Some recent developments in theory of fractional positive and cone linear systems

The overview of some recent developments and new results in the theory of fractional positive and cone one-dimensional (1D) and two-dimensional (2D) linear systems are presented in the article. The state equations and their solution for the fractional continuous-time and discrete-time linear systems are given. Necessary and sufficient conditions for the internal and external positivity of the fractional linear systems are established. The reachability of the fractional linear systems is addressed. A new notation of the cone systems is introduced and methods for computation of such systems are proposed. Positive fractional 2D linear systems are intoduced. Necessary and sufficient conditions for the positivity and reachability are established. The considerations are illustrated with many examples of 1D and 2D linear systems.Запропоновано огляд та деякі нові результати у теорії фракційних додатніх та конусних одновимірних (1D) та двовимірних (2D) лінійних систем. Подано рівняння стану та їхнє розв’язання для фракційних неперервних та дискретних лінійних систем. Встановлено необхідні та достатні умови для внутрішньої та зовнішньої додатності фракційних лінійних систем. Показано їхню досяжність. Запропоновано нову форму запису конусних систем та методи, придатні для комп’ютерного розрахунку таких систем. Представлено додатні фракційні 2D лінійні системи. Встановлено необхідні та достатні умови для додатності та досяжності. Теоретичні виклади проілюстровано чисельними прикладами 1D та 2D лінійних систем.Предложены анализ состояния вопроса и некоторые новые результаты, полученные в теории фракционных положительных и конусных одномерных (1D) и двумерных (2D) линейных систем. Приведены уравнения состояния и их решения для фракционных непрерывных и дискретных линейных систем. Выявлены необходимые и достаточные условия для внутренней и внешней положительности фракционных линейных систем. Показана их достижимость. Предложена новая форма записи конусных систем, а также методы их компьютерного расчета. Представлены положительные фракционные 2D линейные системы. Установлены необходимые и достаточные условия для положительности и достижимости. Теоретические выкладки иллюстрированы численными примерами 1D и 1D линейных систем

A direct method to obtain a realization of a polynomial matrix and its applications

[EN] In this paper we present a Silverman-Ho algorithm-based method to obtain a realization of a polynomial matrix. This method provides the final formulation of a minimal realization directly from a full rank factorization of a specific given matrix. Also, some classical problems in control theory such as model reduction in singular systems or the positive realization problem in standard systems are solved with this method.Work supported by the Spanish DGI grant MTM2017-85669-P-AR.Cantó Colomina, R.; Moll López, SE.; Ricarte Benedito, B.; Urbano Salvador, AM. (2020). A direct method to obtain a realization of a polynomial matrix and its applications. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 114(2):1-15. https://doi.org/10.1007/s13398-020-00819-1S1151142Anderson, B.D.O., Bongpanitlerd, S.: Network Analysis and Synthesis, A Modern Systems Theory Approach. Prentice-Hall Inc., New Jersey (1968)Benvenuti, L., Farina, L.: A tutorial on the positive realization problem. IEEE Trans. Autom. Control 49(5), 651–664 (2004). https://doi.org/10.1109/TAC.2004.826715Bru, R., Coll, C., Sánchez, E.: Structural properties of positive linear time-invariant difference-algebraic equations. Linear Algebra Appl. 349, 1–10 (2002). https://doi.org/10.1016/S0024-3795(02)00277-XCantó, R., Ricarte, B., Urbano, A.M.: Positive realizations of transfer matrices with real poles. IEEE Trans. Circuits Syst. II Expr. Br. 54(6), 517–521 (2007). https://doi.org/10.1109/TCSII.2007.894408Cantó, R., Ricarte, B., Urbano, A.M.: On positivity of discrete-time singular systems and the realization problem. Lect. Notes Control Inf. Sci. 389, 251–258 (2009). https://doi.org/10.1007/978-3-642-02894-6_24Climent, J., Napp, D., Requena, V.: Block Toeplitz matrices for burst-correcting convolutional codes. RACSAM 114, 38 (2020). https://doi.org/10.1007/s13398-019-00744-yDai, L.: Singular Control Systems. Lecture Notes in Control and Information Sciences. Springer-Verlag, New York (1989)Golub, G.H., Van Loan, C.F.: Matrix Computations, Fourth edn. Johns Hopkins University Press, Baltimore (2013)Henrion, D., Šebek, M.: Polynomial and matrix fraction description. In: Control Systems, Robotics and Automation, vol. 7, pp. 211-231, (2009). http://www.eolss.net/Sample-Chapters/C18/E6-43-13-05.pdfHo, B.L., Kalman, R.E.: Effective construction of linear state-variable models from mput/output functions. Regelungstechnik 14(12), 545–548 (1966)Kaczorek, T.: Weakly positive continuous-time linear systems. Lect. Notes Control Inf. Sci. 243, 3–16 (1999)Kaczorek, T.: Positive 1D and 2D Systems, vol. 431. Springer, London (2002)Kaczorek, T.: Externally and internally positive singular discrete-time linear systems. Int. J. Appl. Math. Comput. Sci. 12(2), 197–202 (2002)MATLAB, The Math Works, Inc., Natick, Massachusetts, United States. Official website: http://www.mathworks.comMcCrory, C., Parusinski, A.: The weight filtration for real algebraic varieties II: classical homology. RACSAM 108, 63–94 (2014). https://doi.org/10.1007/s13398-012-0098-ySilverman, L.: Realization of linear dynamical systems. IEEE Trans. Autom. Control 16(6), 554–567 (1971)Virnik, E.: Stability analysis of positive descriptor systems. Linear Algebra Appl. 429(10), 2640–2659 (2008

