Limiting Properties of Linear Control Systems, Synthesis by LQR Methods According to Quadratic Criteria with One Variable Coefficient

Abstract

The work provides a comparative analysis of the properties of optimal control systems for linear stationary single-channel objects, the distinctive feature of which is that the numerator polynomial of the object’s transfer function is Hurwitz. Systems are synthesized by two main methods of the theory of analytical design of optimal regulators (ADOR) — methods the Letov-Kalman and A. A. Krasovsky, which use quality functionals based on an integral criterion containing only two terms: the square of the object’s control signal and the square of its output coordinate with a weighting coefficient q . The limiting properties of the synthesized systems are studied at q q → ∞ . The common known property of these systems, for brevity called the Letov-Kalman and Krasovsky systems, is the property of their stability at the limiting value of the weight coefficient and, accordingly, an infinitely large increase in the total gain of these systems, which for them ensures obtaining a given value static control error. Other analyzed properties of the systems turned out to be significantly different and even mutually opposite. For example, at limiting values q q → ∞ , the coefficients of the optimal Letov-Kalman controller do not depend on the parameters of the object, while the coefficients of the Krasovsky controller, on the contrary, are determined exclusively by the parameters of the object. We also note that, unlike Letov-Kalman control systems, the time of transient processes of Krasovsky systems at q q → ∞ cannot be reduced less than a certain finite value, despite the absence of restrictions on the magnitude of the control signal. In this work, with the aim of combining in one control system the indicated positive, but contradictory properties of the analyzed systems, the so-called combined ADOR method is proposed. Its main idea is to represent the control signal of an object by two terms, which are then sequentially determined by using two main methods of the ADOR theory, and at the first stage of synthesis there is control using the Letov-Kalman method, which ensures the direct application of the combined synthesis method to unstable objects. This synthesis method, when using a quality functional with one variable weight coefficient, allows for the considered control objects of order n m 5 to design systems with given values of the static error and control time, for which overregulation does not exceed 5.2 %.