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Control of Nonlinear Biological Systems by Non-minimal State Variable Feedback

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Abstract

We contrast biostatistical methods for optimal treatment determination with optimal control methodology, originally developed in the engineering literature but now used more widely. We describe non-minimal state space (NMSS) control methods for biological systems, with a particular focus on the use of state-dependent parameter models to represent system nonlinearities. Three examples are considered, namely the control of (i) a nonlinear forced logistic function implemented with a time delay; (ii) athletic horse heart rate with potential application for training improvement; and (iii) a physically-based simulation model for the uptake of CO2 by plant leaves in response to light intensity, with application to closed-environment grow cells. Although all three examples have been extensively studied in the literature, the novelty of the present article is in the NMSS formulation and in the application of a recently developed state-dependent (nonlinear) control algorithm. In the case of the leaf photosynthesis simulation, however, the linear NMSS algorithm yields satisfactory results, illustrating the inherent robustness of feedback.

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Notes

  1. A doctoral studentship on this subject has recently been awarded at Lancaster University (http://www.lancs.ac.uk/staff/taylorcj/projects/grow.htm), and the associated practical results will be reported in future publications.

References

  1. Aerts JM, Van Buggenhout S, Lippens M, Buyse J, Decuypere E, Vranken E, Berckmans D (2003) Active control of the growth trajectory of broiler chickens based on on-line animal responses. Poultry Sci 82:1853–1862

    Article  Google Scholar 

  2. Aerts JM, Gebruers F, Van Camp E, Berckmans D (2008) Controlling horse heart rate as a basis for training improvement. Comput Electron Agric 64:78–84

    Article  Google Scholar 

  3. Alleyne A, Liu R (2000) A simplified approach to force control for electro-hydraulic systems. Control Eng Pract 8:1347–1356

    Article  Google Scholar 

  4. Banks HT, Lewis BM, Tran HT (2007) Nonlinear feedback controllers and compensators: a state-dependent Riccati equation approach. Comput Optim Appl 37:177–218

    Article  MATH  MathSciNet  Google Scholar 

  5. Chalabi ZS (1992) A generalized optimization strategy for dynamic CO2 enrichment in a greenhouse. Eur J Oper Res 59:308–312

    Article  MATH  Google Scholar 

  6. Couroucé A (1999) Field exercise testing for assessing fitness in French standardbred trotters. Vet J 157:112–122

    Article  Google Scholar 

  7. Dorf RC, Bishop RH (2008) Modern control systems, 11th edn. Prentice Hall, New York

    Google Scholar 

  8. Gawthrop PJ, Wang L, Young PC (2007) Continuous-time non-minimal state-space design. Int J Control 80:1690–1697

    Article  MATH  MathSciNet  Google Scholar 

  9. Gonzalez AH, Perez JM, Odloak D (2009) Infinite horizon MPC with non-minimal state space feedback. J Process Control 19:473–481

    Article  Google Scholar 

  10. Huang YM, Hsu MJ, Lin CH, Wei SH, Chang YJ (2010) The non-linear relationship between muscle voluntary activation level and voluntary force measured by the interpolated twitch technique. Sensors 10:796–807

    Article  Google Scholar 

  11. Jovanov E, O’Donnel Lords A, Raskovic D, Cox P, Adhami R, Andrasik F (2003) Stress monitoring using a distributed wireless intelligent sensor system. IEEE Eng Med Biol Mag 22:49–55

    Article  Google Scholar 

  12. Kettlewell PJ, Mitchell MA, Meeks I (1997) An implantable radio-telemetry system for remote monitoring of heart rate and deep body temperature in poultry. Comput Electron Agric 19:161–175

    Article  Google Scholar 

  13. Kirschbaum MUF, Gross LJ, Pearcy RW (1988) Observed and modelled stomatal responses to dynamic light environments in the shade plant Alocasia macrorrhiza. Plant Cell Environ 11:111–121

    Google Scholar 

  14. Leor-Librach RJ, Bobrovsky BZ, Eliash S, Kaplinsky E (1999) Computer-controlled heart rate increase by isoproterenol infusion: mathematical modeling of the system. Am J Physiol 277:H1478–H1483

    Google Scholar 

  15. Massa GD, Kim HH, Wheeler RM, Mitchell C (2008) Plant productivity in response to led lighting. HortScience 43:1951–1956

    Google Scholar 

  16. Murphy S (2003) Optimal dynamic treatment regimes. J R Stat Soc B 65:331–366

    Article  MATH  Google Scholar 

  17. O’Quigley J, Zohar S (2010) Retrospective robustness of the continual reassessment method. J Biopharm Stat 20:1013–1025

