Recursively Feasible Stochastic Model Predictive Control for Time-Varying Linear Systems Subject to Unbounded Disturbances (2410.11107v1)

Abstract: Model predictive control solves a constrained optimization problem online in order to compute an implicit closed-loop control policy. Recursive feasibility -- guaranteeing that the optimal control problem will have a solution at every time step -- is an important property to guarantee the success of any model predictive control approach. However, recursive feasibility is difficult to establish in a stochastic setting and, in particular, in the presence of disturbances having unbounded support (e.g., Gaussian noise). The problem is further exacerbated for time-varying systems, in which case recursive feasibility must be established also in a robust sense, over all possible future time-varying parameter values, as well as in a stochastic sense, over all potential disturbance realizations. This work presents a method for ensuring the recursive feasibility of a convex, affine-feedback stochastic model predictive control problem formulation for systems with time-varying system matrices and unbounded disturbances using ideas from covariance steering stochastic model predictive control. It is additionally shown that the proposed approach ensures the closed-loop operation of the system will satisfy the desired chance constraints in practice, and that the stochastic model predictive control problem may be formulated as a convex program so that it may be efficiently solved in real-time.

• The paper introduces a covariance steering-based SMPC that ensures recursive feasibility with terminal set and covariance constraints.
• It converts the stochastic control problem into a convex formulation enabling efficient real-time solutions for LTV systems.
• Numerical simulations on a kinematic bicycle model validate its robust performance under unbounded disturbances.

Overview of Recursive Feasibility Stochastic Model Predictive Control

The paper "Recursively Feasible Stochastic Model Predictive Control for Time-Varying Linear Systems Subject to Unbounded Disturbances" introduces a robust approach for ensuring recursive feasibility in stochastic model predictive control (SMPC) of linear time-varying (LTV) systems. This research addresses critical challenges related to stochastic disturbances with unbounded support and variability in system parameters.

Technical Contributions

The authors propose a novel method for maintaining recursive feasibility in a convex, affine-feedback SMPC framework for systems influenced by unbounded disturbances, such as Gaussian noise. Recursive feasibility is challenging to ensure due to the stochastic nature of disturbances and variations in system parameters. The authors tackle this by employing a covariance steering approach, which has shown success in finite-horizon optimal control problems.

Their method centers on ensuring that chance constraints are respected during system operation. They leverage terminal constraint sets and covariance matrices to provide guarantees about the feasibility of control actions at each step.

Theoretical Implications

1. Recursive Feasibility: The authors define sufficient conditions for recursive feasibility using a terminal set and covariance constraints. They ensure that these constraints can be computed effectively, thereby allowing the SMPC approach to handle unbounded disturbances reliably.
2. Convex Formulation: By employing an alternative change of variables, the SMPC problem is cast into a convex form, allowing for efficient real-time solutions in practice, which is particularly beneficial when dealing with time-varying dynamics in LTV systems.
3. Adaptive Control: The feedback control policy is fine-tuned through online optimization, handling variations in system matrices and disturbances effectively.

Practical Implications

The proposed method holds substantial promise for practical applications in scenarios where system parameters are not static, such as autonomous vehicle navigation and aerospace control systems. The ability to maintain feasibility under stochastic disturbances implies broader applicability in real-time control systems where computational efficiency and reliability are paramount.

Numerical Validation

The paper validates its theoretical claims through numerical simulations involving a kinematic bicycle model for vehicle lateral control. The results showcase the benefits of robustly designed terminal constraints compared to nominal or absent terminal constraints, highlighting the method's reliability in maintaining system constraints.

Future Developments

The approach establishes a foundation for further exploration into adaptive and robust control strategies in stochastic environments. Potential expansions could involve:

• Extending the method to nonlinear systems
• Integrating learning-based components to adaptively tune model predictions and improve performance over time
• Exploring decentralized control mechanisms for multi-agent systems

Conclusion

By addressing the gap in extending recursive feasibility guarantees to LTV systems with unbounded disturbances, this paper contributes significantly to the field of SMPC. The development of a covariance steering-based SMPC approach presents a practical, computationally efficient solution adaptable to a variety of applications spanning robotics, autonomous vehicles, and more. This work paves the way for future research into more complex system dynamics and integration with data-driven models.