Estimation of Stochastic Systems: Arbitrary System Process with Additive White Noise Observation Errors

Quantum Risk-Sensitive Estimation and Robustness

“…Note that the operators and reduce when to and , where and are given in (28) and (29). This implies that the solution of (42) converges to that of (27) when goes to zero.…”

“…, which is a noncommutative analogue of the classical Kallianpur-Striebel formula [27]. It easily follows from (17) and (24) that satisfies the following difference equation: (25) where the operators and are given by (26) The filter can now be obtained immediately from .…”

“…The unnormalized density matrix is called the unnormalized information density matrix. It is easy to derive a difference equation for : (27) where the operators and are given by In Section V we will formulate risk-sensitive estimation as an optimal control problem for a given cost function, see (32) and (33) below. The idea of risk-sensitive control is to absorb the running cost of the cost function into the dynamics, see (36) below.…”

“…1 (a2). The nominal risk-neutral estimator and the nominal risk-sensitive one are then calculated from (27) with and (44) with , respectively. The risk-sensitive parameters are chosen to be .…”

“…Note that the filter equations include the composition and are driven by the true output data . We compare those two nominal estimators with the ideal true risk-neutral estimator , which is calculated from (27) with . To do this, we use the averaged total estimation errors (50)…”