Monotone dynamical systems are those with solutions preserving specific orderings, relative to the associated initial states. This study shows that, under negative output feedback control, a monotone single-input single-output system with well-defined static characteristics is able to globally asymptotically regulate its solution at desired constant set-points, while the boundedness of all solutions are successfully guaranteed. This valuable result is obtained only through limited amount of qualitative and quantitative data, which may be provided from relatively simple experiments for real applications. The design procedure, in the cast of a new theorem, is mainly derived from the specific version of Small Gain Theorem for autonomous monotone systems; which ensures the global attractivity of the desired set-point. A biological simulation example illustrates the effectiveness and applicability of the proposed control strategy.

References

1. 1)
  - 20. Chisci, L, Falugi, P.: ‘Asymptotic tracking for constrained monotone systems’, IEEE Trans. Autom. Control, 2006, 51, 873–879 (doi: 10.1109/TAC.2006.874994).
2. 2)
  - 18. Wang, Z., Goldsmith, P.: ‘Modified energy-balancing-based control for the tracking problem’, IET Control Theory Applic., 2008, 2, 310–322 (doi: 10.1049/iet-cta:20070124).
3. 3)
  - 7. Angeli, D., De-Leenheer, P., Sontag, E.D.: ‘On predator-prey systems and small-gain theorems’, J. Math. Biosci. Eng., 2005, 2, 25–42.
4. 4)
  - 31. Owens, D.H., Chotai, A.: ‘Controller design for unknown multi-variable systems using monotone modeling errors’, IEE Proc. Control Theory Appl., 1982, 129, 57–69 (doi: 10.1049/ip-d.1982.0012).
5. 5)
  - 14. Angeli, D., De-Leenheer, P., Sontag, E.D.: ‘Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates’, J. Math. Biol., 2010, 61, 581–616 (doi: 10.1007/s00285-009-0309-0).
6. 6)
  - 16. Nazari, R., Seron, M.M., Yetendje, A.: ‘Invariant-set-based fault tolerant control using virtual sensors’, IET Control Theory Appl., 2011, 5, 1092–1103 (doi: 10.1049/iet-cta.2010.0089).
7. 7)
  - 8. Angeli, D., Hirsch, M., Sontag, E.D.: ‘Attractors in coherent systems of differential systems’, J. Differ. Equ., 2007, 246, 3058–3076 (doi: 10.1016/j.jde.2009.01.025).
8. 8)
  - 27. Zames, G.: ‘On the input–output stability of time-varying nonlinear feedback systems. Part I: conditions using concepts of loop’, IEEE Trans. Autom. Control, 1966, 11, 228–238 (doi: 10.1109/TAC.1966.1098316).
9. 9)
  - 26. Safonov, M.: ‘Stability and robustness of multivariable feedback systems’ (MIT Press, Cambridge, Massachusetts, 1980).
10. 10)
  - 24. Angeli, D., Sontag, E.D.: ‘Oscillations in i/o monotone systems under negative feedback’, IEEE Trans. Autom. Control, 2008, 53, 166–176 (doi: 10.1109/TAC.2007.911320).
11. 11)
  - 3. Smith, H.L.: ‘Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems’, Mathematical Surveys and Monographs, vol. 41 (American Mathematical Society, Providence, RI/Ann Arbor, MI, 1995).
12. 12)
  - 6. Sontag, E.D.: ‘Some new directions in control theory inspired by systems biology’, IET Syst. Biol., 2004, 1, 9–18.
13. 13)
  - 12. Enciso, G.A., Sontag, E.D.: ‘Global attractivity, i/o monotone small-gain theorems, and biological delay systems’, Discret. Continuous Dyn. Syst., 2006, 14, 549–578.
14. 14)
  - 22. Angeli, D., Sontag, E.D.: ‘Interconnections of monotone systems with steady-state characteristics’: in de Queiroz, M.S., Malisoff, M., Wolenski, P. (Eds.): ‘Optimal control, stabilization, and nonsmooth analysis’, 2004, (Lecture Notes in Control and Information Sciences, 301), (Springer–Verlag, Berlin, Germany), pp. 135–154.
