⚡ electrical engineering

LMI Optimization Based Multirate Steady-State Kalman Filter Design

This paper proposes an LMI-based design framework for multirate steady-state Kalman filters that overcomes the limitations of semidefinite measurement noise covariances in cyclic reformulations, enabling multi-objective optimization for automotive navigation systems where effective sensor fusion significantly reduces position estimation error below GPS noise levels.

Original authors: Hiroshi Okajima

Published 2026-03-20

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Hiroshi Okajima

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to navigate a car through a foggy city. To know exactly where you are, you have two different helpers:

The GPS: It's like a wise old friend who knows the map perfectly, but they only speak up once every 10 seconds. When they do speak, they give you a very clear location, but they might be slightly off by a few meters.
The Speedometer: It's like a nervous friend who talks 10 times every second. They tell you exactly how fast you are going, but they don't know where you are on the map, only how fast you're moving.

The Problem:
In the old days, engineers tried to combine these two voices into one "super-voice" to guess your location. But they ran into a math problem. Because the GPS is silent for 9 seconds out of 10, the math used to combine the data would sometimes break or crash. It's like trying to bake a cake using a recipe that requires eggs, but the egg carton is empty for most of the time. The standard math tools (called "Riccati equations") simply refused to work when the data was missing or "sparse."

The Solution (The Paper's Big Idea):
This paper introduces a clever new way to bake that cake, even when the ingredients arrive at different times. The author, Hiroshi Okajima, proposes a method called "LMI-based design."

Here is how it works, using a simple analogy:

1. The "Time-Traveling" Trick (Cyclic Reformulation)

Instead of trying to solve the problem second-by-second (which is messy because the GPS is silent sometimes), the author suggests looking at the whole 10-second cycle as a single, giant snapshot.

Imagine you take a photo of the car every second for 10 seconds and stack them into a single, tall tower. Now, instead of a messy, changing puzzle, you have one big, static picture. This turns a "moving target" problem into a "still life" painting problem, which is much easier for computers to solve.

2. The "Safety Net" (Handling Missing Data)

The big breakthrough is how they handle the "missing eggs" (the times when the GPS is silent).

Old Way: The math demanded that every sensor give a perfect, positive number. If a sensor was silent, the number became zero, and the math crashed.
New Way (LMI): The author uses a technique called LMI (Linear Matrix Inequalities). Think of this as a flexible safety net. Instead of demanding a perfect, rigid number, the LMI method says, "Okay, the GPS is silent right now, so the uncertainty is 'zero' or 'flat' instead of 'positive.' That's fine! We can still build a stable bridge over that gap."

It allows the computer to say, "I know the GPS isn't talking right now, so I'll trust the speedometer more, but I'll keep a little bit of 'wiggle room' in my guess just in case."

3. The "Tuning Knobs" (Multi-Objective Design)

The best part of this new method is that it gives engineers extra tuning knobs.

Knob 1 (Speed): Do you want the car to correct its position super fast after a GPS update? You can turn a knob to make the math force the error to shrink quickly, even if it makes the guess slightly less smooth.
Knob 2 (Safety): Do you want to be absolutely sure the car never goes wildly off-course, even in a worst-case storm? You can turn another knob to prioritize safety over perfect average accuracy.

The Real-World Test

The author tested this on a simulated car driving with a GPS (1 Hz) and a wheel speed sensor (10 Hz).

The Result: The new filter was incredibly accurate. Even though the GPS was only 1 meter accurate on its own, the combined system figured out the car's position with an error of only 0.56 meters.
The Proof: They ran the simulation 500 times with different random noise (like driving in different weather conditions). The math proved that the new method's "safety net" was tight and reliable—it never overpromised.

In a Nutshell

This paper is like inventing a new recipe for a multitasking chef.

Old Chef: "I can't cook if I don't have all the ingredients at the exact same time!"
New Chef (This Paper): "No problem! I can organize the ingredients into a big batch, handle the missing ones with a flexible safety net, and even adjust the recipe to be faster or safer depending on what you need."

It allows self-driving cars, robots, and drones to use a mix of slow, accurate sensors and fast, noisy sensors without the math breaking, leading to safer and more precise navigation.

1. Problem Statement

The paper addresses the challenge of state estimation in linear systems equipped with multiple sensors operating at different sampling rates (multirate systems).

Context: Modern applications like autonomous vehicles (fusing GPS at 1–10 Hz with wheel speed/inertial sensors at 50–100 Hz) and industrial process control require fusing asynchronous data.
The Core Difficulty: When sensors operate at different rates, the measurement equation becomes a periodic time-varying system, even if the underlying plant is time-invariant.
The Specific Barrier: Standard Kalman filter design relies on the Discrete-Time Algebraic Riccati Equation (DARE). However, when reformulating multirate systems into a time-invariant form (via cyclic reformulation), the resulting measurement noise covariance matrix ( $\check{R}$ ) becomes positive semidefinite ( $\check{R} \succeq 0$ ) rather than positive definite ( $\check{R} \succ 0$ ).
- This singularity occurs because, at certain time steps within a frame period, specific sensors do not provide measurements (zero rows in the selection matrix).
- Standard DARE solvers require the inversion of $(\check{C}\check{P}\check{C}^T + \check{R})$ , which fails or becomes numerically unstable when $\check{R}$ is singular.

