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Global analysis of piecewise linear systems using impact maps and quadratic surface Lyapunov functions Goncalves, Jorge ; ; in IEEE Transactions on Automatic Control (2003), 48(12), 2089-2106 This paper presents an entirely new constructive global analysis methodology for a class of hybrid systems known as piecewise linear systems (PLS). This methodology infers global properties of PLS solely ... [more ▼] This paper presents an entirely new constructive global analysis methodology for a class of hybrid systems known as piecewise linear systems (PLS). This methodology infers global properties of PLS solely by studying the behavior at switching surfaces associated with PLS. The main idea is to analyze impact maps, i.e., maps from one switching surface to the next switching surface. Such maps are known to be “unfriendly” maps in the sense that they are highly nonlinear, multivalued, and not continuous. We found, however, that an impact map induced by an linear time-invariant flow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state. This representation of impact maps allows the search for surface Lyapunov functions (SuLF) to be done by simply solving a semidefinite program, allowing global asymptotic stability, robustness, and performance of limit cycles and equilibrium points of PLS to be efficiently checked. This new analysis methodology has been applied to relay feedback, on/off and saturation systems, where it has shown to be very successful in globally analyzing a large number of examples. In fact, it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology. Examples analyzed include systems of relative degree larger than one and of high dimension, for which no other analysis methodology could be applied. This success in globally analyzing certain classes of PLS has shown the power of this new methodology, and suggests its potential toward the analysis of larger and more complex PLS. [less ▲] Detailed reference viewed: 133 (1 UL)Global Asymptotic Stability of Oscillations with Sliding Modes Goncalves, Jorge in Proceedings of the 15th IFAC World Congress (2003) This paper explores a new methodology based on quadratic surface Lyapunov functions to globally analyze oscillations with sliding modes in relay feedback systems (RFS). The method consists in efficiently ... [more ▼] This paper explores a new methodology based on quadratic surface Lyapunov functions to globally analyze oscillations with sliding modes in relay feedback systems (RFS). The method consists in efficiently construct quadratic Lyapunov functions on switching surfaces that can be used to show that impact maps, i.e., maps from one switch to the next, are contracting. This, in turn, shows that the system is globally stable. Several classes of piecewise linear systems (PLS) were previously successfully analyzed with this methodology. In this paper, we consider PLS whose trajectories switch between subsystems of different dimensions. We present and discuss distinct relaxations leading to sufficient conditions of different conservatism and computationally complexity. The results in this paper open the door to the analysis of other, more complex classes of PLS. [less ▲] Detailed reference viewed: 72 (0 UL)Global asymptotic stability of the limit cycle in piecewise linear versions of the Goodwin oscillator ; ; Goncalves, Jorge in Proceedings of the 17th IFAC World Congress (2008) Conditions in the form of linear matrix inequalities (LMIs) are used in this paper to guarantee the global asymptotic stability of a limit cycle oscillation for a class of piecewise linear (PWL) systems ... [more ▼] Conditions in the form of linear matrix inequalities (LMIs) are used in this paper to guarantee the global asymptotic stability of a limit cycle oscillation for a class of piecewise linear (PWL) systems defined as the feedback interconnection of a saturation controller with a single input, single output (SISO) linear time-invariant (LTI) system. The proposed methodology extends previous results on impact maps and surface Lyapunov functions to the case when the sets of expected switching times are arbitrarily large. The results are illustrated on a PWL version of the Goodwin oscillator. [less ▲] Detailed reference viewed: 164 (0 UL)Global energy minimization for all crack increment directions in the framework of XFEM ; Bordas, Stéphane Report (2013) Detailed reference viewed: 141 (3 UL)Global Energy Minimization for Multi-crack Growth in Linear Elastic Fracture using the Extended Finite Element Method ; Bordas, Stéphane Presentation (2014, July 24) Detailed reference viewed: 206 (12 UL)Global energy minimization for multi-crack growth in linear elastic fracture using the extended finite element method Bordas, Stéphane Scientific Conference (2014, April 02) We investigate multiple fracture evolution under quasi-static conditions in an isotropic linear elastic solid based on the principle of minimum potential elastic energy in the framework of the extended ... [more ▼] We investigate