Method of synthesis of a reduced-order modal regulator

In the literature devoted to the synthesis of automatic control systems, in the vast majority of cases, the synthesis of PI- and PID-laws of regulation is implied. In this sense, the PID-laws of regulation are called traditional. Such attention to them is primarily due to the fact that until recently, PID-regulators were the only regulators produced by the industry. PID-regulators have proven their effectiveness in controlling objects that are well described by differential equations up to and including the 2nd order. However, on the one hand, the intensive use of computer science and digital automation tools (microprocessor controllers, SCADA systems, etc.) in the control of technological processes in industry allows us to move from traditional PI- and PID-laws of regulation to more complex, for example, modal ones. On the other hand, an increasing number of technological processes (control objects) are described by high-order differential equations (starting from the 3rd). For high-order control objects, the adjustment capabilities of the PID-regulator may not be enough: it is obvious that by increasing the order of the object model, it is necessary to adequately increase the order of the controller. The modal control method allows you to synthesize a regulator for control objects of any order; this method assumes that the control object is described by a linear differential equation of the nth order (where n is any non-negative integer) without delay. The modal regulator is also sought in the form of a linear differential equation. The control quality is defined as a region S on the complex plane that determines the desired location of the poles of the transfer function of a closed system. It has been repeatedly proved in the literature that a modal regulator of the order n-1 and higher provides any given location of the poles of the transfer 18 function of a closed system, and thereby guarantees stability and specified root quality indicators for a closed system. The n-1st order regulator is called a full-order modal regulator. In this article, the problem of lowering the order of the modal regulator is considered. This task is relevant: lowering the order of the regulator will reduce the impact of possible errors in the implementation of the law of regulation, increase the reliability of a closed system (by lowering the order of a closed system), and will save computing resources when calculating the regulator. The article develops a method for justifiably lowering the order of the modal regulator. The method is based on the simplification (reduction) of the original model of the control object and the subsequent synthesis of a simpler regulator. In the reduced model, all the main properties of the original model are preserved: the transmission coefficient, stability and control quality indicators. For this purpose, the modes of the initial model are divided into "dominant dynamics" (subject to regulation) and "structural disturbances" (already satisfying the control goals, and therefore not taken into account when synthesizing the regulator). As a result, the calculation of a modal regulator of a reduced order is reduced to the solution of a system of linear algebraic equations, which allows it to be implemented on a computer. The effectiveness of the method is illustrated by an example. The approach proposed in this paper to lowering the order of a one-dimensional regulator allows generalization to the multidimensional case.

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