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Contribuciones al estudio de sistemas lineales con retardos: el enfoque de funcionales de tipo completo

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Contribuciones al estudio de sistemas lineales con retardos: el enfoque de funcionales de tipo completo

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Mondié, S.; Gomez, M. (2022). Contribuciones al estudio de sistemas lineales con retardos: el enfoque de funcionales de tipo completo. Revista Iberoamericana de Automática e Informática industrial. 19(4):381-393. https://doi.org/10.4995/riai.2022.16828

Por favor, use este identificador para citar o enlazar este ítem: http://hdl.handle.net/10251/187026

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Título: Contribuciones al estudio de sistemas lineales con retardos: el enfoque de funcionales de tipo completo
Otro titulo: Linear time-delay systems: the complete type functionals approach
Autor: Mondié, Sabine Gomez, Marco-Antonio
Fecha difusión:
Resumen:
[EN] Recent results on Lyapunov-Krasovskii functionals of complete type for linear time-delay systems are presented. The main concepts and results are introduced for the single delay system case, and necessary and sufficient ...[+]


[ES] Se introducen resultados recientes del enfoque de funcionales de Lyapunov-Krasovski de tipo completo para sistemas lineales con retardos. Se explican brevemente los principales conceptos y resultados para el caso de ...[+]
Palabras clave: Time-delay systems , Stability analysis , Linear systems , Controller design , Sistemas con retardos , Análisis de estabilidad , Sistemas lineales , Diseño de controladores
Derechos de uso: Reconocimiento - No comercial - Compartir igual (by-nc-sa)
Fuente:
Revista Iberoamericana de Automática e Informática industrial. (issn: 1697-7912 ) (eissn: 1697-7920 )
DOI: 10.4995/riai.2022.16828
Editorial:
Universitat Politècnica de València
Versión del editor: https://doi.org/10.4995/riai.2022.16828
Código del Proyecto:
info:eu-repo/grantAgreement/Conacyt//A1-S-24796
Agradecimientos:
Este trabajo ha sido realizado parcialmente gracias al apoyo del Conacyt, México, Proyecto A1-S-24796.
Tipo: Artículo

References

Alexandrova, I. V., Zhabko, A. P., 2018. A new LKF approach to stability analysis of linear systems with uncertain delays. Automatica 91, 173-178. https://doi.org/10.1016/j.automatica.2018.01.012

Arismendi-Valle, H., Melchor-Aguilar, D., 2019. On the Lyapunov matrices for integral delay systems. Int. J. of Systems Science 50 (6), 1190-1201. https://doi.org/10.1080/00207721.2019.1597943

Bejarano, F. J., 2021. Zero dynamics normal form and disturbance decoupling of commensurate and distributed time-delay systems. Automatica129, 109634. https://doi.org/10.1016/j.automatica.2021.109634 [+]
Alexandrova, I. V., Zhabko, A. P., 2018. A new LKF approach to stability analysis of linear systems with uncertain delays. Automatica 91, 173-178. https://doi.org/10.1016/j.automatica.2018.01.012

Arismendi-Valle, H., Melchor-Aguilar, D., 2019. On the Lyapunov matrices for integral delay systems. Int. J. of Systems Science 50 (6), 1190-1201. https://doi.org/10.1080/00207721.2019.1597943

Bejarano, F. J., 2021. Zero dynamics normal form and disturbance decoupling of commensurate and distributed time-delay systems. Automatica129, 109634. https://doi.org/10.1016/j.automatica.2021.109634

Bejarano, F. J., Zheng, G., 2017. Unknown input functional observability of descriptor systems with neutral and distributed delay effects. Automatica 85, 186-192. https://doi.org/10.1016/j.automatica.2017.07.044

Califano, C., Marquez-Martínez, L. A., Moog, C. H., 2013. Linearization of time-delay systems by input output injection and output transformation. Automatica 49 (6), 1932-1940. https://doi.org/10.1016/j.automatica.2013.03.001

Castaños, F., Estrada, E., Mondié, S., Ramírez, A., 2018. Passivity-based PI control of first-order systems with I/O communication delays: a frequency domain analysis. Int. J. of Control 91 (11), 2549-2562. https://doi.org/10.1080/00207179.2017.1327083