multiple fracture evolution under quasi-static conditions in an isotropic linear elastic solid based on the principle of minimum potential elastic energy in the framework of the extended finite element method. The technique enables a minimization of the potential energy with respect to all crack increment directions. Results show that the maximum hoop stress criterion and the energy minimization approach converge to the same fracture path. It is found that the converged solution lies in between the fracture paths obtained by each criterion for coarser meshes. This presents an opportunity to estimate an upper and lower bound of the true fracture path as well as an error on the crack path [less ▲] Detailed reference viewed: 289 (15 UL)Global energy minimization for multiple fracture growth ; Bordas, Stéphane Report (2013) Detailed reference viewed: 194 (5 UL)A global optimization heuristic for the decomposed static anticipatory network traffic control problem anticipatory network traffic control problem Rinaldi, Marco ; ; Viti, Francesco in Transportation Research Procedia (2017) Developing traffic control strategies taking explicitly into account the route choice behavior of users has been widely recognized irregularities in the solution space shape, such as non-convexity and non ... [more ▼] Developing traffic control strategies taking explicitly into account the route choice behavior of users has been widely recognized irregularities in the solution space shape, such as non-convexity and non-smoothness. In this work, we propose an extended as a very challenging problem. Furthermore, the inclusion of user behavior in optimization based control schemes introduces strong decomposition scheme for the anticipatory traffic control problem, based upon our previous contributions, which aims at i) reducing irregularities in the solution space shape, such as non-convexity and non-smoothness. In this work, we propose an extended the computational complexity of the problem by approaching it in a controller-by-controller fashion, and ii) internalizing specific decomposition scheme for the anticipatory traffic control problem, based upon our previous contributions, which aims at i) reducing constraints in the objective function, guiding the optimization process away from non-significant minima, such as flat regions. the computational complexity of the problem by approaching it in a controller-by-controller fashion, and ii) internalizing specific Through two small scale test networks and different, randomly chosen initial points, we compare how the proposed extension constraints in the objective function, guiding the optimization process away from non-significant minima, such as flat regions. influences optimization results with respect to our previously developed decomposed approach, as well as centralized schemes. Through two small scale test networks and different, randomly chosen initial points, we compare how the proposed extension influences optimization results with respect to our previously developed decomposed approach, as well as centralized schemes. [less ▲] Detailed reference viewed: 29 (2 UL)Global stability analysis of on/off systems Goncalves, Jorge in Proceedings of the 39th IEEE Conference on Decision and Control (2000) This paper considers quadratic surface Lyapunov functions in the study of global stability analysis of on/off systems (OFS), including those OFS with unstable nonlinearity sectors. In previous work ... [more ▼] This paper considers quadratic surface Lyapunov functions in the study of global stability analysis of on/off systems (OFS), including those OFS with unstable nonlinearity sectors. In previous work, quadratic surface Lyapunov functions were successfully applied to prove global asymptotic stability of limit cycles of relay feedback systems. In this work, we show that these ideas can be used to prove global asymptotic stability of equilibrium points of piecewise linear systems (PLS). We present conditions in the form of LMI that, when satisfied, guarantee global asymptotic stability of an equilibrium point. A large number of examples was successfully proven globally stable. These include systems with an unstable affine linear subsystem, systems of relative degree larger than one and of high dimension, and systems with unstable nonlinearity sectors, for which all classical fail to analyze. In fact, existence of an example with a globally stable equilibrium point that could not be successfully analyzed with this new methodology is still an open problem. This work opens the door to the possibility that more general PLS can be systematically globally analyzed using quadratic surface Lyapunov functions. [less ▲] Detailed reference viewed: 96 (1 UL)Global stability of relay feedback systems Goncalves, Jorge ; ; in IEEE Transactions on Automatic Control (2001), 46(4), 550--562 For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence and local stability of limit cycles for these systems are well known. Global ... [more ▼] For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence