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A case study of optimal input-output system with sampled-data control: Ding et al. force and fatigue muscular control model
1. | Le2i Laboratory EA 7508, Dijon, France |
2. | Univ. Bourgogne Franche-Comté and INRIA Sophia Antipolis, Dijon, France |
3. | Univ. Bourgogne Franche-Comté and EPF École Ingénieur-e-s, Troyes, France |
The objective of this article is to make the analysis of the muscular force response to optimize electrical pulses train using Ding et al. force-fatigue model. A geometric analysis of the dynamics is provided and very preliminary results are presented in the frame of optimal control using a simplified input-output model. In parallel, to take into account the physical constraints of the problem, partial state observation and input restrictions, an optimized pulses train is computed with a model predictive control, where a non-linear observer is used to estimate the state-variables.
References:
[1] |
T. Bakir, B. Bonnard and S. Othman, Predictive control based on non-linear observer for muscular force and fatigue model, Annual American Control Conference (ACC), Milwaukee (2018) 2157-2162.
doi: 10.23919/ACC.2018.8430962. |
[2] |
J. Bobet and R. B. Stein,
A simple model of force generation by skeletal muscle during dynamic isometric contractions, IEEE Transactions on Biomedical Engineering, 45 (1998), 1010-1016.
doi: 10.1109/10.704869. |
[3] |
L. Bourdin and E. Trélat,
Optimal sampled-data control, and generalizations on time scales, Math. Cont. Related Fields, 6 (2016), 53-94.
doi: 10.3934/mcrf.2016.6.53. |
[4] |
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
doi: 10.1017/CBO9780511804441.![]() ![]() ![]() |
[5] |
C. R. Cutler and B. L. Ramaker, Dynamic Matrix Control: A Computer Control Algorithm, In Joint automatic control conference, San Francisco, 1981. |
[6] |
J. Ding, S. A. Binder-Macleod and A. S. Wexler,
Two-step, predictive, isometric force model tested on data from human and rat muscles, J. Appl. Physiol., 85 (1998), 2176-2189.
doi: 10.1152/jappl.1998.85.6.2176. |
[7] |
J. Ding, A. S. Wexler and S. A. Binder-Macleod,
Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains, J. Appl. Physiol., 88 (2000), 917-925.
doi: 10.1152/jappl.2000.88.3.917. |
[8] |
J. Ding, A. S. Wexler and S. A. Binder-Macleod,
A predictive model of fatigue in human skeletal muscles, J. Appl. Physiol., 89 (2000), 1322-1332.
doi: 10.1152/jappl.2000.89.4.1322. |
[9] |
J. Ding, A. S. Wexler and S. A. Binder-Macleod,
Mathematical models for fatigue minimization during functional electrical stimulation, J. Electromyogr. Kinesiol., 13 (2003), 575-588.
doi: 10.1016/S1050-6411(03)00102-0. |
[10] |
R. Fletcher, Practical Methods of Optimization, A Wiley-Interscience Publication. John Wiley & Sons, Second edition., Ltd., Chichester, 1987. |
[11] |
J. P. Gauthier, H. Hammouri and S. Othman,
A simple observer for non-linear systems Application to bioreactors, IEEE Trans. Automat. Control, 37 (1992), 875-880.
doi: 10.1109/9.256352. |
[12] |
R. Gesztelyi, J. Zsuga, A. Kemeny-Beke, B. Varga, B. Juhasz and A. Tosaki,
The Hill equation and the origin of quantitative pharmacology, Arch. Hist. Exact Sci., 66 (2012), 427-438.
doi: 10.1007/s00407-012-0098-5. |
[13] |
R. Hermann and J. Krener,
Non-linear controllability and observability, IEEE Transactions on Automatic Control, AC-22 (1977), 728-740.
doi: 10.1109/tac.1977.1101601. |
[14] |
A. Isidori, Non-linear Control Systems, 3rd ed. Berlin, Germany: Springer-Verlag, 1995.
doi: 10.1007/978-1-84628-615-5. |
[15] |
L. F. Law and R. Shields,
Mathematical models of human paralyzed muscle after long-term training, Journal of Biomechanics, 40 (2007), 2587-2595.
|
[16] |
S. Li, K. Y. Lim and D. G. Fisher, A state space formulation for model predictive control, Springer, New York, 35 (1989), 241-249.
doi: 10.1002/aic.690350208. |
[17] |
J. Richalet, A. Rault, J. L. Testud and J. Papon, Model algorithmic control of industrial processes, In IFAC Proceedings, 10 (1977), 103–120.
doi: 10.1016/S1474-6670(17)69513-2. |
[18] |
H. J. Sussmann and V. Jurdjevic,
Controllability of non-linear systems, J. Differential Equations, 12 (1972), 95-116.
doi: 10.1016/0022-0396(72)90007-1. |
[19] |
L. Wang, Model Predictive Control System Design and Implementation Using MATLAB, Springer, London, 2009. |
[20] |
E. Wilson, Force Response of Locust Skeletal Muscle, Southampton University, Ph.D. thesis, 2011. |
show all references
References:
[1] |
T. Bakir, B. Bonnard and S. Othman, Predictive control based on non-linear observer for muscular force and fatigue model, Annual American Control Conference (ACC), Milwaukee (2018) 2157-2162.
doi: 10.23919/ACC.2018.8430962. |
[2] |
J. Bobet and R. B. Stein,
A simple model of force generation by skeletal muscle during dynamic isometric contractions, IEEE Transactions on Biomedical Engineering, 45 (1998), 1010-1016.
doi: 10.1109/10.704869. |
[3] |
L. Bourdin and E. Trélat,
Optimal sampled-data control, and generalizations on time scales, Math. Cont. Related Fields, 6 (2016), 53-94.
