In this study, the author considers systems with trajectories originating in the non-negative orthant and becoming non-negative after some finite time transient. These systems are called eventually positive and the results are based on recent theoretical developments in linear algebra. The author considers dynamical systems (i.e. fully observable systems with no inputs), for which they compute forward-invariant cones and Lyapunov functions. They then extend the notion of eventually positive systems to the input–output system case. The extension is performed in such a manner, that some valuable properties of classical internally positive input–output systems are preserved. For example, their induced norms can be computed using linear programming and the energy functions have non-negative derivatives. The author illustrates the theoretical results on numerical examples.

References

1. 1)
  - 22. Sootla, A., Mauroy, A.: ‘Operator-theoretic characterization of eventually monotone systems’. 2015, Available at:http://arxiv.org/abs/1510.01149.
2. 2)
  - 16. Hirsch, M., Smith, H.: ‘Monotone dynamical systems, Handbook Differential equations: ordinary differential equations, vol. 2 (2005), pp. 239–357.
3. 3)
  - 11. Olesky, D., Tsatsomeros, M., van den Driessche, P.: ‘Mv-matrices: A generalization of M-matrices based on eventually nonnegative matrices’, Electron. J. Linear Algebra, 2009, 18, (1), pp. 29.
4. 4)
  - 5. Rantzer, A.: ‘Scalable control of positive systems’, Eur. J. Control, 2015, 24, pp. 72–80.
5. 5)
  - 15. Sootla, A., Zheng, Y., Papachristodoulou, A.: ‘Block-diagonal solutions to Lyapunov inequalities and generalisations of diagonal dominance’. Proc. IEE Conf. Decision Control, Melbourne, Australia, 2017, pp. 6561–6566.
6. 6)
  - 18. Rantzer, A.: ‘On the Kalman–Yakubovich–Popov lemma for positive systems’, IEEE Trans. Autom. Control, 2016, 61, (5), pp. 1346–1349.
7. 7)
  - 23. Daners, D., Glück, J., Kennedy, J.B.: ‘Eventually positive semigroups of linear operators’, J. Math. Anal. Appl., 2016, 433, (2), pp. 1561–1593.
8. 8)
  - 1. Berman, A., Plemmons, R.J.: ‘Nonnegative matrices in the mathematical sciences’, vol. 9 (SIAM, Philadelphia, PA, 1994).
9. 9)
  - 17. Kaczorek, T.: ‘Externally and internally positive time-varying linear systems’, Appl. Math. Comput. Sci., 2001, 11, (4), pp. 957–964.
10. 10)
  - 6. Tanaka, T., Langbort, C.: ‘The bounded real lemma for internally positive systems and H-infinity structured static state feedback’, IEEE Trans. Autom. Control, 2011, 56, (9), pp. 2218–2223.
11. 11)
  - 4. Briat, C.: ‘Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: L1-gain and L∞-gain characterization’, Int. J. Robust Nonlinear Control, 2013, 23, (17), pp. 1932–1954.
12. 12)
  - 10. Elhashash, A., Szyld, D.: ‘On general matrices having the Perron–Frobenius property’, Electron. J. Linear Algebra, 2008, 17, pp. 389–413.
13. 13)
  - 21. Grussler, C., Damm, T.: ‘A symmetry approach for balanced truncation of positive linear systems’. IEEE Conf Decision Control, Maui, Hi, USA, December 2012, pp. 4308–4313.
14. 14)
  - 2. Leontief, W.: ‘Input–output economics’ (Oxford University Press, Oxford, UK, 1986).
15. 15)
  - 14. Sootla, A., Anderson, J.: ‘On existence of solutions to structured Lyapunov inequalities’. IEEE Proc. of American Control Conf., Boston, MA, 2016, pp. 7013–7018.
16. 16)
  - 19. Mezic, I.: ‘On applications of the spectral theory of the koopman operator in dynamical systems and control theory’. IEEE Conf. Decision Control, Osaka, Japan, 2015, pp. 7034–7041.
17. 17)
  - 9. Altafini, C.: ‘Representing externally positive systems through minimal eventually positive realizations’. 54th IEEE Conference on Decision and Control (CDC), Osaka, Japan, 2015, pp. 6385–6390.
18. 18)
  - 20. Stern, R., Wolkowicz, H.: ‘Exponential nonnegativity on the ice cream cone’, SIAM. J. Matrix Anal. A, 1991, 12, (1), pp. 160–165.
19. 19)
  - 13. Hershkowitz, D., Schneider, H.: ‘Lyapunov diagonal semistability of real H-matrices’, Linear Algebr. Appl., 1985, 71, pp. 119–149.
20. 20)
  - 8. Anderson, B., Deistler, M., Farina, L., et al: ‘Nonnegative realization of a linear system with nonnegative impulse response’, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., 1996, 43, (2), pp. 134–142.
21. 21)
  - 12. Kasigwa, M., Tsatsomeros, M.: ‘Eventual cone invariance’, Electron. J. Linear Algebra, 2017, 32, pp. 204–216.
22. 22)
  - 7. Sootla, A., Rantzer, A.: ‘Scalable positivity preserving model reduction using linear energy functions’. Proc IEEE Conf. Decision Control, Maui, HI, December 2012, pp. 4285–4290.
23. 23)
  - 3. Sontag, E.: ‘Monotone and near-monotone biochemical networks’, Syst. Synth. Biol., 2007, 1, (2), pp. 59–87.

Properties of eventually positive linear input–output systems

References

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