Improving a reaction curve-based analytical identification technique for fractional models

Abstract A new analytical procedure for identifying fractional first-order plus dead-time (FFOPDT) models has recently been proposed. The technique is applicable to systems with S-shaped step responses and involves selecting three specific points on the process response curve for parameter estimation. In a simplified version of the method, the points are symmetrically positioned as $$x_1 = x\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>x</mml:mi> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> , $$x_2 = 50\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>50</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> , and $$x_3 = (100 - x)\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>100</mml:mn> <mml:mo>-</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> , with $$0&lt; x &lt; 50\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>x</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>50</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> , requiring only the optimal position of one point, x , given that the others are set automatically. This study explores the effect of adjusting the value of $$x_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> in the representative points ( x - $$x_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> - $$(100-x)\%)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>100</mml:mn> <mml:mo>-</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>%</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , while preserving symmetry around the center of the interval. Simulations provide insights into the influence of $$x_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> for more accurate estimation, revealing that the accuracy of the identified FFOPDT model is highly sensitive to the position of $$x_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , and an optimal value is proposed to enhance precision. Experimental validation on a thermal-based prototype deployed on a microprocessor confirms the technique’s applicability. This approach provides new insights into selecting the central point $$x_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> and its implications for industrial applications.