Dimension reduction in a feedback loop using the SVD: Results on controllability and stability

This paper presents a method to decrease the number of inputs needed to control a system comprised of a large set of subsystems. The method couples the inputs of the subsystems using a row–column structure, thus reducing the number of inputs from mnmn to m+nm+n. The resulting underactuated system is shown to maintain complete controllability if the subsystems are decoupled. Feedback for the entire system is performed by reducing the dimension of the subspace of the control inputs from a high dimension to one dimension using singular value decomposition. A stability condition for this feedback loop is presented based on the small gain theorem. In addition, the effect of the dimension reduction is explained using a simulation example.