Wei Lin

Editorships

1999-2001 Associate Editor, IEEE Transactions on Automatic Control

2002-2003 Guest Editor, Special Issue on ``New Directions in Nonlinear Control,'' IEEE Trans. Automat. Contr.

2002-2005 Associate Editor, Automatica

2011-Present Associate Editor, Int. J. of Robust and Nonlinear Control

2005-2010 Subject Editor, Int. J. of Robust and Nonlinear Control

2003-2008 Associate Editor, J. of Control Theory and Applications

Invited Plenary Addresses

2016 Plenary Speaker, The 8th Asia-Pacific Power and Energy Engineering Conference (APPEEC'16), Suzhou, China, April 15-17

2011 Plenary Speaker, International Conference on Applied, Computational Mathematics and Optimization, Chiang Mai, Thailand, January 6-10

2009 Plenary Speaker, 2nd International Conference on Intelligent Networks and Systems, Tianjin, China, November 1-3

2009 Semi-Plenary Speaker, 21st Chinese Control and Decision Conference, Guilin, June 17-19

2007 Plenary Speaker, 7th IFAC Symposium on Nonlinear Control Systems (NOLCOS'07), Pretoria, South Africa, August 22-24

2004 Semi-Plenary Speaker, 6th IFAC Symposium on Nonlinear Control Systems (NOLCOS'04), Stuttgart, Germany, September 1-3

2001 Keynote Speaker, 4th Nonlinear Control Network Workshop: Nonlinear and Adaptive Control, Sheffield, U.K., June 25-28

Recent Publications (RP)

[RP60] X. Yu and W. Lin, ``Universal output feedback control of a class of uncertain nonlinear systems with unknown delays in state, input and output'', Automatica, accepted on Nov. 25, 2024 (in press).

[RP59] X. Yu and W. Lin, ``Almost Sure Asymptotic Stabilization of Stochastic Nonlinear Systems by Sampled-Data State and Output Feedback,''  IEEE Trans. on Automat. Contr., vol. 70, no. 2, pp. 1-16, 2025. Published online on August 26, 2024. DOI: 10.1109/TAC.2024.3450350

[RP58] X. Yu and W. Lin, ``Non-identifier based adaptive control of a chain of integrators and perturbations with unknown delays and parameters'', Automatica, vol. 171, Jan. 2025. Article 111969. Published online Oct. 22, 2024. https://doi.org/10.1016/j.automatica.2024.111969

[RP57] X. Yu and W. Lin, ``On input delay tolerance of stochastic nonlinear systems with dominant homogeneity of degree zero,''  IEEE Trans. on Automat. Contr., vol. 69, no. 10, pp. 6935-6950, 2024.  Published online on March 28. DOI: 10.1109/TAC.2024.3382611

[RP56] C. Zhao and W. Lin, ``A nonsmooth analysis and synthesis for semiglobal input delay tolerance of nonlinear systems without LES,''  Automatica, vol. 166,  pp. 1-15, May 2024.  Article 111725.  https://doi.org/10.1016/j.automatica.2024.111725.

[RP55] X. Yu and W. Lin, ``Sampled-data feedback stabilization in mean square for stochastic homogeneous systems,''  IEEE Trans. on Automat. Contr., vol. 69, no. 10, pp. 6805-6820, 2024.  Published online on March 18.  DOI: 10.1109/TAC.2024.3377145

[RP54] X. Yu and W. Lin, "A Lyapunov-Krasovskii method for semiglobal stabilization of a class of time-delay nonlinear systems with uncontrollable linearization by linear feedback" IEEE Trans. on Automat. Contr., vol. 69, no. 6, pp. 4110-4117, 2024.  Published online Dec. 26, 2023.  DOI:  10.1109/TAC.2023.3347340

[RP53] C. Zhao and W. Lin, "Sampled‐data control of time‐delay uncertain systems with uncontrollable/unobservable linearization by memoryless feedback," Special Issue on ``Time-delay systems: recent trend and applications'', Int. J. of Robust and Nonlinear Control, vol. 34, pp. 6532-6549,  February 2024. https://doi.org/10.1002/rnc.7240

[RP52] X. Liu and W. Lin, "Predictor-based observer and controller for nonlinear systems with input/output delays", Syst. & Contr. Lett., vol. 184, Feb. 2024. Article 105723.  https://doi.org/10.1016/j.sysconle.2024.105723

