Article Contents
Article Contents

# A case study of optimal input-output system with sampled-data control: Ding et al. force and fatigue muscular control model

• * Corresponding author: Jérémy Rouot
• 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.

Mathematics Subject Classification: 49K15, 93B07, 92B05.

 Citation:

• Figure 1.  Time evolution of the permanent control (thin continuous line) and sampled-data control for several values of the sampling period $T_s\in \{T/20, T/40, T/200\}$

Figure 2.  Evolution of $K_m$ for different initial conditions (case of $I = 10ms$)

Figure 3.  Relative error of the force for a well known and erroneous $K_m$ initial condition (case of $I = 10ms$)

Figure 4.  General MPC strategy diagram

Figure 5.  (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

Figure 6.  Evolution of $A$ and $\hat{A}$ for $I = 10$, $30\%$ error of $K_m$

Figure 7.  Evolution of $A$ and $\hat{A}$ for $I = 25$, $30\%$ error of $K_m$

Figure 8.  Evolution of $\tau_1$ and $\hat{\tau_1}$ for $I = 10$, $30\%$ error of $K_m$

Figure 9.  Evolution of $\tau_1$ and $\hat{\tau_1}$ for $I = 25$, $30\%$ error of $K_m$

Figure 10.  Evolution of $F$, $\hat{F}$ and $F$ mean value over $I$ for $I = 25$, $30\%$ error of $K_m$, $Fref = 250N$

Figure 11.  Evolution of the force for a reference force of $425N$ and different receding horizons $(3, 5$ and $10)$

Figure 12.  Evolution of the interpulse (control) for a reference force of $425N$ and a preditive horizon of $10$

Figure 13.  Evolution of the amplitude (control) for a reference force of $425N$ and a preditive horizon of $10$

Figure 14.  Evolution of the interpulse (control) for a reference force of 425N and a preditive horizon of 10

Figure 15.  Evolution of the amplitude (control) for a reference force of 425N and a preditive horizon of 3

Table 1.  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 $(A, K_{m}, \tau_{1})$
•  [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.

Figures(15)

Tables(1)