{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T04:07:20Z","timestamp":1768882040398,"version":"3.49.0"},"reference-count":29,"publisher":"Fuji Technology Press Ltd.","issue":"1","funder":[{"DOI":"10.13039\/100017221","name":"State Grid Fujian Electric Power Company","doi-asserted-by":"publisher","award":["52130423000T"],"award-info":[{"award-number":["52130423000T"]}],"id":[{"id":"10.13039\/100017221","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["JACIII","J. Adv. Comput. Intell. Intell. Inform."],"published-print":{"date-parts":[[2026,1,20]]},"abstract":"<jats:p>This study investigated the stability of a self-balancing robot control system under a delayed-feedback proportional-integral-derivative (PID) control scheme. To enhance the effectiveness of the analysis, a control method that explicitly considers time delays within the PID feedback loop was developed. First, a dynamic model of the self-balancing robot was established using a Lagrangian formulation, incorporating the time-delay effects from feedback control. The controller output was defined in terms of the velocity and steering angle of the robot, and a closed-loop control system with time-delayed feedback was derived. The Lyapunov\u2013Krasovskii functional approach was used to analyze the stability of the system. In particular, an augmented Lyapunov functional was constructed and combined with inequality techniques based on auxiliary functions and matrix injection methods to achieve a delay-dependent stability criterion. This criterion enables the quantitative estimation of the admissible delay bounds under system stability. Finally, numerical simulations were conducted to validate the theoretical results. The case study results demonstrated that the proposed method provides a reliable, accurate, and superior stability analysis for PID-controlled self-balancing robots with time-varying delays.<\/jats:p>","DOI":"10.20965\/jaciii.2026.p0132","type":"journal-article","created":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T15:02:06Z","timestamp":1768834926000},"page":"132-143","source":"Crossref","is-referenced-by-count":0,"title":["Delay-Dependent Stability Analysis of Self-Balancing Robot Control Systems with Time-Varying Delays"],"prefix":"10.20965","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-0873-1472","authenticated-orcid":true,"given":"Huixin","family":"Liu","sequence":"first","affiliation":[{"name":"Electric Power Research Institute, State Grid Corporation of China (SGCC), 48 Fuyuan Branch Road, Cangshan District, Fuzhou, Fujian Province 350007, China"}]},{"given":"Yonghua","family":"Lai","sequence":"additional","affiliation":[{"name":"Electric Power Research Institute, State Grid Corporation of China (SGCC), 48 Fuyuan Branch Road, Cangshan District, Fuzhou, Fujian Province 350007, China"}]},{"given":"Hongsong","family":"Lian","sequence":"additional","affiliation":[{"name":"Electric Power Research Institute, State Grid Corporation of China (SGCC), 48 Fuyuan Branch Road, Cangshan District, Fuzhou, Fujian Province 350007, China"}]},{"given":"Guobin","family":"Wang","sequence":"additional","affiliation":[{"name":"Electric Power Research Institute, State Grid Corporation of China (SGCC), 48 Fuyuan Branch Road, Cangshan District, Fuzhou, Fujian Province 350007, China"}]},{"given":"Dongsheng","family":"Zheng","sequence":"additional","affiliation":[{"name":"Electric Power Research Institute, State Grid Corporation of China (SGCC), 48 Fuyuan Branch Road, Cangshan District, Fuzhou, Fujian Province 350007, China"}]}],"member":"8550","published-online":{"date-parts":[[2026,1,20]]},"reference":[{"key":"key-10.20965\/jaciii.2026.p0132-1","doi-asserted-by":"crossref","unstructured":"F. Grasser, A. D\u2019arrigo, S. Colombi, and A. C. Rufer, \u201cJOE: A mobile, inverted pendulum,\u201d IEEE Trans. Ind. Electron., Vol.49, Issue 1, pp. 107-114, 2002. https:\/\/doi.org\/10.1109\/41.982254","DOI":"10.1109\/41.982254"},{"key":"key-10.20965\/jaciii.2026.p0132-2","doi-asserted-by":"crossref","unstructured":"C.