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.