On a stability theorem for nonlinear systems with slowly varying inputs

A robustness analysis with respect to exogenous perturbations for flatness-based exact feedforward linearization

“…Remark that this control structure represents a truly nonlinear control. The advantage of the structure (18) becomes evident in view of (14). On the desired trajectory 9 , one gets…”

“…Moreover, a new value set approach based on the knowledge of at least one stable characteristic polynomial is presented in [15]. As the hypotheses (H1)-(H3) of [14] are satisfied, there exists a ρ > 0, a T > 0, a δ 1 (ρ) > 0 and a δ 2 (ρ, T ) > 0 such that…”

“…Applying the Corollary of [14] in the case Z = Z ∞ , ∀t > t * > 0 implies, in view of the structure of (34), that lim t→∞ e(t) = 0 and e(t * ) lies inside the domain of attraction 13 of the exponentially stable equilibrium given by…”

“…However, the robust stability of the presented control law (28) can be analyzed by making use of a result which was primarily introduced by Kelemen [5], reinterpreted by Khalil and Kokotović [13] and elaborated by Lawrence and Rugh [14]. The version by Lawrence and Rugh [14] (to which the reader is referred for further details) is applied in the sequel. Theorem 5.1: If the initial error e(0), the velocity of the desired trajectory and the velocity of the exogenous parameters Z are not too large, then the augmented tracking error e is uniformly bounded.…”

“…Proof: In order to apply the stability results of [14] based on Lyapunov stability theory, their hypotheses (H1)-(H3) have to be verified:…”