Stabilization of positive linear continuous-time systems by using a Brauer´s theorem

[EN] In this paper we study the stability property of positive linear continuous-time systems. This property is useful to study the asymptotic behavior of a dynamical system and specifically, in positive systems. Stabilization of linear systems using feedbacks has been deeply studied during the last decades. Motivated by some results, in this paper we find conditions on the system such that the eigenvalues of the closed loop system are in the open left half plane of the complex plane C. We do this by applying a Brauer s theorem.The authors would like to thank the referees and the editor for their comments and useful suggestions for improvement of the manuscript. This research was partially supported by Spanish Grant MTM2013-43678-P.Cantó Colomina, B.; Cantó Colomina, R.; Urbano Salvador, AM. (2016). Stabilization of positive linear continuous-time systems by using a Brauer´s theorem. International Journal of Complex Systems in Science. 6(1):23-28. http://hdl.handle.net/10251/81742S23286

Stabilization of positive linear discrete-time systems by using a Brauer's theorem

The stabilization problem of positive linear discrete-time systems (PLDS) by linear state feedback is considered. A method based on a Brauer s theorem is proposed for solving the problem. It allows us to modify some eigenvalues of the system without hanging the rest of them. The problem is studied for the single-input single-output (SISO) and for multi-input multioutput (MIMO) cases and sufficient conditions for stability and positivity of the closed-loop system are proved.The results are illustrated by numerical examples and the proposed method is used in stochastic systems.This work is supported by the Spanish DGI Grant MTM2010-18228.Cantó Colomina, B.; Cantó Colomina, R.; Kostova, S. (2014). Stabilization of positive linear discrete-time systems by using a Brauer's theorem. Scientific World Journal. 2014:1-6. https://doi.org/10.1155/2014/856356S162014Caccetta, L., & Rumchev, V. G. (2000). Annals of Operations Research, 98(1/4), 101-122. doi:10.1023/a:1019244121533Allen, L. J. S., & van den Driessche, P. (2008). The basic reproduction number in some discrete-time epidemic models. Journal of Difference Equations and Applications, 14(10-11), 1127-1147. doi:10.1080/10236190802332308Delchamps, D. F. (1988). State Space and Input-Output Linear Systems. doi:10.1007/978-1-4612-3816-4Méndez-Barrios, C.-F., Niculescu, S.-I., Chen, J., & Maya-Méndez, M. (2013). Output feedback stabilisation of single-input single-output linear systems with I/O network-induced delays. An eigenvalue-based approach. International Journal of Control, 87(2), 346-362. doi:10.1080/00207179.2013.834075Anderson, B. D. O., Ilchmann, A., & Wirth, F. R. (2013). Stabilizability of linear time-varying systems. Systems & Control Letters, 62(9), 747-755. doi:10.1016/j.sysconle.2013.05.003De Leenheer, P., & Aeyels, D. (2001). Stabilization of positive linear systems. Systems & Control Letters, 44(4), 259-271. doi:10.1016/s0167-6911(01)00146-3Fornasini, E., & Valcher, M. E. (2012). Stability and Stabilizability Criteria for Discrete-Time Positive Switched Systems. IEEE Transactions on Automatic Control, 57(5), 1208-1221. doi:10.1109/tac.2011.2173416Bru, R., Cantó, R., Soto, R. L., & Urbano, A. M. (2011). A Brauer’s theorem and related results. Central European Journal of Mathematics, 10(1), 312-321. doi:10.2478/s11533-011-0113-0Soto, R. L., & Rojo, O. (2006). Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem. Linear Algebra and its Applications, 416(2-3), 844-856. doi:10.1016/j.laa.2005.12.026Silva, M. S., & de Lima, T. P. (2003). Looking for nonnegative solutions of a Leontief dynamic model. Linear Algebra and its Applications, 364, 281-316. doi:10.1016/s0024-3795(02)00569-4Mourad, B. (2013). Generalization of some results concerning eigenvalues of a certain class of matrices and some applications. Linear and Multilinear Algebra, 61(9), 1234-1243. doi:10.1080/03081087.2012.746330Pakshin, P. V., & Ugrinovskii, V. A. (2006). Stochastic problems of absolute stability. Automation and Remote Control, 67(11), 1811-1846. doi:10.1134/s0005117906110051Brauer, A. (1952). Limits for the characteristic roots of a matrix. IV: Applications to stochastic matrices. Duke Mathematical Journal, 19(1), 75-91. doi:10.1215/s0012-7094-52-01910-8Perfect, H. (1955). Methods of constructing certain stochastic matrices. II. Duke Mathematical Journal, 22(2), 305-311. doi:10.1215/s0012-7094-55-02232-8Berman, A., & Plemmons, R. J. (1994). Nonnegative Matrices in the Mathematical Sciences. doi:10.1137/1.9781611971262Cantó, B., Cardona, S. C., Coll, C., Navarro-Laboulais, J., & Sánchez, E. (2011). Dynamic optimization of a gas-liquid reactor. Journal of Mathematical Chemistry, 50(2), 381-393. doi:10.1007/s10910-011-9941-1Fieberg, J., & Ellner, S. P. (2001). Stochastic matrix models for conservation and management: a comparative review of methods. Ecology Letters, 4(3), 244-266. doi:10.1046/j.1461-0248.2001.00202.

