Finite-Time $H_{\infty}$ Guaranteed Cost Control for Nonlinear Differential-Algebraic Systems

Regarding the finite time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H_{\infty}$</tex> guaranteed cost stabilization challenge in nonlinear differential-algebraic systems, this study develops a state-based control scheme to guarantee the resulting regulated system achieves finite-time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H_{\infty}$</tex> stability while meeting prescribed performance metrics when subjected to external disturbances. Initially, the performance index function along with the finite-time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H_{\infty}$</tex> guaranteed cost stabilization concept for nonlinear differential-algebraic systems are introduced. Subsequently, necessary and sufficient criteria for the system's impulsive controllability are established, accompanied by a suitable Lyapunov function. Based on this, a finite-time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H_{\infty}$</tex> guaranteed cost controller for the nonlinear differential-algebraic system is developed via state-feedback methodology. Finally, a circuit example is selected, and simulation tests verify the effectiveness of the designed controller.