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Dynamic observers for unknown populations

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  • Dynamic observers are considered in the context of structured-population modeling and management. Roughly, observers combine a known measured variable of some process with a model of that process to asymptotically reconstruct the unknown state variable of the model. We investigate the potential use of observers for reconstructing population distributions described by density-independent (linear) models and a class of density-dependent (nonlinear) models. In both the density-dependent and -independent cases, we show, in several ecologically reasonable circumstances, that there is a natural, optimal construction of these observers. Further, we describe the robustness these observers exhibit with respect to disturbances and uncertainty in measurement.

    Mathematics Subject Classification: Primary: 39A30, 92D40, 93B51, 93B52, 93C05, 93C10, 93D09.


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  • Figure 1.1.  Illustration of a dynamic observer. The dashed lines in the right-hand figure at computed using the measured variable $ y(t) $

    Figure 5.1.  Numerical simulations from Example 5.1. Abundances from the model (LO) are plotted against time. The solid line and dotted lines are the abundance of the eighth stage-class $ x_8(t) $ of the (forced) cheetah population, and its corresponding observer state $ z_8(t) $, respectively. The dashed and dashed-dotted lines are $ \| x(t)\|_1 $ and $ \| z(t)\|_1 $, the total population and its observer estimate, respectively

    Figure 5.2.  Numerical simulations from Example 5.2. In both panels, the solid, dashed, dashed-dotted and dotted lines denote the total population $ \| x(t)\|_1 $, the corresponding estimate $ \| z(t)\|_1 $, the error $ \| x(t) - z(t) \|_1 $ and the unforced equilibrium $ \| x^*\|_1 $, respectively, each plotted against time $ t $. In (a), no forcing terms are present, so $ d = 0 $ and $ v = 0 $, and the error converges to zero. In (b), the forcing and measurement error terms are nonzero, described in the main text

    Figure 5.3.  Numerical simulations from Example 5.3. The solid line denotes $ \|x(t)\|_1 $, and the dashed, dashed-dotted, and dotted lines denote the errors $ \| x(t) - z(t)\|_1 $ for increasing measurement error. See the main text

