Corentin Briat

Positive systems and their applications

Positive systems

Positive systems 1 are a class of systems having their state confined in the nonnegative orthant and which map nonnegative inputs into nonnegative outputs. Since many physical systems naturally involves positive variables, it is hence natural to (try to) represent them as positive dynamical systems. Interesting examples are biological systems [2,3,4], epidemiological systems [2,5], communication networks [6,7,8], etc. Quite surprisingly, linear positive systems of the form

\[ \begin{array}{rcl} \dot{x}(t)&=&Ax(t),\ A\ \mathrm{Metzler}\\ x_0&\ge&0 \end{array} \]

have very interesting properties. Their stability can be exactly characterized by sum-separable Lyapunov functions of the form

\[ \begin{array}{rcl} V_\ell(x)&=&\sum_{i=1}^nv_ix_i, v_i>0,\\ V_q(x)&=&\sum_{i=1}^nd_ix_i^2, d_i>0\\ &=&x^TDx, D=\mathrm{diag}_{i=1}^nd_i>0. \end{array} \]

The former one leads to stability conditions taking the form of a linear programming problem [9] whereas the latter one leads to a semidefinite programming problem 10 (although simpler than usual ones for general linear systems due to the diagonal structure of the Lyapunov matrix D). The use of such Lyapunov functions allows for the design of structured state-feedback in a nonconservative way [11] (a problem for which certain instances are known to be NP-hard). It has also been recently shown that the computation of the \(L_1\)- and the \(L_\infty\)-gains of linear positive can be exactly cast as a linear program as well [12]. This sharply contrasts with the poor tractability of the \(L_\infty\)-gain for general linear systems. Exact robustness results for uncertain linear positive systems in LFT form have also been obtained [12,13,14] and structural results have been obtained in [15].

Linear positive systems with delays

Linear positive with delays [16] take the form

\[ \begin{array}{rcl} \dot{x}(t)&=&Ax(t)+A_hx(t-h)\\ x(s)&=&\phi(s)\ge0, s\in[-h,0] \end{array} \]

where \(A\) is Metzler and \(A_h\) is nonnegative. As for undelayed systems, they have very interesting properties such that the above system is stable for any \(h\ge0\) if and only if \(A+A_h\) is Hurwitz stable; i.e. the system with zero-delay is stable [12,16]. In other words, the worst-case delay value is \(h=0\) which is rather unusual. Results for time-varying delays have also been obtained and coincide with the constant-delay stability conditions suggesting that the worst-case time-varying delay is actually the constant-delay, which goes against intuition as time-varying delays usually tend to be destabilizing [12,17,18,19]. Unifying results have been obtained in [23] using input-output methods for a broad class of linear positive time-delay systems. Results pertaining to linear positive impulsive systems with constant delays have been obtained in [25].

Interval observers

An interesting application of positive systems is in the design of interval-observers [20,21,22]. The goal of such observers is not to estimate the state as closely as possible but instead estimate an upper-bound and lower-bound on the value of the state over time. These observers are hence able to deal with the presence of persistent disturbances that may drive the estimation error away from zero. A state-feedback controller can then be designed using a weighted sum of these bounds; e.g. the mean value.

References

  1. L. Farina and S. Rinaldi, “Positive Linear Systems: Theory and Applications”, John Wiley & Sons, 2000.

  2. J. D. Murray, “Mathematical Biology Part I. An Introduction”, Springer-Verlag, 2002.

  3. U. Müller-Herold, “General mass-action kinetics. Positiveness of concentrations as structural property of Horn's equation”, Chemical Physics Letters, Vol. 33(3), pp. 467-470, 1975.

  4. C. Briat and M. Khammash, “Computer control of gene expression: Robust setpoint tracking of protein mean and variance using integral feedback” (slides), 51st IEEE Conference on Decision and Control, Maui, Hawaii, USA, 2012.

  5. C. Briat and E. I. Verriest, “A New Delay-SIR Model for Pulse Vaccination”, Biomedical Signal Processing and Control, Vol. 4, pp. 272-277, 2009.

  6. S. H. Low, F. Paganini and J. C. Doyle, “Internet congestion control”, IEEE Control Systems Magazine, Vol. 22(1), pp. 28-43, 2002.

  7. R. Shorten, F. Wirth and D. Leith, “A positive systems model of TCP-like congestion control: asymptotic results”, IEEE/ACM Transactions on Networking, Vol. 14(3), pp. 616-629, 2006.

