On minimization of the $\mathcal H_2$ norm of a transfer matrix of delay systems

The $\mathcal H_2$ norm of a transfer matrix plays an important role in the study of dynamical systems. The input signal is usually considered as an external disturbance, therefore it is important to obtain a control that minimizes its influence in the closed-loop system. The rejection level is estimated by the $\mathcal H_2$ norm of a transfer matrix of the system, and the $\mathcal H_2$ norm acts as an optimality criterion. The $\mathcal H_2$ optimal control for the systems of ordinary differential equations is widely discussed. However such systems don't apply to the description of some phenomena like information transmission, taking decisions or populations dynamics. It led to the appearance of a new class of dynamical systems — time-delay systems. The distinctive feature of such systems is that the system's state depends on the previous states. It is necessary to obtain the control law that includes information about the delays in the system. One solution of the $\mathcal H_2$ optimal control problem is the Zubov method of approximations, based on the theory of Lyapunov functions. This theory was extended to the case of time-delay systems using the Lyapunov–Krasovskii functional, and it can be applied to the problem of minimization of the $\mathcal H_2$ norm of a transfer matrix of a time-delay system with commensurate delays considered in this work. Bibliogr. 8.