a-separation-principle-in-d3pc

Code reproducing the numerical results in Section 7 of the research article "Harnessing Uncertainty for a Separation Principle in Direct Data-Driven Predictive Control"

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Authors:

• Prof. Alessandro Chiuso*, University of Padua, alessandro.chiuso@unipd.it
• Dr. Marco Fabris**, University of Padua, marco.fabris.1@unipd.it
• Prof. Valentina Breschi, Eindhoven University of Technology, v.breschi@tue.nl
• Prof. Simone Formentin, Politecnico di Milano, simone.formentin@polimi.it

* A. Chiuso is the main algorithm developer of these MatLab scripts.

** M. Fabris is the main software developer of these MatLab scripts.

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Research Article submitted to the journal "Automatica", ID: 23-1641

Publication history:

• first submission: December 23rd, 2023
• second submission: August 2nd, 2024
• accepted on: October 3rd, 2024
• last submission: October 31st, 2024

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Abstract:

Model Predictive Control (MPC) is a powerful method for complex system regulation, but its reliance on an accurate model poses many limitations in real-world applications. Data-driven predictive control (DDPC) aims at overcoming this limitation, by relying on historical data to provide information on the plant to be controlled. In this work, we present a unified stochastic framework for direct DDPC, where control actions are obtained by optimizing the Final Control Error (FCE), which is directly computed from available data only and automatically weighs the impact of uncertainty on the control objective. Our framework allows us to establish a separation principle for Predictive Control, elucidating the role that predictive models and their uncertainty play in DDPC. Moreover, it generalizes existing DDPC methods, like regularized Data-enabled Predictive Control (DeePC) and $\gamma$-DDPC, providing a path toward noise-tolerant data-based control with rigorous optimality guarantees. The theoretical investigation is complemented by a series of numerical case studies, revealing that the proposed method consistently outperforms or, at worst, matches existing techniques without requiring tuning regularization parameters as other methods do.

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This script and the remaining functions are documented at the beginning of each code fragment. Language used: MatLab (version 2021a).

--- Description of the most relevant files of this repository ---

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(1) MAIN.m

This script starts the numerical simulations presented in this paper, based on a Monte Carlo experiments to test the compared control methods over closed-loop dynamics. Some parameters in it must be selected by the user. Important: to replicate the same results shown in Figures 1 and 2 one has to set sys.kind_of_setup = 0 first (this action reproduces Setups 1 and 2) and then set sys.kind_of_setup = 1 (this action reproduces Setup 3).

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(2) model_selection.m

This subroutine is used to select models, tracking references and optimization parameters. Important: the output reference in this file has to be manually constructed in order to reproduce our numerical results. So, make sure to change as desired any time MAIN.m is launched.

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(3) data_generation.m

This function has two purposes:

• compute and adjust some system parameters;
• generate synthetic data.

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(4) rho_selection.m

This function selects the model order $\hat{\rho}$ at each Monte Carlo run, according to some criterion (e.g. the corrected AIC).

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(5) outerMC.m

This subroutine constitutes the main body of each Monte Carlo run for these simulations.

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(6) innerMC_grids.m

This subroutine constitutes the main body of each Monte Carlo run for the offline grid optimization of regularization parameters.

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(7) cl.m

This function executes the closed-loop dynamics.

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(8) dynamics.m, sys_noise.m

These functions execute the open-loop dynamics or single updates by possibily adding process and/or output noise.

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(9) ol_XXXXX.m / cvx_sol.m

The files ol_XXXXX.m encode the compared predictive control strategies. All these methods can be categorized as data-drive predictive control techniques, except for ol_MPC.m. The latter is a Kalman-based oracle to show the best performance pretending to know the "true" model. Alternatively, one may run cvx_sol.m to solve constrained optimization problems through CVX. Caution: cvx_sol.m could be very inefficient in certain situations. Also, this piece of code has not been sufficiently tested by the authors.

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(10) dpc_ini.m

This function

• provides the first initialization of the initial trajectories for input, output and state;
• it resets parameters used by a Kalman filter to compute run ol_MPC.m;
• it determines the dimensions for partitioning Hankel matrices in gamma-DDPC approaches.

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(11) pbsidopt_ini.m

Check out this subroutine if you wish to dive deep in the computations of the proposed FCE approach. Here the regularized in (37c) is constructed step by step.

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(12) reg_weight.m

This subroutine fulfills a similar task with respect to pbsidopt_ini.m at computing the regularization term according to the approximation discussed in Theorem 3.

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(13) tuning.m, SD_2.m, SD_3.m

These functions are used in the online tuning for beta2 and beta3 of gamma-DDPC approaches.

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(14) hist_rho.m, my_boxplot.m, weird_trajectories.m, track_err_fig3.m

These pieces of code are used to depict Figures 1 and 2.

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(15) table_exe_time.m

This script provides the values in Table 1.

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