⚡ electrical engineering

Optimal Sensor Scheduling and Selection for Continuous-Discrete Kalman Filtering with Auxiliary Dynamics

This paper proposes a gradient-based optimal control framework for jointly optimizing sensor measurement rates and auxiliary dynamics in continuous-discrete Kalman filtering by deriving a differentiable upper bound on the posterior covariance matrix to balance estimation accuracy with resource constraints.

Original authors: Mohamad Al Ahdab, John Leth, Zheng-Hua Tan

Published 2026-07-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohamad Al Ahdab, John Leth, Zheng-Hua Tan

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 keep a perfect mental map of a moving object, like a drone flying through a storm. You can't see it all the time; you only get glimpses through sensors. This is the world of State-Space Models, a mathematical way to guess where something is based on noisy, incomplete clues. The Kalman Filter is the superstar algorithm that does this guessing game. It's like a super-smart detective who constantly updates their theory of "where the suspect is" every time a new piece of evidence arrives.

But here's the twist: in the real world, sensors aren't free. Checking a sensor might drain a battery, overheat a device, or even damage the sensor itself if it's in a dangerous zone. Furthermore, the sensor's own location or temperature might change how good its data is. This creates a tricky puzzle: When should you turn a sensor on, and how often, to get the best picture of reality without running out of juice or breaking your equipment? This paper tackles that puzzle for systems where things move smoothly over time (like a drone) but are only checked at specific, irregular moments.

The Detective's Dilemma: When to Look?

The authors, Mohamad Al Ahdab, John Leth, and Zheng-Hua Tan, are tackling a very specific headache in the world of robotics and signal processing. Imagine you have a robot equipped with two different eyes: one is a high-definition camera that sees clearly but eats up battery power and gets blurry if the robot gets too hot; the other is a radar that works in the dark but is a bit fuzzy and drains energy differently.

The robot needs to track a moving target (like a pollutant cloud or a chemical leak) while also managing its own "auxiliary" state—its battery level, its temperature, or even how much radiation damage it has accumulated. The problem is that the robot can't just stare at the target with both eyes open 24/7. It needs a schedule. It needs to decide: "Should I use the high-def camera now? Should I wait? Should I move to a cooler spot to protect my sensors?"

The Paper's Big Idea: A "Best Guess" Map for Scheduling

The paper proposes a clever way to solve this scheduling puzzle. Instead of guessing randomly or just picking the sensor that seems best at the exact moment (a "greedy" approach), the authors create a mathematical framework that looks ahead.

They treat the arrival of measurements (the "glimpses" of the target) as a Poisson process. Think of this like raindrops hitting a roof. You can't predict exactly when the next drop will fall, but you can control the rate at which they fall. The authors figure out the perfect "rain rate" for each sensor. They ask: "If Sensor 1 drops data 5 times a minute and Sensor 2 drops data 2 times a minute, what happens to our uncertainty about the target's location?"

Here is the magic trick they discovered: They derived a mathematical upper bound on the uncertainty (the "fog" in the detective's mind). This bound is special because it is smooth and differentiable. In plain English, this means they can use standard calculus to slide down a hill of math to find the absolute best schedule. They can tweak the "rain rates" and the robot's movement path simultaneously to minimize the fog while keeping the battery alive.

The "Deterministic" Twist: From Rates to Real Times

Once the math spits out the perfect rates (e.g., "Sensor 1 should ping 3.4 times per second"), the robot still needs actual times to take the measurements. You can't take 3.4 photos in a second.

The authors suggest a deterministic method to turn these rates into specific moments. They use a concept called Wasserstein distance, which is a fancy way of measuring how far apart two distributions are. Imagine you have a bucket of water (the ideal rate) and you need to pour it into a set of cups (the actual measurement times). They arrange the cups so that the water level in the cups matches the bucket as perfectly as possible. This ensures the robot takes measurements at the most "average" and efficient times, rather than relying on a random roll of the dice that might accidentally skip a crucial moment.

