# American Institute of Mathematical Sciences

March 2019, 14(1): 79-100. doi: 10.3934/nhm.2019005

## 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

* Corresponding author: Jérémy Rouot

Received  April 2018 Published  January 2019

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.

Citation: Toufik Bakir, Bernard Bonnard, Jérémy Rouot. A case study of optimal input-output system with sampled-data control: Ding et al. force and fatigue muscular control model. Networks & Heterogeneous Media, 2019, 14 (1) : 79-100. doi: 10.3934/nhm.2019005
##### 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.

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##### 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.
Time evolution of the permanent control (thin continuous line) and sampled-data control for several values of the sampling period $T_s\in \{T/20, T/40, T/200\}$
Evolution of $K_m$ for different initial conditions (case of $I = 10ms$)
Relative error of the force for a well known and erroneous $K_m$ initial condition (case of $I = 10ms$)
General MPC strategy diagram
(left half plane) $E_s$ and force profile for applied amplitude and interpulse stimulation, (right half plane) Predicted $E_s$ and force using single move strategy to be optimized
Evolution of $A$ and $\hat{A}$ for $I = 10$, $30\%$ error of $K_m$
Evolution of $A$ and $\hat{A}$ for $I = 25$, $30\%$ error of $K_m$
Evolution of $\tau_1$ and $\hat{\tau_1}$ for $I = 10$, $30\%$ error of $K_m$
Evolution of $\tau_1$ and $\hat{\tau_1}$ for $I = 25$, $30\%$ error of $K_m$
Evolution of $F$, $\hat{F}$ and $F$ mean value over $I$ for $I = 25$, $30\%$ error of $K_m$, $Fref = 250N$
Evolution of the force for a reference force of $425N$ and different receding horizons $(3, 5$ and $10)$
Evolution of the interpulse (control) for a reference force of $425N$ and a preditive horizon of $10$
Evolution of the amplitude (control) for a reference force of $425N$ and a preditive horizon of $10$
Evolution of the interpulse (control) for a reference force of 425N and a preditive horizon of 10
Evolution of the amplitude (control) for a reference force of 425N and a preditive horizon of 3
Margin settings
 Symbol Unit Value description $C_{N}$ — — Normalized amount of $Ca^{2+}$-troponin complex $F$ $N$ — Force generated by muscle $t_{i}$ $ms$ — Time of the $i^{th}$ pulse $n$ — — Total number of the pulses before time $t$ $i$ — — Stimulation pulse index $\tau_{c}$ $ms$ $20$ Time constant that commands the rise and the decay of $C_{N}$ $R_{0}$ — $1.143$ Term of the enhancement in $C_{N}$ from successive stimuli $A$ $\frac{N}{ms}$ — Scaling factor for the force and the shortening velocity of muscle $\tau_{1}$ $ms$ — Force decline time constant when strongly bound cross-bridges absent $\tau_{2}$ $ms$ $124.4$ Force decline time constant due to friction between actin and myosin $K_{m}$ — — Sensitivity of strongly bound cross-bridges to $C_{N}$ $A_{rest}$ $\frac{N}{ms}$ $3.009$ Value of the variable $A$ when muscle is not fatigued $K_{m, rest}$ — $0.103$ Value of the variable $K_{m}$ when muscle is not fatigued $\tau_{1, rest}$ $ms$ $50.95$ The value of the variable $\tau_{1}$ when muscle is not fatigued $\alpha_{A}$ $\frac{1}{ms^{2}}$ $-4.0 10^{-7}$ Coefficient for the force-model variable $A$ in the fatigue model $\alpha_{K_{m}}$ $\frac{1}{msN}$ $1.9 10 ^{-8}$ Coefficient for the force-model variable $K_{m}$ in the fatigue model $\alpha_{\tau_{1}}$ $\frac{1}{N}$ $2.1 10^{-5}$ Coefficient for force-model variable $\tau_{1}$ in the fatigue model $\tau_{fat}$ $s$ $127$ Time constant controlling the recovery of $(A, K_{m}, \tau_{1})$
 Symbol Unit Value description $C_{N}$ — — Normalized amount of $Ca^{2+}$-troponin complex $F$ $N$ — Force generated by muscle $t_{i}$ $ms$ — Time of the $i^{th}$ pulse $n$ — — Total number of the pulses before time $t$ $i$ — — Stimulation pulse index $\tau_{c}$ $ms$ $20$ Time constant that commands the rise and the decay of $C_{N}$ $R_{0}$ — $1.143$ Term of the enhancement in $C_{N}$ from successive stimuli $A$ $\frac{N}{ms}$ — Scaling factor for the force and the shortening velocity of muscle $\tau_{1}$ $ms$ — Force decline time constant when strongly bound cross-bridges absent $\tau_{2}$ $ms$ $124.4$ Force decline time constant due to friction between actin and myosin $K_{m}$ — — Sensitivity of strongly bound cross-bridges to $C_{N}$ $A_{rest}$ $\frac{N}{ms}$ $3.009$ Value of the variable $A$ when muscle is not fatigued $K_{m, rest}$ — $0.103$ Value of the variable $K_{m}$ when muscle is not fatigued $\tau_{1, rest}$ $ms$ $50.95$ The value of the variable $\tau_{1}$ when muscle is not fatigued $\alpha_{A}$ $\frac{1}{ms^{2}}$ $-4.0 10^{-7}$ Coefficient for the force-model variable $A$ in the fatigue model $\alpha_{K_{m}}$ $\frac{1}{msN}$ $1.9 10 ^{-8}$ Coefficient for the force-model variable $K_{m}$ in the fatigue model $\alpha_{\tau_{1}}$ $\frac{1}{N}$ $2.1 10^{-5}$ Coefficient for force-model variable $\tau_{1}$ in the fatigue model $\tau_{fat}$ $s$ $127$ Time constant controlling the recovery of $(A, K_{m}, \tau_{1})$
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