15. 15)
  - 25. Kunze, H., Siegel, D.: ‘A graph theoretical approach to monotonicity with respect to initial conditions’ in Liu, X., Siegel, D. (Eds.): ‘Comparison methods and stability theory’ (Marcel Dekker, New York, USA, 1994).
16. 16)
  - 10. Scardovi, L., Arcaky, M., Sontag, E.D.: ‘Synchronization of interconnected systems with applications to biochemical networks: an input–output approach’, IEEE Trans. Autom. Control, 2010, 55, 1367–1379 (doi: 10.1109/TAC.2010.2041974).
17. 17)
  - 5. Angeli, D., Sontag, E.D.: ‘Multi-stability in monotone input/output systems’, Syst. Control Lett., 2004, 51, 185–202 (doi: 10.1016/j.sysconle.2003.08.003).
18. 18)
  - 9. Zhu, S., Li, Z., Zhang, C.: ‘Exponential stability analysis for positive systems with delays’, IET Control Theory Appl., 2012, 6, 761–767 (doi: 10.1049/iet-cta.2011.0133).
19. 19)
  - 30. Dashkovskiy, S.N., Efimov, D.V., Sontag, E.D.: ‘Input to state stability and allied system properties’, Autom. Remote Control, 2011, 72, 1579–1614 (doi: 10.1134/S0005117911080017).
20. 20)
  - 29. Dashkovskiy, S.N., Rueffer, B.S., Wirth, F.R.: ‘An ISS small gain theorem for general networks’, Math. Control Signals Syst., 2007, 19, 93–122 (doi: 10.1007/s00498-007-0014-8).
21. 21)
  - 11. Angeli, D., Ferrell, J.E., Sontag, E.D.: ‘Detection of multi-stability, bifurcations, and hysteresis in a large class of biological positive-feedback systems’, Proc. Natl. Acad. Sci. USA, 2004, 101, 1822–1827 (doi: 10.1073/pnas.0308265100).
22. 22)
  - 28. Mareels, I.M.Y., Hill, D.J.: ‘Monotone stability of nonlinear feedback systems’, J. Math. Syst. Estimation, Control, 1992, 2, 275–291.
23. 23)
  - 23. Enciso, G.A., Sontag, E.D.: ‘Monotone systems under positive feedback: multi-stability and a reduction theorem’, Syst. Control Lett., 2005, 54, 159–168 (doi: 10.1016/j.sysconle.2004.08.003).
24. 24)
  - 15. Enciso, G.A., Smith, H.L., Sontag, E.D.: ‘Non-monotone systems decomposable into monotone systems with negative feedback’, J. Differ. Equ., 2006, 224, 205–227 (doi: 10.1016/j.jde.2005.05.007).
25. 25)
  - 19. Chisci, L., Falugi, P.: ‘Tracking control for constrained monotone systems’. Proc. 16th IFAC World Congress (IFAC 2005), Prague, Czech Republic, July 2005.
26. 26)
  - 2. Hirsch, M.: ‘Stability and convergence in strongly monotone dynamical systems’, J. Reine Angew. Math., 1988, 383, 1–53.
27. 27)
  - 13. Ma’ayan, A., Lipshtat, A., Iyengar, R., Sontag, E.D.: ‘Proximity of intracellular regulatory networks to monotone systems’, IET Syst. Biol., 2008, 2, 103–112 (doi: 10.1049/iet-syb:20070036).
28. 28)
  - 1. Smale, S.: ‘On the differential equations of species in competition’, J. Math. Biol., 1976, 3, 5–7 (doi: 10.1007/BF00307854).
29. 29)
  - 17. Caruntu, C.-F., Lazar, C.: ‘Robustly stabilizing model predictive control design for networked control systems with an application to direct current motors’, IET Control Theory Applic., 2012, 6, 943–952 (doi: 10.1049/iet-cta.2011.0103).
30. 30)
  - 4. Angeli, D, Sontag, E.D.: ‘Monotone control systems’, IEEE Trans. Autom. Control, 2003, 48, 1684–98 (doi: 10.1109/TAC.2003.817920).
31. 31)
  - 21. Agahi, H., Yazdanpanah, M.J.: ‘Set-point regulation of constrained strongly monotone systems. In Pre-prints of 18th IFAC World Congress (IFAC 2011), Milan, Italy, August 2011.

Set-point regulation of monotone systems using the monotone small-gain theorem

References

Related content