Showing papers on "Piecewise linear function published in 2003"

A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices

Abstract: In this paper, we present an approach to nonlinear model reduction based on representing a nonlinear system with a piecewise-linear system and then reducing each of the pieces with a Krylov projection. However, rather than approximating the individual components as piecewise linear and then composing hundreds of components to make a system with exponentially many different linear regions, we instead generate a small set of linearizations about the state trajectory which is the response to a "training input." Computational results and performance data are presented for an example of a micromachined switch and selected nonlinear circuits. These examples demonstrate that the macromodels obtained with the proposed reduction algorithm are significantly more accurate than models obtained with linear or recently developed quadratic reduction techniques. Also, we propose a procedure for a posteriori estimation of the simulation error, which may be used to determine the accuracy of the extracted trajectory piecewise-linear reduced-order models. Finally, it is shown that the proposed model order reduction technique is computationally inexpensive, and that the models can be constructed "on the fly," to accelerate simulation of the system response.

Piecewise Linear Control Systems

Abstract: This thesis treats analysis and design of piecewise linear control systems. Piecewise linear systems capture many of the most common nonlinearities in engineering systems, and they can also be used for approximation of other nonlinear systems. Several aspects of linear systems with quadratic constraints are generalized to piecewise linear systems with piecewise quadratic constraints. It is shown how uncertainty models for linear systems can be extended to piecewise linear systems, and how these extensions give insight into the classical trade-offs between fidelity and complexity of a model. Stability of piecewise linear systems is investigated using piecewise quadratic Lyapunov functions. Piecewise quadratic Lyapunov functions are much more powerful than the commonly used quadratic Lyapunov functions. It is shown how piecewise quadratic Lyapunov functions can be computed via convex optimization in terms of linear matrix inequalities. The computations are based on a compact parameterization of continuous piecewise quadratic functions and conditional analysis using the S-procedure. A unifying framework for computation of a variety of Lyapunov functions via convex optimization is established based on this parameterization. Systems with attractive sliding modes and systems with bounded regions of attraction are also treated. Dissipativity analysis and optimal control problems with piecewise quadratic cost functions are solved via convex optimization. The basic results are extended to fuzzy systems, hybrid systems and smooth nonlinear systems. It is shown how Lyapunov functions with a discontinuous dependence on the discrete state can be computed via convex optimization. An automated procedure for increasing the flexibility of the Lyapunov function candidate is suggested based on linear programming duality. A Matlab toolbox that implements several of the results derived in the thesis is presented.

Hierarchical Morse—Smale Complexes for Piecewise Linear 2-Manifolds

Abstract: . We present algorithms for constructing a hierarchy of increasingly coarse Morse—Smale complexes that decompose a piecewise linear 2-manifold. While these complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simulating differentiability. We then simplify Morse—Smale complexes by canceling pairs of critical points in order of increasing persistence.

Morse-smale complexes for piecewise linear 3-manifolds

Abstract: We define the Morse-Smale complex of a Morse function over a 3-manifold as the overlay of the descending and ascending manifolds of all critical points. In the generic case, its 3-dimensional cells are shaped like crystals and are separated by quadrangular faces. In this paper, we give a combinatorial algorithm for constructing such complexes for piecewise linear data.

Approximate explicit constrained linear model predictive control via orthogonal search tree

Abstract: Solutions to constrained linear model predictive control problems can be precomputed off-line in an explicit form as a piecewise linear state feedback on a polyhedral partition of the state-space, avoiding real-time optimization. We suggest an algorithm that can determine an approximate explicit piecewise linear state feedback by imposing an orthogonal search tree structure on the partition. This leads to a real-time computational complexity that is logarithmic in the number of regions in the partition, and the algorithm yields guarantees on the suboptimality, asymptotic stability and constraint fulfillment.

A Comparison of Mixed-Integer Programming Models for Nonconvex Piecewise Linear Cost Minimization Problems

Abstract: We study a generic minimization problem with separable nonconvex piecewise linear costs, showing that the linear programming (LP) relaxation of three textbook mixed-integer programming formulations each approximates the cost function by its lower convex envelope. We also show a relationship between this result and classical Lagrangian duality theory.

Reachability analysis of nonlinear systems using conservative approximation

Abstract: In this paper we present an approach to approximate reachability computation for nonlinear continuous systems. Rather than studying a complex nonlinear system x = g(x), we study an approximating system x = f(x) which is easier to handle. The class of approximating systems we consider in this paper is piecewise linear, obtained by interpolating g over a mesh. In order to be conservative, we add a bounded input in the approximating system to account for the interpolation error. We thus develop a reachability method for systems with input, based on the relation between such systems and the corresponding autonomous systems in terms of reachable sets. This method is then extended to the approximate piecewise linear systems arising in our construction. The final result is a reachability algorithm for nonlinear continuous systems which allows to compute conservative approximations with as great degree of accuracy as desired, and more importantly, it has good convergence rate. If g is a C2 function, our method is of order 2. Furthermore, the method can be straightforwardly extended to hybrid systems.

