Kernel methods are a family of methods that have recently found a constantly increasing interest in the machine learning community. Commonly used methods, such as Gaussian process regression or Tikhonov regularization are in this category. These methods are used to estimate an unknown function from a dataset, and have found diffusion especially in the machine learning community. Following this growing interest, in the last years, the CAL has started several research projects that aim to employ kernel methods for dynamical systems identification. In particular, the CAL is trying to:
- Develop a new type of regularization technique that try to exploit correlations in the regressors space
- Create new kernels that are precisely designed for non-linear dynamical systems
- Investigate the properties of the solutions of kernel-based learning problems
- Apply kernel methods to solve practical problems
New regularization approaches for nonparametric system identification
Nonlinear system identification via data augmentation
The proposed methodology exploits the potential of manifold learning on an artificially augmented dataset, obtained without running new experiments on the plant. The additional data are employed for approximating the manifold where input regressors lie. The knowledge of the manifold acts as a prior information on the system, that induces a proper regularization term on the identification cost. The new regularization term, as opposite to the standard Tikhonov one, enforces local smoothness of the function along the manifold. A graph-based algorithm tailored to dynamical systems is proposed to generate the augmented dataset.
Enhanced kernels for nonparametric identification of a class of nonlinear systems
This work deals with nonparametric nonlinear system identification via Gaussian process regression. We show that, when the system has a particular structure, the kernel recently proposed in  for nonlinear system identification can be enhanced to improve the overall modeling performance. More specifically, we modify the definition of the kernel by allowing different orders for the exogenous and the autoregressive parts of the model. We also show that all the hyperparameters can be estimated by means of marginal likelihood optimization.
 G. Pillonetto, M. H. Quang, and A. Chiuso, “A new kernel-based approach for nonlinearsystem identification,” IEEE Transactions on Automatic Control, vol. 56, pp. 2825–2840, Dec 2011.
A note on the numerical solutions of kernel-based learning problems
In the last decade, kernel-based learning approaches typically employed for classification and regression have shown outstanding performance also in dynamic system identification. For this reason, they are now widely recognized as convenient tools to solve complex model-based control design problems. A key assumption in such learning techniques is that the kernel matrix is non-singular. However, due to limited machine precision, this might not be the case in many practical applications. In this work, we analyze the above problem and show that such an apparent disadvantage actually introduces additional freedom, e.g., to enforce sparsity or to accurately solve ill-conditioned problems such as semi-supervised regression.
Applications of kernel methods
Nuclear particles classification
In this work we used kernel methos for the automatic classification of particles produced by the collision of a heavy ion beam on a target, by focusing on the identification of isotopes of the most energic light charged particles (LCP). In particular, it is shown that the measurement of the particle collision can be traced back to the impulse response of a linear dynamical system and, by employing recent kernel-based approaches, a nonparametric model is found that effectively trades off bias and variance of the model estimate. Then, the smoothened signals can be employed to classify the different types of particles. Experimental results show that the proposed method outperforms the state of the art approaches. All the experiments are carried out with the large detector array CHIMERA (Charge Heavy Ions Mass and Energy Resolving Array) in Catania, Italy.