Advances in Model Predictive Control Under Uncertainty

Abstract

Model Predictive Control (MPC) is an advanced control method that utilizes a model to make predictions about the state evolution for the controlled system, as well as an optimization solver to compute an optimal future control trajectory that satisfies state and control constraints. Inaccuracies in model predictions, caused by modelling errors or external disturbances, significantly hinder the performance of MPC and potentially lead to constraint violations. To address these challenges, Robust MPC (RMPC) explicitly accounts for uncertainties in the model, enhancing the control reliability. Tube-based MPC (TMPC) is a well-known RMPC method, where a nominal MPC generates a reference state trajectory, a tube (i.e., a trajectory of model uncertainties centered around the reference trajectory), and an ancillary control law that ensures the actual states of the system remain within this tube —following the reference trajectory, despite uncertainties.

This thesis presents a critical analysis of current state-of-the-art MPC methods, and proposes novel theoretical developments, architectures, and extensions for effective and computationally efficient handling of uncertainties via MPC . These contributions are rigorously supported by formal proofs. Furthermore, the proposed control frameworks are systematically evaluated through dedicated computer-based simulations, with comparisons drawn against existing methods in terms of optimality, robustness, and computational complexity.

The thesis begins with introducing State-Dependent Dynamic Tube-based MPC (SDD-TMPC), an extension of TMPC designed to more effectively handle the variability of model uncertainties and environmental disturbances. By leveraging available information about state-dependent uncertainties, SDD-TMPC enhances optimality and reduces risks of infeasibility, while maintaining the same level of robustness as TMPC. Although SDD-TMPC demonstrates applicability to systems with varying uncertainties across the state space, its practical implementation is limited by high computational demands.

To mitigate this limitation, Approximate State-Dependent Dynamic Tube-based MPC (ASDDTMPC) is developed. This approach employs Spiking Neural Network (SNN) to approximate the behavior of SDD-TMPC. SNNs were selected for their event-driven processing and biologically inspired efficiency, offering significant advantages for low-power, real-time control. Recognizing that this approximation introduces additional uncertainties, and thus the risk of insufficient robustness to uncertainties, the SDD-TMPC framework is extended to incorporate these approximation errors as additional state-dependent disturbances, thereby preserving robustness. The reduced computational requirements of spiking neural networks enable implementation on resource-constrained platforms, such as small-scale robotic platforms.

Next, the Parent-Child MPC (PC-MPC) architecture is proposed to further reduce computational complexity across a wide range of MPC frameworks. Compatible with both tube-based and deterministic MPC, the PC-MPC architecture decomposes the optimization problem into two linked problems: The Parent MPC (P-MPC) addresses long-term stability and constraint satisfaction, while the Child MPC (C-MPC) focuses on short-term stability and disturbance rejection. P-MPC communicates additional constraints to C-MPC, which determines and executes control strategies. This hierarchical approach, which guarantees robustness and stability, is extendable to systems with complex dynamics and large scales. This is done by incorporating additional Parent layers to further manage computational complexity in such systems.

While robustness is critical, there are environments where maintaining strict constraint satisfaction is infeasible due to the nature and extent of uncertainties. To address this, a new theoretical framework called Fuzzy-Logic-based MPC (FLMPC) is developed, particularly suited for controlling multi-agent systems with imperfect environmental perception operating in unknown environments. FLMPC uses fuzzy vectors to model uncertainties, where each element—being a fuzzy variable—represents the degree to which a region exhibits properties such as “dangerous” or “certain”. Fuzzy maps are constructed by grouping fuzzy vectors. FLMPC performs fuzzy optimization to compute optimal trajectories for all agents, demonstrating superior performance and reduced computational complexity compared to state-of-the-art control methods. This efficiency is achieved by enabling the execution of computationally intensive tasks of fuzzy map generation outside the real-time optimization loop. This is made possible by inheriting the fundamental strength of fuzzy logic, particularly its ability to handle uncertainties over a continuum of values, rather than discrete thresholds. This allows for reliable decision-making based on fuzzy maps within a flexible computational window, rather than being restricted to a specific time step.

The inherent limitation of finite prediction horizons in MPC poses a challenge for exploring tasks in large-scale environments. To improve scalability, a bi-level FLMPC framework—leveraging the PC-MPC architecture in the context of FLMPC — is introduced, with potential for extension to multi-level hierarchies. The Parent Fuzzy-Logic-based MPC (P-FLMPC) formulates a global plan using comprehensive environmental knowledge, while the Child Fuzzy-Logic-based MPC (C-FLMPC) focuses on local enhancement and execution of the plan, retaining flexibility for real-time adaptation.