# 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
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##### References:
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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