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

Previous Article
Stability of metabolic networks via Linear-in-Flux-Expressions
NHM Home
This Issue
Next Article
A network model of immigration: Enclave formation vs. cultural integration

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

Toufik Bakir ^1, , Bernard Bonnard ^2, and Jérémy Rouot ^3,,

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.

Keywords: Muscular force and fatigue model, optimization of sampled-data control, non-linear observer, model predictive control.

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

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. doi: 10.1017/CBO9780511804441. Google Scholar
[5]	C. R. Cutler and B. L. Ramaker, Dynamic Matrix Control: A Computer Control Algorithm, In Joint automatic control conference, San Francisco, 1981. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	R. Fletcher, Practical Methods of Optimization, A Wiley-Interscience Publication. John Wiley & Sons, Second edition., Ltd., Chichester, 1987. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	A. Isidori, Non-linear Control Systems, 3rd ed. Berlin, Germany: Springer-Verlag, 1995. doi: 10.1007/978-1-84628-615-5. Google Scholar
[15]	L. F. Law and R. Shields, Mathematical models of human paralyzed muscle after long-term training, Journal of Biomechanics, 40 (2007), 2587-2595. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[19]	L. Wang, Model Predictive Control System Design and Implementation Using MATLAB, Springer, London, 2009. Google Scholar
[20]	E. Wilson, Force Response of Locust Skeletal Muscle, Southampton University, Ph.D. thesis, 2011. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. doi: 10.1017/CBO9780511804441. Google Scholar
[5]	C. R. Cutler and B. L. Ramaker, Dynamic Matrix Control: A Computer Control Algorithm, In Joint automatic control conference, San Francisco, 1981. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	R. Fletcher, Practical Methods of Optimization, A Wiley-Interscience Publication. John Wiley & Sons, Second edition., Ltd., Chichester, 1987. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	A. Isidori, Non-linear Control Systems, 3rd ed. Berlin, Germany: Springer-Verlag, 1995. doi: 10.1007/978-1-84628-615-5. Google Scholar
[15]	L. F. Law and R. Shields, Mathematical models of human paralyzed muscle after long-term training, Journal of Biomechanics, 40 (2007), 2587-2595. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[19]	L. Wang, Model Predictive Control System Design and Implementation Using MATLAB, Springer, London, 2009. Google Scholar
[20]	E. Wilson, Force Response of Locust Skeletal Muscle, Southampton University, Ph.D. thesis, 2011. Google Scholar

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\} $

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 4. General MPC strategy diagram

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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}) $

Download as excel

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]	Luís Tiago Paiva, Fernando A. C. C. Fontes. Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2335-2364. doi: 10.3934/dcdsb.2019098
[2]	Loïc Bourdin, Emmanuel Trélat. Optimal sampled-data control, and generalizations on time scales. Mathematical Control & Related Fields, 2016, 6 (1) : 53-94. doi: 10.3934/mcrf.2016.6.53
[3]	Torsten Trimborn, Lorenzo Pareschi, Martin Frank. Portfolio optimization and model predictive control: A kinetic approach. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6209-6238. doi: 10.3934/dcdsb.2019136
[4]	João M. Lemos, Fernando Machado, Nuno Nogueira, Luís Rato, Manuel Rijo. Adaptive and non-adaptive model predictive control of an irrigation channel. Networks & Heterogeneous Media, 2009, 4 (2) : 303-324. doi: 10.3934/nhm.2009.4.303
[5]	Rudy R. Negenborn, Peter-Jules van Overloop, Tamás Keviczky, Bart De Schutter. Distributed model predictive control of irrigation canals. Networks & Heterogeneous Media, 2009, 4 (2) : 359-380. doi: 10.3934/nhm.2009.4.359
[6]	Lars Grüne, Marleen Stieler. Multiobjective model predictive control for stabilizing cost criteria. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3905-3928. doi: 10.3934/dcdsb.2018336
[7]	Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-13. doi: 10.3934/jimo.2019099
[8]	Yunfei Lv, Yongzhen Pei, Rong Yuan. On a non-linear size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020053
[9]	Judy Day, Jonathan Rubin, Gilles Clermont. Using nonlinear model predictive control to find optimal therapeutic strategies to modulate inflammation. Mathematical Biosciences & Engineering, 2010, 7 (4) : 739-763. doi: 10.3934/mbe.2010.7.739
[10]	Gregory Zitelli, Seddik M. Djouadi, Judy D. Day. Combining robust state estimation with nonlinear model predictive control to regulate the acute inflammatory response to pathogen. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1127-1139. doi: 10.3934/mbe.2015.12.1127
[11]	Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, 2013, 2013 (special) : 393-406. doi: 10.3934/proc.2013.2013.393
[12]	Laurent Pfeiffer. Optimality conditions in variational form for non-linear constrained stochastic control problems. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2020008
[13]	Paolo Buttà, Franco Flandoli, Michela Ottobre, Boguslaw Zegarlinski. A non-linear kinetic model of self-propelled particles with multiple equilibria. Kinetic & Related Models, 2019, 12 (4) : 791-827. doi: 10.3934/krm.2019031
[14]	James P. Nelson, Mark J. Balas. Direct model reference adaptive control of linear systems with input/output delays. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 445-462. doi: 10.3934/naco.2013.3.445
[15]	Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021
[16]	Tan H. Cao, Boris S. Mordukhovich. Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4191-4216. doi: 10.3934/dcdsb.2019078
[17]	Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[18]	Shui-Nee Chow, Yongfeng Li. Model reference control for SIRS models. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 675-697. doi: 10.3934/dcds.2009.24.675
[19]	Filipe Rodrigues, Cristiana J. Silva, Delfim F. M. Torres, Helmut Maurer. Optimal control of a delayed HIV model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 443-458. doi: 10.3934/dcdsb.2018030
[20]	Jérôme Coville. Nonlocal refuge model with a partial control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1421-1446. doi: 10.3934/dcds.2015.35.1421

2018 Impact Factor: 0.871

Tools

Metrics

Other articles
by authors

on AIMS
Toufik Bakir Bernard Bonnard Jérémy Rouot
on Google Scholar
Toufik Bakir Bernard Bonnard Jérémy Rouot

2018 Impact Factor: 0.871

Tools

Article outline

Figures and Tables

2018 Impact Factor: 0.871

Tools

Figures and Tables

2018 Impact Factor: 0.871

Tools

Metrics

Other articles
by authors

on AIMS
Toufik Bakir Bernard Bonnard Jérémy Rouot
on Google Scholar
Toufik Bakir Bernard Bonnard Jérémy Rouot

[Back to Top]