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This is the peer reviewed version of the following article: Diez, P. [et al.]. Nonlinear dimensionality reduction for parametric problems: a kernel proper orthogonal decomposition. "International journal for numerical methods in engineering", 30 Desembre 2021, vol. 122, núm. 24, p. 7306-7327, which has been published in final form at DOI: 10.1002/nme.6831. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. Reduced-order models are essential tools to deal with parametric problems in the context of optimization, uncertainty quantification, or control and inverse problems. The set of parametric solutions lies in a low-dimensional manifold (with dimension equal to the number of independent parameters) embedded in a large-dimensional space (dimension equal to the number of degrees of freedom of the full-order discrete model). A posteriori model reduction is based on constructing a basis from a family of snapshots (solutions of the full-order model computed offline), and then use this new basis to solve the subsequent instances online. Proper orthogonal decomposition (POD) reduces the problem into a linear subspace of lower dimension, eliminating redundancies in the family of snapshots. The strategy proposed here is to use a nonlinear dimensionality reduction technique, namely, the kernel principal component analysis (kPCA), in order to find a nonlinear manifold, with an expected much lower dimension, and to solve the problem in this low-dimensional manifold. Guided by this paradigm, the methodology devised here introduces different novel ideas, namely, 1) characterizing the nonlinear manifold using local tangent spaces, where the reduced-order problem is linear and based on the neighboring snapshots, 2) the approximation space is enriched with the cross-products of the snapshots, introducing a quadratic description, 3) the kernel for kPCA is defined ad hoc, based on physical considerations, and 4) the iterations in the reduced-dimensional space are performed using an algorithm based on a Delaunay tessellation of the cloud of snapshots in the reduced space. The resulting computational strategy is performing outstandingly in the numerical tests, alleviating many of the problems associated with POD and improving the numerical accuracy. Generalitat de Catalunya, 2017-SGR-1278; Ministerio de Ciencia e Innovación, CEX2018-000797-S; PID2020-113463RB-C32; PID2020-113463RB-C33

© 2022 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ Searching for achieving an ambitious reduction in greenhouse gas emissions, the EU has set as a goal a modal shift in freight transport of 30% by rail for the near future. In this context, it is vital to use modal choice models road versus rail to assess the shippers' acceptance of the actions promoting the use of rail. This paper develops a combined model for jointly evaluating modal split and railway freight ows, addressed to the case when a modal split based on a random utility model is available, and some of its coe -cients may present a non-negligible variability. To this end, after the initial deterministic formulation a robust counterpart of the model is developed. The model, formulated as a non-linear integer programming problem, is oriented to a multi-carrier environment and includes constraints to consider the interactions between the di erent types of ows on the railway network, allowing a good evaluation of the cost types of the carriers and the network capacity. An algorithmic solution based on the outer approximation method is shown to provide accurate solutions in a reasonable computational time for the robust and non-robust versions of the model. Examples centered on a section of the Trans-European Transport Network, the TEN-T Core network corridors, are reported to test the model's applicability. Results show that this model can be a helpful tool for analyzing the possible shippers' response to the di erent railway carriers' services competing with the road.

Renewable and distributed energy sources, with photovoltaic in the first place, are leading the energy transition. However, they alone will hardly find the way to success in this challenge because of their intrinsic limitations, such as their dependency on weather condition and the unpredictability of their production. Demand Response services are considered crucial for the integration of renewable sources in the grid. This study presents a methodology to evaluate potential flexibility in commercial buildings by managing air conditioning and lighting systems of the site. The study proposes two different data-driven approaches to evaluate flexibility depending on the technology considered. For lighting systems, flexibility is evaluated using the XGBoost classification algorithm combined with a piecewise function. The thermal behavior of the building is modeled with a simplified RC equivalent state-space model. For both algorithms, preprocessing showed to be a critical part of the process, which is explained stage-by-stage in the paper. Results show that the lighting flexibility algorithm is very reliable, reaching an average error of less than 1 % during the months considered. The simplified thermal modeling proposed shows promising results, having less than 0.2 o C of RMSE error for the internal temperature forecast. In addition, results show the flexibility potentiality of the building, which is potentially able to shift up to 23 % of its power consumption during certain hours of the day. This opens huge opportunities for taking advantage from dynamic tariffs, increase self-consumption from renewable sources or participate in flexibility markets to reduce buildings' electricity bill and help renewable integration.

