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 $\ell_2$0. The observation equation becomes

$\ell_2$1

where $\ell_2$2 is the sensing matrix and $\ell_2$3 is measurement noise with covariance $\ell_2$4.

If the sampling rates share a rational basis, there exists a period $\ell_2$5 such that $\ell_2$6, leading to a periodic measurement model: $\ell_2$7 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,

$\ell_2$8

where the current $\ell_2$9 is placed in block $x(k+1) = A x(k) + B u(k) + w(k)$0, and similarly for stacked input and noise terms. The block-lifted system evolves as

$x(k+1) = A x(k) + B u(k) + w(k)$1

$x(k+1) = A x(k) + B u(k) + w(k)$2

with block matrices defined by the period $x(k+1) = A x(k) + B u(k) + w(k)$3:

• $x(k+1) = A x(k) + B u(k) + w(k)$4 is block-cyclic with $x(k+1) = A x(k) + B u(k) + w(k)$5 on the final off-diagonal.
• $x(k+1) = A x(k) + B u(k) + w(k)$6, similarly for $x(k+1) = A x(k) + B u(k) + w(k)$7.
• $x(k+1) = A x(k) + B u(k) + w(k)$8.
• $x(k+1) = A x(k) + B u(k) + w(k)$9.

Due to the time steps when a sensor is inactive ($x \in \mathbb{R}^n$0 zeroing rows), the measurement covariance $x \in \mathbb{R}^n$1 is singular ($x \in \mathbb{R}^n$2, but $x \in \mathbb{R}^n$3). 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: $x \in \mathbb{R}^n$4 with cost

$x \in \mathbb{R}^n$5

The optimal control gain $x \in \mathbb{R}^n$6 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 $x \in \mathbb{R}^n$7 and an auxiliary variable $x \in \mathbb{R}^n$8, satisfying the LMI

$x \in \mathbb{R}^n$9

where $u \in \mathbb{R}^p$0 denotes symmetric terms. The filter gain is recovered as $u \in \mathbb{R}^p$1.

To minimize an upper bound on steady-state error covariance, introduce $u \in \mathbb{R}^p$2 via

$u \in \mathbb{R}^p$3

The LMI design problem becomes a semidefinite program (SDP): $u \in \mathbb{R}^p$4

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 $u \in \mathbb{R}^p$5 within a stability disk $u \in \mathbb{R}^p$6, add

$u \in \mathbb{R}^p$7

2. Mixed $u \in \mathbb{R}^p$8/$u \in \mathbb{R}^p$9-Induced Norm: For output $w$0, the induced norm constraint $w$1 is enforced via

$w$2

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 $w$3, $w$4, $w$5, and $w$6 matrices from system and sampling schedule.
2. Selecting objective terms (error covariance minimization, pole placement, $w$7 bounds).
3. Solving the resulting SDP via solvers such as MOSEK or SeDuMi, with computational complexity $w$8.
4. Recovering the filter gain $w$9 and extracting periodic gains $Q \succ 0$0 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 $Q \succ 0$1 s, period $Q \succ 0$2,
• State $Q \succ 0$3 ($Q \succ 0$4, $Q \succ 0$5, $Q \succ 0$6); measurement $Q \succ 0$7 ($Q \succ 0$8, $Q \succ 0$9),
• Plant and sensor model as

$q$0

$q$1

• Measurement selection: at $q$2, $q$3 enables both sensors; for $q$4, $q$5 enables only wheel speed.
• Rank($q$6) $q$7, so $q$8 is PSD but not invertible.

The SDP solution yields

• $q$9,
• $$S_k = \mathrm{diag}(s_{k,1},\dots,s_{k,q}), \quad s_{k,i} \in \{0,1\},$$0.

Periodic Kalman gains are:

$$S_k = \mathrm{diag}(s_{k,1},\dots,s_{k,q}), \quad s_{k,i} \in \{0,1\},$$1	Sensors	$$S_k = \mathrm{diag}(s_{k,1},\dots,s_{k,q}), \quad s_{k,i} \in \{0,1\},$$2 (rounded)
0	GPS+WS	$$S_k = \mathrm{diag}(s_{k,1},\dots,s_{k,q}), \quad s_{k,i} \in \{0,1\},$$3
1–9	WS only	$$S_k = \mathrm{diag}(s_{k,1},\dots,s_{k,q}), \quad s_{k,i} \in \{0,1\},$$4

Simulated performance (20 s, sinusoidal input, Gaussian noise): RMSEs for $$S_k = \mathrm{diag}(s_{k,1},\dots,s_{k,q}), \quad s_{k,i} \in \{0,1\},$$5, $$S_k = \mathrm{diag}(s_{k,1},\dots,s_{k,q}), \quad s_{k,i} \in \{0,1\},$$6, $$S_k = \mathrm{diag}(s_{k,1},\dots,s_{k,q}), \quad s_{k,i} \in \{0,1\},$$7.

Trade-off analysis demonstrates that tighter pole or induced norm constraints increase $$S_k = \mathrm{diag}(s_{k,1},\dots,s_{k,q}), \quad s_{k,i} \in \{0,1\},$$8 (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 $$S_k = \mathrm{diag}(s_{k,1},\dots,s_{k,q}), \quad s_{k,i} \in \{0,1\},$$9,
• 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).