Generic Noninteger Order Controller for Time-Delay Systems

Abstract

<jats:p>This paper focuses on the problem of fractional controller <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M1 >\n <mi>P</mi>\n <mi>I</mi>\n </math>\n </jats:inline-formula> stabilization for a first-order time-delay systems. For this reason, we utilize the Hermite–Biehler and Pontryagin theorems to compute the complete set of the stabilizing <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M2 >\n <mi>P</mi>\n <msup>\n <mrow>\n <mi>I</mi>\n </mrow>\n <mrow>\n <mi>λ</mi>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula> parameters. The widespread industrial utilization of PID controllers and the potentiality of their noninteger order representation justify a timely interest in <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M3 >\n <mi>P</mi>\n <msup>\n <mrow>\n <mi>I</mi>\n </mrow>\n <mrow>\n <mi>λ</mi>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula> tuning techniques. Step responses are calculated through <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M4 >\n <mfenced open= ( close= ) separators= | >\n <mrow>\n <msub>\n <mrow>\n <mi>K</mi>\n </mrow>\n <mrow>\n <mi>p</mi>\n </mrow>\n </msub>\n <mo>,</mo>\n <msub>\n <mrow>\n <mi>K</mi>\n </mrow>\n <mrow>\n <mi>i</mi>\n </mrow>\n </msub>\n <mo>,</mo>\n <mi>l</mi>\n <mi>a</mi>\n <mi>m</mi>\n <mi>b</mi>\n <mi>d</mi>\n <mi>a</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> parameters inside and outside stability region to prove the method efficiency.</jats:p>