Castaños, F., Mondié, S., 2021. Observer-based predictor for a susceptible-infectious-recovered model with delays: an optimal-control case study. Int. J. of Robust and Nonlinear Control 31 (11), 5118-5133. https://doi.org/10.1002/rnc.5522

Cuvas, C., Mondie, S., 2016. Necessary stability conditions for delay systems with multiple pointwise and distributed delays. IEEE Trans. on Automatic Control 61 (7), 1987-1994. https://doi.org/10.1109/TAC.2015.2487478

Cuvas, C., Ramírez, A., Juárez, L., Mondié, S., 2019. Scanning the space of parameters for stability regions of a class of time-delay systems; a Lyapunovmatrix approach. Delays and Interconnections: Methodology, Algorithmsand Applications. https://doi.org/10.1007/978-3-030-11554-8_10

Cuvas, C., Santos-Sánchez, O.-J., Ordaz, P., Rodríguez-Guerrero, L., 2021. Suboptimal control for systems with commensurate and distributed delays of neutral type. Int. J. of Robust and Nonlinear Control n/a (n/a). https://doi.org/10.1002/rnc.5739

Egorov, A. V., 2014. A new necessary and sufficient stability condition for linear time-delay systems. In: Proceedings of the 19th IFAC World Congress. Cape Town, South Africa, pp. 11018-11023. https://doi.org/10.3182/20140824-6-ZA-1003.02677

Egorov, A. V., 2016. A finite necessary and sufficient stability condition for linear retarded type systems. In: Proceedings of the 55th IEEE Conference on Decision and Control. Las Vegas, USA, pp. 3155-3160. https://doi.org/10.1109/CDC.2016.7798742

Egorov, A. V., Cuvas, C., Mondié, S., 2017. Necessary and sufficient stability conditions for linear systems with pointwise and distributed delays. Automatica 80, 218-224. https://doi.org/10.1016/j.automatica.2017.02.034

Egorov, A. V., Mondie, S., 2013. A stability criterion for the single delay equation in terms of the Lyapunov matrix. Vestnik Sankt-Peterburgskogo Universiteta. Prikl. Mat., Inf., Prot. Upr. 1, 106-115.

Egorov, A. V., Mondie, S., 2014. Necessary stability conditions for linear delay systems. Automatica 50 (12), 3204-3208. https://doi.org/10.1016/j.automatica.2014.10.031

Egorov, A. V., Mondie, S., 2015. The delay Lyapunov matrix in robust stabilityanalysis of time-delay systems. In: Proceedings of the 12th IFAC WorkshoponTime Delay Systems. pp. 245-250. https://doi.org/10.1016/j.automatica.2014.10.031

Fragoso-Rubio, V., Velasco-Villa, M., Vallejo-Alarcon, J., Vasquez-Santacruz, M., Hernandez-Perez, M., 2019. Consensus problem for linear time-invariant systems with time-delay. Mathematical Problems in Engineering. https://doi.org/10.1155/2019/1607474

Gomez, M. A., Cuvas, C., Mondie, S., Egorov, A. V., 2016a. Scanning the space of parameters for stability regions of neutral type delay systems: A Lya-punov matrix approach. In: 2016 IEEE 55th Conference on Decision and Control (CDC). pp. 3149-3154. https://doi.org/10.1109/CDC.2016.7798741

Gomez, M. A., Egorov, A. V., Mondie, S., 2018. A new stability criterion for neutral-type systems with one delay. In: Proceedings of the 14th IFAC Workshop on Time Delay Systems. pp. 177-182. https://doi.org/10.1016/j.ifacol.2018.07.219

Gomez, M. A., Egorov, A. V., Mondie, S., 2019a. Lyapunov matrix based necessary and sufficient stability condition by finite number of mathematical operations for retarded type systems. Automatica 108, 108475. https://doi.org/10.1016/j.automatica.2019.06.027

Gomez, M. A., Egorov, A. V., Mondie, S., 2019b. Necessary stability conditions for neutral-type systems with multiple commensurate delays. Int. J. of Control 92 (5), 1155-1166. https://doi.org/10.1080/00207179.2017.1384574