The classification of invariant subspaces is an open problem related to other important ones like the Carlson problem. Here we obtain a reduced form of these invariant subspaces as a new tool to tackle these problems. In particular, it allows us to prove quite easily partial results already known. The key point is assigning to each invariant subspace a marked one (its marked type) in order to partition the set of invariant subspaces in a finite number of subsets (the marked classes), each one containing only one marked subspace. Next, we parametrize (minimally) each marked class by means of the so-called PM reduced families, so that representatives of an invariant subspace (its PM reduced forms) appear in just one of these families.

We tackle the obtaining of canonical forms for classifying linear control systems with regard to changes in the state variables. Although it was solved in the 1980s for controllable systems, it is still an open problem for the general case of multiparametric non-necessarily controllable systems. Here we obtain a general reduced form which is canonical for a class of systems which includes the cases already known (uniparametric systems, controllable systems)

The main objective of this study was to introduce a survival model to contemplate two simultaneous accelerating factors affecting a food product's shelf life: temperature and illumination. A second objective was to consider the case where the same consumer tests different experimental conditions and thus his/her data are not independent. Sample data comprised 108 consumers who evaluated a lemon-flavored juice stored at 24 °C, 37 °C and 45 °C; under conditions of no-illumination and with illumination; with seven different storage times for each of the six experimental conditions. Aiming to estimate the storage time at which a consumer rejects a sample a model including an Arrhenius term for the temperature, a binomial response for illumination (with and without) and the interaction of both was developed. The model also considered that the same consumer tested different experimental conditions. Fil: Garitta, Lorena Garitta. Provincia de Buenos Aires. Dirección General de Cultura y Educación. Instituto Superior Experimental de Tecnología Alimentaria; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Langohr, Klaus. Universidad Politécnica de Catalunya; España Fil: Elizagoyen, Eliana Soledad. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas; Argentina. Provincia de Buenos Aires. Dirección General de Cultura y Educación. Instituto Superior Experimental de Tecnología Alimentaria; Argentina Fil: Gugole Ottaviano, Maria Fernanda. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Provincia de Buenos Aires. Dirección General de Cultura y Educación. Instituto Superior Experimental de Tecnología Alimentaria; Argentina Fil: Gómez, Guadalupe. Universidad Politécnica de Catalunya; España Fil: Hough, Guillermo. Provincia de Buenos Aires. Dirección General de Cultura y Educación. Instituto Superior Experimental de Tecnología Alimentaria; Argentina. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas; Argentina

Large aeronautical structural parts present important distortions after machining. This problem is caused by the presence of residual stresses, which are developed during previous manufacturing steps (quenching). Before being put into service, the nominal geometry is restored by means of mechanical methods. This operation is called reshaping and exclusively depends on the skills of a well-trained and experienced operator. Moreover, this procedure is time consuming and nowadays, it is only based on a trial and error approach. Therefore, there is a need at industrial level to solve this problem with the support of numerical simulation tools. By using a simplification hypothesis, it was found that the springback phenomenon behaves linearly and it allows developing a strategy to implement reshaping at an industrial level. © 2018 Author(s).

It is well known that, when a full rank observable pair (C,A) is slightly perturbed, the new observability indices k′ are majorized by the initial ones k, k≻k′. Conversely, any indices k′ majorized by k can be obtained by perturbing (C,A). The aim of this paper is the explicit construction of perturbations of (C,A) which have the desired indices k′ by means of a sequence of uniparametrical versal perturbations. Even more, using versal deformations we refine this construction in such a way that the perturbation has the maximum possible number of zeros and no parameters in the square part.

Specific algorithms, such as those involving the supremal of the invariant subspaces contained in a suitable subspace, are known to be able to test whether a disturbance decoupling problem (DDP) is solvable. Here, by reducing the system to its Molinari form, we obtain an alternative description of this supremal object and compute its dimension. Hence we have a general result for solving the decoupling provided that a Molinari basis is known. In particular, a necessary numerical condition for it is derived. The same technique is applied to the DDPS, that is, when stability of the decoupled closed loop system is required.

Given a controllable linear control system defined by a pair of constant matrices (A;B), the set of controllability subspaces is an stratified submanifold of the set of (A;B)-invariant subspaces. We parameterize each strata by means of coordinate charts. This parametrization has significant di®erences to that of (A;B) invariant subspaces, showing a more complex geometric structure.

