Constraint Violation Probability Minimization for Norm-Constrained Linear Model Predictive Control

Abstract

Handling uncertainty in model predictive control comes with various challenges, especially when considering state constraints under uncertainty. Most methods focus on either the conservative approach of robustly accounting for uncertainty or allowing a small probability of constraint violation. In this work, we propose a linear model predictive control approach that minimizes the probability that linear state constraints are violated in the presence of additive uncertainty. This is achieved by first determining a set of inputs that minimize the probability of constraint violation. Then, this resulting set is used to define admissible inputs for the optimal control problem. Recursive feasibility is guaranteed and input-to-state stability is proved under assumptions. Numerical results illustrate the benefits of the proposed model predictive control approach.

Minimal Constraint Violation Probability in Model Predictive Control for Linear Systems

“…In contrast to the CVPM-MPC method presented in [5] and [6], the CVPM-MPC approach proposed in this work is more general. Here, we consider general linear constraints for a system with additive uncertainty, whereas [5], [6] are motivated by a vehicle collision avoidance scenario utilizing two dynamics and norm constraints.…”

“…There, the probability of violating a collision avoidance norm constraint in the first prediction step is minimized with CVPM-MPC. An extension of CVPM-MPC where the norm constraint is taken into account for more than one prediction step is presented in [6]. Unlike multiobjective MPC [33], [34], in CVPM-MPC, safety is not part of an objective trade-off, but safety is maximized first before other objectives are considered.…”

“…Unlike multiobjective MPC [33], [34], in CVPM-MPC, safety is not part of an objective trade-off, but safety is maximized first before other objectives are considered. However, the approach in [5] and [6] is designed particularly for obstacle avoidance, i.e., the approach is limited to norm constraints, and only uncertainty affecting the obstacle dynamics is considered.…”

“…This is achieved by guaranteeing constraint satisfaction whenever possible and ensuring minimal constraint violation probability whenever constraint satisfaction is not possible. We generalize, extend, and simplify the results of the CVPM-MPC method from [5] and [6]. In particular, we now replace norm constraints by general linear constraints.…”

“…Whereas RMPC and SMPC partially consider these requirements, both methods are impractical for safety-critical systems due to the above-mentioned issues, e.g., uncertainty bounds must be known in advance or recursive feasibility is not guaranteed in every situation. These considerations resulted in the development of MPC with constraint violation probability minimization (CVPM) [5], [6].…”