and local stability of limit cycles for these systems are well known. Global stability conditions, however, are practically nonexistent. This paper presents conditions in the form of linear matrix inequalities (LMIs) that, when satisfied, guarantee global asymptotic stability of limit cycles induced by relays with hysteresis in feedback with linear time-invariant (LTI) stable systems. The analysis consists in finding quadratic surface Lyapunov functions for Poincaré maps associated with RFS. These results are based on the discovery that a typical Poincaré map induced by an LTI flow between two hyperplanes can be represented as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex subsets of linear manifolds. The search for quadratic Lyapunov functions on switching surfaces is done by solving a set of LMIs. Although this analysis methodology yields only a sufficient criterion of stability, it has proved very successful in globally analyzing a large number of examples with a unique locally stable symmetric unimodal limit cycle. In fact, it is still an open problem whether there exists an example with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology. Examples analyzed include minimum-phase systems, systems of relative degree larger than one, and of high dimension. Such results lead us to believe that globally stable limit cycles of RFS frequently have quadratic surface Lyapunov functions. [less ▲] Detailed reference viewed: 97 (0 UL)Global stability of relay feedback systems Goncalves, Jorge ; ; in Proceedings of the IEEE American Control Conference (2000 ACC) (2000) For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence and local stability of limit cycles for these systems are well known. Global ... [more ▼] For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence and local stability of limit cycles for these systems are well known. Global stability conditions, however, are practically non-existent. The paper presents conditions in the form of linear matrix inequalities (LMIs) that guarantee global asymptotic stability of a limit cycle induced by a relay with hysteresis in feedback with an LTI stable system. The analysis is based on finding global quadratic Lyapunov functions for a Poincare map associated with the RFS. We found that a typical Poincare map induced by an LTI flow between two hyperplanes can be represented as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex. The search for globally quadratic Lyapunov functions is then done by solving a set of LMIs. Most examples of RFS analyzed by the authors were proven globally stable. Systems analyzed include minimum-phase systems, systems of relative degree larger than one, and of high dimension. This leads us to believe that quadratic stability of associated Poincare maps is common in RFS. [less ▲] Detailed reference viewed: 90 (0 UL)Global State Synchronization in Networks of Cyclic Feedback Systems ; ; et al in IEEE Transactions on Automatic Control (2012), 57(2), 478-483 This technical note studies global asymptotic state synchronization in networks of identical systems. Conditions on the coupling strength required for the synchronization of nodes having a cyclic feedback ... [more ▼] This technical note studies global asymptotic state synchronization in networks of identical systems. Conditions on the coupling strength required for the synchronization of nodes having a cyclic feedback structure are deduced using incremental dissipativity theory. The method takes advantage of the incremental passivity properties of the constituent subsystems of the network nodes to reformulate the synchronization problem as one of achieving incremental passivity by coupling. The method can be used in the framework of contraction theory to constructively build a contracting metric for the incremental system. The result is illustrated for a network of biochemical oscillators. [less ▲] Detailed reference viewed: 82 (0 UL)A goal-oriented reduced basis method for the wave equation in inverse analysis ; ; Bordas, Stéphane in International Conference on Computational Mechanics CM13 Proceedings (2013, March) In this paper, we extend the reduced-basis methods developed earlier for wave equations to goal-oriented wave equations with a ne parameter dependence. The essential new ingredient is the dual (or adjoint ... [more ▼] In this paper, we extend the reduced-basis methods developed earlier for wave equations to goal-oriented wave equations with a ne parameter dependence. The essential new ingredient is the dual (or adjoint) problem and the use of its solution in a sampling procedure to pick up “goal-orientedly” parameter samples. First, we introduce the reduced-basis recipe — Galerkin projection onto a space YN spanned by the reduced basis functions which are constructed from the solutions of the governing partial di erential equation at several selected points in parameter space. Second, we propose a new “goal-oriented” Proper Orthogonal Decomposition (POD)–Greedy sampling procedure to construct these associated ba-sis