Matrix pencils under the strict equivalence and matrix pairs under the state feedback equivalence are considered. It is known that a matrix pencil (or a matrix pair) smoothly dependent on parameters can be reduced locally to a special typically more simple form, called the versal deformation, by a smooth change of parameters and a strict equivalence (or feedback equivalence)transformation. We suggest an explicit recurrent procedure for finding the change of parameters and equivalence transformation in the reduction of a given family of matrix pencils (or matrix pairs) to the versal deformation. As an application, this procedure is applied to the analysis of the uncontrollability set in the space of parameters for a one-input linear dynamical system. Explicit formulae for a tangent plane to the uncontrollability set at its regular point and the perturbation of the uncontrollable mode are derived. A physical example is given and studied in detail.

We present a geometric approach to the classification of monogenic invariant subspaces, alternative to the classical algebraic one, which allows us to obtain several matricial canonical forms for each class. Some applications are derived: canonical coordinates of a vector with regard to an endomorphism, and a canonical form for uniparametric linear control systems, not necessarily controllable, with regard to linear changes of state variables. Moreover, the pointwise construction can be extended to differentiable families of changes of basis when differentiable families of equivalent monogenic subspaces are considered.

In this work, a coordinate change of state variables is performed for drift-less systems of dimension m+2 with 2 inputs using Goursat Normal Form. Then, we define a feedback law that will allow us to convert the original system into chained form. Later on, we find the flat outputs and define a new feedback law. Finally, numerical simulations are presented for a planar space robot, a mobile robot with a trailer and a N-trailer.

The relation between original and expanded systems within the Inclusion Principle from the point of view of controllability–observability of both subsystems and composite systems is studied. It is proved that complementary matrices always exist ensuring that the subsystems and the overall expanded system are simultaneously controllable–observable. Two practically important large classes of complementary matrices are identified to offer results computationally attractive. First, the existence of complementary matrices ensuring controllability–observability of decoupled subsystems is proved. Then, using this result, the same property is proved for the composite expanded system.

In this paper we consider finite-dimensional generalized linear discrete-time-invariant systems in the form Ex˙ (k+1) = Ax(k)+Bu(k), y(k) = Cx(k) where E,A 2 M = Mn(C), B 2 Mn×m(C), C 2 Mp×n(C), describ- ing convolutional codes. The notion of output observability is analyzed and a characterization of output observable systems is obtained.

We consider triples of matrices (E; A;B), representing singular linear time invariant systems in the form Ex_ (t) = Ax(t) + Bu(t), with E;A 2 Mp£n(C) and B 2 Mn£m(C), un- der proportional and derivative feedback. Using geometrical techniques we obtain miniversal deformations that permit us to study sensivity and structural stability of singular systems.

We consider pairs of matrices (E;A), representing singular linear time invariant sys- tems in the form Ex_ (t) = Ax with E;A 2 Mp£n(C) under restricted equivalence. In this paper we obtain an alternative collection of invariants that they permit us to deduce the Kronecker canonical reduced form. As application we obtain a reduced form form systems under proportional and derivative feedback as well proportional and derivative output injection.

Las enfermedades cardiovasculares son la primera causa de muerte de la población Europea. Se estima que la mortalidad por enfermedades cardiovasculares es la mitad de la mortalidad total en toda Europa y provoca 4,35 millones de fallecidos cada año en los 52 estados miembros de la Región Europea. Según la Organización Mundial de la Sanidad (OMS) más de 1,9 millones de muertos en la Unión Europea, además de ser la principal causa de deshabilitad y reducción de calidad de vida. Estos tipos de enfermedades afectan sobre todo a la mayor parte de la población, y, sabiendo que el porcentaje de ancianos en la nuestra sociedad sigue subiendo, se puede fácilmente entender que las cardio-patologías puedan constituir siempre un riesgo elevado para la salud de los países occidentales. La cirugía moderna busca cada día soluciones menos invasivas, como por ejemplo a través de cirugía llamada mini-invasiva. La cirugía mini-invasiva es la técnica quirúrgica que consiente acceder a cavidades internas sin operar incisiones (cirugía en abierto). En esta definición, es trivial intuir que entran sobre todo las técnicas endoscópicas que han permitido realizar muchas operaciones de diferente grado de peligrosidad produciendo numerosos beneficios y reduciendo los índices de riesgo. En la cirugía cardiovascular no es raro encontrar problemáticas de tener que insertar aparatos como stent, válvulas cardiacas, sondas y muchas cosas más a través pequeños orificios (como vasos sanguíneos, por ejemplo, la femoral, la aorta la carótida o la subclavia). Estos aparatos van a ser posicionados o conducidos a través de guías (catéteres) que los conducen hasta el sitio destinado. El hecho de utilizar catéteres que tienen que penetrar por estas vías obliga a que estos aparatos respecten dimensiones muy pequeñas. Es interesante entonces estudiar como estas estructuras deben de soportar esfuerzos de compresión y de expansión cuando vienen insertadas en el catéter y luego se sueltan. El ejemplo en el cual enfocaremos nuestro estudio va a ser un soporte metálico intravascular para cualquier aparato necesite ser anclado en el interior de una vena o arteria.

