Disturbance rejection of non-minimum phase MIMO systems: An iterative tuning approach

Document Type : Research Article


1 Department of Electrical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

2 Department of Electrical and Computer Engineering, University of Zanjan, Zanjan, Iran


An iterative tuning method is presented to obtain the multi-input multi-output (MIMO) feedforward controller coefficients to improve disturbance rejection in non-minimum phase (NMP) MIMO systems. In the NMP systems, eliminating the effect of disturbances may cause instability and also can impose extra costs to control the entire system. For this purpose, a simple feedforward controller structure is proposed. The unknown variables of the feedforward controller are calculated using LMIs such that the H norm of the transfer function matrix from disturbance to output is minimized. By taking advantage of the frequency sampling techniques into account and using some iterative algorithms, a new tractable method is constructed to solve the problem. Also, a condition based on the right half plane (RHP) zero direction for the NMP system has been proposed to improve the disturbance rejection property of these systems. To obtain optimal coefficients, the algorithm is repeated several times to reach the best answer. The method employs convex technics and CVX software to perform calculations. The efficiency of the method is shown in various practical examples using different performance indicators such as integral of absolute error (IAE), integral of squared error (ISE), integral of time multiplied by absolute error (ITAE), integral of time multiplied by squared error (ITSE).


Main Subjects

[1] Q. Mei, J. She, Z.-T. Liu, Disturbance rejection and control system design based on a high-order equivalent-input-disturbance estimator, J. Franklin Inst., 358 (2021) 8736-8753.
[2] Y. Du, W. Cao, J. She, M. Wu, M. Fang, Disturbance rejection via feedforward compensation using an enhanced equivalent-input-disturbance approach, J. Franklin Inst., 357 (2020) 10977-10996.
[3] I.M.L.Pataro, J.D. Gil, M.V.A. da Costa, J.L. Guzmán, M. Berenguel, A stabilizing predictive controller with implicit feedforward compensation for stable and time-delayed systems, J. Process Control, 115 (2022) 12-26.
[4]  F. García-añas, J.L. Guzmán, F. Rodríguez, M. Berenguel, T. Hägglund, Experimental evaluation of feedforward tuning rules, Control Eng. Pract., 114 (2021) 104877.
[5]  W.L. Luyben, Comparison of additive and multiplicative feedforward control, J. Process Control, 111 (2022)  1-7.
[6] Y. Hamada, Flight test results of disturbance attenuation using preview feedforward compensation, IFAC-PapersOnLine, 50 (2017) 14188-14193.
[7] D. Carnevale, S. Galeani, M. Sassano, Transient optimization in output regulation via feedforward selection and regulator state initialization, IFACPapersOnLine, 50 (2017) 2405-8963.
[8]  Y. Du, W. Cao, J. She, M. Wu, M. Fang, Disturbance rejection via feedforward compensation using an enhanced equivalent-input-disturbance approach,  J. Franklin Inst., 357 (2020) 10977-10996.
[9]  S. Liu, G. Shi, and D. Li, Active Disturbance Rejection Control Based on Feedforward Inverse System for Turbofan Engines, IFACPapersOnLine, 54 (2021) 376-381.
[10] D. Tena, I. Peñarrocha-Alós, R. Sanchis, Performance, robustness and noise amplification trade-offs in Disturbance Observer Control design, Eur. J. Control, 65 (2022) 100630.
[11] Y. Ashida, M. Obika, Performance, Data-driven Design of a Feed-forward Controller for Rejecting Measurable Disturbance, Comput. Aided Chem. Eng, 49 (2022) 415-420.
[12] S. Wang, Z. Wu, Z.-G. Wu, Performance, Trajectory tracking and disturbance rejection control of random linear systems, Comput. J. Franklin Inst., 359 (2022) 4433-4448.
