### Abstract

We study the performance of stochastic predictive control (SPC) for linear systems with a quadratic performance index and additive and multiplicative uncertainties. Under a finite support assumption, the problem can be cast as a finite-dimensional quadratic program, but the problem becomes quickly intractable as the problem size grows exponentially in the horizon length. SPC aims to compute approximate solutions by solving a sequence of problems with truncated prediction horizons and committing the solution in a receding-horizon fashion. While this approach is widely used in practice, its performance relative to the optimal solution is not well understood. This article reports for the first time a rigorous performance guarantee of SPC: under the standard stabilizability and detectability conditions, the dynamic regret of SPC is exponentially small in the prediction horizon length. Therefore, SPC can achieve near-optimal performance – the expected performance can be made arbitrarily close to the optimal solution – at a substantially reduced computational expense.

Publication

*arXiv preprint arXiv:2210.08599*

###### Postdoc in Mathematics and Computer Science Division

Sungho Shin is a postdoc in the Mathematics and Computer Science Division at Argonne National Laboratory. His research interests include model predictive control, optimization algorithms, and their applications to large-scale energy infrastructures (such as natural gas and power networks).

###### Postdoc in Statistics and ICSI

Sen Na is a postdoctoral scholar in the Statistics department and ICSI at UC Berkeley. His research interests broadly lie in the mathematical foundations of data science, with topics including high-dimensional statistics, graphical models, semiparametric models, optimal control, and large-scale and stochastic nonlinear optimization. He is also interested in applying machine learning methods in biology, neuroscience, and engineering.

###### Professor in Statistics and CAM

Mihai Anitescu is a Professor in the Statistics and CAM departments at the University of Chicago, and is also a senior computational mathematician in the Mathematics and Computer Science Division at Argonne. He works on a variety of topics on control, optimization, and computational statistics.