The effects of the distance to uncontrollability in redundant optimal control

In the paper, the new lower and upper bounds of the distance to uncontrollability are presented by solving the Riccati equations. In redundant optimal control, it is derived that the larger distance to uncontrollability would lead to the smaller quadratic performance index. Due to this result, we obtain some sufficient conditions to strictly increase the distance to uncontrollability with the control input extensions on the basis of the new lower and upper bounds. These results show that how to choose the redundant control inputs to improve the control performance of linear system. Meanwhile, an essential difference is showed from these results between single-input and multi-input control systems. Numerical results illustrate that these conditions work well.