Abstract
In this paper, we propose a novel state-feedback backstepping control design approach for a single-input single-output (SISO) nonlinear system in strict-feedback form. Rational-exponent Lyapunov functions (ReLFs) are employed in the backstepping design, and the Bernoulli inequality is primarily adopted in the stability proof. Semiglobal practical finite-time stability, or global asymptotically stability, is guaranteed by a continuous control law using a commonly used recursive backstepping-like approach. Unlike the inductive design of typical finite-time backstepping controllers, the proposed method has the advantage of reduced design complexity. The virtual control laws are designed by directly canceling the nonlinear terms in the derivative of the specific Lyapunov functions. The terms with exponents are transformed into linear forms 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 concept of practical finite-time stability is discussed. The method can be applied to smoothly extend numerous design methodologies with asymptotic stability with a higher convergence rate near the equilibrium. Two numerical case studies are provided to present the performance of the proposed control.
Highlights
It is known that asymptotic and exponential stabilities imply the system trajectory converges to the equilibrium as time approaches infinity resulting in a slow convergence rate near the equilibrium, i.e., t lim →∞ x (t x0) =where x0 is the initial state
Several Lyapunov stability theorems for global finite-time stabilizability are given in literature, i.e., V + λ1 V γ ≤ 0 [1], V + λ1 V γ + λ2 V ≤ 0 [16], [24], V + λ1 V γ + λ2 V κ ≤ 0 [19], where V is the Lyapunov function, λ1, λ2 > 0, 0 < γ < 1, and κ > 1 are coefficients
Finite-time stability can be extended to stochastic nonlinear systems [22] and switched nonlinear systems [5]
Summary
It is known that asymptotic and exponential stabilities imply the system trajectory converges to the equilibrium as time approaches infinity resulting in a slow convergence rate near the equilibrium, i.e., t lim. Backstepping is a Lyapunov-based recursive design procedure for strict-feedback systems [3], [6], [10]–[13], [15], [17], [20], [25], [27]. The design complexity of the controllers based on finite-time stability is higher than those according to asymptotic stability. We propose a simple and systematic approach to construct practical finite-time trajectory tracking control for the strict-feedback nonlinear system. The existing adaptive backstepping methods on all sorts of system uncertainties and nonlinearities can VOLUME 8, 2020 be extended to the problem of finite-time stabilization smoothly
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