Automatica 44 (2008) 2191–2196 www.elsevier.com/locate/automatica Technical communique On the disturbance response and external stability of a saturating static-feedback-controlled double integrator ✩ Zheng Wen a,∗ , Sandip Roy b , Ali Saberi b a Department of Electrical Engineering, Stanford University, Stanford CA 94305, USA b School of Electrical Engineering and Computer Science, Washington State University, Pullman WA 99164, USA Received 8 July 2006; received in revised form 26 February 2007; accepted 29 November 2007 Available online 7 March 2008 Abstract In this note, we identify a broad class of small but non-vanishing disturbances for which the state of a saturating static-feedback-controlled double integrator remains bounded. c  2008 Elsevier Ltd. All rights reserved. Keywords: Disturbance response; External stability; Actuator saturation; Double integrator 1. Introduction The disturbance responses of feedback-controlled nonlinear plants remain incompletely understood, despite a wealth of work on the external stability of such nonlinear feedback systems. Of particular interest in appropriately defining external stability, is the detailed study in several recent articles concerning the response of a canonical nonlinear feedback system, a double integrator subject to actuator saturation that is controlled by static linear state feedback (e.g. Chitour (2001) and Shi and Saberi (2002)), to external inputs (henceforth called disturbances). Here, we continue this effort to thoroughly characterize the disturbance response of a feedback-controlled double integrator. Specifically, first, we extend the work of Shi and Saberi (2002) by showing that even purely positive disturbances with arbitrarily small magnitude, as well as small disturbances with bounded time derivative, can drive the state of the feedback system away from the origin. We then identify a broad class of sustained disturbances for which the state does remain bounded. This delineation of the disturbance signals for which the state is bounded is significant in that it allows us to ✩ This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Vladimir O. Nikiforov under the direction of Editor Andr´ e Tits. ∗ Corresponding author. E-mail address: zhengwen@stanford.edu (Z. Wen). evaluate precisely the external response of the canonical system to typical disturbances. Precisely, we consider the closed-loop system Σ :  ˙ x 1 = x 2 ˙ x 2 = σ(−K 1 x 1 − K 2 x 2 ) + ω(t ) (1) where the control gains K 1 and K 2 are positive constants, σ is the standard saturation function, ω(t ) is a disturbance signal, and t ∈ R + . The internal-stability properties of the system Σ are well known: the system is globally asymptotically stable (GAS). However, the work of Shi and Saberi (2002) has shown that Σ is not input-to-state stable (ISS). Specifically, they have shown that for any δ > 0, there exists a disturbance with ‖ω(t )‖≤ δ (henceforth denoted by ω(t ) ∈ L ∞ (δ)) that drives the state away from the origin for some initial condition x 0 . In the remainder of this section, we show that even for some more restricted classes of disturbances (e.g., small positive- valued ones), the state of the system can still be driven un- bounded. In Section 2, we develop our main result: specifically, we identify a broad class of small but non-vanishing distur- bances for which the state does remain bounded. 1.1. Some negative results on external stability Let us extend (Shi & Saberi, 2002) by showing that disturbances from some typical and even more restricted classes than L ∞ (δ) can drive the state unbounded. We omit the 0005-1098/$ - see front matter c  2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2007.11.005