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

Toufik Bakir; Bernard Bonnard; Jérémy Rouot

doi:10.3934/nhm.2019005

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2019, Volume 14, Issue 1: 79-100. Doi: 10.3934/nhm.2019005

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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

^* Corresponding author: Jérémy Rouot
^* Corresponding author: Jérémy Rouot

Received: March 31, 2018

Published: January 2019

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Abstract

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.

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

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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\} $

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

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

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Figure 4. General MPC strategy diagram

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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

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

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

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

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

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

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

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

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

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

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

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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}) $

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