Identification of piecewise constant parameters in nonlinear models

Many real-world dynamical systems are characterized by different temporal phases, with sudden changes in the values of the system's parameters in correspondence to variations from one phase to another. Identifying the system's parameters and these switching instants from potentially noisy measurements of the system's states is a relevant problem in several applications. We here propose a novel approach for estimating the time-varying parameters of a broad class of nonlinear dynamical systems from noisy state measurements. We formulate the problem as a mixed-integer quadratic program (MIQP) including a sparsity constraint to enforce the piecewise constant nature of the parameters. Then, we develop a convex relaxation of the problem in the form of a quadratic program (QP). The solution of the relaxed convex QP and/or the sub-optimal solutions of the MIQP returned by the MIQP solvers provide us with computationally-efficient approximations that can be used effectively in those large-dimensional cases in which the solution of the original MIQP is difficult to obtain. After validating our approach in a controlled experiment, we demonstrate its potential on two real-world case studies regarding marketing and epidemiological applications.