Gomez, M. A., Egorov, A. V., Mondie, S., 2020. Necessary and sufficient stability condition by finite number of mathematical operations for time-delay systems of neutral type. IEEE Trans. on Automatic Control 66 (6), 2802-2808. https://doi.org/10.1109/TAC.2020.3008392

Gomez, M. A., Egorov, A. V., Mondie, S., Michiels, W., 2019c. Optimization of the H2 norm for single delay systems, with application to control design and model approximation. IEEE Trans. on Automatic Control 64 (2), 804-811. https://doi.org/10.1109/TAC.2018.2836019

Gomez, M. A., Egorov, A. V., Mondie, S., Zhabko, A. P., 2019d. Computation of the Lyapunov matrix for periodic time-delay systems and its application to robust stability analysis. Systems & Control Letters 132, 104501. https://doi.org/10.1016/j.sysconle.2019.104501

Gomez, M. A., Michiels, W., 2019. Characterization and optimization of the smoothed spectral abscissa for time-delay systems. Int. J. of Robust and Nonlinear Control 29 (13), 4402-4418. https://doi.org/10.1002/rnc.4631

Gomez, M. A., Michiels, W., Mondie, S., 2019e. Design of delay-based output-feedback controllers optimizing a quadratic cost function via the delay Lya-punov matrix. Automatica 107, 146-153. https://doi.org/10.1016/j.automatica.2019.05.045

Gomez, M. A., Ochoa, G., Mondie, S., 2016b. Necessary exponential stability conditions for linear periodic time-delay systems. Int. J. of Robust and Nonlinear Control 26 (18), 3996-4007. https://doi.org/10.1002/rnc.3545

Gonzalez, A., Aragües, R., Lopez-Nicolas, G., Sagues, C., 2020. Predictor-feedback synthesis in coordinate-free formation control under time-varying delays. Automatica 113, 108811. https://doi.org/10.1016/j.automatica.2020.108811

Hernandez-Diez, J.-E., Mendez-Barrios, C.-F., Mondie, S., Niculescu, S.-I., Gonzalez-Galvan, E., 2018. Proportional-delayed controllers design for LTI systems: a geometric approach. Int. J. of Control 91 (4), 907-925. https://doi.org/10.1080/00207179.2017.1299943

Hernandez-Diez, J.-E., Mendez-Barrios, C.-F., Niculescu, S.-I., 2019. Practical guidelines for tuning PD and PI delay-based controllers. In: 15th IFAC Workshop on Time Delay Systems. pp. 61-66. https://doi.org/10.1016/j.ifacol.2019.12.207

Hernandez-Perez, M., Fragoso-Rubio, V., Velasco-Villa, M., del Muro-Cuellar,B., Marquez-Rubio, J., Puebla, H., 2020. Prediction-based control for a class of unstable time-delayed processes by using a modified sequential predictor. J. of Process Control 92, 98-107. https://doi.org/10.1016/j.jprocont.2020.05.014

Jarlebring, E., Vanbiervliet, J., Michiels, W., 2011. Characterizing and computing the H2 norm of time-delay systems by solving the delay Lyapunovequation. IEEE Trans. on Automatic Control 56 (4), 814-825. https://doi.org/10.1109/TAC.2010.2067510

Juarez, L., Alexandrova, I. V., Mondie, S., 2020a. Robust stability analysis for linear systems with distributed delays: A time-domain approach. Int. J. of Robust and Nonlinear Control 30 (18), 8299-8312. https://doi.org/10.1002/rnc.5244

Juarez, L., Mondie, S., Kharitonov, V. L., 2020b. Dynamic predictor for systems with state and input delay: A time-domain robust stability analysis. Int. J. of Robust and Nonlinear Control 30 (6), 2204-2218. https://doi.org/10.1002/rnc.4879

Kharitonov, V. L., 2013. Time-Delay Systems: Lyapunov functionals and matrices. Birkhauser, Basel. https://doi.org/10.1007/978-0-8176-8367-2

Kharitonov, V. L., 2014. An extension of the prediction scheme to the case of systems with both input and state delay. Automatica 50 (1), 211-217. https://doi.org/10.1016/j.automatica.2013.09.042

Kharitonov, V. L., 2015. Predictor-based controls: the implementation problem. Differential Equations 51 (13), 1675-1682. https://doi.org/10.1134/S0012266115130017

Kharitonov, V. L., Zhabko, A. P., 2003. Lyapunov-Krasovskii approach to therobust stability analysis of time-delay systems. Automatica 39 (1), 15-20. https://doi.org/10.1016/S0005-1098(02)00195-4

Krasovskii, N. N., 1963. Stability of motion. Stanford University Press.