functions. Third, based on the assumption of a ne parameter dependence, we use the o ine-online computational procedures developed earlier to split the computational procedure into o ine and online stages. We verify the proposed computational procedure by applying it to a three-dimensional simulation dental implant problem. The good numeri-cal results show that our proposed procedure performs better than the standard POD–Greedy procedure in terms of the accuracy of output functionals. [less ▲] Detailed reference viewed: 258 (3 UL)A gradient weighted extended finite element method (GW-XFEM) for fracture mechanics ; Bordas, Stéphane ; et al in Acta Mechanica (2019), 230 In this study, a gradient weighted extended finite element method (GW-XFEM) is presented for the analysis of fracture problems. For this method, the domain discretization is the same as the standard XFEM ... [more ▼] In this study, a gradient weighted extended finite element method (GW-XFEM) is presented for the analysis of fracture problems. For this method, the domain discretization is the same as the standard XFEM. However, the gradient field is constructed by considering the influences of the element itself and its adjacent elements. Based on the Shepard interpolation, the weighted strain filed can be obtained, which will be utilized to construct the discretized system equations. The validity of the presented method is fully investigated through several numerical examples. From these results, it is shown that compared with standard XFEM, the presented method can achieve much better accuracy, efficiency and higher convergence, when dealing with fracture analysis. [less ▲] Detailed reference viewed: 12 (0 UL)Graft materials provide greater static strength to medial opening wedge high tibial osteotomy than when no graft is included ; Diffo Kaze, Arnaud ; et al in Journal of Experimental Orthopaedics (2019), 6(13), Background The purpose of this study was to compare the stability of medial opening-wedge high tibial osteotomy (MOWHTO) with and without different graft materials. Good clinical and radiological outcomes ... [more ▼] Background The purpose of this study was to compare the stability of medial opening-wedge high tibial osteotomy (MOWHTO) with and without different graft materials. Good clinical and radiological outcomes have been demonstrated when either using or not using graft materials during MOWHTO. Variations in the biomechanical properties of different graft types, regarding the stability they provide a MOWHTO, have not been previously investigated. Methods A 10 mm biplanar MOWHTO was performed on 15 artificial sawbone tibiae, which were fixed using the Activmotion 2 plate. Five bones had OSferion60 wedges (synthetic group), five had allograft bone wedges (allograft group), and five had no wedges (control group) inserted into the osteotomy gap. Static compression was applied axially to each specimen until failure of the osteotomy. Ultimate load, horizontal and vertical displacements were measured and used to calculate construct stiffness and valgus malrotation of the tibial head. Results The synthetic group failed at 6.3 kN, followed by the allograft group (6 kN), and the control group (4.5 kN). The most valgus malrotation of the tibial head was observed in the allograft group (2.6°). The synthetic group showed the highest stiffness at the medial side of the tibial head (9.54 kN·mm− 1), but the lowest stiffness at the lateral side (1.59 kN·mm-1). The allograft group showed high stiffness on the medial side of the tibial head (7.54 kN·mm− 1) as well as the highest stiffness on the lateral side (2.18 kN·mm− 1). Conclusions The use of graft materials in MOWHTO results in superior material properties compared to the use of no graft. The static strength of MOWHTO is highest when synthetic grafts are inserted into the osteotomy gap. Allograft wedges provide higher mechanical strength to a MOWHTO than when no graft used. In comparison to the synthetic grafts, allograft wedges result in the stiffness of the osteotomy being more similar at the medial and lateral cortices. [less ▲] Detailed reference viewed: 79 (1 UL)Graphviz e TikZ Fiandrino, Claudio in Ars TeXnica (2012), 13 Graphviz is a very powerful tool to draw graphs. This article tries to explain how to export such graphs as a TikZ picture in a very simple way. Detailed reference viewed: 103 (8 UL)Greener and larger neighbourhoods make cities more sustainable! A 2D urban economics perspective Caruso, Geoffrey ; ; et al in Computers, Environment & Urban Systems (2015), 54 We analyse urban growth forms by means of a 2D microeconomic model where households value green space at neighbourhood scale. We analytically demonstrate that cities can grow more densely when households ... [more ▼] We analyse urban growth forms by means of a 2D microeconomic model where households value green space at neighbourhood scale. We analytically demonstrate that cities can grow more densely when households have the possibility to enlarge the neighbourhood in which they value green space, thus