-C. Tsai, H.-C. Huang, H.-Y. Chen, C.-C. Hung, and S.-T. Chen, \u201cIntelligent Collision-Free Formation Control of Ball-Riding Robots Using Output Recurrent Broad Learning in Industrial Cyber-Physical Systems,\u201d IEEE Trans. Ind. Cyber-Phys. Syst., Vol.2, pp. 459-470, 2024. https:\/\/doi.org\/10.1109\/TICPS.2024.3416410","DOI":"10.1109\/TICPS.2024.3416410"},{"key":"key-10.20965\/jaciii.2026.p0132-3","doi-asserted-by":"crossref","unstructured":"S. Kwon, S. Kim, and J. Yu, \u201cTilting-Type Balancing Mobile Robot Platform for Enhancing Lateral Stability,\u201d IEEE\/ASME Trans. on Mechatronics, Vol.20, Issue 3, pp. 1470-1481, 2015. https:\/\/doi.org\/10.1109\/TMECH.2014.2364204","DOI":"10.1109\/TMECH.2014.2364204"},{"key":"key-10.20965\/jaciii.2026.p0132-4","doi-asserted-by":"crossref","unstructured":"Y. Liu, H. Mou, H. Jiang, Q. Li, and J. Zhang, \u201cAn Improved Hierarchical Optimization Framework for Walking Control of Underactuated Humanoid Robots Using Model Predictive Control and Whole Body Planner and Controller,\u201d Mathematics, Vol.13, Issue 1, Article No.154, 2025. https:\/\/doi.org\/10.3390\/math13010154","DOI":"10.3390\/math13010154"},{"key":"key-10.20965\/jaciii.2026.p0132-5","doi-asserted-by":"crossref","unstructured":"N. Mao, J. Chen, E. Spyrakos-Papastavridis, and J. S. Dai, \u201cDynamic modeling of wheeled biped robot and controller design for reducing chassis tilt angle,\u201d Robotica, Vol.42, Issue 8, pp. 2713-2741, 2024. https:\/\/doi.org\/10.1017\/S0263574724001061","DOI":"10.1017\/S0263574724001061"},{"key":"key-10.20965\/jaciii.2026.p0132-6","doi-asserted-by":"crossref","unstructured":"I. Politis, S. Brewster, and F. Pollick, \u201cLanguage-based multimodal displays for the handover of control in autonomous cars,\u201d Proc. of the 7th Int. Conf. on Automotive User Interfaces and Interactive Vehicular Applications, pp. 3-10, 2015. https:\/\/doi.org\/10.1145\/2799250.2799262","DOI":"10.1145\/2799250.2799262"},{"key":"key-10.20965\/jaciii.2026.p0132-7","doi-asserted-by":"crossref","unstructured":"T.-J. Ren, T.-C. Chen, and C.-J. Chen, \u201cMotion control for a two-wheeled vehicle using a self-tuning PID controller,\u201d Control Eng. Practice, Vol.16, Issue 3, pp. 365-375, 2008. https:\/\/doi.org\/10.1016\/j.conengprac.2007.05.007","DOI":"10.1016\/j.conengprac.2007.05.007"},{"key":"key-10.20965\/jaciii.2026.p0132-8","doi-asserted-by":"crossref","unstructured":"J. D\u00edaz-T\u00e9llez, R. S. Garc\u00eda-Ram\u00edrez, J. P\u00e9rez-P\u00e9rez, J. Estevez-Carre\u00f3n, and M. A. Carre\u00f3n-Rosales, \u201cROS-based Controller for a Two-Wheeled Self-Balancing Robot,\u201d J. of Robotics and Control, Vol.4, No.4, pp. 491-499, 2023. https:\/\/doi.org\/10.18196\/jrc.v4i4.18208","DOI":"10.18196\/jrc.v4i4.18208"},{"key":"key-10.20965\/jaciii.2026.p0132-9","doi-asserted-by":"crossref","unstructured":"J. Fang, \u201cThe LQR Controller Design of Two-Wheeled Self-Balancing Robot Based on the Particle Swarm Optimization Algorithm,\u201d Math. Probl. Eng., Vol.2014, No.1, Article No.729095, 2014. https:\/\/doi.org\/10.1155\/2014\/729095","DOI":"10.1155\/2014\/729095"},{"key":"key-10.20965\/jaciii.2026.p0132-10","doi-asserted-by":"crossref","unstructured":"L. Guo, S. A. A. Rizvi, and Z. Lin, \u201cOptimal control of a two-wheeled self-balancing robot by reinforcement learning,\u201d Int. J. of Robust Nonlinear Control, Vol.31, Issue 6, pp. 1885-1904, 2021. https:\/\/doi.org\/10.1002\/rnc.5058","DOI":"10.1002\/rnc.5058"},{"key":"key-10.20965\/jaciii.2026.p0132-11","doi-asserted-by":"crossref","unstructured":"J. Wu, W. Zhang, and S. Wang, \u201cA Two-Wheeled Self-Balancing Robot with the Fuzzy PD Control Method,\u201d Math. Probl. Eng., Vol.2012, No.1, Article No.469491, 