Abstract— In this paper we address the problem of vibration reduction of buildings with delayed measurements, where the delays are time-varying and bounded. We focus on a convex optimization approach to the problem of state-feedback ∞ H control design. An appropriate Lyapunov-Krasovskii functional and some free weighting matrices are used to establish some delay-range-dependent sufficient conditions for the design of desired controllers in terms of linear matrix inequalities (LMIs). The controller, which guarantees asymptotic stability and an ∞ H performance, simultaneously, for the closed-loop system of the structure, is then developed. The performance of the controller is evaluated through the simulation of an n-story base-isolated building. I. INTRODUCTION Vibration control has emerged as an important area of scientific and technological development in recent years. Developments in vibration control have allowed successful application of the concept in numerous areas. A variety of control techniques, such as LQR control, sliding mode control, backstepping control, 2 H control, ∞ H control, guaranteed-cost control and multi-objective control have been used in vibration systems (see [1]-[13]). In the field of dynamic systems and control, delays appear either in the state, in the control input, or in the measurements (see for instance the references [14]-[24] and the references therein). The presence of a delay in a system may be the result of some essential simplification of the corresponding process model. Generally, time delay exists inevitably in control systems, which mainly results from the following: (1) the time taken in the online data acquisition from sensors at different locations of the system; (2) the time taken in the filtering and processing of the sensory data for the required control force to the actuator; (3) the time taken by the This work has been partially funded by Norwegian Centre for Offshore Wind Energy (NORCOWE), by the European Union (European Regional Development Fund) and the Ministry of Science and Innovation of Spain through the coordinated research projects DPI2008-06699-C02-01 and by the Government of Catalonia (Spain) through SGR523. M. Zapateiro is grateful to the grant of Juan de la Cierva program of the Ministry of Science and Innovation (Spain). H.R. Karimi is with the Department of Engineering, Faculty of Engineering and Science, University of Agder, N-4898 Grimstad, Norway (e-mail: hamid.r.karimi@uia.no). M. Zapateiro is with the Department of Applied Mathematics III, Universitat Politècnica de Catalunya, Barcelona, Spain (e-mail: mauricio.zapateiro@upc.edu). N. Luo is with the Institute of Informatics and Applications, University of Girona, Girona, Spain (e-mail: ningsu.luo@udg.edu). J.M. Rossell is with Dpt. Mat. Aplicada III, Universitat Politecnica de Catalunya, Spain (e-mail: josep.maria.rossell@upc.edu). actuator to produce the required control force. Therefore, how to analyze and synthesize dynamic systems with delayed arguments is a problem of recurring interest, as the delay may induce complex behaviors (oscillation, instability, bad performances) for the systems concerned (see [24]-[26] and the references therein). It is also worth citing that some appreciable pieces of work have been performed to design different control strategies such as LQR control, sliding mode control, backstepping control, QFT control, 2 H control, ∞ H control for vibration control of a building structure (see [1]-[5], [27]-[31]). However, the system performance and stability, simultaneously, are not investigated for a building structure with time-varying delayed measurements in these works. Up to now, to the best of the authors’ knowledge, no results about a convex optimization method for the delay-range-dependent state- feedback ∞ H control problem of building structures with time-varying delayed measurements are available in the literature, which remains to be important and challenging. This motivates the present study. Fig. 1. Schematic of a base Isolated Structure. In this paper, we further contribute to the development of delay-range-dependent state-feedback aspect of ∞ H control for vibration reduction in a building with delayed Feedback Vibration Control of a Base-Isolated Building with Delayed Measurements Using H ∞ Techniques H.R. Karimi, M. Zapateiro, N. Luo, J.M. Rossell 2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 WeA21.6 978-1-4244-7425-7/10/$26.00 ©2010 AACC 750