[13] G. Shi, S. Liu, D. Li, Y. Ding, Y.Q. Chen, A Controller Synthesis Method to Achieve Independent Reference Tracking Performance and Disturbance Rejection Performance, ACS Omega, 7 (2022) 16164–16186.
[14] Z. Wu, Y. Liu, D. Li, Y.Q. Chen, Multivariable active disturbance rejection control for compression liquid chiller system, Energy, in press (2022) 125344.
[15] Z. Xu, C. Sun, M. Yang, Q. Liu, Active disturbance rejection control for hydraulic systems with full-state constraints and input saturation, Energy, 16 (2022) 1127-1136.
[16] L. Liu, S. Tian, D. Xue, et al, Industrial feedforward control technology: a review, J. Intell. Manuf., 30 (2019) 2819–2833.
[17] S. Tofighi, F. Merrikh-Bayat, A benchmark system to investigate the non-minimum phase behaviour of multi-input multi-output systems, Journal of Control and Decision, 5 (2018) 300-317.
[18] F. Zheng, Q.G. Wang, T.H. Lee, On the design of multivariable PID controllers via LMI approach, Automatica, 38 (2002) 517-526.
[19] S. Boyd, M. Hast, K.J. Astrom, MIMO PID tuning via iterated LMI restriction, Int. J. Robust Nonlinear Control, 26 (2016) 1718-1731.
[20] B. Huang, B. Lu, R. Nagamune, Q. Li, LMI-based linear parameter varying PID control design and its application to an aircraft control system, Aerosp. Syst., 5 (2022) 309–321.
[21] M.N.A. Parlakci, E.M. Jafarov, A robust delay-dependent guaranteed cost PID multivariable output feedback controller design for time-varying delayed systems: An LMI optimization approach, Eur J Control, 61 (2021) 68-79.
[22] Z.Y. Feng, H. Guo, J. She, L. Xu, Weighted sensitivity design of multivariable PID controllers via a new iterative LMI approach, J. Process Control, 110 (2022) 24-34.
[23] J. Sabatier, M. Moze, C.Farges, LMI stability conditions for fractional order systems, Comput. Math Appl., 59 (2010) 1594-1609.
[24] A. Dehak, A.-T. Nguyen, A. Dequidt, L. Vermeiren, M. Dambrine, Reduced-Complexity LMI Conditions for Admissibility Analysis and Control Design of Singular Nonlinear Systems, IEEE Trans. Fuzzy Syst., in press (2022) 1-14.
[25] S. Skogestad, I. Postlethwaite, Multivariable feedback control: analysis and design, 2nd ed., Wiley, New York, 2005.
[26] T. He, G.G. Zhu, X. Chen, A Two-step LMI Scheme for H2- H∞ Control Design, American Control Conference (ACC), Denver, CO, USA, 2020, 1545-1550.
[27] Q. Tran Dinh, S. Gumussoy, W. Michiels, et al, Combining convex-concave decompositions and linearization approaches for solving BMIs, with application to static output feedback, IEEE Trans. Autom. Control, 57 (2012) 1377-1390.
[28] S. Tofighi, F. Merrikh-Bayat, F. Bayat, Robust feedback linearization of an isothermal continuous stirred tank reactor: H mixed-sensitivity synthesis and DK-iteration approaches, Transactions of the Institute of Measurement and Control, 39 (2017) 344-351.
[29] S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in system and control theory, Linear matrix inequalities in system and control theory, 5th ed., SIAM, Philadelphia, 1994.
[30] M. Grant, S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.0 beta, http://cvxr.com/ (2013, accessed March 2022).
[31] R.H. Tutuncu, K. C. Toh, M.J. Todd, Solving semidefinite-quadratic-linear programs using SDPT3, Math Program, 95 (2003) 189-217.
[32] R. Wood, M. Berry, Terminal composition control of a binary distillation column, Chem. Eng. Sci., 28 (1973) 1707-1717.
[33] M. Hovd, M. Skogestad, Simple frequency tools for control system analysis, structure selection, and design, Automatica, 28 (1992) 989–996.
[34] F. Merrikh-Bayat, An iterative LMI approach for H synthesis of multivariable PI/PD controllers for stable and unstable processes, Chem. Eng. Res. Des., 132 (2018) 606–615.