2012. https:\/\/doi.org\/10.1155\/2012\/469491","DOI":"10.1155\/2012\/469491"},{"key":"key-10.20965\/jaciii.2026.p0132-12","doi-asserted-by":"crossref","unstructured":"A. Abdelgawad, T. Shohdy, and A. Nada, \u201cModel- and Data-Based Control of Self-Balancing Robots: Practical Educational Approach with LabVIEW and Arduino,\u201d IFAC-PapersOnLine, Vol.58, Issue 9, pp. 217-222, 2024. https:\/\/doi.org\/10.1016\/j.ifacol.2024.07.399","DOI":"10.1016\/j.ifacol.2024.07.399"},{"key":"key-10.20965\/jaciii.2026.p0132-13","doi-asserted-by":"crossref","unstructured":"C. Liu, K. Qin, G. Xin, C. Li, and S. Wang, \u201cNonlinear Model Predictive Control for a Self-Balancing Wheelchair,\u201d IEEE Access, Vol.12, pp. 28938-28949, 2024. https:\/\/doi.org\/10.1109\/ACCESS.2024.3368853","DOI":"10.1109\/ACCESS.2024.3368853"},{"key":"key-10.20965\/jaciii.2026.p0132-14","doi-asserted-by":"crossref","unstructured":"H. Yu, S. Guan, X. Li, H. Feng, S. Zhang, and Y. Fu, \u201cWhole-Body Motion Generation for Wheeled Biped Robots Based on Hierarchical MPC,\u201d IEEE Trans. Ind. Electron., Vol.72, Issue 8, pp. 8301-8311, 2025. https:\/\/doi.org\/10.1109\/TIE.2025.3531460","DOI":"10.1109\/TIE.2025.3531460"},{"key":"key-10.20965\/jaciii.2026.p0132-15","doi-asserted-by":"crossref","unstructured":"A. Unluturk and O. Aydogdu, \u201cMachine Learning Based Self-Balancing and Motion Control of the Underactuated Mobile Inverted Pendulum with Variable Load,\u201d IEEE Access, Vol.10, pp. 104706-104718, 2022. https:\/\/doi.org\/10.1109\/ACCESS.2022.3210540","DOI":"10.1109\/ACCESS.2022.3210540"},{"key":"key-10.20965\/jaciii.2026.p0132-16","doi-asserted-by":"crossref","unstructured":"A. F. Soliman and B. Ugurlu, \u201cRobust Locomotion Control of a Self-Balancing and Underactuated Bipedal Exoskeleton: Task Prioritization and Feedback Control,\u201d IEEE Robot. Autom. Lett., Vol.6, Issue 3, pp. 5626-5633, 2021. https:\/\/doi.org\/10.1109\/LRA.2021.3082016","DOI":"10.1109\/LRA.2021.3082016"},{"key":"key-10.20965\/jaciii.2026.p0132-17","doi-asserted-by":"crossref","unstructured":"L. Nisar, M. H. Koul, and B. Ahmad, \u201cController Design Aimed at Achieving the Desired States in a 2-DOF Rotary Inverted Pendulum,\u201d IFAC-PapersOnLine, Vol.57, pp. 256-261, 2024. https:\/\/doi.org\/10.1016\/j.ifacol.2024.05.044","DOI":"10.1016\/j.ifacol.2024.05.044"},{"key":"key-10.20965\/jaciii.2026.p0132-18","doi-asserted-by":"crossref","unstructured":"M. Huba, P. Bistak, D. Vrancic, and M. Sun, \u201cPID vs. Model-Based Control for the Double Integrator Plus Dead-Time Model: Noise Attenuation and Robustness Aspects,\u201d Mathematics, Vol.13, Issue 4, Article No.664, 2025. https:\/\/doi.org\/10.3390\/math13040664","DOI":"10.3390\/math13040664"},{"key":"key-10.20965\/jaciii.2026.p0132-19","doi-asserted-by":"crossref","unstructured":"L. Shen and H. Xiao, \u201cDelay-dependent robust stability analysis of power systems with PID controller,\u201d Chin. J. Electr. Eng., Vol.5, Issue 2, pp. 79-86, 2019.","DOI":"10.23919\/CJEE.2019.000014"},{"key":"key-10.20965\/jaciii.2026.p0132-20","doi-asserted-by":"crossref","unstructured":"Y. Huang, H. Chen, and L. Qin, \u201cDesign of self-balancing vehicle based on cascade PID control system,\u201d 2022 4th Int. Conf. on Advances in Computer Technology, Information Science and Communications (CTISC), 2022. https:\/\/doi.org\/10.1109\/CTISC54888.2022.9849694","DOI":"10.1109\/CTISC54888.2022.9849694"},{"key":"key-10.20965\/jaciii.2026.p0132-21","doi-asserted-by":"crossref","unstructured":"S. Wang and K. Shi, \u201cMixed-Delay-Dependent Augmented Functional for Synchronization of Uncertain Neutral-Type Neural Networks with Sampled-Data Control,\u201d Mathematics, Vol.11, Issue 4, Article No.872, 2023. https:\/\/doi.org\/10.3390\/math11040872","DOI":"10.3390\/math11040872"},{"key":"key-10.20965\/jaciii.2026.p0132-22","doi-asserted-by":"crossref","unstructured":"H. Shen, S. Jiao, J. H. Park, and V. Sreeram, \u201cAn