2. Methodology

The author proposes a systematic design framework based on Cyclic Reformulation and Linear Matrix Inequality (LMI) optimization.

A. Cyclic Reformulation

The periodic multirate system is transformed into an equivalent time-invariant system with an expanded state dimension ($Nn$, where $N$ is the frame period and $n$ is the state dimension).

State Augmentation: The state vector is "cycled" to represent the state at every time step within the period simultaneously.
System Matrices: The system matrices ( $\check{A}, \check{B}, \check{C}$ ) are constructed as block-cyclic matrices.
Noise Covariance: The process noise covariance $\check{Q}$ remains block-diagonal and positive definite, but the measurement noise covariance $\check{R}$ becomes block-diagonal with singular blocks corresponding to time steps where sensors are inactive.

B. LMI-Based Filter Design (Dual LQR Formulation)

To overcome the singularity of $\check{R}$ , the paper utilizes the duality between Kalman filtering and Linear Quadratic Regulator (LQR) control.

Dual Problem: The filtering problem is mapped to a dual LQR problem where the measurement noise covariance acts as the input weight.
LMI Formulation: Instead of solving the Riccati equation directly, the problem is cast as a convex Semidefinite Programming (SDP) problem.
- Variables: A Lyapunov matrix $\check{X}$ and a transformed gain variable $\check{Y} = -\check{X}\check{L}$ .
- Key Constraint: The stability and performance conditions are expressed as LMIs (e.g., Eq. 42). Crucially, this formulation does not require the inversion of $\check{R}$ , allowing it to handle semidefinite covariances naturally.
- Objective: Minimize the trace of an upper bound matrix $\check{W}$ (where $\check{W} \succeq \check{X}^{-1} \succeq \check{P}_e$ ) to minimize the estimation error covariance.

C. Multi-Objective Extensions

The LMI framework allows for the simultaneous incorporation of additional design constraints:

Pole Placement: Constraints can be added to force the eigenvalues of the closed-loop error dynamics ( $\check{A} - \check{L}\check{C}$ ) into a specific disk region ( $D_{\bar{r}}$ ), guaranteeing a minimum convergence rate.
$l_2$ -Induced Norm Constraints: To balance average performance (stochastic) with worst-case performance, constraints are added to bound the $l_2$ -induced norm (generalized $H_\infty$ norm for periodic systems) from disturbances to estimation error.

3. Key Contributions

Identification of Semidefinite Covariance: The paper rigorously identifies that cyclic reformulation of multirate systems inherently leads to a singular measurement noise covariance, rendering standard DARE methods inapplicable.
Novel LMI Solution: It introduces a dual LQR-based LMI formulation that naturally handles semidefinite covariances, providing a numerically stable and globally optimal solution (within the convex relaxation).
Stability Guarantees: Theorem 4.1 proves that if the LMI is feasible, the resulting filter guarantees Schur stability of the estimation error dynamics, regardless of the rank of $\check{R}$ .
Multi-Objective Framework: It extends the basic design to include pole placement and $l_2$ -induced norm constraints, offering a unified tool for balancing convergence speed, average accuracy, and worst-case robustness.
Validation: Theoretical bounds are validated against Monte Carlo simulations, proving the tightness of the LMI-derived upper bounds.

4. Results

The methodology was validated using an automotive navigation system example:

Setup: A 3-state system (position, velocity, acceleration) with GPS (1 Hz) and wheel speed sensors (10 Hz).
Simulation: 500 Monte Carlo runs with 200 time steps each.
Performance Metrics:
- Position RMSE: Achieved 0.561 m, significantly below the GPS sensor noise standard deviation (1.0 m), demonstrating effective multirate fusion.
- Theoretical Bounds: The LMI-derived theoretical upper bound (0.587 m) was extremely close to the empirical RMSE (Ratio: 1.046), confirming the bound is tight and non-conservative.
- Stability: The closed-loop spectral radius was 0.9673, confirming stability.
Trade-off Analysis:
- Pole Placement: Enforcing faster convergence (smaller $\bar{r}$ ) increased the error covariance trace non-linearly (e.g., reducing $\bar{r}$ from 0.975 to 0.800 increased the cost by ~9x).
- $l_2$ -Norm: Tightening the worst-case bound ( $\bar{\gamma}$ ) improved stability margins (reduced max eigenvalue) but degraded average performance (increased trace).

5. Significance

Practical Applicability: This framework solves a critical gap in control theory where standard Kalman filters fail for realistic multirate sensor configurations due to singular noise covariances.
Design Flexibility: It moves beyond simple state estimation to a multi-objective design tool, allowing engineers to explicitly tune the trade-off between estimation accuracy, convergence speed, and robustness to worst-case disturbances.
Computational Efficiency: By converting the periodic problem into a static LMI problem, it enables offline computation of periodic gains, which are then applied online in a simple, fixed-gain manner.
Future Impact: The paper lays the groundwork for robust filtering under uncertainty (polytopic systems) and extensions to nonlinear systems, making it a foundational reference for modern sensor fusion in autonomous systems and networked control.