Master Thesis Defense - Brener Gaspar Ferreira
Master Thesis defense of the Graduate Program in Electrical Engineering at UFMG
Abstract:
In recent decades, Nonlinear Model Predictive Control (NMPC) has become a robust and flexible approach for complex dynamic systems, especially in applications that require adherence to operational constraints and safe real-time behavior. Its ability to predict future system behavior based on dynamic models has made it essential for solving problems like reference tracking and interaction with challenging environments. However, despite its advantages, traditional NMPC faces significant limitations when applied to scenarios that require high sampling rates or involve non-convex domains, such as systems with obstacles. The need to solve complex optimization problems at every time step imposes a high computational cost, making it difficult to apply in practical embedded systems or systems with limited computational resources.
Building on tracking NMPC formulations, we adopt the core idea of introducing artificial variables into the optimal control problem. This strategy enables the incorporation of avoidance mechanisms into tracking NMPC, expanding the closed-loop system’s attraction region and ensuring recursive feasibility for references that vary piecewise continuously. In this formulation, control is obtained by solving a finite-horizon optimization problem that minimizes a cost functional with penalties for tracking errors and control effort. This problem includes the system’s dynamic constraints, limits on states and inputs, and can incorporate terminal conditions to ensure stability and feasibility. In scenarios involving obstacles, Control Barrier Functions (CBFs) are introduced to add constraints that avoid collisions and ensure system safety. While this formulation can handle real-time environmental complexity, the need to solve nonlinear optimization problems at every instant can make it impractical for real-world applications, especially in systems with limited hardware or environments that require fast and efficient responses.
To address these challenges, an explicit approach to NMPC (ENMPC) is proposed, eliminating the need for real-time optimization. This approach pre-computes the control laws offline as piecewise affine functions, using state-space partitioning to identify feasible regions and approximating the original multiparametric nonlinear programming (mp-NLP) problem with multiparametric quadratic programming (mp-QP) problems. As a result, each region of the state space is associated with a pre-calculated solution, which can be accessed efficiently at runtime. The methodology also integrates barrier functions to avoid obstacles by excluding infeasible regions during partitioning. Numerical tests on a differential drive robot have shown that ENMPC can track references safely and efficiently while avoiding obstacles, drastically reducing computation time compared to online NMPC. This solution provides a practical and effective alternative for applications requiring high performance in systems with computational constraints.
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