Singular LQR control, impulse-free interconnection and optimal PD controller design

In this paper we consider the LQR control problem with no penalty on the input; this is addressed in the literature as the singular LQR control problem. We show that here the optimal controller is no longer a static controller but a PD controller. We also show that the closed loop system, i.e. the controlled system is a singular state space system. Singular system brings in the concern of existence of inadmissible initial conditions, i.e. initial conditions for which the solution is impulsive. Our main result is that there are no inadmissible initial conditions in the controlled system if and only if states which have relative degree one with respect to the input are penalised. Though the Algebraic Riccati equation is not defined for the singular case, we use the notion of storage function in dissipative systems theory to obtain the optimal cost function explicitly in terms of the initial conditions. We use this to prove that the initial conditions for which states of the autonomous (i.e. closed loop), singular system immediately jump to 0 have optimal cost 0. Our result that the optimal controller is a PD controller underlines a key intuitive statement for the dual problem, namely the Kalman-Bucy filter when measurements are noiseless: the minimum variance estimator differentiates the noiseless measurements. The MIMO case is not dealt in this paper due to space constraints and since it involves controllability indices and Forney indices in the result statements and proofs.