M. V. Sotnikova, A. S. Tomilova, T. A. Lepikhin

Saint-Petersburg State University (St. Petersburg, Russian Federation)

Abstract. Currently, the synthesis of LQR controllers (i.e., linear quadratic regulators) is one of the most significant and widely used method for designing optimal stabilizing controllers. This method guarantees the asymptotic stability of a closed-loop system, which means that the roots of its characteristic polynomial are placed in the open left half-plane of the complex plane. This paper proposes an extension for the capabilities of classical LQR synthesis. Namely, the considered problem is designing an optimal linear controller that minimizes a quadratic performance functional with an additional constraint on the root domain of the closed-loop system's characteristic polynomial. A sector in the left half-plane of the complex plane is introduced as such a region, which makes it possible to influence such performance indices of transient processes in a closed-loop system as the transient time and oscillation. The root domain is parameterized using Euclidean space vectors. It is shown that the parametric synthesis problem of obtaining an optimal controller, taking into account modal constraints, reduces to unconstrained minimization. A computational algorithm for obtaining the optimal controller is developed, which is based on solving a linear Lyapunov matrix equation at each iteration of the optimization method, but does not require solving the nonlinear Riccati equation. Searching for weight matrices of a quadratic functional in the synthesis of an LQR controller without constraints, corresponding to the optimal controller in a problem with modal constraints, is considered. The obtained theoretical results are illustrated with a practical example of vessel course control.

Key Words: linear controller, quadratic functional, modal constraints, Lyapunov matrix equation, unconstrained optimization, marine vessel

For citation: M. V. Sotnikova, A. S. Tomilova, T. A. Lepikhin. Synthesis of optimal linear controllers with constraints on the root domain. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 28:2(2026), 47–61. DOI: https://doi.org/10.15507/2079-6900.28.202602.47-61

Information about the authors:

Margarita V. Sotnikova, D. Sci. (Phys. and Math.), Professor, Department of Computer Applications and Systems, St. Petersburg State University (7-9 Universitetskaya Emb., St. Petersburg, 199034, Russia), ORCID: http://orcid.org/0000-0003-0726-7448, m.sotnikova@spbu.ru

Anastasiia S. Tomilova, Postgraduate Student, Department of Computer Applications and Systems, St. Petersburg State University (7-9 Universitetskaya Emb., St. Petersburg, 199034, Russia), ORCID: http://orcid.org/0000-0001-8074-754X, nt4815@gmail.com

Timur A. Lepikhin, Ph.D. (Phys. and Math.), Assistant, St. Petersburg State University (7-9 Universitetskaya Emb., St. Petersburg, 199034, Russia), ORCID: http://orcid.org/0000- 0001-5218-8250, t.lepihin@spbu.ru