Journal of Computational Applied Mechanics

From Data to Stability: A Novel Approach for Controlling Unknown Linear Time-Invariant Systems with Performance Enhancement

Document Type : Research Paper

Authors

Department of Computer Systems, Tallinn University of Technology, Tallinn, Estonia

Abstract

A novel data-driven control methodology is introduced in this paper, specifically designed for unknown linear time-invariant systems. Schur stability is established through the application of Linear Matrix Inequality (LMI) conditions, and system performance is improved by leveraging the concept of D-stability. Stability and performance are ensured by incorporating LMI features, with reliance solely on a finite set of collected data, eliminating the necessity for system model identification. Hence, the original performance mapping problem undergoes a transformation into a stability issue, incorporating modified system matrices. Then, the stability condition is formulated within the framework of LMI. The effectiveness of our approach is exemplified through two specific examples, highlighting the significant and impactful results obtained. These examples serve to showcase the practical application and outcomes of our methodology within the defined scope, providing a clear demonstration of its performance and efficacy in addressing relevant scenarios.

Keywords

Main Subjects

[1]          C. D. Persis, P. Tesi, Formulas for Data-Driven Control: Stabilization, Optimality, and Robustness, IEEE Transactions on Automatic Control, Vol. 65, No. 3, pp. 909-924, 2020.
[2]          J. C. Willems, P. Rapisarda, I. Markovsky, B. L. M. De Moor, A note on persistency of excitation, Systems & Control Letters, Vol. 54, No. 4, pp. 325-329, 2005/04/01/, 2005.
[3]          F. A. Pellegrino, F. Blanchini, G. Fenu, E. Salvato, Data-driven dynamic relatively optimal control, European Journal of Control, Vol. 74, pp. 100839, 2023/11/01/, 2023.
[4]          B. Pang, Z. P. Jiang, A Data-driven Approach for Constrained Infinite-Horizon Linear Quadratic Regulation, in Proceeding of, 6010-6015.
[5]          F. Dörfler, P. Tesi, C. D. Persis, On the Certainty-Equivalence Approach to Direct Data-Driven LQR Design, IEEE Transactions on Automatic Control, Vol. 68, No. 12, pp. 7989-7996, 2023.
[6]          M. Rotulo, C. D. Persis, P. Tesi, Online Data-driven Stabilization of Switched Linear Systems, in Proceeding of, 300-305.
[7]          J. Berberich, A. Koch, C. W. Scherer, F. Allgöwer, Robust data-driven state-feedback design, in Proceeding of, 1532-1538.
[8]          A. Bisoffi, C. De Persis, P. Tesi, Trade-offs in learning controllers from noisy data, Systems & Control Letters, Vol. 154, pp. 104985, 2021/08/01/, 2021.
[9]          B. Nortmann, T. Mylvaganam, Data-Driven Control of Linear Time-Varying Systems, in Proceeding of, 3939-3944.
[10]        B. Nortmann, T. Mylvaganam, Direct Data-Driven Control of Linear Time-Varying Systems, IEEE Transactions on Automatic Control, Vol. 68, No. 8, pp. 4888-4895, 2023.
[11]        J. Berberich, J. Köhler, M. A. Müller, F. Allgöwer, Robust Constraint Satisfaction in Data-Driven MPC, in Proceeding of, 1260-1267.
[12]        J. Berberich, J. Köhler, M. A. Müller, F. Allgöwer, Data-Driven Model Predictive Control With Stability and Robustness Guarantees, IEEE Transactions on Automatic Control, Vol. 66, No. 4, pp. 1702-1717, 2021.
[13]        A. Alanwar, Y. Stürz, K. H. Johansson, Robust data-driven predictive control using reachability analysis, European Journal of Control, Vol. 68, pp. 100666, 2022/11/01/, 2022.
[14]        L. Deng, Z. Shu, T. Chen, Event-Triggered Robust MPC with Terminal Inequality Constraints: A Data-Driven Approach, IEEE Transactions on Automatic Control, pp. 1-8, 2024.
[15]        X. Dai, C. D. Persis, N. Monshizadeh, P. Tesi, Data-Driven Control of Nonlinear Systems from Input-Output Data, in Proceeding of, 1613-1618.
[16]        S. P. Bhattacharyya, L. H. Keel, ROBUST CONTROL: THE PARAMETRIC APPROACH,  in: A. Ichikawa, K. Furuta, Advances in Control Education 1994, Eds., pp. 49-52, Oxford: Pergamon, 1995.
[17]        M. Ghorbani, A. Tepljakov, E. Petlenkov, Robust D-Stability Analysis of Fractional-Order Controllers, in Proceeding of, 3871-3876.
[18]        D. Peaucelle, D. Arzelier, O. Bachelier, J. Bernussou, A new robust D-stability condition for real convex polytopic uncertainty, Systems & Control Letters, Vol. 40, No. 1, pp. 21-30, 2000/05/15/, 2000.
[19]        F.-H. Hsiao, J.-D. Hwang, S.-P. Pan, D-stability analysis for discrete uncertain time-delay systems, Applied Mathematics Letters, Vol. 11, No. 2, pp. 109-114, 1998/03/01/, 1998.
[20]        J. Zrida, M. H. Bouazizi, Exact robust D-stability analysis for linear dynamical systems with polynomial parameter perturbation, International Journal of Control, Vol. 95, No. 11, pp. 2885-2899, 2022/11/02, 2022.
[21]        A. Zemouche, A. Alessandri, A new LMI condition for decentralized observer-based control of linear systems with nonlinear interconnections, in Proceeding of, 3125-3130.
[22]        G.-R. Duan, H.-H. Yu, 2013, LMIs in control systems: analysis, design and applications, CRC press,
[23]        S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, 1994, Linear matrix inequalities in system and control theory, SIAM,
[24]        D. Q. Mayne, E. C. Kerrigan, E. J. van Wyk, P. Falugi, Tube-based robust nonlinear model predictive control, International Journal of Robust and Nonlinear Control, Vol. 21, No. 11, pp. 1341-1353, 2011.
Journal of Computational Applied Mechanics
Volume 55, Issue 3
July 2024
Pages 451-461
  • Receive Date: 01 December 2023
  • Revise Date: 23 February 2024
  • Accept Date: 23 February 2024