Kuhsner, H. J., Barnea, D., 1970. On the control of a linear functional differential equation with quadratic cost. SIAM J. on Control and Optimization8 (2), 257-272. https://doi.org/10.1137/0308019

Letyagina, O. N., Zhabko, A. P., 2009. Robust stability analysis of linear perio-dic systems with time delay. Int. J. of Modern Physics A 24 (5), 893-907. https://doi.org/10.1142/S0217751X09044371

Lopez-Labra, H.-A., Santos-Sanchez, O.-J., Rodriguez-Guerrero, L., Ordaz-Oliver, J.-P., Cuvas-Castillo, C., 2019. Experimental results of optimal and robust control for uncertain linear time-delay systems. J. of Optimization Theory and Applications 181 (3), 1076-1089. https://doi.org/10.1007/s10957-018-01457-9

Manitius, A., Olbrot, A., 1979. Finite spectrum assignment problem for systems with delays. IEEE Trans. on Automatic Control 24 (4), 541-552. https://doi.org/10.1109/TAC.1979.1102124

Marquez-Martinez, L., Moog, C., 2007. New insights on the analysis of nonlinear time-delay systems: Application to the triangular equivalence. Systems & Control Letters 56 (2), 133-140. https://doi.org/10.1016/j.sysconle.2006.08.004

Melchor-Aguilar, D., Kharitonov, V., Lozano, R., 2010. Stability conditions forintegral delay systems. Int. J. of Robust and Nonlinear Control 20 (1), 1-15. https://doi.org/10.1002/rnc.1405

Michiels, W., Gomez, M. A., 2020. On the dual linear periodic time-delay system: Spectrum and lyapunov matrices, with application to analysis and balancing. Int. J. of Robust and Nonlinear Control 30 (10), 3906-3922. https://doi.org/10.1002/rnc.4970

Mondie, S., 2012. Assessing the exact stability region of the single-delay scalar equation via its Lyapunov function. IMA J. of Mathematical Control andInformation 29 (4), 459-470. https://doi.org/10.1093/imamci/dns004

Mondie, S., Cuvas, C., Ramirez, A., Egorov, A., 2012. Necessary conditions for the stability of one delay systems: a Lyapunov matrix approach. In: Proceedings of the 10th IFAC Workshop on Time Delay Systems. Boston, USA, pp. 13-18. https://doi.org/10.3182/20120622-3-US-4021.00022

Mondie, S., Kharitonov, V., 2005. Exponential estimates for retarded time-delay systems: an LMI approach. IEEE Trans. on Automatic Control 50 (2), 268-273. https://doi.org/10.1109/TAC.2004.841916

Mondie,S.,Melchor-Aguilar, D., 2012. Exponential stability of integral delay systems with a class of analytic kernels. IEEE Trans. on Automatic Control 57(2), 484-489. https://doi.org/10.1109/TAC.2011.2178653

Mondie, S., Michiels, W., 2003. Finite spectrum assignment of unstable time-delay systems with a safe implementation. IEEE Trans. on Automatic Control 48 (12), 2207-2212. https://doi.org/10.1109/TAC.2003.820147

Mondie, S., Ochoa-Ortega, G., Ochoa-Galvan, B., 2011. Instability conditions for linear time delay systems: a Lyapunov matrix function approach. Int. J.of Control 84 (10), 1601-1611. https://doi.org/10.1080/00207179.2011.620632

Najafi, M., Hosseinnia, S., Sheikholeslam, F., Karimadini, M., 2013. Closed-loop control of dead time systems via sequential sub-predictors. Int. J. of Control 86 (4), 599-609. https://doi.org/10.1080/00207179.2012.751627

Neimark, J., 1949. D-subdivisions and spaces of quasi-polynomials. Prikladna-ya Matematika i Mekhanika 13 (5), 349-380.

