Compact Analysis of the Necessity of Padé Approximation for Delayed Continuous-time Models in LQR, H-Infinity and Root Locus Control Strategies

Öz

This paper presents a comprehensive analysis of the need for the Padé approximation for continuous-time models with delays, focusing on its critical role in addressing the control challenges posed by time delays. Time delays, often referred to as dead times, transport delays or time lags, are inherent in a wide range of industrial and engineering processes. These delays introduce phase shifts that degrade control performance by reducing control bandwidth and threatening the stability of closed-loop systems. Accurate modelling and compensation of these delays is essential to maintain system stability and ensure effective control. This paper highlights the difficulties that arise when using advanced control techniques such as root locus (RL), linear quadratic regulator (LQR) and H-infinity (H_∞) control in systems with delays. Representing delays in exponential form leads to an infinite number of state problems, complicating the design and analysis of controllers in such systems. To address these challenges, the Padé approximation is proposed as an effective method for approximating time delays with rational polynomials of appropriate order. This approach allows for more accurate simulation, system analysis and controller design, thereby mitigating the problems caused by delays. The paper also provides a detailed comparative analysis between the Padé approximation and Taylor polynomials, demonstrating the superiority of the former in achieving accurate delay modelling and control performance. The results show that the use of Padé approximation not only improves the accuracy of system models, but also improves the robustness and stability of control strategies such as RL, LQR, and H_∞. These results highlight the importance of the Padé approximation as a valuable tool in the design of delay-affected control systems, offering significant advantages for both theoretical and practical applications.

Anahtar Kelimeler

• Padé approximation
• Time delay systems
• Continuous-time models
• Infinite number of state problem
• Rational polynomial approximation
• Dead time compensation

Kaynakça

1. Abbasspour A, Sargolzaei A, Victorio M, Khoshavi N. 2020. A neural network-based approach for detection of time delay switch attack on networked control systems. Procedia Comput Sci, 168: 279-288.
2. Abdullah HN. 2021. An improvement in LQR controller design based on modified chaotic particle swarm optimization and model order reduction. Int J Intell Eng Syst, 14(1): 157-168.
3. Anh NT. 2020. Control an active suspension system by using PID and LQR controller. Int J Mech Prod Eng Res Dev, 10(3): 7003-7012.
4. Belhamel L, Buscarino A, Fortuna L, Xibilia MG. 2020. Delay independent stability control for commensurate multiple time-delay systems. IEEE Control Syst Lett, 5(4): 1249-1254.
5. Chen N, Zhang P, Dai J, Gui W. 2020. Estimating the state-of-charge of lithium-ion battery using an H-infinity observer based on electrochemical impedance model. IEEE Access, 8: 26872-26884.
6. Conca A, Naldi S, Ottaviani G, Sturmfels B. 2024. Taylor polynomials of rational functions. Acta Mathematica Vietnamica, 49(1): 19-37.
7. De Persis C, Tesi P. 2021. Low-complexity learning of linear quadratic regulators from noisy data. Automatica, 128: 109548.
8. Fridovich-Keil D, Ratner E, Peters L, Dragan AD, Tomlin CJ. 2020. Efficient iterative linear-quadratic approximations for nonlinear multi-player general-sum differential games. In 2020 IEEE Int Conf Robot Autom (ICRA), May 31- August 31, Paris, France, pp: 1475-1481.

9. Gluzman S. 2020. Padé and post-Padé approximations for critical phenomena. Symmetry, 12(10): 1600.
10. Handaya D, Fauziah R. 2021. Proportional-integral-derivative and linear quadratic regulator control of direct current motor position using multi-turn based on LabView. J Robot Control, 2(4): 332-336.
11. Hu J, Yu C, Zhou K. 2024. Padé approximations and irrationality measures on values of confluent hypergeometric functions. Mathematics, 12(16): 2516.
12. Kanokmedhakul Y, Bureerat S, Panagant N, Radpukdee T, Pholdee N, Yildiz AR. 2024. Metaheuristic-assisted complex H-infinity flight control tuning for the Hawkeye unmanned aerial vehicle: A comparative study. Expert Syst Appl, 248: 123428.
13. Khamies M, Magdy G, Ebeed M, Kamel S. 2021. A robust PID controller based on linear quadratic gaussian approach for improving frequency stability of power systems considering renewables. ISA Trans, 117: 118-138.
14. Li X, Yang X, Cao J. 2020. Event-triggered impulsive control for nonlinear delay systems. Automatica, 117: 108981.
15. Luyben WL. 2020. Liquid level control: Simplicity and complexity. J Process Control, 86: 57-64.
16. Maghfiroh H, Nizam M, Anwar M, Ma’Arif A. 2022. Improved LQR control using PSO optimization and Kalman filter estimator. IEEE Access, 10: 18330-18337.
17. Menezes EJ, Araújo AM. 2023. Wind turbine structural control using H-infinity methods. Eng Struct, 286: 116095.
18. Mondié S, Egorov A, Gomez MA. 2022. Lyapunov stability tests for linear time-delay systems. Annu Rev Control, 54: 68-80.
19. Pinheiro RF, Colón D. 2024. On the μ-analysis and synthesis for uncertain time-delay systems with Padé approximations. J Franklin Inst, 361(4): 106643.
20. Priyambodo TK, Dhewa OA, Susanto T. 2020. Model of linear quadratic regulator (LQR) control system in waypoint flight mission of flying wing UAV. J Telecommun Electron Comput Eng, 12(4): 43-49.
21. Pujol-Vazquez G, Mobayen S, Acho L. 2020. Robust control design to the furuta system under time delay measurement feedback and exogenous-based perturbation. Mathematics, 8(12): 2131.
22. Shangguan XC, Zhang CK, He Y, Jin L, Jiang L, Spencer JW, Wu M. 2020. Robust load frequency control for power system considering transmission delay and sampling period. IEEE Trans Ind Informatics, 17(8): 5292-5303.
23. Wei Y, Hu Y, Dai Y, Wang Y. 2016. A generalized Padé approximation of time delay operator. Int J Control Autom Syst, 14(1): 181-187.
24. Werth W, Faller L, Liechtenecker H, Ungermanns C. 2020. Low cost rapid control prototyping–a useful method in control engineering education. In 2020 43rd Int Conv Information, Commun Electron Technol (MIPRO), September 28- October 02, Opatija, Croatia, pp: 711-715.
25. Wu D, Chen Y, Yu C, Bai Y, Teo KL. 2023. Control parameterization approach to time-delay optimal control problems: A survey. J Ind Manag Optim, 19(5): 3750-3783.
26. Yang K, Tang X, Qin Y, Huang Y, Wang H, Pu H. 2021. Comparative study of trajectory tracking control for automated vehicles via model predictive control and robust H-infinity state feedback control. Chinese J Mech Eng, 34: 1-14.
27. Yang T, Bai Z, Li Z, Feng N, Chen L. 2021. Intelligent vehicle lateral control method based on feedforward+ predictive LQR algorithm. Actuators, 10(9): 228.
28. Zhang CK, Long F, He Y, Yao W, Jiang L, Wu M. 2020. A relaxed quadratic function negative-determination lemma and its application to time-delay systems. Automatica, 113: 108764.
29. Zhang JX, Xu KD, Wang QG. 2024. Prescribed performance tracking control of time-delay nonlinear systems with output constraints. IEEE/CAA J Autom Sin, 11(7): 1557-1565.
30. Zhou Y, Ahn S, Wang M, Hoogendoorn S. 2020. Stabilizing mixed vehicular platoons with connected automated vehicles: An H-infinity approach. Transp Res Part B Methodol, 132: 152-170.