A first cubic upper bound on the local reachability index for some positive 2-D systems

[EN] The calculation of the smallest number of steps needed to deterministically reach all local states of an nth-order positive 2-D system, which is called local reachability index (ILR) of that system, was recently tackled bymeans of the use of a suitable composition table. The greatest index ILR obtained in the previous literature was n+3 ([n/2]) 2 for some appropriated values of n. Taking as a basis both a combinatorial approach of such systems and the construction of suitable geometric sets in the plane, an upper bound on ILR depending on the dimension n for a new family of systems is characterized. The 2-D influence digraph of this family of order n = 6 consists of two subdigraphs corresponding to a unique source s. The first one is a cycle involving the first n(1) vertices and is connected to the another subdigraph through the 1-arc (2, n(1) +n(2)), being the natural numbers n(1) and n(2) such that n(1) > n(2) = 2 and n-n(1)-n(2) = 1. The second one has two main cycles, a cycle where only the remaining vertices n(1)+1,..., n appear and a cycle containing only the vertices n(1)+1, n(1)+n(2)-1. Moreover, the last vertices are connected through the 2-arc (n(1) +n(2)-1, n). Furthermore, if n > 12 and is a multiple of 3, for appropriate n(1) and n(2), the ILR of that family is at least cubic, exactly, it must be n(3)+9n(2)+45n+108/27, which shows that some local states can be deterministically reached much further than initially proposed in the literature.We are gratefully thankful to the reviewers for their valuable remarks. This work has been partially supported by the European Union [FEDER funds] and Ministerio de Ciencia e Innovacion through Grants MTM-2013-43678-P and DPI2016-78831-C2-1-R.Bailo Ballarín, E.; Gelonch, J.; Romero Vivó, S. (2019). A first cubic upper bound on the local reachability index for some positive 2-D systems. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 113(4):3767-3784. https://doi.org/10.1007/s13398-019-00699-0376737841134Bailo, E., Gelonch, J., Romero-Vivo, S.: An upper bound on the reachability index for a special class of positive 2-D systems. Electron. J. Linear Algebra 18, 1–12 (2009)Bailo, E., Gelonch, J., Romero-Vivo, S.: Advances on the reachability index of positive 2-D systems. IEEE Trans. Autom. Control 59(8), 2248–2251 (2014)Bartosiewicz, Z.: Reachability and observability graphs for linear positive systems on time scales. IFAC Proc. Vol. 47(3), 3803–3808 (2014)Benvenuti, L., De Santis, A., Farina, L. (eds.): Positive Systems: Theory and Applications. Lecture Notes in Control and Information Sciences, vol. 294. Springer, Berlin (2003)Benvenuti, L.: On the reachable set for third-order linear discrete-time systems with positive control. Syst. Control Lett. 60(9), 690–698 (2011)Benvenuti, L.: On the reachable set for third-order linear discrete-time systems with positive control: the case of complex eigenvalues. Syst. Control Lett. 60(9), 1000–1008 (2011)Benzaouia, A., Hmamed, A., Tadeo, F.: Two-dimensional systems: from introduction to state of the art. In: Studies in Systems, Decision and Control (Vol. 28). Springer, Switzerland, (2016). https://doi.org/10.1007/978-3-319-20116-0Bru, R., Romero-Vivo, S. (eds.): Positive Systems: Theory and Applications. Lecture Notes in Control and Information Sciences, vol. 389. Springer, Berlin (2009)Bru, R., Bailo, E., Gelonch, J., Romero, S.: On the reachability index of positive 2-d systems. IEEE Trans. Circ. Syst. II: Express Brief 53(10), 997–1001 (2006)Bru, R., Coll, C., Romero, S., Sánchez, E.: Reachability indices of positive linear systems. Electron. J. Linear Algebra 11, 88–102 (2004)Bru, R., Romero-Vivó, S., Sánchez, E.: Reachability indices of periodic positive systems via positive shift-similarity. Linear Algebra Appl. 429, 1288–1301 (2008)Bru, R., Cacceta, L., Rumchev, V.G.: Monomial subdigraphs of reachable and controllable positive discrete-time systems. Int. J. Appl. Math. Comput. Sci. 15(1), 159–166 (2005)Bru, R., Romero, S., Sánchez, E.: Canonical forms of reachability and controllability of positive discrete-time control systems. Linear Algebra Appl. 310, 49–71 (2000)Cacace, F., Farina, L., Setola, R., Germani, A. (eds.): Positive Systems: Theory and Applications. Lecture Notes in Control and Information Sciences, vol. 471. Springer, Berlin (2017)Cantó, B., Coll, C., Sánchez, E.: On stability and reachability of perturbed positive systems. Adv. Differ. Equ. 2014(1), 296 (2014). https://doi.org/10.1186/1687-1847-2014-296Coll, C., Fullana, M., Sánchez, E.: Reachability and observability indices of a discrete-time periodic descriptor system. Appl. Math. Comput. 