Detectability subspaces and observer synthesis for two-dimensional systems

The notions of input-containing and detectability subspaces are developed within the context of observer synthesis for two-dimensional (2-D) Fornasini-Marchesini models. Specifically, the paper considers observers which asymptotically estimate the local state, in the sense that the error tends to zero as the reconstructed local state evolves away from possibly mismatched boundary values, modulo a detectability subspace. Ultimately, the synthesis of such observers in the absence of explicit input information is addressed

Significance of the presence of antibiotics on the microbial consortium in wastewater - The case of nitrofurantoin and furazolidone.

Antibiotics in wastewater leads to migration of pollutants and disrupts natural processes of mineralization of organic matter. In order to understand the mechanism of this, research was undertaken on the influence of nitrofurantoin (NFT) and furazolidone (FZD), on the behaviour of a consortium of microorganisms present in a model wastewater in a bioreactor. Our study confirmed biodegradation of the antibiotics by the microbial consortium, with the degradation efficiency within 10 days of 65% for FZD, but only 20% for NFT. The kinetic study proved that the presence of analysed antibiotics had no adverse effect on the microbes, but the consortium behaviour differ significantly with the NFT reducing the consumption of organic carbon in wastewater and increasing the production of extracellular biopolymeric and volatile organic compounds, and the FZD reducing assimilation of other carbon sources to a less extent, at the expense of cellular focus on biodegradation of this antibiotic

Finite-region stabilization via dynamic output feedback for 2-D Roesser models

Finite-region stability (FRS), a generalization of finite-time stability, has been used to analyze the transient behavior of discrete two-dimensional (2-D) systems. In this paper, we consider the problem of FRS for discrete 2-D Roesser models via dynamic output feedback. First, a sufficient condition is given to design the dynamic output feedback controller with a state feedback-observer structure, which ensures the closed-loop system FRS. Then, this condition is reducible to a condition that is solvable by linear matrix inequalities. Finally, viable experimental results are demonstrated by an illustrative example

Uptake of Hydrocarbon by Pseudomonas fluorescens (P1) and Pseudomonas putida (K1) Strains in the Presence of Surfactants: A Cell Surface Modification

