How to derive the stationary Kalman filter predictor?

In its chapter on Kalman filters, my DSP book states, seemingly out of the blue, that the stationary Kalman filter for a system

$$\begin{cases} x(t+1) &= Ax(t) + w(t) \\ y(t) &= Cx(t) + v(t) \end{cases}$$

has the predictor

$$\hat{x}(t+1|t) = (A-A\bar{K}C)\hat{x}(t|t-1) + A\bar{K}y(t)$$

and stationary state vector covariance and Kalman gain

$$\bar{P} = A\bar{P}A^T - A\bar{P}C^T ( C\bar{P}C^T + R )^{-1}C\bar{P}A^T + Q$$ $$\bar{K} = \bar{P}C^T(C\bar{P}C^T+R)^{-1}$$

where $Q$ and $R$ denote the covariances of the input noise $w$ and measurement noise $v$, respectively.

I can't see how to arrive at this from the minimum variance predictor. Could someone explain it to me, or point me to a resource that derives the expression? This is the time-variant minimum-variance filter, which I can derive:

$$\hat{x}(t+1|t) = (A-K(t)C)\hat{x}(t|t-1) + K(t)y(t)$$ $$P(t+1|t) = A\left(P(t|t-1)-P(t|t-1)C^T(CP(t|t-1)C^T + R)^{-1}CP(t|t-1)\right)A^T+Q$$ $$K(t) = AP(t|t-1)C^T(CP(t|t-1)C^T+R)^{-1}$$

I'm just unsure about how to go from here to the stationary filter above.

Update: I can see that substituting $\bar{P} = P(t+1|t) = P(t|t-1)$ and $K(t)=A\bar{K}$ into the time-variant filter results in the stationary filter, but why multiply with $A$? Is this just a symptom of an unfortunate choice of notation, meaning that either $K$ or $\bar{K}$ doesn't really denote the Kalman gain?

Answer

Your derivations are correct.

$\bar P = P(t|t-1)$ and $K(t) = A \bar K$

Is this your confusion:

1. Why didn't they have the term $t|t-1$ in the Kalman Gain and Covariance Matrix Expressions?



2. How can this be "stationary" when your derivation shows that it is time varying?

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1. Bad choice of notation on the book's part

Let's look at the expression: $ \bar P = A\bar PA^T - A\bar P C^T(C\bar P C^T + R)^{-1}C\bar PA^T + Q $. The fact that $\bar P $ is a function of itself shows a recursive relationship. In other words, it uses its past values. So, it is NOT the same for all time instants - It changes at every iteration.

1. Misunderstanding of the word "stationary".

When the author of the book said "stationary" he/she did not mean that $P$ and $K$ have the same value at all time instants. Instead, the author wanted to emphasise that the expressions for those to values are the same for all statistical realisations. Stationarity is a statistical concept which means that the Statistics of the system is same all the time. Look at the expressions for $\bar P$ and $\bar K$ again. They depend only on \

• Previous values of themselves


• Transition matrices $A$ and $C$ which are deterministic and in your case time-invariant ($A$ and $C$ are the same at all times)


• $Q$ and $R$ which are the noise covariance matrices. These 2 matrices describe the statistics of the noises and are the same in all realisations and time instances.

The Kalman gain, $K$, and the state covariance Matrix $P$ will have the same value for all realisations of this random process. (Side Note: None of these 2 terms depend on the measurements, $y$. So they can be computed beforehand.)

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Conclusion:

The "time-variant" equations you derived were equivalent to the ones in the book. Besides, the notational differences, there was a slight misunderstanding on your part regarding what changes and what doesn't.