# Transmissibility operators for state and output estimation in nonlinear systems

Khalil, Abdelrahman (2023) Transmissibility operators for state and output estimation in nonlinear systems. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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## Abstract

Transmissibility operators are mathematical objects that characterize the relationship between two subsets of responses of an underlying system. The importance of transmissiblity operators comes from the fact that these operators are independent on the system inputs. This work develops the transmissibility theory for nonlinear systems for the first time. The system nonlinearities are assumed to be unknown, which gives a wide range of possible engineering applications in different disciplines. Four different methods are developed to deal with these nonlinearities. The first method is by re-constructing the system nonlinearities as independent excitations on the system. This method handles the inherent unmodeled nonlinearities within the system. The second method is by designing a transmissibility-based sliding mode control. This method rejects unwanted nonlinearities such as system faults. The third method is by constructing the system as time-variant linear system, and use recursive least squares to solve it. This method can handle nonlinear systems with time-variant dynamics. The fourth method is by designing a new robust estimation technique called high-gain transmissibility (HGT) that is inspired by high-gain observers. This estimator has the ability to robustly estimate the system states in a high-gain form. The majority of modern fault detection, control systems, and robots localization depend on mathematically estimating the system states and outputs. Transmissibility-based estimation is incorporated in this work with these three theoretical applications. For fault detection, transmissibility operators are used along a set of outputs to estimate the measurements of another set of outputs. Then faults are detected by comparing the estimated and measured outputs with each other. Control approaches use the transmissibility-based estimation to construct the control signal, in which is injected back to the original system. Robots localization fuses the transmissibility-based estimation with real-time sensor measurements to minimize the error in determining the robot displacements. These three theoretical applications are applied on four different systems. The first system is Connected Autonomous Vehicles (CAV) platoons. A CAV platoon is a network of connected autonomous vehicles that communicate together to move in a specific path with the desired velocity. Transmissibilities are proposed along with the measurements from sensors available in CAV platoons to identify transmissibility operators. This will be then developed to mixed autonomous and human-driven vehicle platoons. Besides the wide range of physical and cyber faults in such systems, this is also motivated by the fact that on-road human-drivers’ behaviour is unknown and difficult to be predicted. Transmissibility operators are used here to handle both cyber-physical faults as well as the human-drivers’ behaviour. The platoon faults are then proposed to be mitigated using a transmissibility-based sliding mode controller. Moreover, transmissibilities are integrated with Kalman filter to localize CAV platoons while operating under non-Gaussian environment as unknown nonlinearities. The second system is a multi-actuator micro positioning system that is used in the semi-conductors industry. Transmissibility operators are applied on this system for fault detection and fault-tolerant control. Fault detection is represented in applying the proposed developments to actuator fault detection. Some of the most common actuator faults such as actuator loss of effectiveness and fatigue crack in the connection hinges will be considered. Transmissibilities then will be used for fault detection without knowledge of the dynamics of the system or the excitation that acts on the system. Next, a transmissibility-based sliding mode control will be implemented to mitigate common actuator faults in multi-actuator systems. The third system is flexible structures subjected to unknown and random excitations. Structures used in applications subjected to turbulent fluid flow such as aerospace and underwater applications are subjected to random excitations distributed along the structure. Transmissibility operators are used for the purpose of structural fault detection and localization during the system operation. The fourth system is robotic manipulators with bounded nonlinearities and time-variant parameters. Both parameter variation and system nonlinearities are considered to be unknown. Transmissibility operators are integrated with Recursive Least Squares (RLS) to overcome the unknown variant parameters. RLS identifies transmissibilities used in the structure of noncausal FIR (Finite Impulse Response) models. While parameter variation can be treated as system nonlinearities, the RLS algorithm is used to optimize what time-variant dynamics to include in the transmissibility operator and what dynamics to push to the system nonlinearities over time. The identified transmissibilities are then used for the purpose of fault detection in an experimental robotic arm with variant picked mass.

Item Type: Thesis (Doctoral (PhD)) http://research.library.mun.ca/id/eprint/16043 16043 Includes bibliographical references (pages 128-138) estimation theory, control systems, robotics, mechatronics, fault detection, fault mitigation Engineering and Applied Science, Faculty of March 2023 Submission https://doi.org/10.48336/66QR-5J47 Estimation theory; Robotics; Mechatronics; Fault location (Engineering); Nonlinear systems