  • [1] D. Angeli, A Lyapunov approach to incremental stability properties, IEEE Trans. Automat. Control, 47 (2002), 410-421.  doi: 10.1109/9.989067.
    [2] M. Arcak and P. Kokotovic, Observer-based control of systems with slope-restricted nonlinearities, IEEE Trans. Automat. Control, 46 (2001), 1146-1150.  doi: 10.1109/9.935073.
    [3] A. Berman, M. Neumann and R. J. Stern, Nonnegative Matrices in Dynamic Systems, John Wiley & Sons Inc., New York, 1989.
    [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611971262.
    [5] G. E. P. Box, G. M. Jenkins, G. C. Reinsel and G. M. Ljung, Time Series Analysis, fifth edn., John Wiley & Sons, Inc., Hoboken, 2016.
    [6] H. Caswell, Matrix Population Models : Construction, Analysis and Interpretation, Sinauer, Massachusetts, 2001.
    [7] I. ChadèsE. McDonald-MaddenM. A. McCarthyB. WintleM. Linkie and H. P. Possingham, When to stop managing or surveying cryptic threatened species, Proc. Nat. Acad. Sci., 105 (2008), 13936-13940. 
    [8] T. Chen and B. Francis, Optimal Sampled-Data Control Systems, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1996.
    [9] C. k. Chui and G. Chen, Kalman Filtering, with Real-Time Applications, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-662-02666-3.
    [10] K. R. CrooksM. A. Sanjayan and D. F. Doak, New insights on cheetah conservation through demographic modeling, Conservation Biology, 12 (1998), 889-895. 
    [11] J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970005.
    [12] S. N. DashkovskiyD. V. Efimov and E. D. Sontag., Input-to-state stability and allied system properties, Autom. Remote Control, 72 (2011), 1579-1614.  doi: 10.1134/S0005117911080017.
    [13] N. Dautrebande and G. Bastin, Positive linear observers for positive linear systems, Proceedings of the European Control Conference, 1999, 1092–1095.
    [14] E. A. Eager, Modelling and analysis of population dynamics using Lur'e systems accounting for competition from adult conspecifics, Lett. Biomath., 3 (2016), 41-58.  doi: 10.30707/LiB3.1Eager.
    [15] E. A. Eager and R. Rebarber, Sensitivity and elasticity analysis of a Lur'e system used to model a population subject to density-dependent reproduction, Math. Biosci., 282 (2016), 34-45.  doi: 10.1016/j.mbs.2016.09.016.
    [16] E. A. EagerR. Rebarber and B. Tenhumberg, Global asymptotic stability of plant-seed bank models, J. Math. Biology, 69 (2014), 1-37.  doi: 10.1007/s00285-013-0689-z.
    [17] M. R. EasterlingS. P. Ellner and P. M. Dixon, Size-specific sensitivity: Applying a new structured population model, Ecology, 81 (2000), 694-708. 
    [18] X. Fan and M. Arcak, Observer design for systems with multivariable monotone nonlinearities, Systems Control Lett., 50 (2003), 319-330.  doi: 10.1016/S0167-6911(03)00170-1.
    [19] L. Farina and S.Rinaldi, Positive Linear Systems: Theory and Applications, Wiley-Interscience, New York, 2000. doi: 10.1002/9781118033029.
    [20] D. FrancoC. GuiverH. Logemann and J. Perán, Semi-global persistence and stability for a class of forced discrete-time population models, Phys. D, 360 (2017), 46-61.  doi: 10.1016/j.physd.2017.08.001.
    [21] D. FrancoC. GuiverH. Logemann and J. Perán., Boundedness, persistence and stability for classes of forced difference equations arising in population ecology, J. Math. Biol., 79 (2019), 1029-1076.  doi: 10.1007/s00285-019-01388-7.
    [22] D. FrancoH. Logemann and J. Perán, Global stability of an age-structured population model, Systems Control Lett., 65 (2014), 30-36.  doi: 10.1016/j.sysconle.2013.11.012.
    [23] M. E. Gilmore, C. Guiver and H. Logemann, Stability and convergence properties of forced infinite-dimensional discrete-time Lur'e systems, Int. J. Control, (2019), 1–40.
    [24] J. L. GouzéA. Rapaport and M. Z. Hadj-Sadok, Interval observers for uncertain biological systems, Ecol. Modelling, 133 (2000), 45-56.  doi: 10.1016/S0304-3800(00)00279-9.
