An algebraic approach to the structural properties of positive state control systems

[EN] In this paper, we deal with discrete-time linear control systems in which the state is constrained to lie in the positive orthant R_+^n independently of the inputs involved, that is, the inputs can take negative values. Such (positive state) systems appear, for example, in ecology models where the removal of individuals from a population is described. Controllability and reachability are fundamental properties of a system that show its ability to move in space, which are analyzed from an algebraic point of view throughout the text, paying special attention to the single-input case.This work has been partially supported by Ministerio de Ciencia e Innovacion through Grants DPI-2013-46982-C2-1-R and MTM-2013-43678-P.Ricarte Benedito, B.; Romero Vivó, S. (2018). An algebraic approach to the structural properties of positive state control systems. Mathematical Methods in the Applied Sciences. 41(6):2370-2378. https://doi.org/10.1002/mma.4351S2370237841

Reachability indices of periodic positive systems via positive shift-similarity

AbstractThe problem of the paper is the formulation of the concept of reachability indices of periodic linear positive systems. A positive system is one in which the positive orthant is an invariant subset with respect to positive input signals. The concept of shift-similarity for periodic positive systems is defined and the associated cyclically augmented system is considered. The results are the relation of the positive similarity of both representations, the formulation of the reachability indices, and a detailed example

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/s13398-019-00699-0S376737841134Bailo, 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. 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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. 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Insulin Estimation and Prediction A REVIEW OF THE ESTIMATION AND PREDICTION OF SUBCUTANEOUS INSULIN PHARMACOKINETICS IN CLOSED-LOOP GLUCOSE CONTROL

This work was supported by the Spanish Ministry of Economy and Competitiveness (MINECO) through grant DPI2013-46982-C2-1-R and the EU through FEDER funds.Bondía Company, J.; Romero Vivó, S.; Ricarte Benedito, B.; Diez, J. (2018). Insulin Estimation and Prediction A REVIEW OF THE ESTIMATION AND PREDICTION OF SUBCUTANEOUS INSULIN PHARMACOKINETICS IN CLOSED-LOOP GLUCOSE CONTROL. IEEE Control Systems. 38(1):47-66. https://doi.org/10.1109/MCS.2017.2766312S476638

On the Exact Solution of a Class of Homogeneous Strongly Coupled Mixed Parabolic Problems

[EN] In this paper an exact series solution for homogeneous parabolic coupled systems is constructed using a projection method. An illustrative example is given.This work has been supported by Spanish Ministerio de Economía y Competitividad and the European Regional Development Fund (ERDF) TIN2014-59294-P.Defez Candel, E.; Soler Basauri, V.; Romero Vivó, S.; Verdoy González, JA. (2019). On the Exact Solution of a Class of Homogeneous Strongly Coupled Mixed Parabolic Problems. Filomat. 33(3):897-915. https://doi.org/10.2298/FIL1903897DS89791533

On the Computation of Output Bounds for Compartmental in-Series Models under Parametric Uncertainty

[EN] In this work, the problem of obtaining tight output bounds for compartmental in-series models under parametric uncertainty is addressed. It is well-known that current methods used to compute a solution envelope may produce a significant overestimation. However, monotonicityanalysis enables us to estimate a tight solution envelope. Our main aim is to get an equivalent model to the initial one, which is usually non-monotone, by means of a suitable combination of equations. In this new model the system monotonicity with respect to the uncertain parameters depends on the elimination rate values of the original model. If the equivalent model is monotone, no overestimation occurs in the computation of the output bounds.This work was partially supported by the Spanish Ministerio de Ciencia e Innovacion through Grant DPI-2010- 20764-C02, and by the Generalitat Valenciana through Grant GV/2012/085.De Pereda Sebastián, D.; Romero Vivó, S.; Ricarte Benedito, B.; Bondía Company, J. (2012). On the Computation of Output Bounds for Compartmental in-Series Models under Parametric Uncertainty. Biomath: International Journal on Mathematical Methods and Models in Biosciences. 1(2):1-6. https://doi.org/10.11145/j.biomath.2012.10.043S161

Real-time estimation of plasma insulin concentration from continuous glucose monitor measurements