A novel linear programming approach to fluence map optimization for intensity modulated radiation therapy treatment planning.

Abstract: We present a novel linear programming (LP) based approach for efficiently solving the intensity modulated radiation therapy (IMRT) fluence-map optimization (FMO) problem to global optimality. Our model overcomes the apparent limitations of a linear-programming approach by approximating any convex objective function by a piecewise linear convex function. This approach allows us to retain the flexibility offered by general convex objective functions, while allowing us to formulate the FMO problem as a LP problem. In addition, a novel type of partial-volume constraint that bounds the tail averages of the differential dose-volume histograms of structures is imposed while retaining linearity as an alternative approach to improve dose homogeneity in the target volumes, and to attempt to spare as many critical structures as possible. The goal of this work is to develop a very rapid global optimization approach that finds high quality dose distributions. Implementation of this model has demonstrated excellent results. We found globally optimal solutions for eight 7-beam head-and-neck cases in less than 3 min of computational time on a single processor personal computer without the use of partial-volume constraints. Adding such constraints increased the running times by a factor of 2-3, but improved the sparing of critical structures. All cases demonstrated excellent target coverage (> 95%), target homogeneity (< 10% overdosing and < 7% underdosing) and organ sparing using at least one of the two models.

Efficient digital implementation of the sigmoid function for reprogrammable logic

Abstract: Special attention must be paid to an efficient approximation of the sigmoid function in implementing FPGA-based reprogrammable hardware-based artificial neural networks. Four previously published piecewise linear and one piecewise second-order approximation of the sigmoid function are compared with SIG-sigmoid, a purely combinational approximation. The approximations are compared in terms of speed, required area resources and accuracy measured by average and maximum error. It is concluded that the best performance is achieved by SIG-sigmoid.

Improving a Travel-Time Estimation Algorithm by Using Dual Loop Detectors

Abstract: An algorithm is presented for off-line estimation of route-level travel times for uninterrupted traffic flow facilities, such as motorway corridors, based on time series of traffic-speed observations taken from the sections that constitute a route. The proposed method is an extension of the widely used trajectory method. The novelty of the presented method is that trajectories are based on the assumption of piecewise linear (and continuous at section boundaries) vehicle speeds rather than piecewise constant (and discontinuous at section boundaries) speeds. From these assumptions, mathematical expressions are derived that describe the trajectories within each section. These expressions can be used to replace their existing counterparts in the traditional trajectory methods. A comparison of the accuracy of the new method and of the existing method was carried out by using simulated data. This comparison showed that the root-mean-square error (RMSE) value for the new method is about half the RMSE value for the existing method. When this RMSE is decomposed in a bias and a residual error, it turns out that the existing method significantly overestimates the travel time. However, the largest part of the reduction of the RMSE value is still caused by a reduction of the residual error. In other words, if both methods are corrected for their bias, the new method performs significantly better.

A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems

Abstract: In this study we discuss the problem of Model Order Reduction (MOR) for a class of nonlinear dynamical systems. In particular, we consider reduction schemes based on projection of the original state-space to a lower-dimensional space e.g. by using Krylov methods. In the nonlinear case, however, applying a projection-based MOR scheme does not immediately yield computationally efficient macromodels. In order to overcome this fundamental problem, we propose to first approximate the original nonlinear system with a weighted combination of a small set of linearized models of this system, and then reduce each of the models with an appropriate projection method. The linearized models are generated about a state trajectory of the nonlinear system corresponding to a certain ‘training’ input. As demonstrated by results of numerical tests, the obtained trajectory quasi-piecewise-linear reduced order models are very cost-efficient, while providing superior accuracy as compared to existing MOR schemes, based on single-state Taylor’s expansions. In this dissertation, the proposed MOR approach is tested for a number of examples of nonlinear dynamical systems, including micromachined devices, analog circuits (discrete transmission line models, operational amplifiers), and fluid flow problems. The tests validate the extracted models and indicate that the proposed approach can be effectively used to obtain system-level models for strongly nonlinear devices. This dissertation also shows an inexpensive method of generating trajectory piecewise-linear (TPWL) models based on constructing the reduced models ‘on-the-fly’, which accelerates simulation of the system response. Moreover, we propose a procedure for estimating simulation errors, which can be used to determine accuracy of the extracted trajectory piecewise-linear reduced order models. Finally, we present projection schemes which result in improved accuracy of the reduced order TPWL models, as well as discuss approaches leading to guaranteed stable and passive TPWL reduced-order models. Thesis Supervisor: Jacob K. White Title: Professor of Electrical Engineering and Computer Science

Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions

Abstract: 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.

