Resonant and Antiresonant control of Multiple-Input SecondOrder Systems with Time Delay using Receptance

Abstract

A variety of problems in control engineering field deal with mathematical models with systems of second-order differential equations, resulting from first-principles analysis using discretization (i.e., finite element method) to obtain finite-dimensional models. These models require, in general, an accurate knowledge of the parameters involved, such as, for example, masses, damping and stiffness coefficients in mechanical systems, for the design of controllers with high performance. The need to consider delays in measuring variables or actuation in these types of systems makes the problem more challenging, in view of the transcendent nature of the resulting transfer functions. In this context, the receptance concept emerges as an alternative, enabling the modeling of these systems entirely from experimental data, dispensing with the use of simplifying hypotheses and approximations, as is recurrently done in mathematical models with discrete parameters. Another virtue is highlighted in the waiver of approximations for the exponential term of the delay, which can result in undesirable solutions or very high order transfer functions. Since their introduction, receptance-based models have gained prominence in several works, with emphasis on the active control of vibrations by allocating zeros (anti-resonant frequency) and/or poles (resonant frequency). This work proposes a method for controlling second-order systems with multiple inputs and delays using receptance. The structure of the controllers uses state feedback, and the control objectives include partial placement of zeros and poles with guaranteed stability and robustness, based on the maximum peak of the sensitivity function. Stability is guaranteed from the extended concept of the Nyquist criterion for systems with multiple inputs. The synthesis of the controller is conducted from an optimization problem and the solutions are obtained using an algorithm based on evolutionary theory, where population evaluations are defined according to functions related to the frequency response of the systems. Numerical examples are presented to illustrate and discuss the performance of the solutions using the proposed method.