---
title: Multirate Steady-State Kalman Filters
url: https://www.emergentmind.com/topics/multirate-steady-state-kalman-filters
type: topic
---

# Multirate Steady-State Kalman Filters

Multirate steady-state Kalman filters address state estimation problems for systems where sensors operate at differing sampling rates, resulting in periodic time-varying measurement equations. In such frameworks, Kalman gains converge to periodic steady-state sequences that repeat over a common frame period. Standard algebraic Riccati equation (DARE) solvers are inapplicable because the lifted measurement noise covariance is merely positive semidefinite (PSD), owing to zeroed-out sensor blocks, and thus singular. Recent work utilizes a convex Linear Matrix Inequality (LMI) optimization framework, stemming from a dual Linear Quadratic Regulator (LQR) formulation, enabling the principled design of steady-state multirate Kalman filters even for periodic, degenerate measurement structures. This optimization framework supports multi-objective design, including pole placement for guaranteed convergence and mixed $H_2$/$\ell_2$-induced norm objectives for balanced average and worst-case estimation performance [2602.01537].

## 1. System and Multirate Measurement Model

The process model is a discrete-time, time-invariant state-space system:
\[
x(k+1) = A x(k) + B u(k) + w(k)
\]
where $x \in \mathbb{R}^n$ is the state, $u \in \mathbb{R}^p$ the input, and $w$ is the process noise with covariance $Q \succ 0$.

Measurements are acquired through $q$ scalar sensor channels with heterogenous sampling intervals. Each measurement is modeled via a selection matrix,
\[
S_k = \mathrm{diag}(s_{k,1},\dots,s_{k,q}), \quad s_{k,i} \in \{0,1\},
\]
indicating whether sensor $i$ is active at time $k$. The observation equation becomes
\[
y(k) = S_k C x(k) + S_k v(k)
\]
where $C \in \mathbb{R}^{q \times n}$ is the sensing matrix and $v$ is measurement noise with covariance $R \succ 0$.

If the sampling rates share a rational basis, there exists a period $N$ such that $S_{k+N} = S_k$, leading to a periodic measurement model:
\[
C_k = S_k C, \quad C_{k+N} = C_k.
\]
Periodic structure underlies the necessity for specialized steady-state filter design.

## 2. Cyclic Reformulation and Structure of the Lifted System

The periodic system can be lifted into a time-invariant form via block-cyclic "stacking" of states and measurements across one period. Define the stacked state,
\[
\check{x}(k) \in \mathbb{R}^{Nn}
\]
where the current $x(k)$ is placed in block $(k \bmod N) + 1$, and similarly for stacked input and noise terms. The block-lifted system evolves as
\[
\check{x}(k+1) = \check{A} \check{x}(k) + \check{B} \check{u}(k) + \check{Q}^{1/2} \check{d}_w(k)
\]
\[
\check{y}(k) = \check{C} \check{x}(k) + \check{R}^{1/2} \check{d}_v(k)
\]
with block matrices defined by the period $N$:
- $\check{A}$ is block-cyclic with $A$ on the final off-diagonal.
- $\check{Q} = \mathrm{diag}(Q, \dots, Q)$, similarly for $\check{B}$.
- $\check{C} = \mathrm{diag}(S_0 C, \dots, S_{N-1} C)$.
- $\check{R} = \mathrm{diag}(S_0 R S_0^T, \dots, S_{N-1} R S_{N-1}^T)$.

Due to the time steps when a sensor is inactive ($S_k$ zeroing rows), the measurement covariance $\check{R}$ is singular ($\check{R} \succeq 0$, but $\check{R} \not\succ 0$). This structural singularity precludes standard steady-state Riccati approaches.

## 3. Dual LQR Formulation and LMI Design

The Kalman filter synthesis is recast as a dual LQR problem on the lifted system:
\[
\zeta(k+1) = \check{A}^T \zeta(k) + \check{C}^T u(k)
\]
with cost
\[
J = \sum_{k=0}^\infty \left[\zeta(k)^T \check{Q} \zeta(k) + u(k)^T \check{R} u(k)\right].
\]
The optimal control gain $K^*$ for this problem corresponds via transposition to the Kalman filter gain for the original estimation problem.

Existence of a stabilizing filter is characterized by a Lyapunov matrix $\check{X} \succ 0$ and an auxiliary variable $\check{Y} = -\check{X}\check{L}$, satisfying the LMI
\[
\left[
\begin{array}{cccc}
\check{X} & \check{X}\check{A}+\check{Y}\check{C} & \check{X} \check{Q}^{1/2} & \check{Y} \check{R}^{1/2} \\
* & \check{X} & 0 & 0 \\
* & 0 & I_{Nn} & 0 \\
* & 0 & 0 & I_{Nq}
\end{array}
\right] \succeq 0
\]
where $*$ denotes symmetric terms. The filter gain is recovered as $\check{L} = -\check{X}^{-1}\check{Y}$.

