Model-Based Predictive Control Under Uncertainty

Abstract

This thesis focuses on model-based predictive control under uncertainty, with a specific emphasis on systems affected by uncertainty characterized by worst-case geometric bounds and probability distributions. The research topic is motivated by the necessity for computationally efficient controllers tailored for uncertain dynamical systems, where safety of the controlled systems needs to be guaranteed to an acceptable level. A collection of novel algorithms addressing robust and stochastic model predictive control (MPC) problems has been reported. In contrast to nominal MPC, which is widely employed in many control scenarios, robust and stochastic MPC have encountered fewer real-world applications primarily due to their numerical and theoretical challenges. However, uncertainty is a critical factor in many control scenarios. In MPC community, uncertainty is primarily addressed through two approaches: robust MPC and stochastic MPC. Robust MPC guarantees that all possible future state and control trajectories adhere to constraints while minimizing the “worstcase” cost or some generalized costs. In contrast, stochastic MPC aims to minimize an expectation cost or some generalized costs while incorporating chance (probabilistic) constraints. This approach allows the controlled system to deviate from constraints to an acceptable extent. The primary focus of this thesis is dedicated to the design of these two classes of approaches. Although strictly speaking, robust and stochastic MPC belong to distinct categories, robust MPC tends to yield conservative control policies since it disregards the stochastic information about the bounded uncertainty under consideration, if such stochastic information is available. Additionally, due to the utilization of chance constraints, stochastic MPC is capable of handling stochastic uncertainty with unbounded support. On the other hand, addressing the expectation cost and chance constraints, along with analyzing the control-theoretic properties of the controlled systems, present greater challenges in stochastic MPC compared to robust MPC. Similar to nominal MPC, the key properties to examine when analyzing these MPC approaches include the stability, optimality, and constraint satisfaction of the closed-loop controlled systems. For scenarios where uncertainty is bounded and no further stochastic information is available, this thesis introduces a robust MPC controller using tubes, which is able to exponentially stabilize the controlled system. For scenarios where uncertainty is characterized by stochastic descriptions. Within this context, this thesis introduces two stochastic MPC controllers designed to stabilize the system while adhering to stage-wise (pointwise-in-time) chance constraints, commonly referred to as joint chance constraints in the stochastic MPC community. Subsequently, we investigate mission-wide (dynamic-joint) chance constraints over the state trajectory, which are, in generally, more meaningful in defining safety for engineering systems. In this regard, we offer a characterization of the exact solution to the mission-wide chance-constrained optimal control problems, and subsequently, we approximate these exact solutions using a stochastic MPC approach with shrinking time horizons.