Application peculiarities of operational calculus for dynamic calculations of systems with delay

Abstract

The main mathematical apparatus used for the analysis and synthesis of optimal automatic control algorithms is the operational calculus based on the Laplace transform. It is a very convenient and powerful tool for dynamic calculations of linear stationary systems, based on a universal analytical mathematical apparatus with all known advantages of formulaic (analytical, symbolic) methods over numerical methods. Unfortunately, operational calculus cannot be directly applied in the presence of a delay in the negative feedback circuit, because there is no direct analytical (as a formula) solution to such a problem. However, this case is the main one for practice. It is the delay that is the decisive factor that limits the quality of regulation and the margin of stability. Therefore, practical calculations are inevitably complicated by the use of numerical methods at some stages of calculation with all their typical disadvantages (only the analysis of a specific numerical problem is possible, it is impossible to obtain general conclusions, solve the problem of synthesis, it is necessary to process large numerical arrays, while problems arise with the accumulation of calculation errors, etc.). The purpose of the system of automatic control, including when there is a delay in the feedback loop. It is proposed to replace the irrational transfer function of delay e-p·tau with the approximate rational formula 1/(tau/m·p + 1)m. The value of the order of approximation m is recommended to be 8 – 10 for calculations of optimal operating modes and 80 – 100 for calculations of modes near the stability limit. The estimated error of the transient process from such a replacement does not exceed 2%. The result of the calculation will be obtained in an analytical form (as a high-order formula). Performing calculations in a symbolic formulaic form is a feature of operational calculus. Modern computer tools for automating algebraic transformations (MathCad, MathLab, etc.) allow to operate with formulas of a very high order without problems.