Sobre las Matrices Totalmente No Positivas

[ES] En este trabajo presentamos un procedimiento para construir un tipo de matrices llamadas totalmente no positivas, estudiar sus propiedades y obtener las relaciones que tienen con otra clase de matrices llamadas totalmente no negativas.Este trabajo ha sido financiado por el proyecto MTM2017-85669-P-AR.Cantó Colomina, B.; Cantó Colomina, R.; Urbano Salvador, AM. (2021). Sobre las Matrices Totalmente No Positivas. Compobell. 49-52. http://hdl.handle.net/10251/191304495

Improving the condition number of a simple eigenvalue by a rank one matrix

In this work a technique to improve the condition number si of a simple eigenvalue lambda(i) of a matrix A is an element of C-nxn is given. This technique obtains a rank one updated matrix that is similar to A with the eigenvalue condition number of lambda(i) equal to one. More precisely, the similar updated matrix A + v(i)q*, where Av(i) = lambda(i)v(i) and q is a fixed vector, has s(i) = 1 and the remaining condition numbers are at most equal to the corresponding initial condition numbers. Moreover an expression to compute the vector q, using only the eigenvalue lambda(i) and its eigenvector v(i), is given. (C) 2016 Elsevier Ltd. All rights reserved.Supported by the Spanish DGI grant MTM2013-43678-P.Bru García, R.; Cantó Colomina, R.; Urbano Salvador, AM. (2016). Improving the condition number of a simple eigenvalue by a rank one matrix. Applied Mathematics Letters. 58:7-12. https://doi.org/10.1016/j.aml.2016.01.010S7125

Propiedades de las matrices totalmente no positivas

Una matriz real A se dice que es totalmente (negativa) no positiva si todos sus menores son (negativos) no positivos. En este trabajo veremos la factorización LDU de una matriz totalmente no positiva e invertible a partir del método de eliminación completo de Neville sin intercambio de filas y columnas. Dicha factorización nos permitirá generar de una forma sencilla matrices totalmente (negativas) no positivas del cualquier orden a partir de matrices totalmente positivas.Ministerio de Educación y Ciencia. Dirección General de InvestigaciónUniversidad Politécnica de Valenci

Full rank Cholesky factorization for rank deficient matrices

[EN] Let A be a rank deficient square matrix. We characterize the unique full rank Cholesky factorization LL^T of A where the factor L is a lower echelon matrix with positive leading entries. We compute an extended decomposition for the normal matrix B^TB where B is a rectangular rank deficient matrix. This decomposition is obtained without interchange of rows and without computing all entries of the normal matrix. Algorithms to compute both factorizations are given.This research was supported by the Spanish DGI grant MTM2010-18228 and by the Chilean program FONDECYT 1100029.Cantó Colomina, R.; Peláez, MJ.; Urbano Salvador, AM. (2015). Full rank Cholesky factorization for rank deficient matrices. Applied Mathematics Letters. 40:17-22. https://doi.org/10.1016/j.aml.2014.09.001S17224