Piecewise polynomial nonlinear model reduction

Abstract: We present a novel, general approach towards model-order reduction (MOR) of nonlinear systems that combines good global and local approximation properties. The nonlinear system is first approximated as piecewise polynomials over a number of regions, following which each region is reduced via polynomial model-reduction methods. Our approach, dubbed PWP, generalizes recent piecewise linear approaches and ties them with polynomial-based MOR, thereby combining their advantages. In particular, reduced models obtained by our approach reproduce small-signal distortion and intermodulation properties well, while at the same time retaining fidelity in large-swing and large-signal analyses, e.g., transient simulations. Thus our reduced models can be used as drop-in replacements for time-domain as well as frequency-domain simulations, with small or large excitations. By exploiting sparsity in system polynomial coefficients, we are able to make the polynomial reduction procedure linear in the size of the original system. We provide implementation details and illustrate PWP with an example.

Identification of mechanical properties by displacement field measurement: A variational approach

Abstract: We study a parameter identification problem associated with a two-dimensional mechanical problem. In the first part, the experimental technique of determining the displacement field is briefly presented. The variational method proposed herein is based on the minimization of either a separately convex functional or a convex functional that leads to the reconstruction of the elastic tensor and the stress field. These two reconstructed fields are continuous and piecewise linear on a triangulation of the two-dimensional domain. Some numerical and experimental examples are presented to test the performance of the algorithms.

Multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions

Abstract: Multistability is a property necessary in neural networks in order to enable certain applications (e.g., decision making), where monostable networks can be computationally restrictive. This article focuses on the analysis of multistability for a class of recurrent neural networks with unsaturating piecewise linear transfer functions. It deals fully with the three basic properties of a multistable network: boundedness, global attractivity, and complete convergence. This article makes the following contributions: conditions based on local inhibition are derived that guarantee boundedness of some multistable networks, conditions are established for global attractivity, bounds on global attractive sets are obtained, complete convergence conditions for the network are developed using novel energy-like functions, and simulation examples are employed to illustrate the theory thus developed.

Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method

Abstract: We present a range of mesh-dependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shape-regular (but possibly non-quasi-uniform) meshes. These inequalities involve norms of the form ∥h α u∥ W s,p (Ω) for positive and negative s and α, where h is a function which reflects the local mesh diameter in an appropriate way. The only global parameter involved is N, the total number of degrees of freedom in the finite element space, and we avoid estimates involving either the global maximum or minimum mesh diameter. Our inequalities include new variants of inverse inequalities as well as trace and extension theorems. They can be used in several areas of finite element analysis to extend results – previously known only for quasi-uniform meshes – to the locally refined case. Here we describe applications to (i) the theory of nonlinear approximation and (ii) the stability of the mortar element method for locally refined meshes.

A TBR-based trajectory piecewise-linear algorithm for generating accurate low-order models for nonlinear analog circuits and MEMS

Abstract: In this paper we propose a method for generating reduced models for a class of nonlinear dynamical systems, based on truncated balanced realization (TBR) algorithm and a recently developed trajectory piecewise-linear (TPWL) model order reduction approach. We also present a scheme which uses both Krylov-based and TBR-based projections. Computational results, obtained for examples of nonlinear circuits and a micro-electro-mechanical system (MEMS), indicate that the proposed reduction scheme generates nonlinear macromodels with superior accuracy as compared to reduction algorithms based solely on Krylov subspace projections, while maintaining a relatively low model extraction cost.

Stabilized Finite Elements on Anisotropic Meshes: A Priori Error Estimates for the Advection-Diffusion and the Stokes Problems

Abstract: Stabilized finite elements on strongly anisotropic meshes are considered. The design of the stability coefficients is addressed for both the advection-diffusion and the Stokes problems when using continuous piecewise linear finite elements on triangles. Using the polar decomposition of the Jacobian of the affine mapping from the reference triangle to the current one, K, and from a priori error estimates, a new definition of the stability coefficients is proposed. Our analysis shows that these coefficients do not depend on the element diameter hK but on a characteristic length associated with K via the polar decomposition. A numerical assessment of the theoretical analysis is carried out.

Gradient superconvergence on uniform simplicial partitions of polytopes

Abstract: Superconvergence of the gradient for the linear simplicial finite-element method applied to elliptic equations is a well known feature in one, two, and three space dimensions. In this paper we show that, in fact, there exists an elegant proof of this feature independent of the space dimension. As a result, superconvergence for dimensions four and up is proved simultaneously. The key ingredient will be that we embed the gradients of the continuous piecewise linear functions into a larger space for which we describe an orthonormal basis having some useful symmetry properties. Since gradients and rotations of standard finite-element functions are in fact the rotation-free and divergence-free elements of Raviart-Thomas and Nedelec spaces in three dimensions, we expect our results to have applications also in those contexts.

