# Kernel Identification of Non-Linear Systems with General Structure
## 1 Introduction  
The problem of non-linear system modeling has been intensively examined over the past four decades. Owing to many potential applications (see [1]) and interdisciplinary scope of the topic, both scientists and engineers look for more precise and numerically efficient identification algorithms. First attempts at generalization of linear system identification theory for non-linear models were based on Volterra series representation ([2]). Traditional Volterra series-based approach leads to relatively high numerical complexity, which is often not acceptable from practical point of view. To cope with this problem regularization or tensor network techniques have been proposed recently ([3, 4]). However, strong restrictions are imposed on the system characteristics (e.g., smoothness of nonlinearity, and short memory of the dynamics). Alternatively, the concept of block-oriented models was introduced ([5]). It was assumed that the system can be represented, or approximated with satisfactory accuracy, with the use of structural models including interconnected elementary blocks of two types--linear dynamics and static nonlinearities ([6]). The most popular structures in this class are Hammerstein system (see e.g., [7, 8, 9, 10, 11, 12]) with static nonlinearity followed by linear dynamics, and the Wiener system [13, 14, 15, 16, 17, 18]) including the same blocks connected reversely. Traditionally, identification method needs two kinds of knowledge--set of input-output measurements and a priori parametric formula describing system characteristics and including finitenumber of unknown parameters ([6,19]). Usually, the polynomial model of static characteristics and the difference equation with known orders are assumed. This convention leads to relatively fast convergence of parameter estimates, but the risk of systematic approximation error appears, if the assumed model is not correct. As an alternative, the theory of non-parametric system identification ([20, 21, 22, 23]) was proposed to solve this problem. The algorithms work under mild prior restrictions, such as stability of linear dynamic block and local continuity of static non-linear characteristics. Although the estimates converge to the true characteristics, the rate of convergence is relatively slow in practice, as a consequence of assumptions relaxation. This paper represents the idea of combined, i.e., parametric-non-parametric, approach (see e.g., [24, 25, 26, 27, 28, 29, 30]), in which both parametric and non-parametric algorithms support each other to achieve the best possible results of identification for moderate number of measurements and to guarantee the asymptotic consistency (i.e., convergence to the true system characteristics), when the number of observations tends to infinity. Since the preliminary step of structure selection is treated rather cursorily in the literature, generalizations of the approach towards wider classes of systems seem to be of high importance from practical point of view. The paper was also motivated by the project of Differential Scanning Calorimeter ([31]) developed in the team, and particularly, modeling of heating process for examination the properties of chalcogenide glasses. Main contribution of the paper lays in the following aspects:  
* proposed identification algorithm is run without any prior knowledge about the system structure and parametric representation of nonlinearity,
* non-parametric multi-dimensional kernel regression estimate was generalized for modeling of non-linear dynamic systems, and the dimensionality problem was solved by using special input sequences,
* the scheme elaborated in the paper was successfully applied in Differential Scanning Calorimeter for testing parameters of chalcogenide glasses. The paper is organized as follows. In Section 2 the general class of considered systems and the identification problem is formulated in detail. Next, in Section 3.1, purely non-parametric, regression-based approach is presented, and its disadvantages are discussed.