Fractional-order self-tuning high-gain feedback for a class of linear systems

Abstract This paper presents a new fractional-order self-tuning high-gain control approach designed to stabilize a class of linear time-invariant (LTI) systems with minimum phase and relative degree one. The key contribution of this work lies in the incorporation of a fractional-order integrator into the adaptation law, which ensures that the adaptation gain $$\phi (t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> converges to a desired stabilizing value. This method addresses a critical limitation of classical high-gain adaptive controllers, where the gain increases monotonically, potentially leading to actuator saturation. The stability of the proposed control strategy is rigorously established by extending the classical high-gain lemma, accompanied by a detailed mathematical analysis. To validate the proposed method, we conduct numerical simulations that demonstrate its ability to achieve stable behavior. The results provide strong evidence of the efficacy of the proposed fractional-order self-tuning high-gain controller in ensuring stability for this class of systems while avoiding the risks associated with conventional high-gain strategies.