On the invariance of risk-sensitive LQR gain under input randomization

Abstract

This paper shows that the optimal gain of the risk-sensitive linear quadratic regulator (LQR) problem is invariant under input randomization, i.e., when the controller deliberately injects noise into the nominal control input. This appears counterintuitive at first glance because certainty equivalence does not hold for risk-sensitive LQR and input randomization inflates the effective process noise. Nonetheless, the gain is preserved because the input noise enters not only the system dynamics but also the cost functional, and its total effect on the gain eventually vanishes. Consequently, the optimal gain and its associated Riccati recursion need not be recomputed, and the increment in the optimal cost can be readily evaluated in closed form. This result facilitates the use of risk-sensitive LQR in applications that employ input randomization for privacy or exploration, such as watermarking for replay attack detection, differential privacy, and path integral control.