Abstract
To develop a full battery model in view to accurate battery management, Li-ion cell dynamics is modelled by a capacitor in series with a simplified Randles circuit. The open circuit voltage is the voltage at the capacitor terminals, allowing, in this way, for the dependence of the open circuit voltage on the state-of-charge to be embedded in its capacitance. The Randles circuit is recognised as a trusty description of a cell dynamics. It contains a semi-integrator of the current, known as the Warburg impedance, that is a special case of a fractional integrator. To enable the formulation of a time-domain system identification algorithm, the Warburg impedance impulse response was calculated and normalised, in order to derive a finite order state-space approximation, using the Ho-Kalman algorithm. Thus, this Warburg impedance LTI model, with known parameters (normalised impedance) in series with a gain block, is suitable for system identification, since it has only one unknown parameter. A LTI System identification Algorithm was formulated to estimate the model parameters and the initial values of both the open circuit voltage and the states of the normalised Warburg impedance. The performance of the algorithm was very satisfactory on the whole state-of-charge region and when compared with low order Thevenin models. Once it is understood the parameters variability on the state-of-charge, temperature and ageing, we envisage to continue the work using parameter-varying algorithms.

Abstract
A kernel regressor to estimate a six-degree-of-fredoom non linear model of an autonomous underwater vehicle is proposed. Although this estimator assumes that the model coefficients are linear combinations of basis functions, it circumvents the problem of specifying the basis functions by using the kernel trick. The Gaussian radial basis function is the chosen kernel, with the Kernel matrix being regularized by its principal components. The variance of the Gaussian radial basis function and the number of principal components are hyper-parameters to be determined by the minimisation of a final prediction error criterion and using the training data. A simulated autonomous underwater vehicle is proposed was used as case study.

Abstract
Knowledge of the Kalman filter is very important in machine learning since is the basis for understanding more advanced concepts. Towards this end, control and estimation courses should assure the understanding of the concept and its correct application. A tutorial on the design, implementation and test of the KF to denoise the discharge current of a Li-ion cell is presented in this article. The students are also meant to acquire the discharge data used in the case study - Discharge of a Li-ion cell. The Battery Discharger Board is a low cost device to discharge Li-ion cells with a user programmable current discharge profile. The discharge is controlled and monitored by an external microcontroller connected to a host computer that stores and processes the discharge data. This board has been constructed to help students to gain insight into batteries. The current is measured by ACS712 Hall sensors, which are low cost but also very noisy. To de-noise the current measurements two different KF are used with the current being modelled as the state of a first order integrator. In the first approach, the KF assumes that the system is disturbed by process and measurement noises while in the second it only assumes measurement noise, The operation of the discharge board is illustrated in two experiments: (i) one with a constant discharge current and (ii) the other with a pulsed current. In both experiments, the filters performance was very good. Copyright (c) 2023 The Authors.

Abstract
In this work, the prediction of a time series is formulated as a gaussian process regression, for different levels of noise. The gaussian regressor is translated into lower rank Dynamic Mode Decomposition methods that use kernels (K-DMD) - Kernel regression and Least Squares Support Vector Machines. The presented unified approach delivers an algorithm where the optimisation of the marginal likelihood function can be used to find the parameters of the kernel regression. The viability of the procedure is demonstrated on a chaotic series, with quite good adjustment results being obtained. Copyright (c) 2023 The Authors.

Abstract