[RP51] X. Niu and W. Lin, ``Global state regulation of time-delay minimum-phase systems by delay-free dynamic compensation,'' Automatica, vol. 159, January, 2024. Article 111378. https://doi.org/10.1016/j.automatica.2023.111378

[RP50] W. Lin, "When is a nonlinear system semiglobally asymptotically stabilizable by digital feedback?" IEEE Trans. on Automat. Contr., vol. 65, no. 11, pp. 4584-4599 (2020).  Published online Dec. 9, 2019.  DOI:  10.1109/TAC.2019.2958481

[RP49] Y. J. Wang and W. Lin, "Input delay tolerance of nonlinear systems under smooth feedback: A semiglobal control framework," IEEE. Trans. on Automat. Contr., vol. 67, no. 1, pp. 146-161, 2022. Published online Dec. 23, 2020.  DOI: 10.1109/TAC.2020.3046709

[RP48] X. Yu and W. Lin, "Global input delay tolerance of mimo nonlinear systems under nonsmooth feedback: A homogeneous perspective," IEEE. Trans. on Automat. Contr., vol. 68, no. 7, pp. 3992-4007, 2023. Published online August 9, 2022. DOI: 10.1109/TAC.2022.3197558.

[RP47] X. L. Liu and W. Lin, "A predictor-based linearization approach for local stabilization of nonlinear systems with input delay", IEEE Trans. on Automat. Contr., vol. 68, no. 7, pp. 4297-4304, 2023. Published online August 23, 2022. DOI: 10.1109/TAC.2022.3201007

[RP46]  W. Lin and J. W. Sun, "Using memoryless output feedback to semiglobally stabilize a class of SISO nonlinear systems with input delay", IEEE Trans. on Automat. Contr., vol. 68, no. 3, pp. 1870-1877, 2023. Published online April 19, 2022.  DOI: 10.1109/TAC.2022.3168536.

[RP45] X. Ding, W. Lin, J. Xu, Y.Z. Sun, L. Yao, and B. Mao, ``Coordinated frequency control for isolated power systems with high penetration of DFIG-based wind power'', CSEE Journal of Power and Energy Systems,  vol. 10, no. 4, pp. 1399-1414, July, 2024. Published online May 6, 2022.  DOI: 10.17775/CSEEJPES.2021.08320

[RP44] S. Yang, G. Tao, B. Jiang, Y. Zhang, and W. Lin, ``Modeling and adaptive control of air vehicles with partial nonlinear parametrization," Automatica, vol. 149, March 2023. Article 110805.  https://doi.org/10.1016/j.automatica.2022.110805

[RP43] J. Sun and W. Lin, "Non-identifier based adaptive regulation of feedforward systems with nonlinear parametrization and delays: A saturation control scheme", Special Issue Dedicated to the 80th Birthday of Professor Arthur J. Krener, Syst. & Contr. Lett., vol. 173, March 2023. Article 105456.  https://doi.org/10.1016/j.sysconle.2023.105456

[RP42] X.Niu and W. Lin, "Low gain control of linear growth feedforward systems by delayed output signals and sparse sampling", Syst. & Contr. Lett., vol. 172, Feb. 2023. Article No. 105430. https://doi.org/10.1016/j.sysconle.2022.105430

[RP41] C.R. Zhao and W. Lin, "Homogeneity, forward completeness, and global stabilization of a family of time-delay nonlinear systems by memoryless non-Lipschitz continuous feedback", IEEE Trans. on Automat. Contr., vol. 67, no. 11, pp. 5916-5931, 2022. Published online Nov. 4, 2021. DOI: 10.1109/TAC.2021.3124992   

[RP40] C. Zhao and W. Lin, ``Homogeneous output feedback design for time-delay nonlinear integrators and beyond: An emulation," Automatica, vol. 146, Dec. 2022. Article 110641.   https://doi.org/10.1016/j.automatica.2022.110641

[RP39] Y.J. Wang and W. Lin, "A semiglobal approach for stabilizing nonlinear systems with state delays by memoryless linear feedback", IEEE Trans. on Automat. Contr., vol. 67, no. 11, pp. 6119-6126, 2022.  Published online Nov. 15, 2021.  DOI: 10.1109/TAC.2021.3126773