Continuous glucose monitors can measure interstitial glucose concentration in real time for closed-loop glucose control systems, known as artificial pancreas. These control systems use an insulin feedback to maintain plasma glucose concentration within a narrow and safe range, and thus to avoid health complications. As it is not possible to measure plasma insulin concentration in real time, insulin models have been used in literature to estimate them. Nevertheless, the significant interand intra-patient variability of insulin absorption jeopardizes the accuracy of these estimations. In order to reduce these limitations, our objective is to perform a real-time estimation of plasma insulin concentration from continuous glucose monitoring (CGM). Hovorka s glucose insulin model has been incorporated in an extended Kalman filter in which different selected time-variant model parameters have been considered as extended states. The observability of the original Hovorka s model and of several extended models has been evaluated by their Lie derivatives. We have evaluated this methodology with an in-silico study with 100 patients with Type 1 diabetes during 25 h. Furthermore, it has been also validated using clinical data from 12 insulin pump patients with Type 1 diabetes who underwent four mixed meal studies. Real-time insulin estimations have been compared to plasma insulin measurements to assess performance showing the validity of the methodology here used in comparison with that formerly used for insulin models. Hence, real-time estimations for plasma insulin concentration based on subcutaneous glucose monitoring can be beneficial for increasing the efficiency of control algorithms for the artificial pancreas.This work was partially supported by the Spanish Ministerio de Ciencia e Innovacion through Grant DPI-2010-20764-C02-01 and Grant DPI-2013-46982-C2-1-R, and the European Union through FEDER fund.De Pereda Sebastián, D.; Romero Vivó, S.; Ricarte Benedito, B.; Rossetti, P.; Ampudia Blasco, FJ.; Bondía Company, J. (2015). Real-time estimation of plasma insulin concentration from continuous glucose monitor measurements. Computer Methods in Biomechanics and Biomedical Engineering. Sep:1-9. https://doi.org/10.1080/10255842.2015.1077234S19Se

Nuevos planteamientos en las prácticas de matemáticas con Mathematica en la ETSICCP de Valencia

[ES] En el contexto de los nuevos planes de estudio, los profesores del departamento de Matemática Aplicada de la ETSICCP de Valencia nos planteamos cómo enfocar las prácticas de informática de las dos asignaturas de primer curso que debíamos impartir. Teníamos la experiencia de los otros planes de estudio en los que se habían diseñado prácticas de informática con los programas Derive y Mathematica, en las que los materiales que se aportaban a los alumnos eran, bien textos en papel, o publicados en un libro, o en otros casos materiales más o menos interactivos en un CD multimedia. En esta ocasión hemos optado por diseñar un tutorial de cada práctica, en formato electrónico, que es un archivo del programa Mathematica, que contiene texto, imágenes, instrucciones, ejemplos y ejercicios. El alumno trabaja cada práctica presencial mediante trabajo autónomo, pero con ayuda del profesor y es evaluado mediante un test que se genera desde baterías de preguntas creadas desde la plataforma PoliformaT de la Universidad Politécnica de Valencia. Debemos señalar que estas prácticas podrían ser perfectamente planteadas también como no presencialesLa presente comunicación ha sido financiada por la Universitat Politècnica de València, a través de la Comisión de Evaluación y Seguimiento de Proyectos de Innovación y Convergencia (CESPIe), en dos Proyectos de Innovación y Mejora Educativa (PIME): ¿ Experimentación y validación de estrategias de evaluación en asignaturas de matemáticas en los grados de Obras Públicas e Ingeniería Civil (2010-2011). PIME B011/10. ¿ Necesidades y soluciones para la evaluación de los estudiantes en títulos de grado y máster (2010-2011). PIME A010/10.Company Rossi, R.; García Raffi, LM.; Pérez Peñalver, MJ.; Ponsoda Miralles, E.; Rodríguez López, J.; Romero Vivó, S.; Tirado Peláez, P. (2011). Nuevos planteamientos en las prácticas de matemáticas con Mathematica en la ETSICCP de Valencia. Instituto de Ciencias de la Educación de la Universidad de Alicante. 1058-1069. http://hdl.handle.net/10251/178205S1058106

On a safety set for an epidemic model with a bounded population

[EN] Given a class of non-linear SIRS epidemic model, we analyse some useful conditions on the model parameters to determine a safety set for the containment of an epidemic. In addition, once that set is determined, we find control actions so that the epidemic remains within the security set with infection rates below an allowed amount. More specifically, for every initial state in a certain safety set of the state space there exists an adequate control policy maintaining the state of the system in such safety set. Sufficient conditions for the existence of a solution under a feedback are derived in terms of linear inequalities on the input vectors at the vertices of a polytope.Coll, C.; Romero Vivó, S.; Sánchez, E. (2022). On a safety set for an epidemic model with a bounded population. Mathematical Modelling and Analysis. 27(2):263-281. https://doi.org/10.3846/mma.2022.1458626328127

On a safety set for an epidemic model with a bounded population

Given a class of non-linear SIRS epidemic model, we analyse some useful conditions on the model parameters to determine a safety set for the containment of an epidemic. In addition, once that set is determined, we find control actions so that the epidemic remains within the security set with infection rates below an allowed amount. More specifically, for every initial state in a certain safety set of the state space there exists an adequate control policy maintaining the state of the system in such safety set. Sufficient conditions for the existence of a solution under a feedback are derived in terms of linear inequalities on the input vectors at the vertices of a polytope