{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T05:45:54Z","timestamp":1773207954875,"version":"3.50.1"},"reference-count":47,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,15]],"date-time":"2023-01-15T00:00:00Z","timestamp":1673740800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key R&amp;D Program of China","award":["2019YFB2005804"],"award-info":[{"award-number":["2019YFB2005804"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The stability and convergence analysis of a multivariable stochastic self-tuning system (STC) is very difficult because of its highly nonlinear structure. In this paper, based on the virtual equivalent system method, the structural nonlinear or nonlinear dominated multivariable self-tuning system is transformed into a structural linear or linear dominated system, thus simplifying the stability and convergence analysis of multivariable STC systems. For the control process of a multivariable stochastic STC system, parameter estimation is required, and there may be three cases of parameter estimation convergence, convergence to the actual value and divergence. For these three cases, this paper provides four theorems and two corollaries. Given the theorems and corollaries, it can be directly concluded that the convergence of parameter estimation is a sufficient condition for the stability and convergence of stochastic STC systems but not a necessary condition, and the four theorems and two corollaries proposed in this paper are independent of specific controller design strategies and specific parameter estimation algorithms. The virtual equivalent system theory proposed in this paper does not need specific control strategies, parameters and estimation algorithms but only needs the nature of the system itself, which can judge the stability and convergence of the self-tuning system and relax the dependence of the system stability convergence criterion on the system structure information. The virtual equivalent system method proposed in this paper is proved to be effective when the parameter estimation may have convergence, convergence to the actual value and divergence.<\/jats:p>","DOI":"10.3390\/e25010173","type":"journal-article","created":{"date-parts":[[2023,1,16]],"date-time":"2023-01-16T02:29:55Z","timestamp":1673836195000},"page":"173","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["From Nonlinear Dominant System to Linear Dominant System: Virtual Equivalent System Approach for Multiple Variable Self-Tuning Control System Analysis"],"prefix":"10.3390","volume":"25","author":[{"given":"Jinghui","family":"Pan","sequence":"first","affiliation":[{"name":"School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kaixiang","family":"Peng","sequence":"additional","affiliation":[{"name":"School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0047-0558","authenticated-orcid":false,"given":"Weicun","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,15]]},"reference":[{"key":"ref_1","unstructured":"Goodwin, G.C., and Sin, K.S. (1984). Adaptive Filtering, Prediction and Control, Prentice Hall."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"829","DOI":"10.1137\/0319052","article-title":"Discrete time adaptive control","volume":"19","author":"Goodwin","year":"1981","journal-title":"SIAM J. Control Optim."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.conengprac.2018.06.017","article-title":"A self-tuning robust full-state feedback control design for the magnetic levitation system","volume":"78","author":"Bidikli","year":"2018","journal-title":"Control Eng. Pract."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1364","DOI":"10.1016\/j.cjche.2015.01.009","article-title":"Pole-placement self-tuning control of nonlinear Hammerstein system and its application to pH process control","volume":"23","author":"Zou","year":"2015","journal-title":"Chin. J. Chem. Eng."},{"key":"ref_5","first-page":"905","article-title":"Stability and Optimality of Self-tuning Regulator","volume":"9","author":"Guo","year":"1991","journal-title":"Sci. China (Ser. A)"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"802","DOI":"10.1109\/9.85060","article-title":"The Astrom-Wittenmark self-tuning regulator revisited and ELS-based adaptive trackers","volume":"36","author":"Guo","year":"1991","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1109\/9.481609","article-title":"Self-convergence of weighted least-squares with applications to stochastic adaptive control","volume":"41","author":"Guo","year":"1996","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1109\/9.553686","article-title":"Indirect adaptive pole-placement control of MIMO stochastic systems: Self-tuning results","volume":"42","author":"Ren","year":"1997","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"102552","DOI":"10.1016\/j.simpat.2022.102552","article-title":"Clonal selection algorithm based control for two-wheeled self-balancing mobile robot","volume":"118","author":"Yagmur","year":"2022","journal-title":"Simul. Model. Pract. