A polynomial optimization approach to gray-box system identification

System identification consists in the study of techniques allowing to estimate parameters of models for dynamical systems starting from input and output data experimentally collected. This problem is strongly motivated in the context of control system design and by a number of other practical situations such as the determination of useful system parameters in system’s testing or analysis. Among all the system identification algorithms, it is possible to recognize two big classes: • Black-box identification: allows to estimate parameters of a model whose structure is selected according to some general a-priori information which does not require any knowledge of the physics of the systems. • Gray-box identification: allows to estimate parameters of a so-called gray-box model, which is a dynamical model built by means of the first principles of physics in which some parameters may be unknown. At the current state of art most of the system identification techniques are developed under the assumption that the model to be estimated belongs to the LTI systems class. Moreover most of the system identification algorithms deals with the problem of identifying black-box models, mostly described as a discrete-time transfer function in the Z-domain. On the contrary when dealing with gray-box identification the most natural choice is to consider continuous-time models having the form of state-space models. This is because when modeling a system thanks to the first principles of physics the involved equations are differential ones. This thesis arises from the analysis of a recent article: ’Combining linear algebra and numerical optimization for gray-box affine state-space model identification’, O. Prot and G. Mercère, IEEE Trans. Autom. Control, 2020. In this paper the authors, propose a linear algebra approach for grey-box system identification. Differently from previous literature, this approach has the advantage of not relying on non-convex optimization, thus it does not get trapped at local minima. Nevertheless, it has the drawback of not envisaging the presence of noise on the data. In this thesis, we propose a new mathematical formulation of the gray-box identification problem that takes into account the possible presence of noise and relaxes the technical conditions considered in the above mentioned paper. In particular, this formalization of the problem is carried out in a set-membership framework, i.e. under the assumption that all variables involved in the problem belongs to some bounded set. In fact, the only assumption on the noise is that it is bounded and solving the problem we look for upper and lower bounds on each parameter we want to estimate, obtaining therefore an estimate giving information about the quality of the result. The problem of finding the bounds of interest is formulated as a set of Polynomial Optimization Problems (POPs), which are solved for global optimal by means of the tool of convex SDP relaxation, implemented in software as sparsePOP, SeDuMi and Mosek. Such software has been used in order to perform a validation of the proposed solutions with numerical examples. Finally further relaxations of conditions under which we developed this new solution are discussed, in particular gray-box models with polynomial dependency from the parameters and MIMO systems are handled.