Nuño, E., Arteaga-Pérez, M., Espinosa-Pérez, G., 2018. Control of bilateral teleoperators with time delays using only position measurements. Int. J. of Robust and Nonlinear Control 28 (3), 808-824. https://doi.org/10.1002/rnc.3903

Nuño, E., Ortega, R., 2018. Achieving consensus of Euler Lagrange agents with interconnecting delays and without velocity measurements via passivity-based control. IEEE Trans. on Control Systems Technology 26 (1), 222-232. https://doi.org/10.1109/TCST.2017.2661822

Ochoa-Ortega, G., Kharitonov, V., Mondie, S., 2013. Critical frequencies and parameters for linear delay systems: A Lyapunov matrix approach. Systems & Control Letters 62 (9), 781-790. https://doi.org/10.1016/j.sysconle.2013.05.010

Ochoa-Ortega, G., Villafuerte-Segura, R., Ramirez-Neria, M., Vite-Hernandez, L., 2019. σ-stabilization of a flexible joint robotic arm via delayed controllers. Complexity, 7289689. https://doi.org/10.1155/2019/7289689

Ordaz, J., Salazar, S., Mondie, S., Romero, H., Lozano, R., 2013. Predictor-based position control of a quad-rotor with delays in GPS and vision measurements. J. of Intelligent and Robotic Systems 70 (4), 13-26. https://doi.org/10.1007/s10846-012-9714-5

Ortega-Martinez, J., Santos-Sanchez, O., Rodriguez-Guerrero, L., Romero-Trejo, H., Ordaz-Oliver, J.-P., 2018. Adaptive nonlinear optimal control for a banana dehydration process. Int. J. of Innovative Computing, Informationand Control 14 (6), 2055-2069.

Ortega-Martinez, J.-M., Santos-Sanchez, O.-J., Mondie, S., 2021. Comments on the Bellman functional for linear time-delay systems. Optimal Control Applications and Methods 42 (5), 1531-1540. https://doi.org/10.1002/oca.2726

Ortiz, R., Del Valle, S., Egorov, A. V., Mondie, S., 2020. Necessary stability conditions for integral delay systems. IEEE Trans. on Automatic Control 65 (10), 4377-4384. https://doi.org/10.1109/TAC.2019.2955962

Ortiz, R., Egorov, A. V., Mondie, S., 2021. Necessary and sufficient stabilityconditions for integral delay systems. Int. J. of Robust and Nonlinear Con-trol 24 (12), 1760-1771. https://doi.org/10.1002/rnc.2962

Ramirez, A., Mondie, S., Garrido, R., Sipahi, R., 2016. Design of proportional-integral-retarded (PIR) controllers for second-order LTI systems. IEEE Trans. on Automatic Control 61 (6), 1688-1693. https://doi.org/10.1109/TAC.2015.2478130

Ramirez, A., Sipahi, R., 2019. Single-delay and multiple-delay Proportional-Retarded (PR) protocols for fast consensus in a large-scale network. IEEE Trans. on Automatic Control 64 (5), 2142-2149. https://doi.org/10.1109/TAC.2018.2866444

Ramirez, A., Sipahi, R., Mendez-Barrios, C.-F., Leyva-Ramos, J., 2021. Derivative-dependent control of a fuel cell system with a safe implementation: An artificial delay approach. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering. https://doi.org/10.1177/09596518211012784

Ramirez, L. F., Saldivar, B., Avila Vilchis, J. C., Montes de Oca, S., 2018. Lyapunov-Krasovskii approach to the stability analysis of the milling process. IET Control Theory & Applications 12 (9), 1332-1339. https://doi.org/10.1049/iet-cta.2017.1252

Ramirez, M., Villafuerte, R., Gonzalez, T., Bernal, M., 2015. Exponential estimates of a class of time-delay non linear systems with convex representations. Int. J. of Applied Mathematics and Computer Science 25 (4), 815-826. https://doi.org/10.1515/amcs-2015-0058

Ramirez Jeronimo, L. F., Zenteno Torres, J., Saldivar, B., Davila, J., Avila Vilchis, J. C., 2020. Robust stabilisation of linear time-invariant time-delay systems via first order and super-twisting sliding mode controllers. IET Control Theory & Applications 14 (1), 175-186. https://doi.org/10.1049/iet-cta.2018.6434

