A Decentralized and Spatial Approach to the Robust Vibration Control of Structures

Designing active controllers forminimizingmechanical vibration of structures is a challenging task which presents several levels of difficulties. Due to the continuous nature of the structures, they have an infinite number of degrees of freedom which leads to infinite vibration modes. This requires a model reduction and modal truncation considering the controller objectives, in order to achieve a viable numerical model which may allow the designed controller to perform satisfactorily within the frequency range of interest (Zhou & Doyle, 1997). But for real structures, even for truncated models, it may be expected a significant number of vibration modes to consider, conducting to mathematical and computational issues, besides the natural consequences of the reduction of the model order leading to unexpected behavior due to the controller feedback. Considering the now vast literature in the vibration control area, there is no consensus regarding the most suitable control design method. Several techniques seem to give similar results, as shown in the works of Baz & Chen (2000); Bhattacharya et al. (2002); Hurlebaus et al. (2008). Linear matrix inequalities methods, due to powerful yet simple formulation and computational solution to implement the theory of robust control, present nowadays a slight predominance (Boyd et al., 1994; Zhou & Doyle, 1997). Several recent works that use this approachmay be cited such as Barrault et al. (2007; 2008); Cheung & Wong (2009); Halim et al. (2008). The H∞ control technique emerged in the last decades as a robust control technique in the context of multiple-inputs and multiple-outputs (MIMO) feedback problems. The usual formulation involves the minimization of the H∞ norm from the disturbances inputs to the performance outputs, corresponding to the minimization of the worst possible response. Vibration control of structures is a well reported application using this approach (Gawronski, 2004). Usually, performance outputs are selected based on the interest points distributed over the structure, and taken for the formulation of the objective function in the minimization problem. However, the control problem, stated as the transfer matrix between the vibration actuator and sensor positions, has a known drawback. Because it does not clearly impose the desired behavior on the whole structure, it is not possible to guarantee the vibration level minimization beyond the sensor isolated position points. This approach may present acceptable reduction levels for simple structures, but more comprehensive methods are needed to achieve good results with real engineering structures, guaranteeing a vibration reduction through regions of the structure instead of isolated points. 7