Uncertainty Propagation in Bilinear and Polynomial System for Probabilistic Threshold Detection

Abstract

Uncertainty can be defined as imperfect or unknown information arising in a stochastic environment. Due to the very limited knowledge, it is difficult to propagate and quantify various uncertainties affecting the system to its next step. As a result, it has been a challenge to consider multiple uncertainties affecting the system in various fields, such as Fault Detection and Diagnosis. Fault detection has been an essential part of a large industrial and manufacturing system to take a proper corrective measure into account in a case of unexpected behavior. However, determining a robust threshold bound for fault detection is a big challenge. Several uncertainties affect the system (such as parametric uncertainties, experimental uncertainties, process noise, measurement noise, etc.). Ignoring the effects of various uncertainties (i.e., Deterministic Bound) can lead to a false alarm.

Therefore, to design a robust threshold probabilistic-based technique is used where all unknown parameters are taken into account. However, the major problem lies in propagating these unknown parameters into the next time step with their limited knowledge. A novel Message Passage Bilinear Uncertainty Propagation (MPBUP) algorithm is being proposed, which is used to quantify and propagate various uncertainties affecting the dynamical system into the next time step. The main aim of the algorithm is to quantify and propagate various uncertain parameters affecting the system at each time step. Uncertainty is propagated in terms of mean and covariance at each algorithm iteration to find the overall effect. Therefore the primary research is to design and validate the proposed algorithm and to check the algorithm can be used to determine a probabilistic threshold.

In the report, a detailed explanation of the algorithm for a trivial example is presented. The algorithm is then developed and implemented in MATLAB. Next, validation of the output generated through the algorithm is performed using Monte Carlo simulation. Finally, various analyses based on the MC simulation are discussed to support the results generated through the algorithm. An innovative approach is also discussed to extend the polynomial system algorithm as the proposed algorithm is limited to the bilinear system.

Further, a detailed explanation is given on applying the algorithm on a general state-space model in terms of mathematics involved, which is further extended on applying the algorithm on a real-time application such as the Four Tank System. After successfully implementing the algorithm on the Four Tank System, the algorithm is used to propagate uncertainty into the dynamical system to determine the robust threshold. A robust threshold is found considering the effect of various uncertainty. The threshold found using the proposed algorithm is dynamic as it evolves based upon the state dynamics and satisfies the required condition. As a result presented algorithm satisfy all the requirement and can also be used in other applications. The algorithm was analyzed based upon various criteria to conclude this thesis, and a comparative study is conducted.