Abstract
In this paper, a loop shaping controller design methodology for single input and a single output (SISO) system is proposed. The theoretical background for this approach is based on complex elliptic functions which allow a flexible design of a SISO controller considering that elliptic functions have a double periodicity. The gain and phase margins of the closed-loop system can be selected appropriately with this new loop shaping design procedure. The loop shaping design methodology consists of implementing suitable filters to obtain a desired frequency response of the closed-loop system by selecting appropriate poles and zeros by the Abel theorem that are fundamental in the theory of the elliptic functions. The elliptic function properties are implemented to facilitate the loop shaping controller design along with their fundamental background and contributions from the complex analysis that are very useful in the automatic control field. Finally, apart from the filter design, a PID controller loop shaping synthesis is proposed implementing a similar design procedure as the first part of this study.
Highlights
The loop shaping design procedure and synthesis for several kinds of single-input and single-output (SISO) systems have been implemented during recent years due to the selection of the frequency domain characteristics of the open and the closed-loop systems, this means, designing the control strategy based on the gain and phase margin specifications of the open-loop plant and the specifications of the closed-loop system [1,2,3,4,5,6]
The H∞ loop shaping robust controller design was studied in recent years to synthesize controllers for SISO plants with unmodeled dynamics and/or uncertainties due to the importance of the controller synthesis in the process control field
In the SISO system, the H-infinity controller is found by synthesizing the performance index to find the desired controller meeting the robust stability and robust performance conditions
Summary
The loop shaping design procedure and synthesis for several kinds of single-input and single-output (SISO) systems have been implemented during recent years due to the selection of the frequency domain characteristics of the open and the closed-loop systems, this means, designing the control strategy based on the gain and phase margin specifications of the open-loop plant and the specifications of the closed-loop system [1,2,3,4,5,6]. It is important to remark that the main contribution and opposition to other research studies found in the literature is that in the proposed control approach, the weighting functions for the loop shaping control system consider the maximum advantages of complex modular function by taking into consideration that these functions are double-periodic. This is an advantage because the gain and frequency crossover frequencies can be selected independently to shape the frequency domain characteristics of the open-loop plant for optimal closed-loop performance.
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