Modified Butterworth Filters in Solving the Inverse Problem of Analytical Design of Optimal Controllers

In this work, for linear stationary one-dimensional control objects, the inverse problem of analytical design of optimal controllers (ADOC) is considered, which consists in determining the weight coefficients of the quadratic functional of the optimality of the control process that provide the closed-loop control system with the specified values of the time of transient processes and overshoot. The time of the transient process (regulation time) of the synthesized system is understood in the sense of the classical theory of automatic control and is determined using a " tube" , the value of which is taken, in contrast to known works, equal to the required (desired) small value of the system overshoot of a few percent (2—5 %). The equality of the percentage values characterizing the " tube" and the desired overshoot of the synthesized system is a necessary condition for the maximum response rate of linear dynamic systems and, accordingly, ensures the unambiguity of the solution of the considered inverse ADOC problem in the class of fast-response systems. The proposed solution method provides for the transformation of the ADOC problem to the canonical form, in which the control object is described by a matrix differential equation in the Frobenius form, and the quality functional is defined as the integral of the sum of the products of the object’s canonical phase coordinates, as well as the square of the control signal with appropriate weight coefficients. It is shown that the solution of the inverse canonical ADOC problem is determined by the values of only three nonzero weight coefficients of the criterion, and one of them has a single value. The values of the other two coefficients are proposed to be found in the process of modeling the synthesized optimal control system from the conditions of providing for it a given control time and a given overshoot. To obtain numerical estimates of the two main weight coefficients of the quadratic quality criterion, the solution of the ADOC problem is considered with the limiting increase in the values of these weight coefficients. By the limiting solution of the ADOC problem, the transfer functions of dynamic systems with the limiting (maximum) speed are determined, which have a given overshoot of 4.321 %. The dynamical systems described by these transfer functions are called modified Butterworth filters due to the fact that the well-known Butterworth filters are obtained as their special case with a zero value of a certain constant. The parameters and indicators of the dynamics of these filters up to the sixth order are presented in the table. Using the indicators of Butterworth filters, numerical estimates of the weight coefficients of the quadratic quality criterion are established. Transfer functions of modified Butterworth filters are recommended to be used as reference transfer functions of synthesized high-speed control systems.