# Time Distributed Optimization for Model Predictive Control: Stability, Robustness, and Constraint Satisfaction

## TL;DR

This paper introduces a systems theoretic framework for analyzing time distributed optimization in model predictive control, focusing on stability, robustness, and constraint satisfaction, and extends theoretical results for real-time iteration schemes.

## Contribution

It provides a general stability analysis framework for time distributed optimization in MPC, extending existing theory for real-time iteration methods.

## Key findings

- The framework guarantees stability and constraint satisfaction under certain conditions.
- Numerical simulations validate the effectiveness of the proposed approach.
- The analysis enhances understanding of robustness in time distributed optimization schemes.

## Abstract

Time distributed optimization is an implementation strategy that can significantly reduce the computational burden of model predictive control by exploiting its robustness to incomplete optimization. When using this strategy, optimization iterations are distributed over time by maintaining a running solution estimate for the optimal control problem and updating it at each sampling instant. The resulting controller can be viewed as a dynamic compensator which is placed in closed-loop with the plant. This paper presents a general systems theoretic analysis framework for time distributed optimization. The coupled plant-optimizer system is analyzed using input-to-state stability concepts and sufficient conditions for stability and constraint satisfaction are derived. When applied to time distributed sequential quadratic programming, the framework significantly extends the existing theoretical analysis for the real-time iteration scheme. Numerical simulations are presented that demonstrate the effectiveness of the scheme.