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"108976","DOI":"10.1016\/j.ymssp.2022.108976","article-title":"Pole-zero assignment by the receptance method: Multi-input active vibration control","volume":"172","author":"Dario","year":"2022","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1109\/TAC.1985.1103799","article-title":"Global adaptive pole positioning","volume":"30","author":"Anderson","year":"1985","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1109\/TAC.1985.1103954","article-title":"Global stability of adaptive pole placement algorithms","volume":"30","author":"Elliott","year":"1985","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1109\/9.273338","article-title":"Adaptive pole placement without excitation probing signals","volume":"39","author":"Lozano","year":"1994","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1080\/00207178908953394","article-title":"Convergence of adaptive pole-zero placement controller for stable non-minimum phase systems","volume":"50","author":"Chan","year":"1989","journal-title":"Int. J. Control"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"898","DOI":"10.1109\/TAC.1986.1104138","article-title":"Extended least squares and their applications to adaptive control and prediction in linear systems","volume":"31","author":"Lai","year":"1986","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1137\/0325031","article-title":"Asymptotically optimal adaptive control with consistent parameter estimates","volume":"25","author":"Chen","year":"1987","journal-title":"SIAM J. Control. Optim."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1993","DOI":"10.1080\/00207178708934029","article-title":"Adaptive decoupling of multivariable systems","volume":"46","author":"Wittenmark","year":"1987","journal-title":"Int. J. Control"},{"key":"ref_18","first-page":"432","article-title":"The global convergence analysis of a multivariable decoupling self-tuning controller","volume":"15","author":"Chai","year":"1989","journal-title":"Acta Autom. Sin."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1080\/00207179008934103","article-title":"Direct adaptive decoupling control for general stochastic multivariable systems","volume":"51","author":"Chai","year":"1990","journal-title":"Int. J. Control"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1080\/00207179208934246","article-title":"Globally convergent multivariable adaptive decoupling controller and its application to a binarydistillation column","volume":"55","author":"Chai","year":"1992","journal-title":"Int. J. Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1002\/acs.1027","article-title":"Stability of self-tuning control based on Lyapunov function","volume":"22","author":"Patete","year":"2008","journal-title":"Int. J. Adapt. Control. Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1080\/00207170903487421","article-title":"On the stability and convergence of self-tuning control\u2013virtual equivalent system approach","volume":"83","author":"Zhang","year":"2010","journal-title":"Int. J. Control"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1002\/acs.912","article-title":"Issues, progress and new results in robust adaptive control","volume":"20","author":"Fekri","year":"2006","journal-title":"Int. J. Adapt. Control. Signal Process."},{"key":"ref_24","first-page":"4","article-title":"MPPT control of photovoltaic power generation system based on fuzzy parameter self-tuning PID method","volume":"28","author":"Wang","year":"2008","journal-title":"Electr. Power Autom. Equip."},{"key":"ref_25","first-page":"1516","article-title":"The Control Technology of Self-correction for Intelligent AC Contactors","volume":"35","author":"Tang","year":"2015","journal-title":"Proc. CSEE"},{"key":"ref_26","unstructured":"Chen, S., Wu, J., Yang, B., and Ma, J. (2020). A Self-Tuning Control Method for Simulation Turntable Based on Precise Identification of Model Parameters. (201710271289.4), CN Patent."