What They Found (and What They Didn't)

The authors tested their method in two simulated scenarios:

A Robot with Energy Constraints: A robot trying to map a pollutant cloud while managing its battery. The robot had to move to a charging station when low on power.
A Radioactive Zone: A robot entering a dangerous area where taking measurements actually damages the sensors over time, making them less accurate.

In these simulations, their "Optimized" approach outperformed three other strategies:

Random: Just picking times randomly.
Greedy: Making a decision based only on the immediate next second without looking ahead.
M-Optimized: Sampling many random possibilities and picking the best one (which is computationally expensive).

The results showed that their method kept the uncertainty (the "fog") lower than the greedy and random methods. It also managed the robot's energy and sensor damage much better. For instance, in the radioactive scenario, their method kept the total sensor damage significantly lower than the random approach.

However, it is important to note that these results come from simulations. The paper suggests that this framework works well for these specific mathematical models, but it hasn't been proven on a real robot in a real radioactive disaster zone yet. The authors also note that their method relies on certain mathematical assumptions (like the cost functions being "concave" or "convex"), which holds true for many systems but might need adjustment for others.

Why This Matters

This paper doesn't just say "turn sensors on and off." It provides a unified way to think about the robot's movement, its battery, its sensor health, and the quality of its data all at once. By turning a chaotic scheduling problem into a smooth, solvable math equation, the authors give engineers a powerful tool to design smarter, more efficient autonomous systems. Whether it's a satellite monitoring ocean temperatures or a drone searching for survivors, the ability to balance "how much we know" against "how much it costs to know" is a giant leap forward in making machines truly autonomous.

Technical Summary: Optimal Sensor Scheduling and Selection for Continuous-Discrete Kalman Filtering with Auxiliary Dynamics

1. Problem Statement

The paper addresses the challenge of optimal sensor scheduling and selection within a Continuous-Discrete Kalman Filter (CD-KF) framework. Unlike traditional setups where state evolution is continuous but measurements are discrete, this work considers scenarios where the measurement process is coupled with auxiliary dynamics. These auxiliary states ( $\xi$ ) represent variables such as a sensor's spatial trajectory, energy consumption, or temperature, which directly influence measurement costs, accuracy, and feasibility.

The core problem involves a multi-sensor system where:

Continuous-time state dynamics are observed via multiple sensors providing discrete, potentially irregular measurements.
Measurement rates and sensor selection are constrained by auxiliary dynamics (e.g., a satellite's orbital position affecting sensor accuracy, or a robot's energy depleting with each measurement).
The goal is to jointly optimize the measurement rates ( $\lambda_s(t)$ ) for each sensor and the control inputs ( $u$ ) governing the auxiliary dynamics to minimize estimation uncertainty (posterior covariance) while satisfying resource constraints.

Existing literature largely focuses on either continuous-time measurements or discrete-time formulations, often neglecting the coupling between sensor scheduling and general auxiliary state dynamics (e.g., trajectory-dependent accuracy or energy constraints).

2. Methodology

The authors propose a framework that transforms the stochastic scheduling problem into a deterministic optimal control problem through the following steps:

A. Modeling Measurement Arrival
Measurement arrivals for each sensor $s$ are modeled as independent Poisson processes with time-varying intensities $\lambda_s(t)$ . This allows the stochastic nature of measurement timing to be represented by continuous rate functions.

B. Derivation of Deterministic Bounds
To enable gradient-based optimization, the authors derive upper bounds on the system's stochastic behavior:

Covariance Bound (Proposition 6.1): An upper bound $\hat{\Sigma}(t)$ on the mean posterior covariance matrix is derived, conditioned on the mean auxiliary state. This bound satisfies a deterministic differential equation where the stochastic jump terms (Dirac deltas) are replaced by the measurement rates $\lambda_s(t)$ . Crucially, this bound is continuously differentiable with respect to $\lambda_s(t)$ .
Auxiliary State Bound (Proposition 6.2): Under the assumption that the auxiliary dynamics functions are concave (or convex) with respect to the perturbed state components, a deterministic trajectory $\hat{\xi}(t)$ is derived that bounds the true mean auxiliary state.