On the characterization of totally nonpositive matrices

The final publication is available at Springer via http://dx.doi.org/10.1007/s40324-016-0073-1[EN] A nonpositive real matrix $A= (a_{ij})_{1 \leq i, j \leq n}$ is said to be totally nonpositive (negative) if all its minors are nonpositive (negative) and it is abbreviated as t.n.p. (t.n.). In this work a bidiagonal factorization of a nonsingular t.n.p. matrix $A$ is computed and it is stored in an matrix represented by $\mathcal{BD}_{(t.n.p.)}(A)$ when $a_{11}< 0$ (or $\mathcal{BD}_{(zero)}(A)$ when $a_{11}= 0$). As a converse result, an efficient algorithm to know if an matrix $\mathcal{BD}_{(t.n.p.)}(A)$ ($\mathcal{BD}_{(zero)}(A)$) is the bidiagonal factorization of a t.n.p. matrix with $a_{11}<0$ ($a_{11}= 0$) is given. Similar results are obtained for t.n. matrices using the matrix $\mathcal{BD}_{(t.n.)}(A)$, and these characterizations are extended to rectangular t.n.p. (t.n.) matrices. Finally, the bidiagonal factorization of the inverse of a nonsingular t.n.p. (t.n.) matrix $A$ is directly obtained from $\mathcal{BD}_{(t.n.p.)}(A)$ ($\mathcal{BD}_{(t.n.)}(A)$).This research was supported by the Spanish DGI grant MTM2013-43678-P and by the Chilean program FONDECYT 1100029Cantó Colomina, R.; Pelaez, MJ.; Urbano Salvador, AM. (2016). On the characterization of totally nonpositive matrices. SeMA Journal. 73(4):347-368. doi:10.1007/s40324-016-0073-1S347368734Ando, T.: Totally positive matrices. Linear Algebra Appl. 90, 165–219 (1987)Alonso, P., Peña, J.M., Serrano, M.L.: Almost strictly totally negative matrices: an algorithmic characterization. J. Comput. Appl. Math. 275, 238–246 (2015)Bapat, R.B., Raghavan, T.E.S.: Nonnegative Matrices and Applications. Cambridge University Press, New York (1997)Cantó, R., Koev, P., Ricarte, B., Urbano, A.M.: $$LDU$$ L D U -factorization of nonsingular totally nonpositive matrices. SIAM J. Matrix Anal. Appl. 30(2), 777–782 (2008)Cantó, R., Ricarte, B., Urbano, A.M.: Full rank factorization in echelon form of totally nonpositive (negative) rectangular matrices. Linear Algebra Appl. 431, 2213–2227 (2009)Cantó, R., Ricarte, B., Urbano, A.M.: Characterizations of rectangular totally and strictly totally positive matrices. Linear Algebra Appl. 432, 2623–2633 (2010)Cantó, R., Ricarte, B., Urbano, A.M.: Quasi- $$LDU$$ L D U factorization of nonsingular totally nonpositive matrices. Linear Algebra Appl. 439, 836–851 (2013)Cantó, R., Ricarte, B., Urbano, A.M.: Full rank factorization in quasi- $$LDU$$ L D U form of totally nonpositive rectangular matrices. Linear Algebra Appl. 440, 61–82 (2014)Fallat, S.M., Van Den Driessche, P.: On matrices with all minors negative. Electron. J. Linear Algebra 7, 92–99 (2000)Fallat, S.M.: Bidiagonal factorizations of totally nonnegative matrices. Am. Math. Mon. 108(8), 697–712 (2001)Fallat, S.M., Johnson, C.R.: Totally Nonnegative Matrices. Princeton University Press, New Jersey (2011)Gasca, M., Micchelli, C.A.: Total positivity and applications. Math. Appl. 359, Kluwer Academic Publishers, Dordrecht (1996)Gasca, M., Peña, J.M.: Total positivity, $$QR$$ Q R factorization and Neville elimination. SIAM J. Matrix Anal. Appl. 4, 1132–1140 (1993)Gasca, M., Peña, J.M.: A test for strict sign-regularity. Linear Algebra Appl. 197(198), 133–142 (1994)Gasca, M., Peña, J.M.: A matricial description of Neville elimination with applications to total positivity. Linear Algebra Appl. 202, 33–53 (1994)Gassó, M., Torregrosa, J.R.: A totally positive factorization of rectangular matrices by the Neville elimination. SIAM J. Matrix Anal. Appl. 25, 86–994 (2004)Huang, R., Chu, D.: Total nonpositivity of nonsingular matrices. Linear Algebra Appl. 432, 2931–2941 (2010)Huang, R., Chu, D.: Relative perturbation analysis for eigenvalues and singular values of totally nonpositive matrices. SIAM J. Matrix Anal. Appl. 36(2), 476–495 (2015)Karlin, S.: Total Nonpositivity. Stanford University Press, Stanford (1968)Koev, P.: Accurate eigenvalues and SVDs of totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 27(1), 1–23 (2005)Koev, P.: Accurate computations with totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 29(3), 731–751 (2007)Parthasarathy, T.: $$N$$ N -matrices. Linear Algebra Appl. 139, 89–102 (1990)Peña, J.M.: Test for recognition of total positivity. SeMA J. 62(1), 61–73 (2013)Pinkus, A.: Totally Positive Matrices. Cambridge Tracts in Mathematics, vol. 181. Cambridge University Press (2009)Saigal, R.: On the class of complementary cones and Lemke’s algorithm. SIAM J. Appl. Math. 23, 46–60 (1972