To minimize an upper bound on steady-state error covariance, introduce $\check{W} \succeq \check{X}^{-1}$ via
\[
\begin{bmatrix} \check{W} & I \\ I & \check{X} \end{bmatrix} \succeq 0
\]
The LMI design problem becomes a semidefinite program (SDP):
\[
\begin{aligned}
\min_{\check{X} \succ 0,\, \check{Y},\, \check{W}} \quad & \operatorname{trace}(\check{W}) \\
\text{subject to}\quad & \text{LMI feasibility constraints above.}
\end{aligned}
\]

## 4. Multi-Objective Performance Criteria

The framework naturally accommodates performance and robustness objectives by the inclusion of further LMIs:
1. **Pole Placement**: To constrain the eigenvalues of $(\check{A}-\check{L}\check{C})$ within a stability disk $|z|<\bar{r}$, add
   \[
   \left[
   \begin{array}{cc}
   \bar{r}^2 \check{X} & \check{X}\check{A}+\check{Y}\check{C} \\
   * & \check{X}
   \end{array}
   \right] \succ 0
   \]
2. **Mixed $H_2$/$\ell_2$-Induced Norm**: For output $z(k) = C_z e(k)$, the induced norm constraint $\|G_{d \rightarrow z}\|_{\ell_2/\ell_2} < \gamma$ is enforced via
   \[
   \left[
   \begin{array}{cccc}
   \check{X} & \check{X}\check{A}+\check{Y}\check{C} & \check{X}\check{Q}^{1/2} & \check{Y}\check{R}^{1/2} \\
   * & \check{X} - C_z^T C_z & 0 & 0 \\
   * & 0 & \gamma^2 I & 0 \\
   * & 0 & 0 & \gamma^2 I
   \end{array}
   \right] \succ 0
   \]
All objectives can be synthesized in a unified SDP, allowing trade-offs among estimation covariance, convergence speed, and disturbance amplification.

## 5. Design and Algorithmic Workflow

The synthesis procedure for LMI-optimized multirate steady-state Kalman filters consists of:
1. Building $\check{A}$, $\check{C}$, $\check{Q}^{1/2}$, and $\check{R}^{1/2}$ matrices from system and sampling schedule.
2. Selecting objective terms (error covariance minimization, pole placement, $\ell_2$ bounds).
3. Solving the resulting SDP via solvers such as MOSEK or SeDuMi, with computational complexity $O((Nn)^{3.5})$.
4. Recovering the filter gain $\check{L}$ and extracting periodic gains $L_k$ for each phase in the cycle.
5. Verifying stability, pole constraints, and induced norm as required.

## 6. Numerical Example: Automotive Navigation

A canonical test case demonstrates filter performance. For an automotive navigation system with GPS (1 Hz) and wheel speed (10 Hz) sensors,
- Sampling period $\Delta t = 0.1$ s, period $N = 10$,
- State $n = 3$ ($p$, $v$, $a$); measurement $q = 2$ ($p$, $v$),
- Plant and sensor model as
  \[
  A=\begin{bmatrix}1&0.1&0.005\\0&1&0.1\\0&0&0.8\end{bmatrix}, \quad
  C=\begin{bmatrix}1&0&0\\0&1&0\end{bmatrix},
  \]
  \[
  Q=\mathrm{diag}(0.01,0.1,0.5),\quad R=\mathrm{diag}(1.0,0.1)
  \]
- Measurement selection: at $k=0$, $S_0$ enables both sensors; for $k=1,\dots,9$, $S_k$ enables only wheel speed.
- Rank($\check{R}$) $= 11 < 20$, so $\check{R}$ is PSD but not invertible.

The SDP solution yields
- $\operatorname{trace}(\check{W}) = 18.07$,
- $\max|\lambda(\check{A}-\check{L}\check{C})| = 0.9673$.

Periodic Kalman gains are:

| $k \,\mathrm{mod}\, 10$ | Sensors      | $L_k$ (rounded)                       |
|-------------------------|--------------|---------------------------------------|
| 0                       | GPS+WS       | $\begin{bmatrix}0.283&0.102\\0.004&0.698\\0.006&0.376\end{bmatrix}$ |
| 1–9                     | WS only      | $\begin{bmatrix}0&0.110\\0&0.698\\0&0.376\end{bmatrix}$            |

Simulated performance (20 s, sinusoidal input, Gaussian noise): RMSEs for $\mathrm{pos}=0.600\,\mathrm{m}$, $\mathrm{vel}=0.268\,\mathrm{m/s}$, $\mathrm{acc}=1.165\,\mathrm{m/s}^2$.

Trade-off analysis demonstrates that tighter pole or induced norm constraints increase $\operatorname{trace}(\check{W})$ (mean estimation error), but yield improved convergence rates and bounded worst-case error gain.

## 7. Implications and Extensions

The LMI dual-LQR methodology enables rigorous steady-state Kalman filter design for multirate periodic systems by addressing the degeneracy in lifted measurement covariances. It provides:
- Direct handling of $\check{R}\succeq0$,
- Global convexity and numerical robustness,
- Systematic synthesis of multi-objective filters via simple LMI addition.

Potential extensions include nonlinear/extended-Kalman variants, adaptive noise tuning, and distributed or asynchronous settings. The convex LMI approach offers a unified platform for steady-state filter synthesis under diverse performance and structural requirements in multirate sensor environments [2602.01537].

Source: https://www.emergentmind.com/topics/multirate-steady-state-kalman-filters