212 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 1, JANUARY 2011 Exponential Stabilization of the Uncertain Wave Equation via Distributed Dynamic Input Extension Yury Orlov, Alessandro Pisano, and Elio Usai Abstract—The wave equation is studied under the effect of a per- sistent external perturbation. A dynamical distributed controller is suggested, based on an infinite-dimensional generalization of second-order sliding-mode control techniques, that provides the exponential attainment of a sufficiently smooth arbitrary reference of the state trajectory. The control system comprises of both feedforward and feedback parts, the latter being a discontinuous term directly connected to the plant input through a dynamical filter that augments the system state smoothing out the discontinuity of the feedback control loop. As a result, a continuous input is applied to the plant. A constructive Lyapunov-based proof of con- vergence of the proposed control algorithm is carried out and supporting numerical results are presented. Index Terms—Distributed control systems, second-order sliding modes, uncertain systems, wave equation. I. INTRODUCTION Sliding-mode control has long been recognized as a powerful con- trol method to counteract non-vanishing external disturbances and un- modelled dynamics when controlling dynamical systems of finite and infinite dimension [15]. Presently, the discontinuous control synthesis in the infinite-dimen- sional setting is well documented [9]–[12], [14] and it is generally shown to retain the main robustness features as those possessed by its finite-dimensional counterpart. Other robust control paradigms have been fruitfully applied in the infinite dimensional setting such as adap- tive and model-reference control [4], [6], [7], geometric and Lyapunov- based design [2], and LMI-based design [5]. It should be noted that the latter paradigms are capable of attenuating vanishing disturbances only, whereas the former is additionally capable of rejecting persistent disturbances with an a priori known bound on their norm. In the present technical note we consider a popular example of hy- perbolic infinite dimensional dynamics, the wave equation, under the effect of a persistent external smooth disturbance. Even in the finite dimensional setting, and with a perfectly known linear time-invariant controlled plant, the asymptotic rejection of persistent disturbances is difficult to achieve when the unique information is the existence of a priori known upper bounds on the disturbance and its time derivative. Despite structured perturbations (e.g., constant, or sinusoidal ones with known frequency) can be easily rejected by linear control techniques, the problem remains open when arbitrarily shaped disturbances need to Manuscript received April 02, 2010; revised July 09, 2010; accepted September 30, 2010. Date of publication October 21, 2010; date of current version January 12, 2011. This work was supported by the University of Cagliari and Region of Sardinia in the framework of the 2009/2010 “Long Stay” Visiting Professor Program and by the Consejo Nacional de Ciencia y Tecnologia de Mexico. Recommended by Associate Editor K. Morris. Y. Orlov is with the Department of Electrical and Electronic Engineering, University of Cagliari, Cagliari I-09123, Italy and is also with the CICESE, San Diego, CA 92143-4944 USA (e-mail: yorlov@cicese.mx; yorlov@cicese.mx). A. Pisano and E. Usai are with the Department of Electrical and Electronic Engineering, University of Cagliari, Cagliari I-09123, Italy (e-mail address: pisano@diee.unica.it; pisano@ieee.org; eusai@diee.unica.it). Color versions of one or more of the figures in this technical note are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2010.2089380 be taken into account. The only known solution appears to be the dis- continuous sliding-mode control feedback [15], [16], which is effec- tively implemented in the infinite dimensional setup via the so-called (distributed) “unit-vector” control [10], [11]. In order to attenuate the practical drawbacks affecting the discontin- uous control systems while retaining similar properties of robustness as those offered by the unit-vector approach, here we put the further speci- fication that the distributed control input must be a continuous function. To do that, we make a dynamic input extension by passing it through a first-order dynamical filter. We cast a problem of exponential state tracking, and we show that an appropriate infinite-dimensional gen- eralization of a second-order sliding mode controller (presently recog- nized as “twisting” algorithm [8]), which is properly complemented by a feedforward term depending on the reference trajectory, is endowed by the following properties: • it provides the exponential convergence to the desired smooth ref- erence trajectory; • it exponentially rejects the effect of the external persistent distur- bance having almost no prior information on it; • it gives rise to a continuous distributed control input; • the suggested controller is easy to implement and is characterized by very simple parameter tuning rules for the two constant gain parameters. To facilitate the exposition the present investigation is confined to wave equations with a scalar spatial variable, however, the extension to the case of two- and three-dimensional spatial variables seems possible. The rest of the technical note is structured as follows. Some nota- tions are introduced in the remainder of the Introduction. Section II presents the problem formulation and the distributed sliding manifold design. Section III contains the controller design along with a construc- tive Lyapunov-based convergence analysis demonstrating the exponen- tial stability of the error system. Section IV presents some numerical simulation results. Finally, Section V gives some concluding remarks and draws possible direction of improvement of the proposed result. Notation The notation used throughout is fairly standard (see [3] for details). with stands for the Hilbert space of square integrable functions , equipped with the norm (1) stands for the Sobolev space of absolutely continuous scalar functions on with square integrable derivatives of the order . denotes the Banach space of essentially bounded functions with values almost for all and such that . II. PERTURBED WAVE EQUATION We consider a class of uncertain infinite-dimensional systems, de- fined in the Hilbert space , which is governed by a perturbed version of the hyperbolic PDE commonly referred to as the “Wave Equation” (2) where and are the state variables, is the monodimensional (1-D) space variable, and is time. The coefficient represents the squared value of the wave velocity, 0018-9286/$26.00 © 2010 IEEE