The Effects of Adding Input Redundancies in Linear Quadratic Regulator Problems

This paper investigates the effects of adding input redundancies repeatedly into linear quadratic regulator (LQR) problems. As the number of input redundancies increases, three equivalent conditions are stated to guarantee a strict decrease of the minimum cost, which is constrained by a pair of lower and upper bounds. The contribution of a new added input redundancy to reduce the minimum cost will diminish after more redundancies are added. Moreover, the minimum cost will converge to zero as the number of input redundancies goes toward infinity, which is proven by transferring the LQR problem into a cheap control problem.