Showing papers on "Piecewise linear function published in 1998"

Quadratic stabilization and control of piecewise-linear systems

Abstract: We consider analysis and controller synthesis of piecewise-linear systems. The method is based on constructing quadratic and piecewise-quadratic Lyapunov functions that prove stability and performance for the system. It is shown that proving stability and performance, or designing (state-feedback) controllers, can be cast, as convex optimization problems involving linear matrix inequalities that can be solved very efficiently. A couple of simple examples are included to demonstrate applications of the methods described.

Bifurcation Sets of Continuous Piecewise Linear Systems with Two Zones

Abstract: Planar continuous piecewise linear vector fields with two zones are considered. A canonical form which captures the most interesting oscillatory behavior is obtained and their bifurcation sets are drawn. Different mechanisms for the creation of periodic orbits are detected, and their main characteristics are emphasized.

Long wave runup on piecewise linear topographies

Abstract: We study long-wave evolution and runup on piecewise linear one- and two-dimensional bathymetries analytically and experimentally with the objective of understanding certain coastal effects of tidal waves. We develop a general solution method for determining the amplification factor of different ocean topographies consisting of linearly varying and constant-depth segments to study how spectral distributions evolve over bathymetry, and apply our results to study the evolution of solitary waves. We find asymptotic results which suggest that solitary waves often interact with piecewise linear topographies in a counter-intuitive manner. We compare our analytical predictions with numerical results, with results from a new set of laboratory experiments from a physical model of Revere Beach, and also with the data on wave runup around an idealized conical island. We find good agreement between our theory and the laboratory results for the time histories of free-surface elevations and for the maximum runup heights. Our results suggest that, at least for simple piecewise linear topographies, analytical methods can be used to calculate effectively some important physical parameters in long-wave runup. Also, by underscoring the effects of the topographic slope at the shoreline, this analysis qualitatively suggests why sometimes predictions of field-applicable numerical models differ substantially from observations of tsunami runup.

Piecewise Linear Modeling and Analysis

Abstract: From the Publisher: Piecewise Linear Modeling and Analysis explains in detail all possible model descriptions to efficiently store piecewise linear functions starting with the Chua descriptions. Detailed explanation on how the model parameter can be obtained for a given mapping is provided and demonstrated by examples. The models are ranked to compare them and to show which model can handle the largest class of PL mappings. To analyse PL electrical networks a simulator is mandatory. Piecewise Linear Modeling and Analysis provides a detailed outline of a possible PL simulator, including pseudo-programming code. Several simulation domains like transcient, AC and distortion are discussed. The book explains the attractive features of PL simulators with respect to mixed-level and mixed-signal simulation while paying due regard also to hierarchical simulation. Piecewise Linear Modeling and Analysis shows in detail how many existing components in electrical networks can be modeled. These range from digital logic and analog basic elements such as transistors to complex systems like Phase-Locked Loops and detection systems. Simulation results are also provided. The book concludes with a discussion on how to find multiple solutions for PL functions or networks. Again, the most common techniques are outlined using clear examples. Piecewise Linear Modeling and Analysis is an indispensable guide for researchers and designers interested in network theory, network synthesis and network analysis.

Visualizing nonlinear vector field topology

Abstract: We present our results on the visualization of nonlinear vector field topology. The underlying mathematics is done in Clifford algebra, a system describing geometry by extending the usual vector space by a multiplication of vectors. We started with the observation that all known algorithms for vector field topology are based on piecewise linear or bilinear approximation, and that these methods destroy the local topology if nonlinear behavior is present. Our algorithm looks for such situations, chooses an appropriate polynomial approximation in these areas, and, finally, visualizes the topology. This overcomes the problem, and the algorithm is still very fast because we are using linear approximation outside these small but important areas. The paper contains a detailed description of the algorithm and a basic introduction to Clifford algebra.

New computational results on the discrete time/cost trade-off problem in project networks

Abstract: We describe a new exact procedure for the discrete time/cost trade-off problem in deterministic activity-on-the-arc networks of the CPM type, where the duration of each activity is a discrete, nonincreasing function of the amount of a single resource (money) committed to it. The objective is to construct the complete and efficient time/cost profile over the set of feasible project durations. The procedure uses a horizon-varying approach based on the iterative optimal solution of the problem of minimising the sum of the resource use over all activities subject to the activity precedence constraints and a project deadline. This optimal solution is derived using a branch-and-bound procedure which computes lower bounds by making convex piecewise linear underestimations of the discrete time/cost trade-off curves of the activities to be used as input for an adapted version of the Fulkerson labelling algorithm for the linear time/cost trade-off problem. Branching involves the selection of an activity in order to partition its set of execution modes into two subsets which are used to derive improved convex piecewise linear underestimations. The procedure has been programmed in Visual C ++ under Windows NT and has been validated using a factorial experiment on a large set of randomly generated problem instances.

Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method

Abstract: The numerical solution of a parabolic equation with memory is considered. The equation is first discretized in time by means of the discontinuous Galerkin method with piecewise constant or piecewise linear approximating functions. The analysis presented allows variable time steps which, as will be shown, call then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data. The combination with finite element discretizat in space is also studied.

An Algorithm for Single-Item Capacitated Economic Lot Sizing with Piecewise Linear Production Costs and General Holding Costs

Abstract: We consider the Capacitated Economic Lot Size Problem with piecewise linear production costs and general holding costs, which is an NP-hard problem but solvable in pseudo-polynomial time. A straightforward dynamic programming approach to this problem results in an O(n2cd) algorithm, where n is the number of periods, and d and c are the average demand and the average production capacity over the n periods, respectively. However, we present a dynamic programming procedure with complexity O(n2qd), where q is the average number of pieces required to represent the production cost functions. In particular, this means that problems in which the production functions consist of a fixed set-up cost plus a linear variable cost are solved in O(n2d) time. Hence, the running time of our algorithm is only linearly dependent on the magnitude of the data. This result also holds if extensions such as backlogging and startup costs are considered. Moreover, computational experiments indicate that the algorithm is capable of solving quite large problem instances within a reasonable amount of time. For example, the average time needed to solve test instances with 96 periods, 8 pieces in every production cost function, and average demand of 100 units is approximately 40 seconds on a SUN SPARC 5 workstation.

Fuzzy piecewise multilinear and piecewise linear systems as universal approximators in Sobolev norms

Abstract: A piecewise multilinear and a piecewise linear model for fuzzy systems are introduced and their approximation capabilities investigated. Classical results on the approximation of regular functions are improved showing that fuzzy systems may keep their semantic structure while approximating to any degree of accuracy not only sufficiently regular functions, but also their derivatives.

Genetic algorithms for generation of class boundaries

Abstract: A method is described for finding decision boundaries, approximated by piecewise linear segments, for classifying patterns in /spl Rfr//sup N/,N/spl ges/2, using an elitist model of genetic algorithms. It involves generation and placement of a set of hyperplanes (represented by strings) in the feature space that yields minimum misclassification. A scheme for the automatic deletion of redundant hyperplanes is also developed in case the algorithm starts with an initial conservative estimate of the number of hyperplanes required for modeling the decision boundary. The effectiveness of the classification methodology, along with the generalization ability of the decision boundary, is demonstrated for different parameter values on both artificial data and real life data sets having nonlinear/overlapping class boundaries. Results are compared extensively with those of the Bayes classifier, k-NN rule and multilayer perceptron.

Finding all solutions of piecewise-linear resistive circuits using linear programming

Abstract: An efficient algorithm is proposed for finding all solutions of piecewise-linear resistive circuits. This algorithm is based on a new test for nonexistence of a solution to a system of piecewise-linear equations f/sub i/(x)=0(i=1.2,/spl middot//spl middot//spl middot/,n) in a super-region. Unlike the conventional sign test, which checks whether the solution surfaces of the single piecewise-linear equations exist or not in a super-region, the new test checks whether they intersect or not in the super-region. Such a test can be performed by using linear programming. It is shown that the simplex method can be performed very efficiently by exploiting the adjacency of super-regions in each step. The proposed algorithm is much more efficient than the conventional sign test algorithms and can find all solutions of large scale circuits very efficiently. Moreover, it can find all characteristic curves of piecewise-linear resistive circuits.

Stochastic Shortest Path Problems with Piecewise-Linear Concave Utility Functions

Abstract: This paper considers a stochastic shortest path problem where the arc lengths are independent random variables following a normal distribution. In this problem, the optimal path is one that maximizes the expected utility, with the utility function being piecewise-linear and concave. Such a utility function can be used to approximate nonlinear utility functions that capture risk averse behaviour for a wide class of problems. The principal contribution of this paper is the development of exact algorithms to solve large problem instances. Two algorithms are developed and incorporated in labelling procedures. Computational testing is done to evaluate the performance of the algorithms. Overall, both algorithms are very effective in solving large problems quickly. The relative performance of the two algorithms is found to depend on the "curvature" of the piecewise linear utility function.

