Algorithms for the synthesis of the optimum observer for stationary linear systems with delay

The article examines the problem of constructing an observer for a stationary linear system that minimizes the state estimation error. The research is aimed at determining the elements of the observer amplification matrix in such a way that the system’s eigenvalues take on given values, and the quality indicator is minimal. The problem lies in choosing the optimal observer structure and minimizing the quadratic error functionality. The method of quadratic optimization with the introduction of weight coefficients and the Jordan form of the observer matrix is used. The developed algorithm ensures the stability of estimates, increases the accuracy of state restoration, and improves the dynamic characteristics of the system. The work also investigates the task of synthesizing an observer for linear systems with delay, which is relevant for real technological processes where signal delays worsen the quality of control and assessment. The problem lies in ensuring stability and minimal sensitivity to parametric disturbances with temporary delays. Modal and Jordan approaches are used to solve problems, as well as radiating transformations that allow for the implementation of feedback control. The proposed algorithm forms a combined controller and observer system that satisfies the separation principle. It increases the accuracy of state assessment and resistance to parameter variations and delays.