[RP38] C. R. Zhao and W. Lin, ``Memoryless homogeneous control of time-delay nonlinear integrators and beyond," Automatica, vol. 141, July 2022. Article 110299.   https://doi.org/10.1016/j.automatica.2022.110299

[RP37]  Y. J. Wang and W. Lin, "Stability preservation of nonlinear systems with sampled-delayed input", IEEE Control Systems Letters, vol. 6, pp. 2761-2766, May 17, 2022.  DOI: 10.1109/LCSYS.2022.3175767

[RP36]  X.R. Niu and W. Lin, "Sampled-data delayed output feedback control of a class of nonlinear systems", IEEE Control Systems Letters, vol. 6, pp. 2906-2911, June 1, 2022. DOI: 10.1109/LCSYS.2022.3179671

[RP35] X.L. Liu, W. Lin, C.R. Zhao and Y.H. Hu, "Digital control of a family of time-delay feedforward systems with sparse sampling", Syst. Contr. Lett., vol. 163, May 2022. Article 105200. https://doi.org/10.1016/j.sysconle.2022.105200

[RP34] X.L. Liu, W. Lin, C.R. Zhao and Y.H. Hu, "Sampled-data control of a class of time-delay nonlinear systems via memoryless feedback", Syst. & Contr. Lett., vol. 157, Nov. 2021. Article 105048.  https://doi.org/10.1016/j.sysconle.2021.105048

[RP33] W. Lin, Y.J. Wang and X.L. Liu, ``Asymptotic stabilization of nonlinear systems with long input delay via memoryless feedback: A linearization method," Automatica, vol. 130, August 2021. Article 109731. https://doi.org/10.1016/j.automatica.2021.109731                               

[RP32]  X. Niu, W. Lin and X.W. Gao, "Static output feedback control of a chain of integrators with input constraints using multiple saturations and delays," Automatica, vol. 125, March 2021.  Art. no. 109457.   https://doi.org/10.1016/j.automatica.2020.109457

[RP31] W. Lin and J. W. Sun, "New results and examples in semiglobal asymptotic stabilization of nonaffine systems by sampled-data output feedback," Syst. & Contr. Lett., vol. 148, Feb. 2021. Art. no. 104850.  https://doi.org/10.1016/j.sysconle.2020.104850

[RP30] J.W. Sun and W. Lin, "A dynamic gain-based saturation control strategy for feedforward systems with long delays in state and input", IEEE Trans. on Automat. Contr., vol. 66, no. 9, pp. 4357-4364, 2021. Published online Oct. 23, 2020.  DOI: 10.1109/TAC.2020.3030744

[RP29] C. Zhao and W. Lin, "Global stabilization by memoryless feedback for nonlinear systems with a limited input delay and large state delays", IEEE Trans. on Automat. Contr., vol. 66, no. 8, pp. 3702-3709, 2021.  Published online Sept. 1, 2020.  DOI:  10.1109/TAC.2020.3021053

[RP28] W. Lin, X.R. Niu, X.W. Gao and G. Tao, "Adaptive regulation of discrete-time nonaffine systems with parametric uncertainty," IEEE Trans. on Automat. Contr., vol. 66, no. 5, pp. 2365-2371, 2021. DOI: 10.1109/TAC.2020.3006855

[RP27] J.W. Sun, W. Lin, M. G. Hong and K. Loparo, "Voltage regulation of DC-microgrid with PV and battery," IEEE Trans. on Smart Grid,  vol. 11, no. 6, pp. 4662-4675 (2020).  DOI:  10.1109/TSG.2020.3005415 

[RP26] W. Lin and X. Zhang, "A dynamic feedback framework for control of time-delay nonlinear systems with unstable zero dynamics," IEEE Trans. on Automat. Contr., vol. 65, no. 8, pp. 3317-3332 (2020).  Published online Sept. 11, 2019.  DOI: 10.1109/TAC.2019.2940865

[RP25] Y.J. Wang and W. Lin, "Semiglobal asymptotic stabilization of nonlinear systems with triangular zero dynamics by linear feedback," Automatica, vol. 115, May 2020. Art. no. 108870.  https://doi.org/10.1016/j.automatica.2020.108870