153, 485–496 (2004)Commault, C.: A simple graph theoretic characterization of reachability for positive linear systems. Syst. Control Lett. 52(3–4), 275–282 (2004)Commault, C.: On the reachability in any fixed time for positive continuous-time linear systems. Syst. Control Lett. 56(4), 272–276 (2007)Commault, C., Marchand, N. (eds.): Positive Systems: Theory and Applications. Lecture Notes in Control and Information Sciences, vol. 341. Springer, Berlin (2006)Coxson, P.G., Shapiro, H.: Positive reachability and controllability of positive systems. Linear Algebra Appl. 94, 35–53 (1987)Coxson, P.G., Larson, L.C., Schneider, H.: Monomial patterns in the sequence $${A}^kb$$. Linear Algebra Appl. 94, 89–101 (1987)De la Sen, M.: On the reachability and controllability of positive linear time-invariant dynamic systems with internal and external incommensurate point delays. Rocky Mt J Math 40(1), 177–207 (2010)Fanti, M.P., Maione, B., Turchiano, B.: Controllability of linear single-input positive discrete time systems. Int J Control 50, 2523–2542 (1989)Fanti, M.P., Maione, B., Turchiano, B.: Controllability of multi-input positive discrete time systems. Int J Control 51, 1295–1308 (1990)Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. Pure and Applied Mathematics. Wiley, New York (2000)Fornasini, E., Marchesini, G.: State-Space realization of two-dimensional filters. IEEE Trans. Autom. Control AC–21(4), 484–491 (1976)Fornasini, E., Marchesini, G.: Doubly indexed dynamical systems. Math. Syst. Theory 12, 59–72 (1978)Fornasini, E., Valcher, M.E.: On the positive reachability of 2D positive systems. In: Farina, L., Benvenuti, L., De Santis, A. (eds.) Positive Systems. Lecture Notes in Control and Information Sciences, pp. 297–304. Springer, Berlin (2003)Fornasini, E., Valcher, M.E.: Controllability and reachability of 2-d positive systems: a graph theoretic approach. IEEE Trans. Circuits Syst. I Regul. Pap. 52(3), 576–585 (2005)Hrynlów, K., Markowski, K.A.: Experimental evaluation of upper bounds of reachability index for set of solutions of 2-D positive system. In: 17th International Carpathian Control Conference (ICCC), Tatranska Lomnica, pp. 248–252 (2016). https://doi.org/10.1109/CarpathianCC.2016.7501103Kaczorek, T.: Reachability and controllability of 2D positive linear systems with state feedback. Control Cybern. 29(1), 141–151 (2000)Kaczorek, T.: Positive 1D and 2D Systems. Springer, London (2002)Kaczorek, T.: Reachability and minimum energy control of positive 2D systems with delays. Control Cybern 34(2), 411–423 (2005)Kaczorek, T.: New reachability and observability tests for positive linear discrete-time systems. Bull. Polish Acad. Sci. Tech. Sci. 55(1), 19–21 (2007)Kaczorek, T.: Reachability of linear hybrid systems described by the general model. J. Arch. Control Sci. 20(2), 199–207 (2010)Kaczorek, T., Borawski, K.: Existence of reachable pairs (A, B) of discrete-time linear systems. In: Proceedings of 21st International Conference on Methods and Models in Automation and Robotics (MMAR). IEEE, pp. 702–707 (2016). https://doi.org/10.1109/MMAR.2016.7575222Kostova, S.P.: A PLDS model of pollution in connected water reservoirs. In: Benvenuti, L., De Santis, A., Farina, L. (eds.) Positive Systems. Lecture Notes in Control and Information Science, vol. 294, pp. 257–263. Springer, Berlin (2003)Marszalek, W.: Two-dimensional state space discrete models for hyperbolic partial differential equations. Appl. Math. Model. 8(1), 11–14 (1984)Markowski, K.A.: Determination reachability index space of positive two-dimensional linear system using digraph-based theory. In: 19th International Conference on System Theory, Control and Computing (ICSTCC), Cheile Gradistei, pp. 533–538 (2015). https://doi.org/10.1109/ICSTCC.2015.7321348Moysis, L., Mishra, V.: Existence of reachable and observable triples of linear discrete-time descriptor systems. Circ. Syst. Signal Process. 3, 1–13 (2018). https://doi.org/10.1007/s00034-018-0922-5Pereira, R., Rocha, P., Simões, R.: Characterizations of global reachability of 2D structured systems. Multidimens. Syst. Signal Process. 24, 1–14 (2011). https://doi.org/10.1007/s11045-011-0154-3Rumchev, V.G., James, D.J.G.: Reachability and controllability of time-variant discrete-time positive linear systems. Control Cybern. 33(1), 87–93 (2004)Rumchev, V., Chotijah, S.: The minimum energy problem for positive discrete-time linear systems with fixed final state. In: Bru, R., Romero-Vivo, S. (eds.) Positive Systems. Lecture notes in control and information sciences, vol. 389, pp. 141–149. Springer, Berlin (2009)Valcher, M.E.: Controllability and reachability criteria for discrete time positive systems. Int. J. Control 65(3), 511–536 (1996)Valcher, M.E.: Reachability properties of continuous-time positive systems. IEEE Trans. Autom. Control 54(7), 1586–1590 (2009)Valcher, M.E.: Reachability analysis for different classes of positive systems. In: Bru, R., Romero-Vivo, S. (eds.) Positive Systems. Lecture notes in control and information sciences, vol. 389, pp. 29–41. Springer, Berlin (2009