Finite-time Backstepping of a Nonlinear System in Strict-feedback Form: Proved by Bernoulli Inequality

Abstract

In this paper, we propose a novel state-feedback backstepping control design approach fora single-input single-output (SISO) nonlinear system in strict-feedback form. Rational-exponent Lyapunovfunctions (ReLFs) are employed in the backstepping design, and the Bernoulli inequality is primarily adoptedin the stability proof. Semiglobal practical finite-time stability, or global asymptotically stability, is guaran-teed by a continuous control law using a commonly used recursive backstepping-like approach. Unlike theinductive design of typical finite-time backstepping controllers, the proposed method has the advantage ofreduced design complexity. The virtual control laws are designed by directly canceling the nonlinear termsin the derivative of the specific Lyapunov functions. The terms with exponents are transformed into linearforms as their bases. The stability proof is simplified by applying several inequalities in the final proof,instead of in each step. Furthermore, the singularity problem no longer exists. The weakness of the conceptof practical finite-time stability is discussed. The method can be applied to smoothly extend numerous designmethodologies with asymptotic stability with a higher convergence rate near the equilibrium. Two numericalcase studies are provided to present the performance of the proposed control.