[RP24] C.R. Zhao and W. Lin, "Memoryless linear feedback control for a class of upper-triangular systems with large delays in the state and input", Syst. & Contr. Lett., vol. 139, May 2020. Art. no. 104679.  https://doi.org/10.1016/j.sysconle.2020.104679

[RP23] S. W. Su and W. Lin, "Global asymptotic tracking for nonminimum‐phase nonlinear systems in output feedback form," Int. J. of Robust and Nonlinear Control, vol. 30, pp. 4485-4502  (2020).   http://dx.doi.org/10.1002/rnc.4991

[RP22] X. Zhang and W. Lin, "Nonidentifier‐based adaptive control for nonlinearly parameterized systems with measurement uncertainty," Int. J. of Robust and Nonlinear Control, vol. 30, pp. 3055-3072 (2020).  https://doi.org/10.1002/rnc.4921

[RP21] C. Zhang, W. Lin, D. P. Ke and Y.Z. Sun, "Smoothing tie-line power fluctuations for industrial microgrids by demand side control: An output regulation approach,"  IEEE Trans. on Power Systems,  vol. 34, no. 5, pp. 3716-3728 (2019).  DOI: 10.1109/TPWRS.2019.2906252   

[RP20]  W. Lin, R. Kanya and P. Radom, "L_gV-type adaptive controllers for uncertain non-affine systems and application to a DC-microgrid with PV and battery," IEEE Trans. on Automat. Contr.,  vol. 64, no. 5, pp. 2182-2189 (2019).  DOI: 10.1109/TAC.2018.2864268 

[RP19] X. Zhang and W. Lin, "A K-filter-based adaptive control for nonlinear systems with unknown parameters in state and output equations," Automatica, vol. 105, pp. 186-197 (2019).  https://doi.org/10.1016/j.automatica.2019.03.013

[RP18] W. Lin and W. Wei, "Semiglobal asymptotic stabilization of lower triangular systems by digital output feedback," IEEE Trans. on Automat. Contr., vol. 64, no. 5, pp. 2135-2141(2019). Published online 8/8/2018. DOI:  10.1109/TAC.2018.2864259  

[RP17] X. Zhang, W. Lin and Y. Lin, "Iterative changing supply rates, dynamic state feedback, and adaptive stabilization of time-delay systems," IEEE Trans. on Automat. Contr., vol. 64, no. 2, pp. 751-758 (2019). DOI: 10.1109/TAC.2018.2838319  

[RP16] X. Zhang and W. Lin, "Robust output feedback control of polynomial growth nonlinear systems with measurement uncertainty," Int. J. of Robust and Nonlinear Control, vol. 29, pp. 4562-4576 (2019).  https://doi.org/10.1002/rnc.4638

[RP15] H.Y. Zhao, M.G. Hong, W. Lin and K. A. Loparo, ``Voltage and frequency regulation of microgrid with battery energy storage systems,''  IEEE Trans. on Smart Grid, vol. 10, no. 1, pp. 414-424 (2019).  DOI: 10.1109/TSG.2017.2741668 

[RP14] Y. Zhang, G. Tao, M. Chen, W. Lin and Z. Q. Zhang, ``Relative degrees and implicit function-based control of discrete-time noncanonical form neural network systems,'' IEEE Trans. on Cybernetics, vol. 50, no.2, pp. 514-524 (2020). DOI: 10.1109/TCYB.2018.2869335 

[RP13] K. Rattanamongkhonkun, R. Pongvuthithum and W. Lin, ``Nonsmooth feedback stabilization of a class of nonlinear systems with unknown control direction and time delay,''  Int. J. of Robust and Nonlinear Control, vol. 28, pp. 5358-5374 (2018).  https://doi.org/10.1002/rnc.4317

[RP12] X. Zhang, W. Lin and Y. Lin, "Adaptive control of time-delay cascade systems with unknown parameters by partial state feedback," Automatica, vol. 94, pp. 45-54 (2018). https://doi.org/10.1016/j.automatica.2018.04.007

[RP11] R. Pongvuthithum, K. Rattanamongkhonkun and W. Lin, "Asymptotic regulation of time-delay nonlinear systems with unknown control directions," IEEE Trans. on Automat. Contr., vol. 63, no. 5, pp. 1495-1502 (2018).  DOI: 10.1109/TAC.2017.2748898 

