{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,21]],"date-time":"2025-11-21T18:02:17Z","timestamp":1763748137463,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,13]],"date-time":"2020-02-13T00:00:00Z","timestamp":1581552000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000921","name":"European Cooperation in Science and Technology","doi-asserted-by":"publisher","award":["CA15225"],"award-info":[{"award-number":["CA15225"]}],"id":[{"id":"10.13039\/501100000921","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Due to unpredictable and fluctuating conditions in real-world control system applications, disturbance rejection is a substantial factor in robust control performance. The inherent disturbance rejection capacity of classical closed loop control systems is limited, and an increase in disturbance rejection performance of single-loop control systems affects the set-point control performance. Multi-loop control structures, which involve model reference control loops, can enhance the inherent disturbance rejection capacity of classical control loops without degrading set-point control performance; while the classical closed Proportional Integral Derivative (PID) control loop deals with stability and set-point control, the additional model reference control loop performs disturbance rejection control. This adaptive disturbance rejection, which does not influence set-point control performance, is achieved by selecting reference models as transfer functions of real control systems. This study investigates six types of multi-loop model reference (ML-MR) control structures for PID control loops and presents straightforward design schemes to enhance the disturbance rejection control performance of existing PID control loops. For this purpose, linear and non-linear ML-MR control structures are introduced, and their control performance improvements and certain inherent drawbacks of these structures are discussed. Design examples demonstrate the benefits of the ML-MR control structures for disturbance rejection performance improvement of PID control loops without severely deteriorating their set-point performance.<\/jats:p>","DOI":"10.3390\/a13020038","type":"journal-article","created":{"date-parts":[[2020,2,14]],"date-time":"2020-02-14T10:09:57Z","timestamp":1581674997000},"page":"38","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Multi-Loop Model Reference Proportional Integral Derivative Controls: Design and Performance Evaluations"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5238-6433","authenticated-orcid":false,"given":"Baris Baykant","family":"Alagoz","sequence":"first","affiliation":[{"name":"Department of Computer Engineering, Inonu University, 44000 Malatya, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aleksei","family":"Tepljakov","sequence":"additional","affiliation":[{"name":"Department of Computer Systems, Tallinn University of Technology, 12616 Tallinn, Estonia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2167-6280","authenticated-orcid":false,"given":"Eduard","family":"Petlenkov","sequence":"additional","affiliation":[{"name":"Department of Computer Systems, Tallinn University of Technology, 12616 Tallinn, Estonia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6106-2374","authenticated-orcid":false,"given":"Celaleddin","family":"Yeroglu","sequence":"additional","affiliation":[{"name":"Department of Computer Systems, Tallinn University of Technology, 12616 Tallinn, Estonia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,13]]},"reference":[{"key":"ref_1","unstructured":"Fossard, A.J., and Normand-Cyrot, D. (1995). Nonlinear Systems Modeling and Estimation, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Tepljakov, A. (2017). Fractional-Order Modeling and Control of Dynamic Systems, Springer.","DOI":"10.1007\/978-3-319-52950-9"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Lanchier, N. (2017). Stochastic Modeling, Springer.","DOI":"10.1007\/978-3-319-50038-6"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2522","DOI":"10.1049\/iet-cta.2015.0175","article-title":"Implicit disturbance rejection performance analysis of closed loop control systems according to communication channel limitations","volume":"9","author":"Alagoz","year":"2015","journal-title":"IET Control Theory A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1109\/37.16756","article-title":"Model reference adaptive control of a direct-drive DC motor","volume":"9","author":"Butler","year":"1989","journal-title":"IEEE Control Syst. Mag."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"112","DOI":"10.2514\/3.8544","article-title":"Model reference adaptive control of large structural systems","volume":"6","author":"Barkana","year":"1983","journal-title":"J. Guid. Control Dyn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1002\/acs.4480030106","article-title":"A new approach to model reference adaptive control","volume":"3","author":"Duarte","year":"1989","journal-title":"Int. J. Adapt. Control."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1002\/acs.2411","article-title":"Simple adaptive control\u2013a stable direct model reference adaptive control methodology\u2013brief survey","volume":"28","author":"Barkana","year":"2014","journal-title":"Int. J. Adapt. Control"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1109\/TCST.2008.921808","article-title":"Design of a data-driven PID controller","volume":"17","author":"Yamamoto","year":"2008","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s11071-004-3763-7","article-title":"Tuning of PID controllers based on Bode\u2019s ideal transfer function","volume":"38","author":"Barbosa","year":"2004","journal-title":"Nonlinear Dyn."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Wellenreuther, A., and Gambier, A. (2006, January 4\u20138). Multi-loop Controller Design for a Heat Exchanger. Proceedings of the 2006 IEEE International Conference on Control Applications, Munich, Germany.","DOI":"10.1109\/CCA.2006.286190"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"514","DOI":"10.1016\/j.isatra.2012.03.003","article-title":"Design of PID controllers in double feedback loops for SISO systems with set-point filters","volume":"51","author":"Vijayan","year":"2012","journal-title":"ISA Trans."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1631","DOI":"10.1002\/aic.10166","article-title":"Analytical Design of Multiloop PID Controllers for Desired Closed-Loop Responses","volume":"50","author":"Lee","year":"2004","journal-title":"AIChE J."