  8. C. Briat, E. A. Yavuz, H. Hjalmarsson, K.-H. Johansson, U. T. Jönsson, G. Karlsson and H. Sandberg, “The conservation of information, towards an axiomatized modular modeling approach to congestion control”, IEEE Transactions on Networking, Vol. 23(3), pp. 851-865, 2015.

  9. W. M. Haddad and V. Chellaboina, “Stability and dissipativity theory for nonnegative dynamic systems: a unified analysis framework for biological and physiological systems”, Nonlinear Analysis: Real World Applications, Vol.6(1), pp. 35-65, 2005.

  10. R. Shorten, O. Mason and D. Leith, “An alternative proof of the Barker, Berman, Plemmons result on diagonal stability and extensions”, Linear Algebra and Its Applications, Vol.430, pp. 34-40, 2009.

  11. M. Ait Rami and F. Tadeo, “Controller synthesis for positive linear systems with bounded controls”, IEEE Transactions on Circuits and Systems – II. Express Briefs, Vol. 54(2), pp. 151-155, 2007.

  12. C. Briat, “Robust stability and stabilization of uncertain linear positive systems via Integral Linear Constraints: \(L_1\)- and \(L_\infty\)-gains characterization”, International Journal of Robust and Nonlinear Control, Vol. 23(17), pp. 1932-1954, 2013.

  13. Y. Ebihara, D. Peaucelle and D. Arzelier, “\(L_1\) Gain Analysis of Linear Positive Systems and Its Application”, 50th IEEE Conference on Decision and Control, 2011.

  14. M. Colombino and R. S. Smith, “Convex characterization of robust stability analysis and control synthesis for positive linear systems”, 53rd IEEE Conference on Decision and Control, 2014.

  15. C. Briat, “Sign properties of Metzler matrices with applications”, Linear Algebra and Its Applications, Vol. 515, pp. 53-86, 2017.

  16. W. M. Haddad and V. Chellaboina, “Stability theory for nonnegative and compartmental dynamical systems with time delay”, Systems & Control Letters, Vol. 51(5), pp. 355-361, 2004.

  17. M. Ait Rami, “Stability Analysis and Synthesis for Linear Positive Systems with Time-Varying Delays”, In "Positive systems - Proceedings of the 3rd Multidisciplinary International Symposium on Positive Systems: Theory and Applications, pp. 205-216, 2009.

  18. J. Shen and J. Lam, “ \(\ell_\infty/L_\infty\) Gain Analysis for Positive Linear Systems with Unbounded Time-Varying Delays”, IEEE Transactions on Automatic Control, Vol. 60(3), pp. 857-862, 2015.

  19. J. Zhu and J. Chen, “Stability of systems with time-varying delays: An \(L_1\) small-gain perspective”, Automatica, Vol. 52, pp. 260-265, 2015.

  20. J. L. Gouzé, A. Rapaport and M. Z. Hadj-Sadok, “Interval observers for uncertain biological systems”, Ecological modelling, Vol. 133, pp. 45-56, 2000.

  21. F. Mazenc and O. Bernard, “Interval observers for linear time-invariant systems with disturbances”, Automatica, Vol. 47, pp. 140-147, 2011.

  22. C. Briat and M. Khammash, “Interval peak-to-peak observers for continuous- and discrete-time systems with persistent inputs and delays”, Automatica, Vol. 74, pp. 206-213, 2016.

  23. C. Briat, “Stability and performance analysis of linear positive systems with delays using input-output methods”, International Journal of Control, Vol. 71(7), pp. 1669-1692, 2018.

  24. C. Briat, “Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems”, Nonlinear Analysis: Hybrid Systems, Vol. 24, pp. 198-226, 2017.

  25. C. Briat, “Stability and \(L_1\times\ell_l\)-to-\(L_1\times\ell_1\) performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays”, International Journal of Control, Vol. 93(11), pp. 2634-2652, 2020.

  26. C. Briat, “\(L_1\times\ell_l\)-to-\(L_1\times\ell_1\) analysis of linear positive impulsive systems with application to the \(L_1/\times\ell_l\)-to-\(L_1\times\ell_1\) interval observation of linear impulsive and switched systems”, Nonlinear Analysis: Hybrid Systems, Vol. 34, pp. 1-17, 2019.

  27. C. Briat, “Co-design of aperiodic sampled-data min-jumping rules for linear impulsive, switched impulsive and sampled-data systems”, Systems & Control Letters, Vol. 130, pp. 32-42, 2019.

  28. C. Briat, “A class of \(L_1\)-to-\(L_1\) and \(L_\infty\)-to-\(L_\infty\) interval observers for (delayed) Markov jump linear systems”, IEEE Control Systems Letters, Vol. 3(2), pp. 410-415, 2019