Efficient algorithms for function approximation with piecewise linear sigmoidal networks

Abstract: This paper presents a computationally efficient algorithm for function approximation with piecewise linear sigmoidal nodes. A one hidden layer network is constructed one node at a time using the well-known method of fitting the residual. The task of fitting an individual node is accomplished using a new algorithm that searches for the best fit by solving a sequence of quadratic programming problems. This approach offers significant advantages over derivative-based search algorithms (e.g., backpropagation and its extensions). Unique characteristics of this algorithm include: finite step convergence, a simple stopping criterion, solutions that are independent of initial conditions, good scaling properties and a robust numerical implementation. Empirical results are included to illustrate these characteristics.

Hyperchaos from a 4-D manifold piecewise-linear system

Abstract: A piecewise-linear 4-D autonomous system with hyperchaotic behavior is proposed. The system is analyzed by constructing a 2-D return map. We describe this map explicitly and give theoretical evidence for hyperchaos generation. An implementation example is also provided, and its chaotic behavior is demonstrated.

On stability and passivity of a class of hybrid systems

Abstract: The paper deals with the stabilizability and passifiability properties of a class of hybrid dynamical systems. The systems under consideration are composed of a continuous time invariant plant and discrete event controller. An algebraic criterion for existence of a Lyapunov function for a piecewise linear system is given. Based on these results some passifiability issues are considered.

On period-adding sequences of attracting cycles in piecewise linear maps

Abstract: We study numerically bifurcations in a family of bimodal three-piecewise linear continuous one-dimensional maps. Attention is paid to the attracting cycles arising after the bifurcation ‘from unimodal map to bimodal map’. It is found that this type of bifurcation is accompanied by the appearance of period-adding cascades of attracting cycles γ ( a 11 + a 12 k )/( a 21 + a 22 k ) which are characterized by ρ k = ( a 11 + a 12 k )/( a 21 + a 22 k ), k = 0, 1, …

Robust Control Theory and Applications

Abstract: : This AASERT program supported research in robust simulation, hierarchical uncertainty representation, and novel methods for robustness analysis of uncertain systems. In the context of this program, robust simulation means simulating simultaneously sets of initial conditions and disturbance or noise signals. Thus sets of state space must be propagated by the dynamics of the model. Initial investigations have focused on piecewise linear discrete time systems, which map polyhedra to polyhedra at each time step. Linear programming can be used to refine the resulting bounds. This is important if the potentially exponential growth in set descriptions is to be overcome. Hierarchical uncertainty modeling is a new framework to include explicit representation of uncertainty in component modeling. The focus has been on LFTs and implicit (DAE) representations. A variety of examples including parasitics and non linearities illustrate the key ideas. Finally, this report describes new bounds on a spherical mu problem that allows for correlations between uncertainties in an LFT framework. Interestingly, this setting provides quite elegant bounds and simplified computation.

Piecewise linear (pre-)wavelets on non-uniform meshes

Abstract: In this paper, an explicit construction of compactly supported prewavelets on linear finite element spaces is introduced on non-uniform meshes on polyhedron domains and on boundaries of such domains. The obtained basas are stable in the Sobolev spaces Hr for |r| < 3/2. The only condition we need is that of uniform refinements. Compared to existing prewavelets bases on uniform meshes, with our construction the basis transformation from wavelet- to nodal basis (the wavelet transform) can be implemented more efficiently.

Stabilizing and destabilizing control for a piecewise-linear circuit

Abstract: This paper considers a basic approach to generalize techniques of controlling chaos. First, we propose a novel manifold piecewise-linear (MPL) system as a control object. The novel MPL can generate both chaotic attractors and periodic attractors. Second, we propose a novel occasional proportional feedback (OPF) method and apply it to the MPL. The OPF can change the form of the return map of the MPL-it can change chaos into a periodic attractor and change a periodic attractor into chaos. The OPF functions can be guaranteed theoretically. Third, we propose an implementation example of the OPF and confirm its function in the laboratory.

Efficient admission control of piecewise linear traffic envelopes at EDF schedulers

Abstract: We present algorithms for flow admission control at an earliest deadline first link scheduler when the flows are characterized by piecewise linear traffic envelopes. We show that the algorithms have very low computational complexity and, thus, practical applicability. The complexity can be further decreased by introducing the notion of discretized admission control. Through discretization, the range of positions for the end points of linear segments of the traffic envelopes is restricted to a finite set. Simulation experiments show that discretized admission control can lead to two orders of magnitude decrease in the amount of computation needed to make admission control decisions over that incurred when using exact (nondiscrete) admission control, with the additional benefit that this amount of computation no longer depends on the number of flows. We examine the relative performance degradation (in terms of the number of flows admitted) incurred by the discretization and find that it is small.