    [25] C. GuiverD. Hodgson and S. Townley, Positive state controllability of positive linear systems, Systems Control Lett., 65 (2014), 23-29.  doi: 10.1016/j.sysconle.2013.12.002.
    [26] C. GuiverC. EdholmY. JinM. MuellerJ. PowellR. RebarberB. Tenhumberg and S. Townley, Simple adaptive control for positive linear systems with applications to pest management, SIAM J. Appl. Math, 76 (2016), 238-275.  doi: 10.1137/140996926.
    [27] C. GuiverH. LogemannR. RebarberA. BillB. TenhumbergD. Hodgson and S. Townley, Integral control for population management, J. Math. Biol., 70 (2005), 1015-1063.  doi: 10.1007/s00285-014-0789-4.
    [28] C. Guiver, H. Logemann and B. Rüffer, Small-gain stability theorems for positive Lur'e inclusions, Positivity, 23 (2019), 249–289. doi: 10.1007/s11117-018-0605-2.
    [29] W. M. Haddad, V. Chellaboina and Q. Hui, Nonnegative and Compartmental Dynamical Systems, Princeton University Press, Princeton, 2010. doi: 10.1515/9781400832248.
    [30] M. Z. Hadj-Sadok and J. L. Gouzé, Estimation of uncertain models of activated sludge processes with interval observers, J. Process Control, 11 (2001), 299-310.  doi: 10.1016/S0959-1524(99)00074-8.
    [31] E. Halfon (ed), Theoretical Systems Ecology, Academic Press, New York, 1979.
    [32] H. R. Heinimann, A concept in adaptive ecosystem management — An engineering perspective, Forest Ecol. Manag., 259 (2010), 848-856.  doi: 10.1016/j.foreco.2009.09.032.
    [33] D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I, Springer–Verlag, Berlin, 2005. doi: 10.1007/b137541.
    [34] D. Hinrichsen and N. K. Son, Stability radii of positive discrete-time systems, Internat. J. Robust Nonlinear Control, 8 (1995).
    [35] S. Ibrir, Circle-criterion approach to discrete-time nonlinear observer design, Automatica, 43 (2007), 1432-1441.  doi: 10.1016/j.automatica.2007.01.012.
    [36] R. E. Kalman, A new approach to linear filtering and prediction problems, J. Basic Engineering (ASME), 82 60), 34–45. doi: 10.1115/1.3662552.
    [37] I. Karafyllis and Z.-P. Jiang, Stability and Stabilization of Nonlinear Systems, Springer-Verlag, London, 2011. doi: 10.1007/978-0-85729-513-2.
    [38] N. Keyfitz and H. Caswell, Applied Mathematical Demography, Springer Science+Business Media, Inc., 2005.
    [39] R KleinN. A. ChaturvediJ. ChristensenJ. AhmedR. Findeisen and A. Kojic, Electrochemical model based observer design for a lithium-ion battery, IEEE Trans. Control Syst. Technol., 21 (2013), 289-301.  doi: 10.1109/TCST.2011.2178604.
    [40] A. J. Krener and A. Isidori, Linearization by output injection and nonlinear observers, Systems Control Lett., 3 (1983), 47-52.  doi: 10.1016/0167-6911(83)90037-3.
    [41] T. Liao and N. Huang, An observer-based approach for chaotic synchronization with applications to secure communications, IEEE Trans. Circuits Syst. I, 46 (1999), 1144–1150. doi: 10.1109/81.788817.
    [42] L. Ljung, System Identification: Theory for the User, Prentice Hall Information and System Sciences Series. Prentice Hall, Inc., Englewood Cliffs, NJ, 1987.
    [43] D. Luenberger, An introduction to observers, IEEE Trans. Automat. Control, 16 (1971), 596-602.  doi: 10.1109/TAC.1971.1099826.
    [44] D. Luenberger, Observing the state of a linear system, IEEE Trans. Mil. Electronics, 8 (1964), 74-80.  doi: 10.1109/TME.1964.4323124.
    [45] C. MerowJ. P. DahlgrenC. J. E. MetcalfD. Z. ChildsM. E. EvansE. JongejansS. RecordM. ReesR. Salguero-Gómez and S. M. McMahon, Advancing population ecology with integral projection models: A practical guide, Methods Ecol. Evol., 5 (2014), 99-110.  doi: 10.1111/2041-210X.12146.
    [46] W. F. Morris and D. F. Doak, Quantitative Conservation biology: Theory and Practice of Population Viability Analysis, Sinauer Associates Sunderland, Massachusetts, USA, 2002.
    [47] M. Müller and C. A. Sierra, Application of input to state stability to reservoir models, Theor. Ecol., 10 (2017), 451–475. doi: 10.1007/s12080-017-0342-3.
