INVESTIGATIONS ON DYNAMICAL SYSTEMS WITH TIME SCALE SEPARATION

Abstract

The control of nonlinear dynamical systems exhibiting multi-time scale behaviors and discontinuities remains a significant challenge in the field of control theory. This thesis addresses two such classes of complex systems, singularly perturbed systems (SPS) and singularly perturbed switched systems (SPSS) and proposes advanced nonlinear control strategies to enhance their stability and robustness in the presence of uncertainties and external disturbances. A novel saturated controller is developed for singularly perturbed systems that exhibit a time-scale separation between the slow and fast dynamics. These systems often suffer performance degradation due to inherent uncertainties and external disturbances. To mitigate these effects, a nonlinear saturated controller is proposed by leveraging the singular perturbation technique. The control scheme incorporates a filter to manage fast dynamics and a high-gain disturbance observer (HGDO) to estimate and reject unknown disturbances effectively. A contraction-theoretic framework is employed to establish the convergence of all closed-loop system states, thereby ensuring robust performance even under singular perturbations and model uncertainties. The proposed method is validated through the pitch angle control of a Twin Rotor MIMO System (TRMS), which serves as a practical benchmark for demonstrating the effectiveness of the controller in replicating real-time operational conditions. Hereafter, the focus shifts to switched nonlinear systems, where discontinuities arise due to switching among subsystems, as commonly encountered in chemical processes, robotic systems, and multi-agent networks. A time-scale redesign-based robust filter backstepping controller is introduced for a class of uncertain switched nonlinear systems expressed in strict feedback form. Here, the singular perturbation technique plays a dual role: first, in the design of high-gain filters to address the "explosion of complexity" typically associated with backstepping methods; second, in the construction of high-gain disturbance observers to counteract unknown disturbances and unmodeled vi dynamics. The interplay between system discontinuities, high-gain design elements, and disturbance rejection results in a three-time scale switched nonlinear closed-loop system. The stability of this system is rigorously analyzed using a Lyapunov-based average dwell time approach. To demonstrate real-world applicability, the proposed control strategy is implemented on a single-link robotic manipulator, and the results confirm its efficacy in achieving robust performance and tracking accuracy under switching and uncertain conditions. A filtered backstepping controller integrated with HGDO is specifically designed for SPSSs with frequent mode transitions. Contraction theory is employed to guarantee stability and convergence within a bounded domain. The approach is applied to a robotic manipulator, showing strong robustness against switching and external disturbances. A neural network-based disturbance observer (NNDO) within the filtered backstepping framework. Exploiting the approximation capabilities of neural networks, the NNDO enables real-time estimation and compensation of unknown disturbances. Combined with a high-gain filter, this structure improves robustness while reducing computational burden. Stability is ensured using average dwell-time analysis, and simulations on a robotic manipulator confirm the approach’s practical viability. A neural backstepping controller is developed for singularly perturbed switched systems to effectively handle system nonlinearities and unknown disturbances. The design incorporates a high-gain filter to manage the complexity typically associated with recursive backstepping, while a high-gain disturbance observer (HGDO) is employed for real-time disturbance rejection. The proposed control scheme is tested on a single-link robotic manipulator that includes actuator dynamics, demonstrating its strong robustness and practical applicability in dynamic and uncertain environments. Overall, this thesis presents a unified framework for robust control of singularly perturbed and switched nonlinear systems by incorporating backstepping, high-gain observers, and rigorous stability tools.