Let (E;A1;A2;B) be a quadruple of matrices representing a two-order generalized time-invariant linear system, E¨x = A1 ˙ x + A2x + Bu. We study the controllability character under an algebraic point of view.

Given a controllable linear control system defined by a pair of constant matrices (A,B), the set of controllability subspaces is a stratified submanifold of the set of (A,B)-invariant subspaces. We parametrize each strata by means of coordinate charts. This parametrization has significant differences to that of (A,B) invariant subspaces, showing a more complex geometric structure.

A natural equivalence relation can be considered in the generalized flag manifold. First we give a complete set of invariants of it as well as a canonical matrix description of the classes. Next we consider parametric flags. We give a miniversal deformation for the above canonical form and we use it to characterize the stable flags.

Given an orbit space $M/\Gamma$ and an equivalence relation defined in it by means of the action of a group $G$, we obtain a miniversal deformation of an orbit through a miniversal deformation in $M$ with regard to a suitable group action of $G\times \Gamma$. We show some applications to the perturbations of $m$-tuples of subspaces and $(C,A)$-invariant subspaces.

V. I. Arnold constructed smooth generic families of matrices with respect to similarity transformations depending smoothly on the entries of matrices with a small number of parameters. We Solve the same problem for a family of triples of matrices (E; A;B), E;A 2 Mn(C), B 2 Mn£m(C), representing generalized linear systems in the form E ˙ x = Ax+Bu, with respect to feedback and derivative feedback equivalence.

We extend some known results about the smooth stratification of the set of conditioned invariant subspaces for a general pair $(C,A)\in\w C^n\times\w C^{n+m}$ without any assumption on the observability. More precisely we prove that the set of $(C,A)$-conditioned invariant subspaces having a fixed Brunovsky-Kronecker structure is a submanifold of the corresponding Grassman manifold, with a vector bundle structure relating the observable and nonobservable part, and we compute its dimension. We also prove that the set of all $(C,A)$-conditioned invariant subspaces having a fixed dimension is connected, provided that the nonobservable part of $(C,A)$ has at most one eigenvalue (this condition is in general necessary).

For a given system $\Sigma(A,B)$ and a subspace $\maths$, the cover problem consists of finding all $(A,B)$-invariant subspaces containing $\maths$. For controllable systems, the set of these subspaces can be suitably stratified. In this paper, necessary and sufficient conditions are given for the cover problem to have a solution on a given strata. Then the geometry of these solutions is studied. In particular, the set of the solutions is provided with a differentiable structure and a parameterization of all solutions is obtained through a coordinate atlas of the corresponding smooth manifold.

Given a quadruple of matrices $(E,A,B,C)$ representing a generalized discrete time-invariant system $Ex(k+1)=Ax(k)+Bu(k), y(k)=Cx(k)$, we obtain necessary and sufficient conditions for which the system can be standardized by means of a discrete derivative feedback. Also we analyze the controllability and observability of the system.

Given an orbit space of matrices $M/\Gamma$ and an equivalence relation defined in it by means of the action of a group $G$, we obtain a miniversal deformation of an orbit through a minivesal deformation in $M$ with regard to a suitable group action of $G\times \Gamma$. We show some applications to the perturbations of $m$-tuples of subspaces and $(C,A)$-invariant subspaces.

The possible observability indices of an observable pair of matrices, when supplementary subpairs are prescribed, are characterized when the “quotient” one is nilpotent. The geometric techniques used are also valid in the classical Carlson problem for square matrices.

Given an observable system defined by a pair of matrices $(C,A)$ we obtain a cellular decomposition of the manifold of $(C,A)$-conditioned invariant subspaces having the restricted system fixed observability indices and we apply this decomposition to compute their homology groups.

We consider quadruples of matrices (E,A,B,C) defining generalized linear multivariable time-invariant dynamical systems, with E,A square matrices and B, C rectangular matrices. Using geometrical techniques we present upper bounds and lower bounds for the distances between a quadruple and the nearest structurally unstable, uncontrollable and/or unobservable one, in terms of the singular values of matrices associated to the quadruple.

