IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i9p1399-d799576.html
   My bibliography  Save this article

Observer-Based PID Control Strategy for the Stabilization of Delayed High Order Systems with up to Three Unstable Poles

Author

Listed:
  • César Cruz-Díaz

    (Instituto Politécnico Nacional, ESIME Culhuacán, Av. Santa Ana 1000, Col. San Francisco Culhuacán, Mexico City 04440, Mexico)

  • Basilio del Muro-Cuéllar

    (Instituto Politécnico Nacional, ESIME Culhuacán, Av. Santa Ana 1000, Col. San Francisco Culhuacán, Mexico City 04440, Mexico)

  • Gonzalo Duchén-Sánchez

    (Instituto Politécnico Nacional, ESIME Culhuacán, Av. Santa Ana 1000, Col. San Francisco Culhuacán, Mexico City 04440, Mexico)

  • Juan Francisco Márquez-Rubio

    (Instituto Politécnico Nacional, ESIME Culhuacán, Av. Santa Ana 1000, Col. San Francisco Culhuacán, Mexico City 04440, Mexico)

  • Martín Velasco-Villa

    (Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Electrical Engineering Department, Mechatronics Section, Av. IPN 2508, Col. San Pedro Zacatenco, Mexico City 07360, Mexico)

Abstract

In this paper, a new method to manage the stabilization and control problems of n -dimensional linear systems plus dead time, which includes one, two, or three unstable poles, is proposed. The control methodology proposed in this work is an Observer-based Proportional-Integral-Derivative (PID) strategy, where an observer and a PID controller are used to relocate the original unstable open-loop poles to stabilize the resultant closed-loop system. The observer provides an adequate estimation of the delayed-free variables and the PID uses the delay-free variables estimated by the proposed observer. Also, step-tracking is achieved in the overall control scheme. Necessary and sufficient conditions are presented to ensure closed-loop stability based on the open loop parameters of the system. The observer-based PID strategy considers five to seven constant parameters to obtain a stable closed-loop system. A general procedure to implement the proposed control strategy is presented and its performance is evaluated by means of numerical simulations.

Suggested Citation

  • César Cruz-Díaz & Basilio del Muro-Cuéllar & Gonzalo Duchén-Sánchez & Juan Francisco Márquez-Rubio & Martín Velasco-Villa, 2022. "Observer-Based PID Control Strategy for the Stabilization of Delayed High Order Systems with up to Three Unstable Poles," Mathematics, MDPI, vol. 10(9), pages 1-17, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1399-:d:799576
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/9/1399/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/9/1399/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Chen, Wei-Ching, 2008. "Dynamics and control of a financial system with time-delayed feedbacks," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 1198-1207.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Veronika NOVOTNA & Stanislav Å KAPA & Bernard NEUWIRTH, 2019. "Analysis Of A Non-Linear Dynamic Financial System," Proceedings of the INTERNATIONAL MANAGEMENT CONFERENCE, Faculty of Management, Academy of Economic Studies, Bucharest, Romania, vol. 13(1), pages 288-297, November.
    2. Ding, Yuting & Jiang, Weihua & Wang, Hongbin, 2012. "Hopf-pitchfork bifurcation and periodic phenomena in nonlinear financial system with delay," Chaos, Solitons & Fractals, Elsevier, vol. 45(8), pages 1048-1057.
    3. Çalış, Yasemin & Demirci, Ali & Özemir, Cihangir, 2022. "Hopf bifurcation of a financial dynamical system with delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 201(C), pages 343-361.
    4. Harshavarthini, S. & Sakthivel, R. & Ma, Yong-Ki & Muslim, M., 2020. "Finite-time resilient fault-tolerant investment policy scheme for chaotic nonlinear finance system," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    5. Shi, Jianping & He, Ke & Fang, Hui, 2022. "Chaos, Hopf bifurcation and control of a fractional-order delay financial system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 348-364.
    6. Wu, Jianjun & Xia, Lu, 2024. "Double well stochastic resonance for a class of three-dimensional financial systems," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    7. Vasily E. Tarasov, 2019. "On History of Mathematical Economics: Application of Fractional Calculus," Mathematics, MDPI, vol. 7(6), pages 1-28, June.
    8. Jin, Maolin & Chang, Pyung Hun, 2009. "Simple robust technique using time delay estimation for the control and synchronization of Lorenz systems," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2672-2680.
    9. Son, Woo-Sik & Park, Young-Jai, 2011. "Delayed feedback on the dynamical model of a financial system," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 208-217.
    10. Changjin Xu & Maoxin Liao & Peiluan Li & Qimei Xiao & Shuai Yuan, 2019. "Control Strategy for a Fractional-Order Chaotic Financial Model," Complexity, Hindawi, vol. 2019, pages 1-14, April.
    11. Qijia Yao & Hadi Jahanshahi & Larissa M. Batrancea & Naif D. Alotaibi & Mircea-Iosif Rus, 2022. "Fixed-Time Output-Constrained Synchronization of Unknown Chaotic Financial Systems Using Neural Learning," Mathematics, MDPI, vol. 10(19), pages 1-14, October.
    12. Xinggui Li & Ruofeng Rao & Xinsong Yang, 2022. "Impulsive Stabilization on Hyper-Chaotic Financial System under Neumann Boundary," Mathematics, MDPI, vol. 10(11), pages 1-18, May.
    13. Baogui Xin & Tong Chen & Junhai Ma, 2010. "Neimark-Sacker Bifurcation in a Discrete-Time Financial System," Discrete Dynamics in Nature and Society, Hindawi, vol. 2010, pages 1-12, September.
    14. Ruofeng Rao, 2019. "Global Stability of a Markovian Jumping Chaotic Financial System with Partially Unknown Transition Rates under Impulsive Control Involved in the Positive Interest Rate," Mathematics, MDPI, vol. 7(7), pages 1-15, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1399-:d:799576. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.