Improved Result on\n                      H\n                      \u221e\n                      Load Frequency Control for Power Systems with Time Delays,\u201d IEEE Systems J., Vol.15, Issue 3, pp. 3238-3248, 2021. https:\/\/doi.org\/10.1109\/JSYST.2020.3014936","DOI":"10.1109\/JSYST.2020.3014936"},{"key":"key-10.20965\/jaciii.2026.p0132-23","doi-asserted-by":"crossref","unstructured":"F. Zheng, X. Wang, and X. Cheng, \u201cStability of Stochastic Networks with Proportional Delays and the Unsupervised Hebbian-Type Learning Algorithm,\u201d Mathematics, Vol.11, Issue 23, Article No.4755, 2023. https:\/\/doi.org\/10.3390\/math11234755","DOI":"10.3390\/math11234755"},{"key":"key-10.20965\/jaciii.2026.p0132-24","doi-asserted-by":"crossref","unstructured":"K. Ma, C. Ren, T. Zhang, B. Sun, Z. Sun, L. Li, and Y. Zhang, \u201cDynamic response and stability analysis of vehicle systems with double time-delay feedback control,\u201d Int. J. of Control, Vol.98, Issue 5, pp. 1085-1099, 2025. https:\/\/doi.org\/10.1080\/00207179.2024.2380745","DOI":"10.1080\/00207179.2024.2380745"},{"key":"key-10.20965\/jaciii.2026.p0132-25","doi-asserted-by":"crossref","unstructured":"Z. Zhang, C. Lin, and B. Chen, \u201cNew stability criteria for linear time-delay systems using complete LKF method,\u201d Int. J. Syst. Sci., Vol.46, Issue 2, pp. 377-384, 2015. https:\/\/doi.org\/10.1080\/00207721.2013.794906","DOI":"10.1080\/00207721.2013.794906"},{"key":"key-10.20965\/jaciii.2026.p0132-26","doi-asserted-by":"crossref","unstructured":"V. Dev Deepak, N. K. Arun, and K. V. Shihabudheen, \u201cAn investigation into the stability of helicopter control system under the influence of time delay using an improved LKF,\u201d J. Franklin Inst., Vol.361, Issue 10, Article No.106913, 2024. https:\/\/doi.org\/10.1016\/j.jfranklin.2024.106913","DOI":"10.1016\/j.jfranklin.2024.106913"},{"key":"key-10.20965\/jaciii.2026.p0132-27","doi-asserted-by":"crossref","unstructured":"P. Park, W.-I. Lee, and S. Y. Lee, \u201cAuxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems,\u201d J. Franklin Inst., Vol.352, Issue 4, pp. 1378-1396, 2015. https:\/\/doi.org\/10.1016\/j.jfranklin.2015.01.004","DOI":"10.1016\/j.jfranklin.2015.01.004"},{"key":"key-10.20965\/jaciii.2026.p0132-28","doi-asserted-by":"crossref","unstructured":"Y.-L. Fan, C.-K. Zhang, Y.-F. Liu, Y. He, and Q.-G. Wang, \u201cStability analysis of systems with time-varying delays for conservatism and complexity reduction,\u201d Syst. Control Lett., Vol.193, Article No.105948, 2024. https:\/\/doi.org\/10.1016\/j.sysconle.2024.105948","DOI":"10.1016\/j.sysconle.2024.105948"},{"key":"key-10.20965\/jaciii.2026.p0132-29","doi-asserted-by":"crossref","unstructured":"Y. He, C.-K. Zhang, H.-B. Zeng, and M. Wu, \u201cAdditional functions of variable-augmented-based free-weighting matrices and application to systems with time-varying delay,\u201d Int. J. of Systems Science, Vol.54, Issue 5, pp. 991-1003, 2023. https:\/\/doi.org\/10.1080\/00207721.2022.2157198","DOI":"10.1080\/00207721.2022.2157198"}],"container-title":["Journal of Advanced Computational Intelligence and Intelligent Informatics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.fujipress.jp\/main\/wp-content\/themes\/Fujipress\/hyosetsu.php?ppno=jacii003000010012","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T15:02:52Z","timestamp":1768834972000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.fujipress.jp\/jaciii\/jc\/jacii003000010132"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,20]]},"references-count":29,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1,20]]},"published-print":{"date-parts":[[2026,1,20]]}},"URL":"https:\/\/doi.org\/10.20965\/jaciii.2026.p0132","relation":{},"ISSN":["1883-8014","1343-0130"],"issn-type":[{"value":"1883-8014","type":"electronic"},{"value":"1343-0130","type":"print"}],"subject":[],"published":{"date-parts":[[2026,1,20]]}}}