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Project Overview

The code in this project underlies the numerical experiments from the article
"Optimal Instrumental Variable Selection for Closed-loop Data-Driven Predictive Control" by
Rogier Dinkla1, Tom Oomen1,2, Sebastiaan P. Mulders1, and Jan-Willem van Wingerden1.

Affiliations:
1 Delft Center for Systems and Control, Faculty of Mechanical Engineering, Delft University of Technology, The Netherlands
2 Control Systems Technology Group, Department of Mechanical Engineering, Eindhoven University of Technology, The Netherlands

The above work derives an optimal instrumental variable (IV) for closed-loop Data-Driven Predictive Control (DDPC) that minimizes the asymptotic variance of employed system estimates that are asymptotically unbiased. Furthermore, approximations of optimal IVs are provided that either do or do not rely on controller knowledge.

The primary intent of the code is to facilitate reproduction of results presented in the paper, which stem from a batch of Monte Carlo simulations that perform a parameteric sweep over a range of innovation noise variances (Re). The code can furthermore be used to generate results from a wider variety of systems and initial controllers, and perform batches of Monte Carlo simulations that sweep over the past window length (p) or number of Hankel data matrix columns (N). Since the analysis of results that were not presented in the paper is not the primary purpose of this repository, manual adjustments may be needed to visualize other data.

Main Components

1. Monte Carlo Simulation Engine (src/MC_sim/)

  • main_MC.m — Orchestrates a full Monte Carlo run: generates data, computes IVs, builds predictors, runs closed-loop simulation
  • CaseDefinitions.m — Defines 17 controllers (14 types of IV-DDPC + 3 benchmarks)
  • IV_4_DDPC.m — class to efficiently store the similarly structured IVs
  • get_Z.m — Creates IV matrices for all 14 IV-DDPC controllers.
  • get_Lf_Cz.m — Generates (estimates of) the matrix $L_f$, and uses this to generate an SPC-type of controller by means of Lf_2_SPC.m.
  • get_solver.m — function to find how the optimal input from unconstrained SPC depends on $L_f$, past IO data, and future reference signals.
  • approx_IV_no_controller_info.m, approx_IV_controller_info.m — Respectively Algorithms 1 and 2 from the paper. Used to approximate optimal IVs.

2. Simulation Entry Points (src/)

  • main.m — Single Monte Carlo simulation with configurable parameters (Re, N, p, f, seed, sys, ...).
  • main_dRe.m — Batch experiments sweeping innovation noise variance (Re). Single MC runs performed by run_Re.m.
  • main_dp.m — Batch experiments sweeping past horizon length (p). Single MC runs performed by run_p.m.
  • main_dN.m — Batch experiments sweeping number of Hankel matrix columns (N). Single MC runs performed by run_N.m.

3. Plant Models & Controllers (src/systems/)

  • get_sys_info.m — Route to the selected plant model & initial controller based on sys parameter
  • init_sims.m — Initialize simulation: select plant model and initial controller using get_sys_info.m and construct initial closed-loop system.
  • tune_Cz0.m — Tune initial controller Cz0 via H-infinity mixed-sensitivity design
  • Details of the system configurations are available in directories that are named after the used plant model:
    sys Plant/Directory Initial Controller (Cz0) Notes
    1 Landau1995 [24,25] Cz0_Landau1995_D0.mat Results presented in paper.
    4th-order system with input-output delay,
    5th order Cz0 without direct feedthrough
    2 Landau1995 [24,25] Cz0_Landau1995_D0_n50.mat 50th order Cz0 without direct feedthrough
    3 Landau1995 [24,25] Cz0_Landau1995.mat 5th order Cz0 with direct feedthrough ($D_c$ ≠ 0)
    4 Bemporad2002 [30] Cz0_Bemporad2002.mat 2nd-order unstable system, 4th order Cz0
    5 Favoreel1999 [31] Cz0_Favoreel1999.mat Marginally stable 5th-order system, 7th order Cz0
    6 Wang2023 [6] Cz0_Wang2023.mat Unstable 3rd-order plant, 5th order Cz0
    7 Wang2023 [6] Cz0_Wang2023_provided.mat Unstable 3rd-order plant, Cz0 of 2nd order provided by authors and .mat file generated from model_Wang2023.m

