Fault detection observer design for T–S fuzzy systems based on membership function-dependent mixed <i> H ₋ /H _∞ </i> performance in the finite-frequency domain

This paper focuses on the design of a fault detection observer for T–S fuzzy systems with faults and unknown disturbances, based on a membership function-dependent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="false">/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> performance index. Currently, common methods for disturbance suppression and fault detection are based on traditional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="false">/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> performance criteria. Traditional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="false">/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> performance indices impose the same constraints on the entire system, which does not align with the characteristic that T–S fuzzy systems do not operate continuously in all linear subsystems. Therefore, aiming to improve the performance of the linear subsystems where the system operates for long periods, this paper proposes a membership function-dependent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="false">/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> performance index and designs a fault detection observer based on this performance index. In addition, considering that the frequency of actual system faults and disturbances is not across the entire frequency domain, the design of the fault detection observer based on the membership function-dependent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="false">/</mml:mo> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> performance index is achieved through the generalized Kalman-–Yakubovich-–Popov lemma in a finite-frequency domain.