Observations on Nonlinear Risk-Sensitive Control

This paper clariies the relationship between risk-sensitive and robust control. This topic has received attention in the recent literature. We show that the formulations are related through an appropriate limit that is more natural than the small noise limit considered previously. The relationship we discuss is more closely related to that which has been shown in the linear systems setting. 1 System Description and Problem Formulation We consider a discrete-time stochastic system de-ned on a nite time horizon, k 2 f0 where at time k, x k 2 X is the state of the system, u k 2 U is the control, y k 2 Y is the observed output, z k 2 < is the controlled output, and where w N?1 =1) are sequences of random variables which represent disturbances to the system. We assume the initial state of the system x 0 has a known probability distribution 0 on the set of states X. The total cost Z incurred in the system's evolution is given by the sum of the controlled outputs at each time, that is: Z = N?1 X k=0 z k (4) The total cost will have a probability distribution induced by the policy , and can be parametrized by the initial distribution 0 on x 0. Based on this distribution , we can consider a number of criteria for the optimal control of the system.