Automatica 45 (2009) 2169–2171 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Technical communique Finite-time output stabilization with second order sliding modes ✩ Francesco Dinuzzo a,∗ , Antonella Ferrara b a Department of Mathematics, University of Pavia, Via Ferrata, 1, 27100 Pavia, Italy b Department of Computer Engineering and Systems Science, University of Pavia, Via Ferrata, 1, 27100 Pavia, Italy article info Article history: Received 17 November 2008 Received in revised form 9 March 2009 Accepted 20 May 2009 Available online 1 July 2009 Keywords: Sliding-mode control Robust control of nonlinear systems Uncertain dynamic systems Variable-structure control abstract In this note, a class of discontinuous feedback laws that switch over branches of parabolas in the auxiliary state plane is analyzed. Conditions are provided under which controllers belonging to this class are second order sliding-mode algorithms: they ensure uniform global finite-time output stability for uncertain systems of relative degree two. © 2009 Elsevier Ltd. All rights reserved. 1. Preliminaries Consider a SISO (single input, single output) dynamic system affine in the input  x(0) = x 0 ˙ x = a(x) + b(x)u, y = c (x), (1) where x ∈ AC (R; Ω) (Ω ⊂ R n , bounded) is an absolutely continuous state trajectory satisfying the differential equation almost everywhere (a.e.), u ∈ L ∞ (R; R) is an essentially bounded scalar input, and a, b ∈ C 1 (Ω; R n ). The output y(t ) is obtained by applying the function c ∈ C 2 (Ω; R) to the state. We require the following: Assumption 1. Dynamical system (1) is complete in Ω meaning that x(t ) ∈ Ω and, for each initial state x 0 and each control u, x(t ) is defined for almost all t ∈ R + . Moreover, system (1) has a uniform relative degree equal to 2 and admits a global normal form in Ω. For a background on the concepts of relative degree and normal forms, see Isidori (1995) and Liberzon, Morse and Sontag (2002). ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Ilya V. Kolmanovsky under the direction of Editor André L. Tits. ∗ Corresponding author. E-mail addresses: francesco.dinuzzo@unipv.it (F. Dinuzzo), antonella.ferrara@unipv.it (A. Ferrara). Assumption 1 also states that there exists a global diffeomorphism of the form Φ : Ω → R n , Φ(x) =  Ψ (x) c (x) b(x) ·∇ (a(x) ·∇c (x))  =  z ξ  , Ψ : Ω → R n−2 , z ∈ R n−2 ,ξ =  y ˙ y  ∈ R 2 . such that,          ˙ z = f 0 (z ,ξ) ξ(0) = ξ 0 ˙ ξ 1 = ξ 2 , ˙ ξ 2 = f 1 (z ,ξ) + f 2 (z ,ξ)u y = ξ 1 , where f 0 (z ,ξ) = dΨ dx (Φ −1 (z ,ξ))a(Φ −1 (z ,ξ)) f 1 (z ,ξ) = a(Φ −1 (z ,ξ)) ·∇ ( a(Φ −1 (z ,ξ)) ·∇c (Φ −1 (z ,ξ)) ) , f 2 (z ,ξ) = b(Φ −1 (z ,ξ)) ·∇ ( a(Φ −1 (z ,ξ)) ·∇c (Φ −1 (z ,ξ)) ) . f 2 (z ,ξ)  = 0, ∀(z ,ξ) ∈ Φ(Ω). Recall that u is essentially bounded. Since f 0 , f 1 , f 2 are continuous functions and Φ(Ω) is a bounded set, we have ∃α> 0 :|u(t )|≤ α a.e., ∃β> 0 :|f 1 (z ,ξ)|≤ β ∀ (z ,ξ) ∈ Φ(Ω), ∃δ> 0 : f 2 (z ,ξ) ≤ δ ∀ (z ,ξ) ∈ Φ(Ω). 0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.05.015