4. Data Processing (src/processing/)

  • main_processing.m — Aggregates raw Monte Carlo batch results into summary statistics, saves these to processed_data.mat files.
  • process_dX.m — Batch aggregation logic (computes m1, mLf, m4 metrics across seeds)
  • util_fun/ — Helper functions for processing

5. Visualization (src/plot_figs/)

  • plot_figs_paper.m — Generates the figures from the paper.
  • Fig_IV_approx.m, Fig_Lf_estimates.m, Fig_prediction_quality.m, Fig_sim_example.m — Functions to help plot individual figures.

6. Utility Functions (src/util/)

  • blk_toeplitz_mean.m — Performs averaging over block-diagonals corresponding to a block-Toeplitz structure.
  • make_Hankel.m — Hankel matrix construction from data.
  • get_subdir1.m — Helps navigation to the right data directory. Called by main_processing.m.
  • make_blk_tril.m — Block-lower-triangularize provided matrix.
  • make_blk_tril_toeplitz.m — Block Toeplitz/triangular matrix operations.
  • make_ext_ctrb.m, make_ext_obsv.m — Extended controllability/observability matrices.
  • make_reference.m — Replicates reference from [25]
  • plant2ABCDK.m — Gets A, B, C, D, and K matrices from the specified plant.
  • ss2lag.m — Computes the lag of a state-space system.

Simulated Controllers (17 cases)

ID Type Description
iv1 Baseline Open-loop IV
iv2a, iv2b, iv2c Optimal (exact) Optimal IV with 0, 1, 2 refinement iterations
iv3a, iv3c LCF-IV [6] IV based on the left coprime factorization
iv4a, iv4b, iv4c Approx. (no Cz0 info) Approximated optimal IV without controller knowledge
iv5a, iv5b, iv5c Approx. (with Cz0 info) Approximated optimal IV with controller knowledge
iv6a, iv6c Reference-based Future reference as IV
CLSPC Benchmark Standard closed-loop SPC
actLf Oracle True transfer matrix (upper bound)
TrPred Transient Transient predictor

IVs with "b" suffix incorporate (approximations of) future denoised outputs $\tilde{Y}_{\mathrm{f}}$
IVs with "c" suffix are variants that apply 2SLS (two-stage least squares) to the IV preceding it in the table.

License

This code is released under the MIT License (see LICENSE.md).

References

[6] Y. Wang, Y. Qiu, M. Sader, D. Huang, and C. Shang, "Data-Driven Predictive Control Using Closed-Loop Data: An Instrumental Variable Approach," IEEE Control Systems Letters, vol. 7, pp. 3639–3644, 2023, doi: 10.1109/LCSYS.2023.3340444.
[24] I. D. Landau, D. Rey, A. Karimi, A. Voda, and A. Franco, "A Flexible Transmission System as a Benchmark for Robust Digital Control," European Journal of Control, vol. 1, no. 2, pp. 77–96, Jan. 1995, doi: 10.1016/S0947-3580(95)70011-5.
[25] A. Chiuso, M. Fabris, V. Breschi, and S. Formentin, "Harnessing uncertainty for a separation principle in direct data-driven predictive control," Automatica, vol. 173, p. 112070, Mar. 2025, doi: 10.1016/j.automatica.2024.112070.
[30] A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos, "The explicit linear quadratic regulator for constrained systems," Automatica, vol. 38, no. 1, pp. 3–20, Jan. 2002, doi: 10.1016/S0005-1098(01)00174-1.
[31] W. Favoreel, B. De Moor, M. Gevers, and P. Van Overschee, "Closed-Loop Model-Free Subspace-Based LQG-Design," in Proceedings of the Mediterranean Conference on Control and Automation, Jan. 1999.