Analytical Solution for Nonlinear Unknown Input Observability and Reconstruction

Core Concepts

This paper provides the general analytical solution to the problem of nonlinear unknown input observability, including an automatic procedure to determine the observability properties of a system even when some of its inputs are unknown.

Abstract

The paper presents a comprehensive solution to the problem of nonlinear unknown input observability. It starts by characterizing the general class of nonlinear systems considered, which include known and unknown inputs, as well as time-invariant and time-variant systems.
The key contributions are:

A thorough analysis of the concept of canonicity with respect to unknown inputs, including new definitions of unknown input reconstructability matrix, unknown input degree of reconstructability, canonic system, and canonical form.

An algorithm (Algorithm 5.1) that provides the general analytical solution for the observability properties of nonlinear systems, even if they are not in canonical form with respect to the unknown inputs and not even canonizable. This algorithm automatically computes the observability codistribution, which contains the gradients of all the observable functions.

New convergence criteria for the algorithms that solve the problem in the special cases of systems in canonical form (Algorithms 3.1 and 4.1), which are simpler and more general than previous solutions.

The extension of the observability analysis to the problem of unknown input reconstruction.

Illustration of the implementation of the general algorithm (Algorithm 5.1) on a nonlinear visual-inertial sensor fusion system, which is not in canonical form.

The paper provides a complete and automatic solution to the long-standing problem of nonlinear unknown input observability, which has important applications in various domains such as robotics, biology, and control theory.

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Nonlinear Unknown Input Observability and Unknown Input Reconstruction: The General Analytical Solution

by Agostino Mar... at arxiv.org 09-11-2024

https://arxiv.org/pdf/2201.07610.pdf

Deeper Inquiries

Potential Applications of the Proposed Nonlinear Unknown Input Observability Solution Beyond Visual-Inertial Sensor Fusion

The proposed nonlinear unknown input observability solution has a wide range of potential applications across various fields. Some notable examples include:

Robotics: In robotic systems, especially those operating in unstructured environments, the dynamics may be influenced by unknown inputs such as external disturbances (e.g., wind, terrain variations). The observability solution can enhance state estimation and control strategies, enabling robots to navigate and perform tasks more effectively.

Autonomous Vehicles: For self-driving cars, the ability to observe and reconstruct the state of the vehicle in the presence of unknown inputs (like road conditions or other vehicles' behaviors) is crucial for safe navigation and decision-making. The proposed solution can improve the robustness of state estimation algorithms in these scenarios.

Aerospace Engineering: In aerospace applications, such as satellite or drone operations, the dynamics can be affected by unknown atmospheric conditions. The observability solution can be applied to enhance the estimation of the vehicle's state, improving mission performance and safety.

Biological Systems: In biological modeling, where systems may be influenced by unmeasured factors (e.g., environmental changes), the observability framework can help in understanding the dynamics of biological processes and improving the design of experiments.

Economics and Finance: In economic models where certain inputs (like market conditions) are unknown or difficult to measure, the observability solution can assist in reconstructing the underlying state of the economy, aiding in better policy-making and forecasting.

Control Systems: In industrial control systems, where disturbances can affect system performance, the observability solution can be utilized to design more effective observers that can estimate the state of the system despite the presence of unknown inputs.

Extension of the Analytical Solution to Handle Stochastic or Partially Observable Systems

To extend the analytical solution to handle stochastic or partially observable systems, several approaches can be considered:

Incorporating Stochastic Models: The current deterministic framework can be augmented by introducing stochastic elements into the system dynamics. This could involve modeling the unknown inputs as stochastic processes, allowing for the application of stochastic calculus and filtering techniques (e.g., Kalman filters) to estimate the state.

Partially Observable Markov Decision Processes (POMDPs): The observability solution can be integrated into the framework of POMDPs, where the state is not fully observable. By defining a belief state that represents the probability distribution over possible states, the observability conditions can be reformulated to account for the uncertainty in state estimation.

Robust Control Techniques: The analytical solution can be combined with robust control strategies that account for uncertainties in the system. This would involve designing observers that can handle both known and unknown inputs while ensuring stability and performance under stochastic disturbances.

Adaptive Algorithms: Developing adaptive algorithms that can adjust the observability criteria based on real-time data could enhance the applicability of the solution in dynamic environments where the nature of unknown inputs may change over time.

Simulation-Based Approaches: Utilizing Monte Carlo simulations to explore the impact of unknown inputs and stochastic variations on observability can provide insights into the robustness of the proposed solution and guide further refinements.

Limitations or Assumptions in the Proposed Approach That Could Be Relaxed in Future Work

While the proposed nonlinear unknown input observability solution presents significant advancements, there are several limitations and assumptions that could be relaxed in future work:

Assumption of Smoothness: The current framework assumes that the vector fields and functions involved are smooth. Future research could explore the implications of discontinuities or non-smooth dynamics, which are common in real-world systems.

Known Input Requirement: The solution primarily focuses on systems with known inputs alongside unknown inputs. Future work could investigate scenarios where all inputs are unknown, potentially leading to new observability criteria.

Canonical Form Requirement: The solution is designed for systems that can be transformed into a canonical form with respect to unknown inputs. Research could focus on developing methods for systems that are not canonizable, broadening the applicability of the observability solution.

Finite-Dimensional Systems: The current approach is tailored for finite-dimensional systems. Extending the solution to infinite-dimensional systems, such as those described by partial differential equations, could open new avenues for application in fields like fluid dynamics and control of distributed systems.

Computational Complexity: The algorithms presented may have computational limitations in terms of scalability for high-dimensional systems. Future work could focus on optimizing these algorithms or developing approximation techniques to handle larger systems efficiently.

By addressing these limitations and assumptions, future research can enhance the robustness and applicability of the nonlinear unknown input observability solution across a broader range of complex systems.