# Kalman Filter with Missing Measurement: UB and LB on error covariance matrix

I consider 2 linear Kalman filters (KF) that follow periodic cycles as shown here: KF(1) : Receive 1 measurement at time $$k$$, No measurements at time $$k+1$$, Receive 1 measurement at time $$k+2$$, No measurements at time $$k+3$$, etc.

KF(2) : Receive measurements at every time step.

In KF(1), whenever a measurement is missing, I simply apply time-update and ignore the measurement update step. Simulations show that in the steady-state, tr($$P_{k|k}$$) (the trace of the a-posteriori error covariance matrix computed after the reception of a measurement) is slightly greater in KF(1) than in KF(2), which intuitively makes sense. I would like to quantify this difference, or at least find upper and lower bounds for their ratio. Any leads or suggestions on where to look at or how to proceed?

Because of time variable measurements for system one, its error covariance won't become constant. Though, it should reach a "steady" state where it oscillates between values. Probably the easiest way to find those is by combining two time steps, such that one can obtained an augmented model which does have a measurement each effective time step.

Assuming that your initial model can be described by

\begin{align} x[k+1] &= A\,x[k] + B\,u[k] + w[k], \\ y[k] &= C\,x[k] + D\,u[k] + v[k], \end{align}

with $$w[k]$$ and $$v[k]$$ covariance matrices $$W$$ and $$V$$ respectively and $$y[k]$$ only measured when $$k$$ is even.

Substituting two time steps yields the following

\begin{align} x[k+2] ​&= A^2\,x[k] + \begin{bmatrix} A\,B & B \end{bmatrix} \begin{bmatrix} u[k] \\ u[k+1] \end{bmatrix} + \begin{bmatrix} A & I \end{bmatrix} \begin{bmatrix} w[k] \\ w[k+1] \end{bmatrix}, \\ y[k] &= C\,x[k] + D\,u[k] + v[k], \end{align}

with $$k$$ only being even.

This is equivalent to the following augmented model

\begin{align} z[n+1] ​&= A^2\,z[n] + \begin{bmatrix} A\,B & B \end{bmatrix} \hat{u}[n] + \hat{w}[n], \\ y[n] &= C\,z[n] + \begin{bmatrix}D & 0\end{bmatrix}\hat{u}[n] + v[k], \end{align}

with $$k=2\,n$$, $$z[n]=x[2\,n]$$, $$\hat{u}[n]=\begin{bmatrix}u[2\,n]^\top & u[2\,n+1]^\top\end{bmatrix}^\top$$ and $$\hat{w}[n]$$ has covariance $$A\,W\,A^\top + W$$.

The steady state error covariance can be found by solving the discrete algebraic Riccati equation associated with the Kalman filter using this augmented model.

The error covariance in this steady state when no correction step can be performed can be calculated using $$A\,P\,A^\top + W$$, with $$P$$ the solution to the discrete algebraic Riccati equation.