Date of Award
2025
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Electrical and Computer Engineering
Committee Chair
Avimanyu Sahoo
Committee Member
Laurie Joiner
Committee Member
Aubrey N. Beal
Committee Member
Farbod Fahimi
Committee Member
Shangbing Ai
Research Advisor
Avimanyu Sahoo
Subject(s)
System identification, Nonlinear systems, Chebyshev polynomials, Spectral theory (Mathematics)
Abstract
Nonlinear system identification plays a crucial role in developing effective control for various systems, including autonomous vehicles, robotics, and industrial process control. Acquiring the system model with minimal sensing and computation can significantly reduce control costs in these applications. Traditional adaptive control techniques often rely on uniform or high-frequency sampling, which may become impractical or computationally expensive for rapidly changing or high-dimensional systems. While Chebyshev-based pseudospectral (PS) methods have proven highly effective in offline contexts, their direct online implementation faces several challenges. Successfully approximating system dynamics online requires prior information about the approximation intervals to compute Chebyshev nodes for data collection, as well as the basis polynomial order or node count to ensure the desired approximation accuracy. Furthermore, accurate measurement or estimation of state derivatives is critical for reliably computing Chebyshev coefficients. This dissertation presents a suite of online pseudospectral methods that utilize Chebyshev polynomial basis functions for nonlinear system identification. It employs non-uniform (aperiodic) sampling at Chebyshev nodes to take advantage of the minimax error property, ensuring the desired approximation accuracy. These methods aim to reduce sampling frequency while adhering to strict theoretical limits on parameter estimation errors in real-time scenarios. In the first approach, the nonlinear system dynamics are approximated using a state-based Chebyshev polynomial basis. To apply the PS method in real time, the time horizon is divided into intervals, allowing for the determination of sampling points based on Chebyshev nodes within a moving time window. A least-squares (LS) approach is employed to estimate the coefficients of the Chebyshev basis for accurate approximation. The state measurements used to construct the Chebyshev basis, in the first approach, are not the nodes of the polynomial. Therefore, the approximation does not guarantee an optimal error bound. To overcome this limitation and ensure the desired approximation accuracy, in the second approach, the Chebyshev polynomial basis is formulated as a function of time. The state and its derivative are measured and computed, respectively, at the Chebyshev time nodes, preserving the minimax property of the polynomials. Since this basis function results in a single-variable format, it significantly reduces computational overhead. The Chebyshev coefficients were estimated using the LS-based method. This approach still required state derivatives, as in the first approach, which necessitated additional samples near each node, increasing the sampling frequency. To address the need for state derivative computation, which is known to amplify noise, the proposed online PS approach is extended to first approximate the system dynamics in the third approach. The system dynamics are recovered analytically from the state approximation. To further improve computational efficiency in parameter estimation, the Fast Chebyshev Transform (FCT) was adapted to the moving window scheme. To guarantee the desired approximation accuracy in each proposed method, an adaptive node selection scheme based on the corresponding average approximation error was implemented. Rigorous analysis, based on Lyapunov arguments and bounded-error proofs, confirmed that both parameter error and function error remained finite over time. Each approach was validated through simulation studies, demonstrating a reduced sensing frequency while maintaining approximation accuracy. Overall, these online Chebyshev pseudospectral methods represent a novel approach for real-time, resource-constrained environments, offering a promising avenue for future research and industrial deployment where high accuracy and minimal sensing are equally essential.
Recommended Citation
Yousefian, Arian, "Online Chebyshev-based pseudospectral methods for nonlinear system identification" (2025). Dissertations. 461.
https://louis.uah.edu/uah-dissertations/461