New discrete-time robust H2/H∞ algorithm for vibration control of smart structures using linear matrix inequalities

In real structural systems, such as a building structure or a mechanical system, due to inherent structural modeling approximations and errors, and changeable and unpredictable environmental loads, the structural response unavoidably involves uncertainties. These uncertainties can reduce the performance of a control algorithm significantly and possibly make it unstable. In this paper, based on the theories of the Bounded Real Lemma and the linear matrix inequalities (LMI), a novel discrete-time robust H2/H∞ control algorithm is presented which not only reduces the structural peak response caused by external dynamic forces but also is robust and stable in the presence of parametric uncertainties which is always the case in real-life structures. To facilitate practical implementation, the uncertainties of structural parameters are considered in the time domain as opposed to the frequency domain. Compared with traditional H∞ control methods, the new control algorithm proposes a convenient design procedure to facilitate practical implementations of active control of complex and large structural systems through the use of a quadratic performance index and the LMI-based solution method. The effectiveness of the new discrete-time robust H2/H∞ adaptive control algorithm is demonstrated using a three-story frame with active bracing systems (ABS) and a ten-story frame with an active tuned mass damper (ATMD).