emphasizing the importance of neighbourhood planning in particular for facilitating short trips and views of green amenities. We also show by simulation that the size and form of the city, relative to the size and form of neighbourhoods, impact on the decision of households to leapfrog land or not, thus impacting on the emergence of scattered urbanisation patterns. We conclude that carefully addressing the spatial arrangement of green space and buildings and facilitating trips within neighbourhood units constitute an effective policy lever and an attractive way to deliver more sustainable cities. We further argue that our theoretical experiment with complementary analytical and computer-based simulation provides micro-economic reasoning to the main elements of the Garden City and neighbourhood unit planning concepts. [less ▲] Detailed reference viewed: 328 (47 UL)Greener and larger neighbourhoods make cities more sustainable! A 2D urban economics perspective Caruso, Geoffrey ; ; et al in Computers, Environment & Urban Systems (2015) We analyse urban growth forms by means of a 2D microeconomic model where households value green space at neighbourhood scale. We analytically demonstrate that cities can grow more densely when households ... [more ▼] We analyse urban growth forms by means of a 2D microeconomic model where households value green space at neighbourhood scale. We analytically demonstrate that cities can grow more densely when households have the possibility to enlarge the neighbourhood in which they value green space, thus emphasising the importance of neighbourhood planning in particular for facilitating short trips and views of green amenities. We also show by simulation that the size and form of the city, relative to the size and form of neighbourhoods, impact on the decision of households to leapfrog land or not, thus impacting on the emergence of scattered urbanisation patterns. We conclude that carefully addressing the spatial arrangement of green space and buildings and facilitating trips within neighbourhood units constitute an effective policy lever and an attractive way to deliver more sustainable cities. We further argue that our theoretical experiment with complementary analytical and computer-based simulation provides micro-economic reasoning to the main elements of the Garden City and neighbourhood unit planning concepts. [less ▲] Detailed reference viewed: 138 (5 UL)h- and p-adaptivity driven by recovery and residual-based error estimators for PHT-splines applied to time-harmonic acoustics ; ; et al in Computers and Mathematics with Applications (2018), 77(9), 2369-2395 In this work, we demonstrate the application of PHT-splines for time-harmonic acoustic problems, modeled by the Helmholtz equation. Solutions of the Helmholtz equation have two features: global ... [more ▼] In this work, we demonstrate the application of PHT-splines for time-harmonic acoustic problems, modeled by the Helmholtz equation. Solutions of the Helmholtz equation have two features: global oscillations associated with the wave number and local gradients caused by geometrical irregularities. We show that after a sufficient number of degrees of freedom is used to approximate global oscillations, adaptive refinement can capture local features of the solution. We compare residual-based and recovery-based error estimators and investigate the performance of -refinement. The simulations are done in the context of recently introduced Geometry Independent Field approximaTion (GIFT), where PHT-splines are only used to approximate the solution, while the computational domain is parameterized with NURBS. This approach builds on the natural adaptation ability of PHT-splines and avoids the re-parameterization of the NURBS geometry during the solution refinement process. [less ▲] Detailed reference viewed: 15 (0 UL)H2 Norm Based Network Volatility Measures ; ; Goncalves, Jorge et al in The proceedings of the American Control Conference (2014) Motivated by applications in biology and economics, we propose new volatility measures based on the H2 system norm for linear networks stimulated by independent or correlated noise. We identify critical ... [more ▼] Motivated by applications in biology and economics, we propose new volatility measures based on the H2 system norm for linear networks stimulated by independent or correlated noise. We identify critical links in a network, where relatively small improvements can lead to large reductions in network volatility measures. We also examine volatility measures of individual nodes and their dependence on the topological position in the network. Finally, we investigate the dependence of the volatility on different network interconnections, weights of the edges and other network properties. Hence in an intuitive and efficient way, we can identify critical links, nodes and interconnections in network which can shed light in the network design to make it more robust. [less ▲] Detailed reference viewed: 103 (5 UL) |
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