Design of Robust D-Stable IIR Filters Using Genetic Algorithms With Embedded Stability Criterion

Abstract

This paper proposes a novel evolution strategy for a genetic algorithm (GA). This new algorithm is then applied to design robust <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> (alpha, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> )-stable infinite-impulse-response (IIR) filters. Unlike existing research on designing IIR filters by using GA, in which the stability of IIR filters is tested by trial and error after the evolution of each generation of a GA, the stability criterion in this paper is embedded within the evolution of each generation. Consequently, the stability of this system can be guaranteed without the need for any other checks of the stability criterion in the evolution of each generation. Numerical experimental results are discussed to illustrate the soundness of the proposed evolution strategy. The robustness of the IIR filters is achieved by ensuring that all poles of the filters are located inside a disk <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> (alpha, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> ) contained in the unit circle, in which <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">alpha</i> is the center, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> is the radius of the disk and IalphaI +r < 1 . So, in this paper, a <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> (alpha, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> )-stability criterion will be first derived and then embedded in the GA for the design of robust IIR filters. Finally, two examples will be presented to show that the designed filters remain <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> (alpha, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> )-stable during the evolution of the GA and will provide satisfactory results.