A framework for randomized time-splitting in linear-quadratic optimal control - Veldman, Daniel; Zuazua Iriondo, Enrique (Numerische Mathematik, 2022)
Inspired by the successes of stochastic algorithms in the training of deep neural networks and the simulation of interacting particle systems, we propose and analyze a framework for randomized time-splitting in linear-quadratic optimal control. In our proposed framework, the linear dynamics of the original problem is replaced by a randomized dynamics. To obtain the randomized dynamics, the system matrix is split into simpler submatrices and the time interval of interest is split into subintervals. The randomized dynamics is then found by selecting randomly one or more submatrices in each subinterval. We show that the dynamics, the minimal values of the cost functional, and the optimal control obtained with the proposed randomized time-splitting method converge in expectation to their analogues in the original problem when the time grid is refined. The derived convergence rates are validated in several numerical experiments. Our numerical results also indicate that the proposed method can lead to a reduction in computational cost for the simulation and optimal control of large-scale linear dynamical systems.
Keywords: Random Batch Method; Operator Splitting; Optimal Control; Model Predictive Control

Local Stability and Convergence of Unconstrained Model Predictive Control - Veldman, Daniel; Zuazua Iriondo, Enrique (2024)
The local stability and convergence for Model Predictive Control (MPC) of unconstrained nonlinear dynamics based on a linear time-invariant plant model is studied. Based on the long-time behavior of the solution of the Riccati Differential Equation (RDE), explicit error estimates are derived that clearly demonstrate the influence of the two critical parameters in MPC: the prediction horizon T and the control horizon τ. In particular, if the MPC-controller has access to an exact (linear) plant model, the MPC-controls and the corresponding optimal state trajectories converge exponentially to the solution of an infinite-horizon optimal control problem when T-τ → ∞. When the difference between the linear model and the nonlinear plant is sufficiently small in a neighborhood of the origin, the MPC strategy is locally stabilizing and the influence of modeling errors can be reduced by choosing the control horizon τ smaller. The obtained convergence rates are validated in numerical simulations.
https://cris.fau.de/converis/publicweb/Publication/276378997
Keywords: Convergence; Model Predictive Control; Receding Horizon Control; Stability

Stability and Convergence of a Randomized Model Predictive Control Strategy - Veldman, Daniel; Borkowski, Alexandra; Zuazua, Enrique (IEEE Transactions on Automatic Control, 2024)
RBM-MPC is a computationally efficient variant of Model Predictive Control (MPC) in which the Random Batch Method (RBM) is used to speed up the finite-horizon optimal control problems at each iteration. In this paper, stability and convergence estimates are derived for RBM-MPC of unconstrained linear systems. The obtained estimates are validated in a numerical example that also shows a clear computational advantage of RBM-MPC.
CRIS-Team Scopus Importer:2024-03-22
Keywords: Approximation algorithms; Computational efficiency; Convergence; Error Estimates; Model Predictive Control; Numerical stability; Random Batch Method; Read only memory; Receding Horizon Control; Stability; Stability criteria; Vectors

Stability and Convergence of a Randomized Model Predictive Control Strategy - Veldman, Daniel; Borkowski, Alexandra; Zuazua Iriondo, Enrique (2025)
This paper is concerned with a combination of Random Batch Methods (RBMs) and Model Predictive Control (MPC) called RBM-MPC. In RBM-MPC, the RBM is used to speed up the solution of the finite horizon optimal control problems that need to be solved in MPC. We analyze our algorithm in the linear quadratic setting and obtain explicit error estimates that characterize the stability and convergence of the proposed method. The obtained estimates are validated in numerical experiments that also demonstrate the effectiveness of RBM-MPC.<br
https://cris.fau.de/converis/publicweb/Publication/285034893
Keywords: Convergence; Model Predictive Control; Random Batch Method; Receding Horizon Control; Stability