Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 30K / c

Table of contents
- Solution c

Solution

The minimizer in the quadratic expression from part (b) is

u_{t - 1}^{*} = - \frac{P _{t}}{1 + P _{t}} x_{t - 1} .

Relabelling the time index,

u_{t}^{*} = - K_{t} x_{t}, K_{t} = \frac{P _{t + 1}}{1 + P _{t + 1}} = \frac{1}{T - t + 1} .

If

Var (x_{0})

or the observation-noise variances change, the Kalman filter gains and the sequence

V_{t}

change. The control gain

K_{t}

does not change, because the state dynamics and quadratic cost are unchanged. This is the separation principle: estimation uncertainty changes the additive constants

d_{t}

, while the optimal linear feedback gain is the full-information gain.

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