Ramirez-Neria, M., Ochoa-Ortega, G., Luviano-Juarez, A., Lozada-Castillo,N., Trujano-Cabrera, M. A., Campos-Lopez, J. P., 2019. Proportional Retarded control of robot manipulators. IEEE Access 7, 13989-13998. https://doi.org/10.1109/ACCESS.2019.2892414

Rocha, E., Mondie, S., Di Loreto, M., 2018. Necessary stability conditions forlinear difference equations in continuous time. IEEE Transactions on Automatic Control 63 (12), 4405-4412. https://doi.org/10.1109/TAC.2018.2822667

Rodriguez-Guerrero, L., Kharitonov, V. L., Mondie, S., 2016. Robust stability of dynamic predictor based control laws for input and state delay systems. Systems & Control Letters 96, 95-102. https://doi.org/10.1016/j.sysconle.2016.07.006

Ross, D. W., Flugge-Lotz, I., 1969. An optimal control problem for systemswith differential difference equation dynamics. SIAM J. on Control and Optimization 7 (4), 609-623. https://doi.org/10.1137/0307044

Santos, O., Mondie, S., Kharitonov, V. L., 2009. Linear quadratic suboptimal control for time-delays systems. Int. J. of Control 82 (1), 147-154. https://doi.org/10.1080/00207170802027401

Santos-Sanchez, N.-F., Raul, S.-C., Santos-Sanchez, O.-J., Romero-Trejo, H.,Garrido-Aranda, E., 2016. On the effects of the temperature control at the performance of a dehydration process: energy optimization and nutrients retention. The Int. J. of Advanced Manufacturing Technology 9 (12), 3157-3171. https://doi.org/10.1007/s00170-016-8481-z

Santos-Sanchez, O.-J., Mondie, S., Rodriguez-Guerrero, L., Carmona-Rosas, J.-C., 2019. Delays compensation for an atmospheric sliced tomatoes dehydration process via state predictors. J. of the Franklin Institute 356 (18), 11473-11491. https://doi.org/10.1016/j.jfranklin.2019.09.036

Santos-Sanchez, O.-J., Velasco-Rebollo, R.-E., Rodriguez-Guerrero, L., Ordaz-Oliver, J.-P., Cuvas-Castillo, C., 2021. Lyapunov redesign for input and state delays systems by using optimal predictive control and ultimate bound approaches: theory and experiments. IEEE Trans. on Industrial Electronics 68 (12), 12575-12583. https://doi.org/10.1109/TIE.2020.3040678

Sumacheva, V. A., Kharitonov, V. L., 2014. Computation of the H2 norm of the transfer matrix of a neutral type system. Differential equations 50 (13), 1752-1759. https://doi.org/10.1134/S0012266114130060

Velasco-Villa, M., Cruz-Morales, R., Rodriguez-Angeles, A., Dominguez-Ortega, C., 2021. Observer-based time-variant spacing policy for a platoon of non-holonomic mobile robots. Sensors 21 (11). https://doi.org/10.3390/s21113824

Villafuerte, R., Mondie, S., Poznyak, A., 2011. Practical stability of time-delaysystems: LMI's approach. European Journal of Control 17 (2), 127-138. https://doi.org/10.3166/ejc.17.127-138

Villafuerte, R., Saldivar, B., Mondie, S., 2013. Practical stability and stabilization of a class of nonlinear neutral type time delay systems with multipledelays: a BMI approach. Int. J. of Control, Automation and Systems 11 (5), 859-867. https://doi.org/10.1007/s12555-013-0083-z

Vite, L., Gomez, M. A., Mondie, S., Michiels, W., 2021a. Stabilization of distributed time-delay systems: a smoothed spectral abscissa optimization approach. Int. J. of Control, 1-29. https://doi.org/10.1080/00207179.2021.1943759

Vite, L., Gomez, M. A., Morales, J., Mondie, S., 2020. A new control schemefor time-delay compensation for structural vibration. In: 21st IFAC World Congress. pp. 4804-4809. https://doi.org/10.1016/j.ifacol.2020.12.1025

Vite, L., Júarez, L., Gomez, M. A., Mondi ́e, S., 2021b. Dynamic predictor-based adaptive cruise control. J. of The Franklin Institute, submitted

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