Let a time-invariant linear system $\left .\aligned \dot x(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)\endaligned \right \}$ corresponding to a realization of a prescribed transfer function matrix can be represented by triples of matrices $(A,B,C)$. The permitted transformations of basis changes in the space state on the systems can be seen in the space of triples of matrices as similarity equivalence. In this paper we give a geometric characteriaztion of controllable and observable systems as orbits under a Lie group action. As a corollary we obtain a lower bound of the distance between a controllable and observable triple and the nearest uncontrollable one.

Starting from the existence of a $C^{p}$-basis for any $C^{p}$-family of subspaces having constant dimension, we construct a Brunovsky basis of class $C^{p}$ for a $C^{p}$-family of pairs of matrices having constant Brunovsky type. We derive a global pole assignment theorem for such kind of pairs. In all the cases we assume that the manifold of parameters is contractible

En este artículo se analizan los problemas planteados en la estimación en tiempo real de los parámetros de sistemas variantes con el tiempo, con el objeto de definir los requisitos que debe verificar un estimador de este tipo. Seguidamente se realiza un análisis crítico de las técnicas de estimación adaptativa más usuales, incidiendo en los aspectos fundamentales: la seguridad de funcionamiento del método, su capacidad de adaptación, la facilidad de uso del mismo y su coste computacional. Por último, en base a los análisis anteriores se desarrollan dos nuevos algoritmos que verifican todos los requisitos anteriores. In this paper, the problems originated by the real time estimation of time-varying systems parameters are analyzed so that the requirements to be verified by an estimator of this kind can be defined. Next, a critical analysis of the more usual adaptive estimation techniques is accomplished, incising in the following basic aspects: the operating security of the method, its capability of adaptation, its use facility and its computational cost. Finally, based on the previous analysis, new methods, which verify all previous requirements, are developed.

We find explicit miniversal deformations of flags in the generalized flag manifolds, with regard a natural equivalence relation defined by the group action that keeps fixed the reference flag.

We study the partition of the set of pairs of matrices according to the Brunovsky- Kronecker type. We show that it is a constructible stratification, and that it is Whitney regular when the second matrix is a column matrix. We give an application to the obtention of bifurcation diagrams for few-parameter generic families of linear systems.

We consider the topology of the set of conditioned invariant subspaces of an observable pair $(C,A)$ of a fixed dimension. By fixing the observability indices of the restricted system, a stratification by finitely many smooth manifolds is obtained, termed Brunovsky strata. It is shown that each Brunovsky stratum is homotopy equivalent to a generalized flag manifold. From this description an effective formula for the Betti numbers of the Brunovsky strata can be derived.

Let $(A,B,C)$ be a triple of matrices representing a time-invariant linear system $\left .\aligned \dot x(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)\endaligned \right \}$ under similarity equivalence, corresponding to a realization of a prescribed transfer function matrix. In this paper we measure the distance between a irreducible realization, that is to say a controllable and observable triple of matrices $(A,B,C)$ and the nearest reducible one that is to say uncontrollable or unobservable one. Different upper bounds are obtained in terms of singular values of the controllability matrix $C(A,B,C)$, observability matrix $O(A,B,C)$ and controllability and observability matrix $CO(A,B,C)$ associated to the triple.

Estudiar la influencia de diversos parámetros, relacionados con los modelos de comportamiento de vehículos, en los resultados producidos por el simulador A1MSUN2.El proyecto constará de dos partes: · en la primera realizaremos, sobre diversos modelos interurbanos ficticios, los experimentos de simulación necesarios para determinar la influencia de algunos parámetros de entrada en los outputs del simulador. Técnicas estadísticas más relevantes: diseño de experimentos y el análisis de resultados. · y en la segunda modelizaremos la red de tráfico de Can Vidalet (Esplugues de Llob.) con el fin de aplicar los conocimientos sobre calibración adquiridos anteriormente. Técnicas estadísticas más relevantes: recogida de datos y comparación de datos reales/simulados.

The classification of the invariant subspaces of an endomorphism has been an open problem for a long time, and it is a ”wild” problem in the general case. Here we obtain a full classification for the monogenic ones. Some applications are derived: in particular, canonical forms for uniparametric linear control systems, non necessarily controllable, with regard to linear changes of state variables

Proper Generalized Decomposition (PGD) is devised as a computational method to solve high-dimensionalboundary value problems(where many dimensions are associated with the space of parameters defining the problem). The PGD philosophy consists in providing a separated representation of the multidimensional solution using agreedy approachcombined with an alternated directions scheme to obtain the successive rank-one terms. This paper presents an algorithmic approach to high-dimensional tensorseparation based on solving theLeast Squares approximationin a separable format of multidimensional tensor using PGD. This strategy is usually embedded in a standard PGD code in order to compress the solution (reduce the number of terms and optimize the available storage capacity), but it stands also as an alternative and highly competitive method for tensor separation.

