Equivalence of SS-based MPC and ARX-based MPC

TL;DR

This paper demonstrates the theoretical equivalence between state-space and ARX-based MPC models, introducing a first-principle approach to interpret and transform models for improved control performance.

Contribution

It proposes a first-principle-based paradigm for ARX model identification and introduces new SS-to-ARX transformations, linking interpretability with model adaptability in MPC.

Findings

SS and ARX models can be transformed into each other with specific methods.

Choosing ARX model order based on process noise improves closed-loop control.

Transformations' robustness to noise varies across methods.

Abstract

Two kinds of control-oriented models used in MPC are the state-space (SS) model and the input-output model (such as the ARX model). The SS model has interpretability when obtained from the modeling paradigm, and the ARX model is black-box but adaptable. This paper aims to introduce interpretability into ARX models, thereby proposing a first-principle-based modeling paradigm for acquiring control-oriented ARX models, as an alternative to the existing data-driven ARX identification paradigm. That is, first to obtain interpretative SS models via linearizing the first-principle-based models at interesting points and then to transform interpretative SS models into their equivalent ARX models via the SS-to-ARX transformations. This paper presents the Cayley-Hamilton, Observer-Theory, and Kalman Filter based SS-to-ARX transformations, further showing that choosing the ARX model order should depend on the process noise to achieve a good closed-loop performance rather than the fitting criteria in data-driven ARX identification paradigm. An AFTI-16 MPC example is used to illustrate the equivalence of SS-based MPC and ARX-based MPC problems and to investigate the robustness of different SS-to-ARX transformations to noise.