Realizaciones positivas de determinados sistemas singulares

En este trabajo se estudian los sistemas singulares lineales de control a partir de las propiedades obtenidas para los sistemas estándares. Se obtienen realizaciones positivas de ciertas matrices de trasferencia con polos reales, analizando las condiciones para que la dimensión de la realización positiva sea minimal

Full rank factorization in quasi-LDU form of totally nonpositive rectangular matrices

Let A = (a(ij)) is an element of R-nxm be a totally nonpositive matrix with rank(A) = r <= min{n, m} and a(11) = 0. In this paper we obtain a characterization in terms of the full rank factorization in quasi-LDU form, that is, A = (L) over tilde DU where (L) over tilde is an element of R-nxr is a block lower echelon matrix, U is an element of R-rxm is a unit upper echelon totally positive matrix and D is an element of R-rxr is a diagonal matrix, with rank((L) over tilde) = rank(U) = rank(D) = r. We use this quasi-LDU decomposition to construct the quasi-bidiagonal factorization of A. Moreover, some properties about these matrices are studied. (C) 2013 Elsevier Inc. All rights reserved.This research was supported by the Spanish DGI grant MTM2010-18228.Cantó Colomina, R.; Ricarte Benedito, B.; Urbano Salvador, AM. (2014). Full rank factorization in quasi-LDU form of totally nonpositive rectangular matrices. Linear Algebra and its Applications. 440:61-82. https://doi.org/10.1016/j.laa.2013.11.002S618244

Aplicaciones de las matrices totalmente positivas

[ES] Se dice que una matriz es totalmente positiva (se denota por TP) si el determinante de cada una de sus submatrices cuadradas es mayor o igual que cero, y es estrictamente totalmente positiva (denotada por STP) si todos los determinantes de sus submatrices cuadradas son positivos. En este trabajo vamos a considerar la teoría de la total positividad con considerables consecuencias y aplicaciones en diferentes áreas, como son en Biología, Física, Economía, etc. y también en la propia ciencia matemática.Este trabajo ha sido financiado por el proyecto DGI MTM2013-43678-P.Cantó Colomina, R.; Ricarte Benedito, B.; Urbano Salvador, AM. (2014). Aplicaciones de las matrices totalmente positivas. Compobell, S.L. http://hdl.handle.net/10251/74277