C. Finite-Horizon Optimal Control Problem (OCP)
Using these bounds, the authors formulate a finite-horizon optimal control problem (Eq. 12). The objective is to minimize a cost function (e.g., the trace of the covariance matrix) subject to:

The deterministic dynamics of the covariance bound $\hat{\Sigma}$ .
The deterministic dynamics of the auxiliary state bound $\hat{\xi}$ .
Running and terminal constraints (e.g., energy limits, return-to-base requirements).
Non-negativity of measurement rates.

D. Deterministic Measurement Scheduling
Once the optimal rates $\lambda_s(t)$ and inputs $u(t)$ are computed, the paper proposes a deterministic method to select specific measurement time instances. This is formulated as an optimal quantization problem. The method minimizes the Wasserstein-2 distance between the continuous intensity measure defined by $\lambda_s(t)$ and a discrete measure representing the selected measurement times. This ensures the selected times closely match the statistical average of the optimized Poisson process.

3. Key Contributions

The paper explicitly summarizes its contributions as follows:

Derivation of a Differentiable Bound: The authors derive an upper bound on the mean posterior covariance matrix of the CD-KF conditioned on the mean auxiliary state. This bound is continuously differentiable with respect to sensor-specific measurement rates, enabling efficient gradient-based optimization.
Joint Optimization Framework: A finite-horizon optimal control framework is proposed to jointly optimize measurement rates and auxiliary-state dynamics (e.g., vehicle trajectory).
Deterministic Scheduling Strategy: A deterministic method is introduced to convert optimized Poisson rates into specific measurement time events. This method minimizes the Wasserstein distance between the optimized rate distribution and the empirical distribution of measurement events.

4. Experimental Results

The approach is validated through two scenarios involving a robot collecting data for dynamic temporal Gaussian Process (GP) regression (equivalent to Kalman smoothing):

Scenario 1 (Standard): A robot with two sensors (one high-accuracy/high-cost, one lower-accuracy/low-cost) must navigate to a target and return while managing energy. The optimization balances measurement frequency, trajectory, and energy consumption.
Scenario 2 (Radioactive Environment): An extension where sensors degrade over time due to accumulated radiation damage, which increases measurement noise. The auxiliary state includes radiation damage levels.

Comparative Performance:
The proposed "Optimized" method was compared against:

Random: Uniform discretization with constant Poisson rates.
Greedy: Step-wise selection based on immediate cost reduction.
M-Optimized: Sampling multiple Poisson realizations and selecting the best one.

Findings:

The proposed method achieved the lowest mean covariance trace (1.902 vs. 2.608 for Greedy), indicating superior estimation accuracy.
It maintained positive energy reserves (Mean $\eta \approx 21.5$ ), whereas Random and Greedy strategies resulted in negative energy (failure to return to base).
The deterministic quantization method performed comparably to the computationally expensive "M-Optimized" (sampling multiple realizations) but without the sampling overhead.
The planned upper bounds ( $\hat{\Sigma}, \hat{\eta}$ ) closely tracked the true simulated trajectories, validating the utility of the derived bounds.

5. Significance and Claims

The paper claims to fill a gap in the literature by formalizing optimal sensor scheduling for continuous-discrete SSMs with auxiliary dynamics, a setup not previously explored directly.

Novelty: The work unifies continuous-time state evolution with stochastic measurement scheduling via Poisson processes and integrates auxiliary state dynamics into a single optimal control framework.
Practicality: By deriving differentiable bounds, the method enables the use of standard gradient-based optimization tools for complex, coupled problems that were previously intractable.
Efficiency: The deterministic scheduling strategy avoids the computational cost of Monte Carlo sampling while maintaining performance close to the best-case stochastic realization.
Generality: The framework is applicable to various domains where sensor quality is coupled with operational states, such as satellite monitoring, medical glucose tracking, and environmental robotics.

The authors remain modest regarding the theoretical guarantees, noting that for fully nonlinear systems violating specific concavity assumptions, the proposed bounds serve as heuristic approximations, though empirical results suggest acceptable performance in such cases.