Deterministic stochastic resonance in a piecewise linear chaotic map

Abstract: The phenomenon of Stochastic Resonance (SR) is observed in a completely deterministic setting - with thermal noise being replaced by one-dimensional chaos. The piecewise linear map investigated in the paper shows a transition from symmetry-broken to symmetric chaos on increasing a system parameter. In the latter state, the chaotic trajectory switches between the two formerly disjoint attractors, driven by the map's inherent dynamics. This chaotic switching rate is found to `resonate' with the frequency of an externally applied periodic perturbation (multiplicative or additive). By periodically modulating the parameter at a specific frequency $\omega$, we observe the existence of resonance where the response of the system (in terms of the residence-time distribution) is maximum. This is a clear indication of SR-like behavior in a chaotic system.

Dynamically scheduled MPC of nonlinear processes using hinging hyperplane models

Abstract: This article deals with the issues associated with designing scheduled model predictive controllers for nonlinear systems within the multiple-linear-model-based control framework. The issues of model set generation from empirical data and closed-loop application of the generated model set are considered. A method of hinging hyperplanes is proposed as a way to construct a piecewise linear dynamic model, conducive to dynamic scheduling of linear MFC controllers. The design and implementation of dynamically scheduled MFC using a hinge function model are discussed, as well as its advantages. Alternate MFC formulations considered here require more computation, but utilize the hinge function model as a global model. Simulated examples of isothermal CSTRs and a batch fermenter are also presented to illustrate the proposed methodologies.

Local scale controlled anisotropic diffusion with local noise estimate for image smoothing and edge detection

Abstract: A novel local scale controlled piecewise linear diffusion for selective smoothing and edge detection is presented. The diffusion stops at the place and time determined by the minimum reliable local scale and a spatial variant, anisotropic local noise estimate. It shows anisotropic, nonlinear diffusion equation using diffusion coefficients/tensors that continuously depend on the gradient is not necessary to achieve sharp, distorted, stable edge detection across many scales. The new diffusion is anisotropic and asymmetric only at places it needs to be, i.e., at significant edges. It not only does not diffuse across significant edges, but also enhances edges. It advances geometry-driven diffusion because it is a piecewise linear model rather than a full nonlinear model, thus it is simple to implement and analyze, and avoids the difficulties and problems associated with nonlinear diffusion. It advances local scale control by introducing spatial variant, anisotropic local noise estimation, and local stopping of diffusion. The original local scale control was based on the unrealistic assumption of uniformly distributed noise independent of the image signal. The local noise estimate significantly improves local scale control.

Finite element approximation of a model for phase separation of a multi-component alloy with a concentration-dependent mobility matrix

Abstract: We consider a model for phase separation of a multi-component alloy with a concentration-dependent mobility matrix and logarithmic free energy. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions. Finally numerical experiments with three components in one space dimension are presented.

Nonlinear image estimation using piecewise and local image models

Abstract: We introduce a new approach to image estimation based on a flexible constraint framework that encapsulates meaningful structural image assumptions. Piecewise image models (PIMs) and local image models (LIMs) are defined and utilized to estimate noise-corrupted images, PIMs and LIMs are defined by image sets obeying certain piecewise or local image properties, such as piecewise linearity, or local monotonicity. By optimizing local image characteristics imposed by the models, image estimates are produced with respect to the characteristic sets defined by the models. Thus, we propose a new general formulation for nonlinear set-theoretic image estimation. Detailed image estimation algorithms and examples are given using two PIMs: piecewise constant (PICO) and piecewise linear (PILI) models, and two LIMs: locally monotonic (LOMO) and locally convex/concave (LOCO) models. These models define properties that hold over local image neighborhoods, and the corresponding image estimates may be inexpensively computed by iterative optimization algorithms. Forcing the model constraints to hold at every image coordinate of the solution defines a nonlinear regression problem that is generally nonconvex and combinatorial. However, approximate solutions may be computed in reasonable time using the novel generalized deterministic annealing (GDA) optimization technique, which is particularly well suited for locally constrained problems of this type. Results are given for corrupted imagery with signal-to-noise ratio (SNR) as low as 2 dB, demonstrating high quality image estimation as measured by local feature integrity, and improvement in SNR.