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In this study, we consider an inverse mathematical modeling problem for dynamical systems with a single output. Generally, the final solution of this problem is an approximation of a system transient process and a system state at some time point. Only those classes of models, which describe the transient process properly, can portray the system behavior and can be applicable for prediction and optimal control problems. One of possible mathematical representations of dynamical systems is differential equations, in particular, linear differential equations for linear systems. While solving the inverse problem, we aim to identify a differential equation order and parameters, an initial system state. Since all the parameters are interrelated, we propose to identify them by solving a two-criterion optimization problem, which includes the model adequacy (i.e. a distance between model outputs and observations) and the closeness of the initial value estimation to the observation data. To solve this complex optimization problem, we apply a Real-valued Cooperative MultiObjective Evolutionary Algorithm which effectiveness has been proved on the set of high-dimensional test problems. We investigate the dependency between the considered criteria by depicting the Pareto front approximation. Then, having the same amount of computational resources, we vary the system order, the number of control inputs and the initial state to analyze changes in the algorithm effectiveness based on each criterion and estimate basic limitations. Finally, we conclude that the optimization problem considered is quite challenging and it might be used for testing and comparing various heuristics.

Multi-objective Evolutionary Approach in the Linear Dynamical System Inverse Modeling