Shape of stiffest controlled structures under unknown loads

This paper discusses the optimal geometry for maximum stiffness of controlled truss-type structures subjected to a class of unknown disturbances, under a constant volume constraint. The stiffness is defined by a quadratic function of the worst distortion at the controlled degrees of freedom after applying optimal control. The disturbance is arbitrary but is limited by a quadratic bound. Herein, the design variables are the spatial coordinates of a predetermined set of nodes. Based on earlier publications, it is indicated that if the structure is controlled by Nc ideal actuators this min–max problem is equivalent to minimizing the Nc+1th singular value of the disturbance influence matrix. This implies that the first Nc singular modes are taken care of by the control system. Consequently the designed structures are often highly unstable without control. A methodology is presented to design sub-optimal structures which addresses this problem. In order to alleviate the effects of a possible failure of the control system, limits on the levels of the lower singular values are incorporated in the design process. Numerical examples illustrate the approach and indicate clearly the beneficial effect of this technique which increases the stiffness of the structure while maintaining stability in the case of a breakdown of the control system.