Report on Current Developments in Nonlinear Systems Control
General Direction of the Field
The field of nonlinear systems control is currently witnessing a significant shift towards data-driven and robust methodologies, driven by advancements in optimization techniques and the increasing availability of computational resources. Recent developments are focused on integrating data-driven approaches with traditional model-based control strategies to enhance the performance and robustness of control systems. This integration is particularly evident in the design of controllers for nonlinear systems, where data-driven techniques are being used to approximate complex system behaviors and to design controllers that can regulate tracking errors more effectively.
One of the key trends is the use of semi-definite programming (SDP) and semi-infinite programming (SIP) to formulate and solve complex control problems. These optimization techniques are enabling the development of more sophisticated control algorithms that can handle uncertainties and disturbances more effectively. For instance, the incorporation of distributionally robust optimization (DRO) in model predictive control (MPC) is gaining traction, especially for systems with multiplicative noise, which is common in applications like mathematical finance.
Another notable trend is the application of fixed-time control techniques, particularly in the context of disturbance estimation and robust trajectory tracking. These methods aim to ensure that the system's response converges to the desired state within a predetermined time, regardless of initial conditions. This is particularly useful in applications like quadrotor control, where precise and timely trajectory tracking is critical.
The field is also seeing a growing interest in the control of hybrid systems, such as Takagi-Sugeno fuzzy Markovian jump systems (FMJSs). These systems require adaptive control strategies that can handle mode-switching dynamics and imperfect premise matching. The development of dynamic prediction optimization (DPO)-MPC frameworks is addressing these challenges by balancing control performance with computational efficiency.
Noteworthy Developments
Data-driven approximate output regulation of nonlinear systems: This work introduces a novel data-dependent SDP approach that significantly improves upon existing methods for linear systems, demonstrating the potential of data-driven techniques in nonlinear control.
Fixed-time Disturbance Observer-Based MPC for Quadrotor Control: The proposed fixed-time disturbance observer and robust MPC algorithm for quadrotor trajectory tracking is a notable advancement, ensuring convergence within a fixed time and demonstrating robustness against disturbances.
Model Predictive Control for T-S Fuzzy Markovian Jump Systems: The DPO-MPC framework for FMJSs is a significant contribution, providing a balance between control performance and computational burden, and ensuring recursive feasibility and mean-square stability.
Data-driven distributionally robust MPC for systems with multiplicative noise: The introduction of a semi-infinite SDP approach for DRMPC in systems with multiplicative noise is a pioneering effort, particularly relevant for applications in mathematical finance.
Robust MPC exploiting monotonicity properties: The proposed approach for robust MPC in monotone systems, leveraging mixed-monotonicity for non-monotone systems, demonstrates a less conservative strategy with potential applications in high-dimensional nonlinear systems.
