Multirate Steady-State Kalman Filters

• Multirate steady-state Kalman filters are estimation methods designed for systems where sensors operate at different sampling rates, resulting in periodic measurement models.
• They leverage a dual LQR formulation and convex LMI optimization to address singular measurement covariances, ensuring robust and principled filter design.
• The approach supports multi-objective criteria such as pole placement and mixed H2/ℓ2 norm constraints to balance convergence speed with average and worst-case estimation performance.

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 (Okajima, 2 Feb 2026).

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 (Okajima, 2 Feb 2026).