[RP10] W. Lin and W. Wei, ``Robust control of a family of uncertain nonminimum‐phase systems via continuous‐time and sampled‐data output feedback,''  Int. J. of Robust and Nonlinear Control, vol 28, pp. 1440-1455 (2018).   https://doi.org/10.1002/rnc.3963

[RP9] Z. Dai and W. Lin, ``Adaptive estimation of three-phase grid voltage parameters under unbalanced faults and harmonic disturbances,'' IEEE Trans. on Power Electronics, vol. 32, pp. 5613-5627 (2017). DOI: 10.1109/TPEL.2016.2605131 

[RP8] W. Lin, Z.T. Lu and W. Wei, "Asymptotic tracking control for wind turbines in variable speed mode, " IEEE/CAA Journal of Automatic Sinica, A Special Issue on Control and Optimization in Renewable Energy Systems, vol. 4, No. 3, pp. 569-576 (2017). DOI: 10.1109/JAS.2017.7510568

[RP7] X. Zhang, W. Lin and Y. Lin, "Dynamic partial state feedback control of cascade systems with time-delay," Automatica, vol. 77, no. 3, pp. 370-379 (2017).  https://doi.org/10.1016/j.automatica.2016.09.037

[RP6] X. Zhang, W. Lin and Y. Lin, "Nonsmooth feedback control of time-delay nonlinear systems: A dynamic gain based approach," IEEE Trans. on Automat. Contr., vol. 62, no. 1, pp. 438-444 (2017).  DOI: 10.1109/TAC.2016.2562059

[RP5] Z.J. Liu, W. Lin, Y.L. Geng and P. Yang, "Intent pattern recognition of lower-limb motion based on mechanical sensors, " IEEE/CAA Journal of Automatic Sinica, vol. 4, no. 4, pp. 651-660 (2017). DOI: 10.1109/JAS.2017.7510619

[RP4] T. Cui, W. Lin, Y.Z. Sun, J. Xu and H. Zhang, "Excitation voltage control for emergency frequency regulation of island power systems with voltage-dependent loads," IEEE Trans. on Power Systems, vol. 31, no. 2, pp. 1204-1217 (2016). DOI: 10.1109/TPWRS.2015.2415556 

[RP3] W. Lin, W. Wei and G.Q. Ye, "Global stabilization of a class of nonminimum-phase nonlinear systems by sampled-data output feedback," IEEE Trans. on Automat. Contr., vol. 61, no. 10, pp. 3076-3082 (2016).  DOI: 10.1109/TAC.2015.2498703 

[RP2] W. Lin and J. F. Wei, "Output feedback control of a class of nonminimum‐phase nonlinear systems," Int. J. of Robust and Nonlinear Control, vol. 26, pp. 3631-3644 (2016). https://doi.org/10.1002/rnc.3522

[RP1] Z.Y. Dai, W. Lin and H. Lin, "Estimation of single-phase grid voltage parameters with zero steady-state error," IEEE Trans. on Power Electronics, vol. 31, no. 5, pp. 3867-3879 (2016).  DOI: 10.1109/TPEL.2015.2461665 

Selected Papers (SP)

[SP20] W. Lin and H. Lei, "Taking advantage of homogeneity: A unified framework for output feedback control of nonlinear systems," Proc. of the 7th IFAC Nonlinear Control Systems Symposium, Pretoria, South Africa, August 22-24, pp. 27-38, 2007 (Plenary Lecture).  IFAC Proceedings Vol. 40 (12), 1191-1202.   https://doi.org/10.3182/20070822-3-ZA-2920.00197

[SP19] C. Qian and W. Lin, "A continuous feedback approach to global strong stabilization of nonlinear systems," IEEE Trans. on Automat. Contr., Vol. 46, No. 7, pp. 1061-1079 (2001). DOI: 10.1109/9.935058

[SP18] C. J. Qian and W. Lin, "Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems," IEEE Trans. on Automat. Contr., Vol. 51, No. 9, pp. 1457-1471 (2006). DOI: 10.1109/TAC.2006.880955

[SP17] B. Yang and W. Lin, "Robust output feedback stabilization of uncertain nonlinear systems with uncontrollable and unobservable linearization," IEEE Trans. on Automat. Contr., Vol. 50, No. 5, pp. 619-630 (2005). DOI: 10.1109/TAC.2005.847084