    [48] R. A. Myers, Stock and recruitment: Generalizations about maximum reproductive rate, density dependence, and variability using meta-analytic approaches, ICES J. Marine Science, 58 (2001), 937-951.  doi: 10.1006/jmsc.2001.1109.
    [49] I. M. Navon, Data assimilation for numerical weather prediction: A review, in Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications, S.K. Park and L. Xu eds. Springer, Berlin, 2009. doi: 10.1007/978-3-540-71056-1_2.
    [50] N. Poppelreiter, Dynamic Observers for Unknown Populations, Phd Thesis, University of Nebraska, Lincoln, 2019.
    [51] M. A. Rami and F. Tadeo, Positive observation problem for linear discrete positive systems, in Proceedings of the 45th IEEE Conference on Decision and Control, IEEE, (2006). doi: 10.1109/CDC.2006.377749.
    [52] P. Reichert and M. Omlin, On the usefulness of overparameterized ecological models, Ecol. Mod., 95 (1997), 289-299.  doi: 10.1016/S0304-3800(96)00043-9.
    [53] K. E. RoseS. M. Louda and M. Rees, Demographic and evolutionary impacts of native and invasive insect herbivores on cirsium canescens, Ecology, 86 (2005), 453-465.  doi: 10.1890/03-0697.
    [54] M. de la Sen, Non-periodic and adaptive sampling. A tutorial review, Informatica, 7 (1996), 175-228. 
    [55] H. L. Smith and H. R. Thieme, Persistence and global stability for a class of discrete time structured population models, Discrete Contin. Dyn. Syst., 33 (2013), 4627-4646.  doi: 10.3934/dcds.2013.33.4627.
    [56] T. Söderström and P. Stoica, System Identification, Prentice-Hall, Inc., London, 1989.
    [57] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.
    [58] E. D. Sontag, Input-to-state stability: Basic concepts and results, in Nonlinear and Optimal Control Theory, 163–220, Lecture Notes in Math., 1932, Springer, Berlin, 2008. doi: 10.1007/978-3-540-77653-6_3.
    [59] E. D. Sontag, Mathematical Control Theory, second ed., Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.
    [60] M. Soroush, Nonlinear state-observer design with application to reactors, Chemical Engineering Science, 52 (1997), 387-404.  doi: 10.1016/S0009-2509(96)00391-0.
    [61] I. StottS. TownleyD. Carslake and D. J. Hodgson, On reducibility and ergodicity of population projection matrix models, Methods Ecol. Evol., 1 (2010), 242-252.  doi: 10.1111/j.2041-210X.2010.00032.x.
    [62] F. Tadeo and M. Rami, Selection of time-after-injection in bone scanning using compartmental observers, in Proceeding of World Engineering Congress, WCE (2010).
    [63] B. TenhumbergS. LoudaJ. Eckberg and M. Takahashi, Monte carlo analysys of parameter uncertainy in matrix models for the weed, J. Appl. Ecol., 45 (2008), 439-447. 
    [64] S. TownleyR. Rebarber and B. Tenhumberg, Feedback control systems analysis of density dependent population dynamics, Systems Control Lett., 61 (2012), 309-315.  doi: 10.1016/j.sysconle.2011.11.014.
    [65] H. L. Trentelman, A. A. Stoorvogel and M. Hautus, Control Theory for Linear Systems, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 2001. doi: 10.1007/978-1-4471-0339-4.
    [66] J. M. Van Den Hof, Positive linear observers for linear compartmental systems, SIAM J. Control Optim., 36 (1998), 590-608.  doi: 10.1137/S036301299630611X.
    [67] X.-H. Xia and W.-B. Gao., Nonlinear observer design by observer error linearization, SIAM J. Control Optim., 27 (1989), 199-216.  doi: 10.1137/0327011.
    [68] J. I. Yuz and G. C. Goodwin, Sampled-Data Models for Linear and Nonlinear Systems, Communications and Control Engineering Series, Springer, London, 2014. doi: 10.1007/978-1-4471-5562-1.
    [69] J. ZhangX. ZhaoR. Zhang and Y. Chen, Improved controller design for uncertain positive systems and its extension to uncertain positive switched systems, Asian J. Control, 20 (2018), 159-173.  doi: 10.1002/asjc.1553.
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