Piecewise linear differential equations and integrate-and-fire neurons: insights from two-dimensional membrane models.

Abstract: We derive and study two-dimensional generalizations of integrate-and-fire models which can be found from a piecewise linear idealization of the FitzHugh-Nagumo or Morris-Lecar model. These models give rise to new properties not present in one-dimensional integrate-and-fire models. A detailed analytical study of the models is presented. In particular, (i) for the piecewise linear FitzHugh-Nagumo model, we determine analytically the bistability regime between stationary solutions and oscillations, that is, typical for class-II models. (ii) In the piecewise Morris-Lecar model, we find a noncanonical class-I transition from a stationary state to oscillations with logarithmic dependence similar to that found for leaky integrate-and-fire models. (iii) Furthermore, we establish a relation to the recently proposed resonate-and-fire model and show that a short input current pulse can trigger several spikes.

A digital hardware pulse-mode neuron with piecewise linear activation function

Abstract: This paper proposes a new type of digital pulse-mode neuron that employs piecewise-linear function as its activation function. The neuron is implemented on field programmable gate array (FPGA) and tested by experiments. As well as theoretical analysis, the experimental results show that the piecewise-linear function of the proposed neuron is programmable and robust against the change in the number of input signals. To demonstrate the effect of piecewise-linear activation function, pulse-mode multilayer neural network with on-chip learning is implemented on FPGA with the proposed neuron, and its learning performance is verified by experiments. By approximating the sigmoid function by the piecewise-linear function, the convergence rate of the learning and generalization capability are improved.

A novel FDTD formulation for dispersive media

Abstract: A novel FDTD formulation for dispersive media called piecewise linear JE recursive convolution (PLJERC) finite-different time-domain (FDTD) method is derived using the piecewise linear approximation and the recursive convolution relationship between the current density J and the electric field E. The high accuracy and efficiency of the PLJERC method is confirmed by computing the reflection coefficients of an electromagnetic wave through a collision plasma slab in one dimension.

A dynamic programming approach for the airport capacity allocation problem

Abstract: In most of the optimization models developed to manage airports operations, arrivals and departures capacities are treated as independent variables: that is the number of flights allowed to take off does not affect the number of landings in any unit of time, and vice versa. This assumption is seldom verified in most of the congested airports, where many interactions between arrivals and departures take place. In this paper, we face the problem of finding the optimal trade‐off between the number of arrivals and departures in order to reduce a delay function of all the flights, using a more realistic representation of the airport capacity, i.e. the capacity envelope. Under the assumption of piecewise linear convex capacity envelopes and of the exact interpolation of all the Pareto‐optimal operational points, we show that the problem can be formulated as a linear programming model. For general airport capacity envelopes, we propose a dynamic programming formulation with a corresponding backward solution algorithm, which is robust, easy to implement and has a linear computational complexity. The algorithm performances are evaluated on different realistic scenarios, and the optimal solutions are compared with those computed by a greedy algorithm, which can be seen as an approximation of the current decision procedures. The percentage deviation of the cost of these two solutions ranges from 3.98 to 35.64%

Scheduling jobs with piecewise linear decreasing processing times

Abstract: We study the problems of scheduling a set of nonpreemptive jobs on a single or multiple machines without idle times where the processing time of a job is a piecewise linear nonincreasing function of its start time. The objectives are the minimization of makespan and minimization of total job completion time. The single machine problems are proved to be NP-hard, and some properties of their optimal solutions are established. A pseudopolynomial time algorithm is constructed for makespan minimization. Several heuristics are derived for both total completion time and makespan minimization. Computational experiments are conducted to evaluate their efficiency. NP-hardness proofs and polynomial time algorithms are presented for some special cases of the parallel machine problems. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 531–554, 2003

Fault Detection and Optimal Sensor Location in Vehicle Suspensions

Abstract: A statistical system identification methodology is applied for performing parametric identification and fault detection studies in nonlinear vehicle systems. The vehicle nonlinearities arise due to the function of the suspension dampers, which assume a different damping coefficient in tension than in compression. The suspension springs may also possess piecewise linear characteristics. These lead to models with parameter discontinuities. Emphasis is put on investigating issues of unidentifiability arising in the system identification of nonlinear systems and the importance of sensor configuration and excitation characteristics in the reliable estimation of the model parameters. A methodology is proposed for designing the optimal sensor configuration (number and location of sensors) so that the corresponding measured data are most informative about the condition of the vehicle. The effects of excitation characteristics on the quality of the measured data are systematically explored. The effectiveness of th...