[SP16] B. Yang and W. Lin, "Homogeneous observers, iterative design, and global stabilization of high-order nonlinear systems by smooth output feedback," IEEE Trans. on Automat. Contr., Vol. 49, No. 7, pp. 1069-1080 (2004). DOI: 10.1109/TAC.2004.831186

[SP15] C. Qian and W. Lin, "Practical output tracking of nonlinear systems with uncontrollable unstable linearization," IEEE Trans. on Automat. Contr., Vol. 47, No.1, pp. 21-36 (2002). DOI: 10.1109/9.981720

[SP14] W. Lin and C. Qian, "Adaptive control of nonlinearly parameterized systems: the smooth feedback case," IEEE Trans. on Automat. Contr., Vol. 47, No. 8, pp. 1249-1266 (2002). DOI: 10.1109/TAC.2002.800773

[SP13] W. Lin and C. Qian, "Adaptive control of nonlinearly parameterized systems: a nonsmooth feedback framework," IEEE Trans. on Automat. Contr., Vol. 47, No. 5, pp. 757-774 (2002). DOI: 10.1109/TAC.2002.1000270

[SP12] W. Lin and C. Qian, "Adding one power integrator: a tool for global stabilization of high-order lower-triangular systems," Systems and Control Letters, Vol. 39, No. 5, pp. 339-351 (2000). https://doi.org/10.1016/S0167-6911(99)00115-2

[SP11] H. Lei and W. Lin, "Universal adaptive control of nonlinear systems with unknown growth rate by output feedback,"  Automatica, Vol. 42, pp. 1783-1789 (2006).  https://doi.org/10.1016/j.automatica.2006.05.006

[SP10] X. Huang, W. Lin and B. Yang, "Global finite-time stabilization of a class of uncertain nonlinear systems," Automatica, Vol. 41, pp. 881-888 (2005). https://doi.org/10.1016/j.automatica.2004.11.036

[SP9] C. Qian and W. Lin, "Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm," IEEE Trans. on Automat. Contr., Vol. 47, No. 10, pp. 1710-1715 (2002). DOI: 10.1109/TAC.2002.803542

[SP8] W. Lin and R. Pongvuthithum, "Adaptive output tracking of inherently nonlinear systems with nonlinear parameterization," IEEE Trans. on Automat. Contr., Vol. 48, pp. 1737-1749 (2003). DOI: 10.1109/TAC.2003.817922

[SP7] W. Lin and T. Shen, "Robust passivity and feedback design for minimum-phase nonlinear systems with structureal uncertainty," Automatica, Vol. 35, No. 1, pp. 35-47 (1999). https://doi.org/10.1016/S0005-1098(98)00120-4

[SP6] A. Isidori and W. Lin, ``Global L2-gain design for a class of nonlinear systems," Systems and Control Letters, Vol. 34, No.5, pp. 295-302 (1998).  https://doi.org/10.1016/S0167-6911(98)00010-3

[SP5] W. Lin, "Global asymptotic stabilization of general nonlinear systems with stable free dynamics via passivity and bounded feedback," Automatica, Vol. 32, pp. 915-924 (1996). https://doi.org/10.1016/0005-1098(96)00013-1

[SP4] W. Lin and C. I. Byrnes, "H-infinity control of discrete-time nonlinear systems," IEEE Trans. on Automat. Contr., Vol. 41, pp. 494-510 (1996). DOI: 10.1109/9.489271

[SP3] W. Lin, "Feedback stabilization of general nonlinear control systems: a passive system approach," Systems and Control Letters, Vol. 25, pp. 41-52 (1995).  https://doi.org/10.1016/0167-6911(94)00056-2

[SP2] W. Lin and C. I. Byrnes, "Discrete-time nonlinear H∞ control with measurement feedback," Automatica, Vol. 31, pp. 419-434 (1995).  https://doi.org/10.1016/0005-1098(94)00116-Z

[SP1] C. I. Byrnes and W. Lin, "Losslessness, feedback equivalence, and the global stabilization of discrete-time nonlinear systems," IEEE Trans. on Automat. Contr.,Vol. 39, pp. 83-98 (1994). DOI: 10.1109/9.273341