access icon free Algebraic analysis of the structural properties of parametric linear time-invariant systems

By using techniques borrowed from algebraic geometry, some tests are proposed to certify structural properties (such as reachability, observability, controllability, constructibility, stabilisability, and detectability) of discrete-time and continuous-time linear time-invariant systems. These results are then generalised for linear time-invariant systems depending polynomially on some real parameters, by exploiting the notion of Gröbner cover. The main innovation of the proposed results with respect to existing techniques is that they provide exact certificates for the structural properties of linear time-invariant systems. Examples of application are given all throughout the study to illustrate and validate the theoretical results.

Inspec keywords: polynomials; continuous time systems; geometry; discrete time systems; algebra; linear systems; multidimensional systems

Other keywords: certify structural properties; algebraic analysis; algebraic geometry; parametric linear time-invariant systems

Subjects: Linear control systems; Stability in control theory; Discrete control systems; Control system analysis and synthesis methods; Algebra; Distributed parameter control systems

References

    1. 1)
      • 24. Jury, E.I.: ‘A simplified stability criterion for linear discrete systems’, Proc. IRE, 1962, 50, (6), pp. 14931500.
    2. 2)
      • 17. Cox, D., Little, J., O'Shea, D.: ‘Ideals, varieties, and algorithms, undergraduate texts in mathematics’ (Springer-Verlag, Springer International Publishing Switzerland, 2015).
    3. 3)
      • 14. Habets, L.C.G.J.M.: ‘A reachability test for systems over polynomial rings using Gröbner bases’. American Control Conf., San Francisco, CA, USA, 1993, pp. 226230.
    4. 4)
      • 6. Kalman, R.E., Falb, P.L., Arbib, M.A.: ‘Topics in mathematical system theory’ (McGraw-Hill, New York, 1969).
    5. 5)
      • 21. Brunat, J.M., Montes, A.: ‘Computing the canonical representation of constructible sets’, Math. Comput. Sci., 2016, 10, (1), pp. 165178.
    6. 6)
      • 3. Popov, V.M.: ‘Hyperstability of control systems’ (Springer-Verlag, Berlin Heidelberg, 1973).
    7. 7)
      • 5. Glover, K., Silverman, L.: ‘Characterization of structural controllability’, IEEE Trans. Autom. Control, 1976, 21, (4), pp. 534537.
    8. 8)
      • 7. Zabczyk, J.: ‘Mathematical control theory: an introduction’ (Birkhäuser Basel, 2009).
    9. 9)
      • 4. Gilbert, E.G.: ‘Controllability and observability in multivariable control systems’, J. Soc. Ind. Appl. Math. A, Control, 1963, 1, (2), pp. 128151.
    10. 10)
      • 1. Belevitch, V.: ‘Classical network theory’ (Holden-day, San Francisco, 1968).
    11. 11)
      • 8. Sontag, E.D.: ‘Mathematical control theory: deterministic finite dimensional systems’ (Springer Science & Business Media, New York, 2013).
    12. 12)
      • 9. Hespanha, J.P.: ‘Linear systems theory’ (Princeton University Press, Princeton, NJ, USA, 2018).
    13. 13)
      • 10. Belevitch, V.: ‘Classical network theory’ (Holden-Day, San Francisco, 1968).
    14. 14)
      • 15. Menini, L., Possieri, C., Tornambe, A.: ‘Algebraic certificates for the structural properties of parametric linear systems’. IFAC-PapersOnLine (1st Virtual IFAC World Congress, Berlin, Germany, 2020.
    15. 15)
      • 11. Possieri, C., Teel, A.R.: ‘Structural properties of a class of linear hybrid systems and output feedback stabilization’, IEEE Trans. Autom. Control, 2016, 62, (6), pp. 27042719.
    16. 16)
      • 20. Abánades, M.Á., Botana, F., Montes, A., et al: ‘An algebraic taxonomy for locus computation in dynamic geometry’, Comput.-Aided Des., 2014, 56, pp. 2233.
    17. 17)
      • 19. Kapur, D., Sun, Y., Wang, D.: ‘A new algorithm for computing comprehensive Gröbner systems’. Int. Symp. on Symbolic and Algebraic Computation, Munich Germany, 2010, pp. 2936.
    18. 18)
      • 22. Decker, W., Greuel, G.-M., Pfister, G., et al (2019). ‘Singular 4-1-2 – A computer algebra system for polynomial computations’. http://www.singular.uni-kl.de.
    19. 19)
      • 18. Montes, A., Wibmer, M.: ‘Gröbner bases for polynomial systems with parameters’, J. Symb. Comput., 2010, 45, (12), pp. 13911425.
    20. 20)
      • 12. Hao, Y., Duan, Z., Chen, G., et al: ‘New controllability conditions for networked, identical LTI systems’, IEEE Trans. Autom. Control, 2019, 64, (10), pp. 42234228.
    21. 21)
      • 13. Zhou, T.: ‘Topology and subsystem parameter based verification for the controllability/observability of a networked dynamic system’. 58th Conf. on Decision and Control, Nice, France, 2019, pp. 35753580.
    22. 22)
      • 25. Hurwitz, A.: ‘Ueber die bedingungen, unter welchen eine gleichung nur wurzeln mit negativen reellen theilen besitzt’, Math. Ann., 1895, 46, (2), pp. 273284.
    23. 23)
      • 2. Hautus, M.L.: ‘Controllability and observability conditions of linear autonomous systems’, Indag. Math., 1969, 72, pp. 443448.
    24. 24)
      • 16. Reinschke, K.J.: ‘Multivariable control: a graph-theoretic approach, vol. 41’ (Springer, Springer-Verlag Berlin Heidelberg, 1988).
    25. 25)
      • 23. Meyer, C.D.: ‘Matrix analysis and applied linear algebra’ (SIAM, Philadelphia, PA, USA, 2000).
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