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2077","DOI":"10.1021\/ie0612360","article-title":"IMC-PID Controller Design for Improved Disturbance Rejection of Time-Delayed Processes","volume":"46","author":"Shamsuzzoha","year":"2007","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1016\/j.jprocont.2003.09.003","article-title":"A new design for a PID plus feedforward controller","volume":"14","author":"Visioli","year":"2004","journal-title":"J. Process Control"},{"key":"ref_16","first-page":"237","article-title":"Model Reference-Based Adaptive PID Controller for Robot Motion Control of Not Explicitly Known Systems","volume":"12","author":"Su","year":"2007","journal-title":"IJICS"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1896","DOI":"10.1016\/j.cnsns.2007.03.025","article-title":"Direct model reference adaptive control (MRAC) design and simulation for the vibration suppression of piezoelectric smart structures","volume":"13","author":"Trajkov","year":"2008","journal-title":"Commun. Nonlinear Sci."},{"key":"ref_18","unstructured":"Landau, I.D. (1979). Adaptive Control the Model Reference Approach, Marcel Dekker."},{"key":"ref_19","unstructured":"Astrom, K.J., and Wittenmark, B. (1995). Adaptive Control, Addison-Wesley."},{"key":"ref_20","unstructured":"Butler, H. (1992). Model-Reference Adaptive Control-From Theory to Practice, Prentice-Hall."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1023\/A:1016504620249","article-title":"Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control","volume":"29","author":"Vinagre","year":"2001","journal-title":"Nonlinear Dyn."},{"key":"ref_22","first-page":"84","article-title":"Implementation of model reference adaptive controller with fractional order adjustment rules for coaxial rotor control test system","volume":"4","author":"Kavuran","year":"2016","journal-title":"BAJECE"},{"key":"ref_23","first-page":"101","article-title":"An Experimental Study on Model Reference Adaptive Control of TRMS by Error-Modified Fractional Order MIT Rule","volume":"19","author":"Kavuran","year":"2017","journal-title":"Control Eng. Appl. Inf."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1049\/ip-cta:20020100","article-title":"Backstepping-based adaptive PID control","volume":"149","author":"Benaskeur","year":"2002","journal-title":"IET Control Theory A"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Alagoz, B.B., Tepljakov, A., Petlenkov, E., and Yeroglu, C. (2017, January 5\u20137). Multi-Loop model reference adaptive control of fractional-order PID control systems. Proceedings of the 2017 40th International Conference on Telecommunications and Signal Processing (TSP), Barcelona, Spain.","DOI":"10.1109\/TSP.2017.8076078"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1850176","DOI":"10.1142\/S0218126618501761","article-title":"Model reference adaptive control scheme for retuning method-based fractional-order PID control with disturbance rejection applied to closed-loop control of a magnetic levitation system","volume":"27","author":"Tepljakov","year":"2018","journal-title":"J. Circuits Syst. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Rajesh, R., and Deepa, S.N. (2019). Design of direct MRAC augmented with 2 DoF PIDD controller: An application to speed control of a servo plant. J. King Saud Univ. Eng. Sci.","DOI":"10.1016\/j.jksues.2019.02.005"},{"key":"ref_28","first-page":"276","article-title":"Multi-loop Model Reference Adaptive PID Control for Fault-Tolerance","volume":"7","author":"Alagoz","year":"2019","journal-title":"BAJECE"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Tavakoli, S., and Tavakoli, M. (2003, January 12). Optimal tuning of PID controllers for first order plus time delay models using dimensional analysis. Proceedings of the 2003 4th International Conference on Control and Automation Proceedings, Montreal, QC, Canada.","DOI":"10.23919\/ECC.2003.7085293"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.isatra.2014.09.013","article-title":"Disturbance rejection performance analyses of closed loop control systems by reference to disturbance ratio","volume":"55","author":"Alagoz","year":"2015","journal-title":"ISA Trans."},{"key":"ref_31","unstructured":"Bendjeghaba, O., and Boushaki, S.I. (2013, January 25\u201327). Optimal Tuning of PID Controller in Automatic Voltage Regulator System using Improved Harmony Search Algorithm. Proceedings of the 7th Global Conference on Power Control and Optimization, Prague, Czech."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Zhang, P. (2010). Industrial control system simulation routines. Advanced Industrial Control Technology, Elsevier.","DOI":"10.1016\/B978-1-4377-7807-6.10019-1"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"4807","DOI":"10.1021\/ie010756m","article-title":"PI\/PID Controller Design Based on Direct Synthesis and Disturbance Rejection","volume":"41","author":"Chen","year":"2002","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"798","DOI":"10.1016\/j.conengprac.2007.08.006","article-title":"Tuning and auto-tuning of fractional order controllers for industry applications","volume":"16","author":"Monje","year":"2008","journal-title":"Control Eng. Pract."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s11071-017-3658-z","article-title":"New alternatives for analog implementation of fractional-order integrators, differentiators and PID controllers based on integer-order integrators","volume":"90","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1520","DOI":"10.1007\/s00034-018-0944-z","article-title":"Practical design of RC approximants of constant phase elements and their implementation in fractional-order PID regulators using CMOS voltage differencing current conveyors","volume":"38","author":"Domansky","year":"2019","journal-title":"Circuit Syst. Signal Process."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1016\/j.mechatronics.2013.05.001","article-title":"Auto-tuning of PID controller according to fractional-order reference model approximation for DC rotor control","volume":"23","author":"Alagoz","year":"2013","journal-title":"Mechatronics"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1016\/j.conengprac.2011.12.003","article-title":"Constrained output feedback model predictive control for nonlinear systems","volume":"20","author":"Rahideh","year":"2012","journal-title":"Control Eng. Pract."},{"key":"ref_39","unstructured":"Twin Rotor MIMO System Control Experiments 33-949S, Feedback Instruments Ltd."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/13\/2\/38\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:57:23Z","timestamp":1760173043000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/13\/2\/38"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,13]]},"references-count":39,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2]]}},"alternative-id":["a13020038"],"URL":"https:\/\/doi.org\/10.3390\/a13020038","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2020,2,13]]}}}