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"7479","DOI":"10.1109\/TII.2021.3057832","article-title":"Tracking Control of a Linear Motor Positioner Based on Barrier Function Adaptive Sliding Mode","volume":"17","author":"Shao","year":"2021","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Nassiri-Toussi, K., and Ren, W. (1995, January 21\u201323). A unified analysis of stochastic adaptive control: Potential self-tuning. Proceedings of the American Control Conference, Seattle, DC, USA.","DOI":"10.1007\/978-1-4419-8568-2_12"},{"key":"ref_29","unstructured":"Nassiri-Toussi, K., and Ren, W. (1995, January 13\u201315). A unified analysis of stochastic adaptive control: Asymptotic self-tuning. Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, LA, USA."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1002","DOI":"10.1109\/9.58528","article-title":"Towards a unified theory of parameter adaptive control: Tunability","volume":"35","author":"Morse","year":"1990","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1109\/9.109635","article-title":"Towards a unified theory of parameter adaptive control-part II: Certainty equivalence and implicit tuning","volume":"37","author":"Morse","year":"1992","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/j.arcontrol.2006.01.001","article-title":"Trends in systems and signals","volume":"30","author":"Katayama","year":"2006","journal-title":"Annu. Rev. Control"},{"key":"ref_33","first-page":"56","article-title":"Adaptive control","volume":"7","author":"Li","year":"1999","journal-title":"Comput. Autom. Meas. Control"},{"key":"ref_34","unstructured":"Li, Q.Q. (1990). Adaptive Control System Theory, Design and Application, Science Press."},{"key":"ref_35","unstructured":"Astr\u00f6m, K.J., and Wittenmark, B. (1995). Adaptive Control, Addison-Wesley."},{"key":"ref_36","unstructured":"Ioannou, P.A., and Sun, J. (1996). Robust Adaptive Control, Prentice-Hall."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1109\/9.52293","article-title":"Convergence of adaptive control schemes using least-squares parameter estimates","volume":"35","author":"Kumar","year":"1990","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2178","DOI":"10.1109\/9.333764","article-title":"Tuning of Gaussian stochastic control systems","volume":"39","year":"1994","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Zhang, W.C. (2009, January 15\u201318). The convergence of parameter estimates is not necessary for a general self-tuning control system-stochasticplant. Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, China.","DOI":"10.1109\/CDC.2009.5400884"},{"key":"ref_40","unstructured":"Zhang, W.C., Li, X.L., and Choi, J.Y. (2008, January 6\u201311). A unified analysis of switching multiple model adaptive control\u2014Virtual equivalent system approach. Proceedings of the 17th IFAC World Congress, Seoul, Republic of Korea."},{"key":"ref_41","first-page":"107","article-title":"Virtual equivalent system theory for self-tuning control","volume":"46","author":"Zhang","year":"2014","journal-title":"J. Harbin Inst. Technol."},{"key":"ref_42","unstructured":"Feng, C., and Shi, W. (1986). Adaptive Control, Publishing House of Electronics Industry."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1137\/040619429","article-title":"Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov functions","volume":"45","author":"Chatterjee","year":"2006","journal-title":"SIAM J. Control. Optim."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Chatterjee, D., and Liberzon, D. (2004, January 14\u201317). On stability of stochastic switched systems. Proceedings of the 43rd Conference on Decision and Control, Nassau, Bahamas.","DOI":"10.1109\/CDC.2004.1429398"},{"key":"ref_45","unstructured":"Prandini, M. (2002, January 27\u201331). Switching control of stochastic linear systems: Stability and performance results. Proceedings of the 6th Congress of SIMAI, Chia Laguna, Cagliari, Italy."},{"key":"ref_46","unstructured":"Prandini, M., and Campi, M.C. (2001, January 4\u20137). Logic-based switching for the stabilization of stochastic systems in presence of unmodeled dynamics. Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, USA."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1360\/N112017-00282","article-title":"Virtual equivalent system theory for adaptive control and simulation verification","volume":"48","author":"Zhang","year":"2018","journal-title":"Sci. Sin. Inf."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/173\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:06:33Z","timestamp":1760119593000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/173"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,15]]},"references-count":47,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["e25010173"],"URL":"https:\/\/doi.org\/10.3390\/e25010173","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,15]]}}}