We present a geometric approach to the study of time-invariant systems, represented by quadruples of matrices (A, B, C, D). Our main goal is, given a differentiable family of such quadruples having constant Kronecker type, the obtention of a differentiable family of nonsingular matrices such that pointwise reduces each quadruple to its Kronecker reduced form. The key point is a geometrical construction of Kronecker reducing bases.

Probe-based technologies are proliferating as a means of inferring traffic states. Technological companies are interested in traffic data for computing the best routes in a traffic-aware manner and they also provide real-time traffic information with certain temporal accuracy. This paper analyses and evaluates how data provided by a fleet of probe cars can be used to develop a navigation service and how the penetration rate of this service affects a set of city-scale KPIs (Key Performance Indicators) and driver KPIs. The case study adopts a model-driven approach in which microscopic simulation emulates real-size fleets of probe vehicles that provide positions and speed data. What is noteworthy about the modelling behaviour is that drivers are segmented according to their knowledge of network conditions for selected trips: experts, regular drivers, and tourists. The paper presents and discusses the modelling approach and the results obtained from an experimental Barcelona CBD model designed to evaluate the penetration rates of probe vehicles and route guidance. An analysis of the simulation experiments reveals remarkable links among city-scale KPIs, which—from a multivariate point of view—is a novelty. A simulation-based framework for results analysis and visualization is also introduced in order to simplify the simulation results analysis and easily visualize OD paths for driver segments. Throughout this work, the authors have benefited from the support of inLab FIB team at UPC and the suggestions of Jaume Barceló (Emeritus Professor at UPC). This research was funded by Spanish R+D Programs, TRA2016-76914-C3-1-P, and by Secretaria d’Universitats-i-Recerca-Generalitat de Catalunya, 2017-SGR-1749.

New sustainable mobility concepts and smart resilient ideas are arising every day. However, there is not an easy way to bring these ideas into reality, or to test how good they are as mobility solutions. Virtual Mobility Lab offers the opportunity to evaluate the impact of new mobility concepts before taking them to the real world. In this work, a multimodal macroscopic traffic simulation model of the Barcelona Metropolitan Area is developed, including both public and private transport network. This paper explains the remarkable features developed for this model, such as the network hierarchy and the multimodal public network interchangers, allowing demand to exchange between public transportation modes along their origin-destination paths.

Recent advances in communication and computing have made the control and coordination of dynamic network agents to become an area of multidisciplinary research at the intersection of the theory of control systems, communication and linear algebra. The advances of the research in multi-agent systems are strongly supported by their critical applications in different areas as for example in consensus problem of communication networks, or formation control of mobile robots. Mainly, the consensus problem has been studied from the point of view of stability. Nevertheless, recently some researchers have started to analyze the controllability problems. The study of controllability is motivated by the fact that the architecture of communication network in engineering multi-agent systems is usually adjustable. Therefore, it is meaningful to analyze how to improve the controllability of a multi-agent system. In this work we analyze the stability and controllability of multiagent systems consisting of k + 1 agents with dynamics x¿i = Aixi + Biui, i = 0, 1, . . . , k

Let $(A,B)$ be a pair of matrices representing a time-invariant linear system $\dot x(t)=Ax(t)+Bu(t)$ under block-similarity equivalence. In this paper we measure the distance between a controllable pair of matrices $(A,B)$ and the nearest uncontrollable one. A bound is obtained in terms of singular values of the controllability matrix $C(A,B)$ associated to the pair. This bound is not simply based on the smallest singular value of $C(A,B)$ contrary to what one may expect. Also a lower bound is obtained using geometrical techniques expressed in terms of the singular values of a matrix representing the tangent space of the orbit of the pair $(A,B)$.

Given a (` + 1)-ple of matrices (A`¡1; : : : ;A0;B) representing `- order time-invariant linear systems, x(`) = A`¡1x(`¡1) + : : : + A0x(0) + Bu, we analyze conditions in such a way that changing the control u by u1 = u ¡ F`x(`) ¡ : : : F0x(0) the system obtained has a stable solution.