅, we estimate g 0 as the solution of the unconstrained problem.…”

“…In this framework, the identification problem is formulated as a regularized regression in a reproducing kernel Hilbert space (RKHS) (Berlinet & Thomas-Agnan, 2011) where the regularization term, based on the norm of RKHS, penalizes solutions not compatible with the side-information. By a suitable choice of the kernel function, or by imposing appropriate constraints in the regression problem, one can incorporate a variety of side-information such as stability, resonant frequencies, smoothness of the impulse response, steady-state gain, and, internal or external positivity of the system (Chen, 2018;Khosravi & Smith, 2019, 2021dMarconato, Schoukens, & Schoukens, 2016;Zheng & Ohta, 2021). Moreover, employing a Tikhonov-like regularization in this framework leads to an improvement in the bias-variance trade-off (Pillonetto et al, 2014).…”

Kernel-Based Impulse Response Identification With Side-Information on Steady-State Gain

“…Remark 1: Appendix H in [30] provides ϕ 0 and ϕ 0 2 for the standard kernels introduced in (11), (12), and (13), when T = Z + .…”

The existence and uniqueness of solutions for kernel-based system identification

“…Moreover, the kernel-based scheme allows the incorporation of various forms of side-information in the identification problem by designing appropriate kernel functions or imposing suitable constraints to the regression problem. The forms of this sideinformation, studied to date, include stability, relative degree, smoothness of the impulse response, resonant frequencies, external positivity, oscillatory behaviors, steady-state gain, internal positivity, exponential decay of the impulse response, structural properties, internal low-complexity, frequency domain features, and the presence of fast and slow poles (Chen, 2018b;Chen, Ohlsson, & Ljung, 2012;Darwish, Pillonetto, & Tóth, 2018;Everitt, Bottegal, & Hjalmarsson, 2018;Fujimoto, Maruta, & Sugie, 2017;Fujimoto & Sugie, 2018;Khosravi & Smith, 2019, 2021b, 2021cMarconato et al, 2016;Pillonetto, Chen, Chiuso, Nicolao, & Ljung, 2016;Prando, Chiuso, & Pillonetto, 2017;Risuleo, Bottegal, & Hjalmarsson, 2017;Risuleo, Lindsten, & Hjalmarsson, 2019;Zheng & Ohta, 2021). While kernel-based system identification has enjoyed considerable progress in the past decade, it is still a thriving area of research with state-of-theart results and recent studies (Bisiacco & Pillonetto, 2020a, 2020bPillonetto, Chiuso, & De Nicolao, 2019;Pillonetto & Scampicchio, 2021;Scandella, Mazzoleni, Formentin, & Previdi, 2020, 2021.…”