The objective of this research was the evaluation of the effects of exogenous added surfactants on hydrocarbon biodegradation and on cell surface properties. Crude oil hydrocarbons are often difficult to remove from the environment because of their insolubility in water. The addition of surfactants enhances the removal of hydrocarbons by raising the solubility of these compounds. These surfactants cause them to become more vulnerable to degradation, thereby facilitating transportation across the cell membrane. The obtained results showed that the microorganism consortia of bacteria are useful biological agents within environmental bioremediation. The most effective amongst all, as regards biodegradation, were the consortia of Pseudomonas spp. and Bacillus spp. strains. The results indicated that the natural surfactants (rhamnolipides and saponins) are more effective surfactants in hydrocarbon biodegradation as compared to Triton X-100. The addition of natural surfactants enhanced the removal of hydrocarbon and diesel oil from the environment. Very promising was the use of saponins as a surfactant in hydrocarbon biodegradation. This surfactant significantly increases the organic compound biodegradation. In the case of those surfactants that could be easily adsorbed on cells of strains (e.g., rhamnolipides), a change of hydrophobicity to ca. 30–40% was noted. As the final result, an increase in hydrocarbon biodegradation was observed

An algorithm to study the nonnegativity, regularity and stability via state-feedbacks of singular systems of arbitrary index

This paper deals with singular systems of index k ≥ 1. Our main goal is to find a state-feedback such that the closed-loop system satis- fies the regularity condition and it is nonnegative and stable. In order to do that, the core-nilpotent decomposition of a square matrix is applied to the singular matrix of the system. Moreover, if the Drazin projector of this matrix is nonnegative then the previous decomposition allows us to write the core-part of the matrix in a specific block form. In addition, an algorithm to study this kind of systems via a state-feedback is designed.This paper has been partially supported by Ministry of Education of Spain [grant number DGI MTM2010-18228].Herrero Debón, A.; Francisco J. Ramírez; Thome, N. (2014). An algorithm to study the nonnegativity, regularity and stability via state-feedbacks of singular systems of arbitrary index. Linear and Multilinear Algebra. 1-11. https://doi.org/10.1080/03081087.2014.904559S11

On some aspects of the geometry of differential equations in physics

In this review paper, we consider three kinds of systems of differential equations, which are relevant in physics, control theory and other applications in engineering and applied mathematics; namely: Hamilton equations, singular differential equations, and partial differential equations in field theories. The geometric structures underlying these systems are presented and commented. The main results concerning these structures are stated and discussed, as well as their influence on the study of the differential equations with which they are related. Furthermore, research to be developed in these areas is also commented.Comment: 21 page

Improving the vibration suppression capabilities of a magneto-rheological damper using hybrid active and semi-active control

This paper presents a new hybrid active & semi-active control method for vibration suppression in flexible structures. The method uses a combination of a semi-active device and an active control actuator situated elsewhere in the structure to suppress vibrations. The key novelty is to use the hybrid controller to enable the magneto-rheological damper to achieve a performance as close to a fully active device as possible. This is achieved by ensuring that the active actuator can assist the magneto-rheological damper in the regions where energy is required. In addition, the hybrid active & semi-active controller is designed to minimize the switching of the semi-active controller. The control framework used is the immersion and invariance control technique in combination with sliding mode control. A two degree-of-freedom system with lightly damped resonances is used as an example system. Both numerical and experimental results are generated for this system, and then compared as part of a validation study. The experimental system uses hardware-in-the-loop to simulate the effect of both the degrees-of-freedom. The results show that the concept is viable both numerically and experimentally, and improved vibration suppression results can be obtained for the magneto-rheological damper that approach the performance of an active device

Geometric techniques for implicit two-dimensional systems

Geometric tools are developed for two-dimensional (2-D) models in an implicitFornasini–Marchesini form. In particular, the structural properties of controlled and conditionedinvariance are defined and studied. These properties are investigated in terms ofquarter-plane causal solutions of the implicit model given compatible boundary conditions.The definitions of controlled and conditioned invariance introduced, along with the correspondingoutput-nulling and input-containing subspaces, are shown to be richer than theone-dimensional counterparts. The analysis carried out in this paper establishes necessaryand sufficient conditions for the solvability of 2-D disturbance decoupling problems andunknown-input observation problems. The conditions obtained are expressed in terms ofoutput-nulling and input-containing subspaces, which can be computed recursively in a finitenumber of steps