Algebraic Method for the Synthesis of Astatic Continuous-Time Control Systems

An algebraic method for the synthesis of astatic continuous-time control systems is considered. The method is based on the construction of the desired transfer function (DTF) from given performance indicators (setting time, overshoot, etc.) and a given plant transfer function. The construction of DTF is based on the use of the desired normalized transfer function (NTF). The desired NTF is the transfer function whose denominator is a monic polynomial with unit free term and whose performance indicators, except for the setting time, coincide with those of the DTF. Therefore, one can obtain the DTF by constructing the desired NTF and then by applying the inverse transform with transformation ratio equal to the ratio of the setting time of the system to be synthesized to that of the system with the desired NTF. The desired NTF is assembled from standard NTFs. There are various standard NTFs: binomial, arithmetic, and geometric. The type of an NTF is determined by its characteristic polynomial; an NTF is said to be binomial if its characteristic polynomial is the Newton binomial and arithmetic or geometric if the roots of its characteristic polynomial form an arithmetic or a geometric progression, respectively. When constructing the desired NTF, three conditions must be met: the physical feasibility of the controller, solvability, and robustness. These three conditions determine the degrees of the characteristic equation of the system to be synthesized and the degrees of the unknown polynomials that are introduced in the synthesis process. After that, according to the given performance indicators, the type of the desired NTF is determined. Here we find only the denominator of the desired NTF. If the system to be synthesized is rth-order astatic and the plant does not contain right poles and zeros, then the numerator of the desired NTF is equal to the sum of the last r terms of the characteristic polynomial. After the system DTF has been obtained, the transfer function of the controller is determined by equating the transfer function of the closed-loop system with the DTF.

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