doi: 10.3934/mcrf.2016.6.53. |
[4] |
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
doi: 10.1017/CBO9780511804441.![]() ![]() ![]() |
[5] |
C. R. Cutler and B. L. Ramaker, Dynamic Matrix Control: A Computer Control Algorithm, In Joint automatic control conference, San Francisco, 1981. |
[6] |
J. Ding, S. A. Binder-Macleod and A. S. Wexler,
Two-step, predictive, isometric force model tested on data from human and rat muscles, J. Appl. Physiol., 85 (1998), 2176-2189.
doi: 10.1152/jappl.1998.85.6.2176. |
[7] |
J. Ding, A. S. Wexler and S. A. Binder-Macleod,
Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains, J. Appl. Physiol., 88 (2000), 917-925.
doi: 10.1152/jappl.2000.88.3.917. |
[8] |
J. Ding, A. S. Wexler and S. A. Binder-Macleod,
A predictive model of fatigue in human skeletal muscles, J. Appl. Physiol., 89 (2000), 1322-1332.
doi: 10.1152/jappl.2000.89.4.1322. |
[9] |
J. Ding, A. S. Wexler and S. A. Binder-Macleod,
Mathematical models for fatigue minimization during functional electrical stimulation, J. Electromyogr. Kinesiol., 13 (2003), 575-588.
doi: 10.1016/S1050-6411(03)00102-0. |
[10] |
R. Fletcher, Practical Methods of Optimization, A Wiley-Interscience Publication. John Wiley & Sons, Second edition., Ltd., Chichester, 1987. |
[11] |
J. P. Gauthier, H. Hammouri and S. Othman,
A simple observer for non-linear systems Application to bioreactors, IEEE Trans. Automat. Control, 37 (1992), 875-880.
doi: 10.1109/9.256352. |
[12] |
R. Gesztelyi, J. Zsuga, A. Kemeny-Beke, B. Varga, B. Juhasz and A. Tosaki,
The Hill equation and the origin of quantitative pharmacology, Arch. Hist. Exact Sci., 66 (2012), 427-438.
doi: 10.1007/s00407-012-0098-5. |
[13] |
R. Hermann and J. Krener,
Non-linear controllability and observability, IEEE Transactions on Automatic Control, AC-22 (1977), 728-740.
doi: 10.1109/tac.1977.1101601. |
[14] |
A. Isidori, Non-linear Control Systems, 3rd ed. Berlin, Germany: Springer-Verlag, 1995.
doi: 10.1007/978-1-84628-615-5. |
[15] |
L. F. Law and R. Shields,
Mathematical models of human paralyzed muscle after long-term training, Journal of Biomechanics, 40 (2007), 2587-2595.
|
[16] |
S. Li, K. Y. Lim and D. G. Fisher, A state space formulation for model predictive control, Springer, New York, 35 (1989), 241-249.
doi: 10.1002/aic.690350208. |
[17] |
J. Richalet, A. Rault, J. L. Testud and J. Papon, Model algorithmic control of industrial processes, In IFAC Proceedings, 10 (1977), 103–120.
doi: 10.1016/S1474-6670(17)69513-2. |
[18] |
H. J. Sussmann and V. Jurdjevic,
Controllability of non-linear systems, J. Differential Equations, 12 (1972), 95-116.
doi: 10.1016/0022-0396(72)90007-1. |
[19] |
L. Wang, Model Predictive Control System Design and Implementation Using MATLAB, Springer, London, 2009. |
[20] |
E. Wilson, Force Response of Locust Skeletal Muscle, Southampton University, Ph.D. thesis, 2011. |










Symbol | Unit | Value | description |
— | — | Normalized amount of | |
— | Force generated by muscle | ||
— | Time of the |
||
— | — | Total number of | |
the pulses before time |
|||
— | — | Stimulation pulse index | |
Time constant that commands | |||
the rise and the decay of |
|||
— | Term of the enhancement | ||
in |
|||
— | Scaling factor for the force and | ||
the shortening velocity | |||
of muscle | |||
— | Force decline time constant | ||
when strongly bound | |||
cross-bridges absent | |||
Force decline time constant | |||
due to friction between actin | |||
and myosin | |||
— | — | Sensitivity of strongly bound | |
cross-bridges to |
|||
Value of the variable |
|||
when muscle is not fatigued | |||
— | Value of the variable |
||
when muscle is not fatigued | |||
The value of the variable |
|||
when muscle is not fatigued | |||
Coefficient for the force-model | |||
variable |
|||
model | |||
Coefficient for the force-model | |||
variable |
|||
model | |||
Coefficient for force-model | |||
variable |
|||
model | |||
Time constant controlling the | |||
recovery of |
Symbol | Unit | Value | description |
— | — | Normalized amount of | |
— | Force generated by muscle | ||
— | Time of the |
||
— | — | Total number of | |
the pulses before time |
|||
— | — | Stimulation pulse index | |
Time constant that commands | |||
the rise and the decay of |
|||
— | Term of the enhancement | ||
in |
|||
— | Scaling factor for the force and | ||
the shortening velocity | |||
of muscle | |||
— | Force decline time constant | ||
when strongly bound | |||
cross-bridges absent | |||
Force decline time constant | |||
due to friction between actin | |||
and myosin | |||
— | — | Sensitivity of strongly bound | |
cross-bridges to |
|||
Value of the variable |
|||
when muscle is not fatigued | |||
— | Value of the variable |
||
when muscle is not fatigued | |||
The value of the variable |
|||
when muscle is not fatigued | |||
Coefficient for the force-model | |||
variable |
|||
model | |||
Coefficient for the force-model | |||
variable |
|||
model | |||
Coefficient for force-model | |||
variable |
|||
model | |||
Time constant controlling the | |||
recovery of |
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