    Article  MathSciNet  Google Scholar 

  18. Pearcy RW, Gross LJ, He D (1997) An improved dynamic model of photosynthesis for estimation of carbon gain in sunfleck light regimes. Plant Cell Environ 20:411–424

    Article  Google Scholar 

  19. Rao RR, Aufderheide B, Bequette BW (2003) Experimental studies on multiple-model predictive control for automated regulation of hemodynamic variables. IEEE Trans Biomed Eng 50:277–288

    Article  Google Scholar 

  20. Rietmann TR, Stuart AEA, Bernasconi P, Stauffacher M, Auer JA, Weishaupt MA (2004) Assessment of mental stress in warmblood horses: Heart rate variability in comparison to heart rate and selected behavioural parameters. Appl Anim Behav Sci 121–136

  21. Ross SM (1995) Introduction to stochastic dynamic programming. Academic Press, San Diego

    Google Scholar 

  22. Summers D, Cranford JG, Healey BP (2000) Chaos in periodically forced discrete-time ecosystem models. Chaos Solitons Fractals 11:2331–2342

    Article  MATH  MathSciNet  Google Scholar 

  23. Taylor CJ, Seward DW (2010) Control of a dual-arm robotic manipulator. Nucl Eng Int 55:24–26

    Google Scholar 

  24. Taylor CJ, Shaban EM (2006) Multivariable proportional-integral-plus (PIP) control of the ALSTOM nonlinear gasifier simulation. IEE Proc, Control Theory Appl 153(3):277–285

    Article  Google Scholar 

  25. Taylor CJ, Young PC, Chotai A (1996) PIP optimal control with a risk sensitive criterion. In: UKACC international conference (Control 1996), Exeter, UK

    Google Scholar 

  26. Taylor CJ, Chotai A, Young PC (2000) State space control system design based on non-minimal state-variable feedback: further generalisation and unification results. Int J Control 73:1329–1345

    Article  MATH  MathSciNet  Google Scholar 

  27. Taylor CJ, Chotai A, Young PC (2001) Design and application of proportional integral plus (PIP) controllers: robust control of the IFAC93 benchmark. Trans Inst Meas Control 3(23):183–200

    Article  Google Scholar 

  28. Taylor CJ, Pedregal DJ, Young PC, Tych W (2007) Environmental time series analysis and forecasting with the captain Toolbox. Environ Model Softw 22(6):797–814

    Article  Google Scholar 

  29. Taylor CJ, Shaban EM, Stables MA, Ako S (2007) Proportional-Integral-Plus (PIP) control applications of state dependent parameter models. Proc Inst Mech Eng, Part I, J Syst Control Eng 221(17):1019–1032

    Article  Google Scholar 

  30. Taylor CJ, Chotai A, Young PC (2009) Nonlinear control by input–output state variable feedback pole assignment. Int J Control 82:1029–1044

    Article  MATH  MathSciNet  Google Scholar 

  31. Taylor CJ, Chotai A, Burnham KJ (2011) Controllable forms for stabilising pole assignment design of generalised bilinear systems. Electron Lett 47:437–439

    Article  Google Scholar 

  32. Taylor CJ, Young PC, Chotai A (2013) True digital control: statistical modelling and non-minimal state space design. Wiley, Chichester

    Book  Google Scholar 

  33. Thornley JHM (1976) Mathematical models in plant physiology. Academic Press, London

    Google Scholar 

  34. Van Loon K, Guiza F, Meyfroidt G, Aerts JM, Ramon J, Blockeel H, Bruynooghe M, Van den Berghe G, Berckmans D (2010) Prediction of clinical conditions after coronary bypass surgery using dynamic data analysis. J Med Syst 34:229–239

    Article  Google Scholar 

  35. Young PC (2011) Recursive estimation and time series analysis: an introduction for the student and practitioner. Springer, Berlin

    Book  Google Scholar 

  36. Young PC, Behzadi MA, Wang CL, Chotai A (1987) Direct digital and adaptive control by input–output, state variable feedback pole assignment. Int J Control 46:1867–1881

    Article  MATH  MathSciNet  Google Scholar 

  37. Young PC, McKenna P, Bruun J (2001) Identification of nonlinear stochastic systems by state dependent parameter estimation. Int J Control 74:1837–1857

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The statistical tools for this analysis have been assembled as the CAPTAIN Toolbox [28] for Matlab® (http://www.engineering.lancs.ac.uk/tdc).

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Correspondence to C. James Taylor.

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Taylor, C.J., Aerts, JM. Control of Nonlinear Biological Systems by Non-minimal State Variable Feedback. Stat Biosci 6, 290–313 (2014). https://doi.org/10.1007/s12561-013-9098-5

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