Showing papers on "Nonlinear system published in 2005"
TL;DR: The current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness, are described.
Abstract: SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordinary differential or differential-algebraic equations. The basic versions of these codes are called KINSOL, CVODE, and IDA, respectively. The codes are written in ANSI standard C and are suitable for either serial or parallel machine environments. Common and notable features of these codes include inexact Newton-Krylov methods for solving large-scale nonlinear systems; linear multistep methods for time-dependent problems; a highly modular structure to allow incorporation of different preconditioning and/or linear solver methods; and clear interfaces allowing for users to provide their own data structures underneath the solvers. We describe the current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness. We also describe how the codes stem from previous and widely used Fortran 77 solvers, and how the codes have been augmented with forward and adjoint methods for carrying out first-order sensitivity analysis with respect to model parameters or initial conditions.
2,124 citations
01 Feb 2005
TL;DR: The theory of linear difference equations applied to population growth and the applications of nonlinear difference equations to population biology are explained.
Abstract: Part I. Discrete Process in Biology: 1. The theory of linear difference equations applied to population growth 2. Nonlinear difference equations 3. Applications of nonlinear difference equations to population biology Part II. Continuous Processes and Ordinary Differential Equations: 4. An introduction to continuous models 5. Phase-plane methods and qualitative solutions 6. Applications of continuous models to population dynamics 7. Models for molecular events 8. Limit cycles, oscillations, and excitable systems Part III. Spatially Distributed Systems and Partial Differential Equation Models: 9. An introduction to partial differential equations and diffusion in biological settings 10. Partial differential equation models in biology 11. Models for development and pattern formation in biological systems Selected answers Author index Subject index.
1,925 citations
18 Apr 2005
TL;DR: This paper attempts to examine the claim that PSO has the same effectiveness (finding the true global optimal solution) as the GA but with significantly better computational efficiency by implementing statistical analysis and formal hypothesis testing.
Abstract: Particle Swarm Optimization (PSO) is a relatively recent heuristic search method whose mechanics are inspired by the swarming or collaborative behavior of biological populations. PSO is similar to the Genetic Algorithm (GA) in the sense that these two evolutionary heuristics are population-based search methods. In other words, PSO and the GA move from a set of points (population) to another set of points in a single iteration with likely improvement using a combination of deterministic and probabilistic rules. The GA and its many versions have been popular in academia and the industry mainly because of its intuitiveness, ease of implementation, and the ability to effectively solve highly nonlinear, mixed integer optimization problems that are typical of complex engineering systems. The drawback of the GA is its expensive computational cost. This paper attempts to examine the claim that PSO has the same effectiveness (finding the true global optimal solution) as the GA but with significantly better computational efficiency (less function evaluations) by implementing statistical analysis and formal hypothesis testing. The performance comparison of the GA and PSO is implemented using a set of benchmark test problems as well as two space systems design optimization problems, namely, telescope array configuration and spacecraft reliability-based design.
1,221 citations
TL;DR: An algorithm for computing the set of reachable states of a continuous dynamic game based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time-dependent Hamilton-Jacobi-Isaacs partial differential equation.
Abstract: We describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game. The algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time-dependent Hamilton-Jacobi-Isaacs partial differential equation. While alternative techniques for computing the reachable set have been proposed, the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters. Because the time-dependent equation's solution is continuous and defined throughout the state space, methods from the level set literature can be used to generate more accurate approximations than are possible for formulations with potentially discontinuous solutions. A numerical implementation of our formulation is described and has been released on the web. Its correctness is verified through a two vehicle, three dimensional collision avoidance example for which an analytic solution is available.
1,107 citations
TL;DR: A backstepping based control design for a class of nonlinear systems in strict-feedback form with arbitrary uncertainty is developed and is able to eliminate the problem of "explosion of complexity" inherent in the existing method.
Abstract: The dynamic surface control (DSC) technique was developed recently by Swaroop et al. This technique simplified the backstepping design for the control of nonlinear systems in strict-feedback form by overcoming the problem of "explosion of complexity." It was later extended to adaptive backstepping design for nonlinear systems with linearly parameterized uncertainty. In this paper, by incorporating this design technique into a neural network based adaptive control design framework, we have developed a backstepping based control design for a class of nonlinear systems in strict-feedback form with arbitrary uncertainty. Our development is able to eliminate the problem of "explosion of complexity" inherent in the existing method. In addition, a stability analysis is given which shows that our control law can guarantee the uniformly ultimate boundedness of the solution of the closed-loop system, and make the tracking error arbitrarily small.
1,079 citations
TL;DR: It is shown that the constrained optimal control law has the largest region of asymptotic stability (RAS) and the result is a nearly optimal constrained state feedback controller that has been tuned a priori off-line.
1,045 citations
TL;DR: This paper shows that many kernel methods can be equivalently formulated as minimum enclosing ball (MEB) problems in computational geometry and obtains provably approximately optimal solutions with the idea of core sets, and proposes the proposed Core Vector Machine (CVM) algorithm, which can be used with nonlinear kernels and has a time complexity that is linear in m.
Abstract: Standard SVM training has O(m3) time and O(m2) space complexities, where m is the training set size. It is thus computationally infeasible on very large data sets. By observing that practical SVM implementations only approximate the optimal solution by an iterative strategy, we scale up kernel methods by exploiting such "approximateness" in this paper. We first show that many kernel methods can be equivalently formulated as minimum enclosing ball (MEB) problems in computational geometry. Then, by adopting an efficient approximate MEB algorithm, we obtain provably approximately optimal solutions with the idea of core sets. Our proposed Core Vector Machine (CVM) algorithm can be used with nonlinear kernels and has a time complexity that is linear in m and a space complexity that is independent of m. Experiments on large toy and real-world data sets demonstrate that the CVM is as accurate as existing SVM implementations, but is much faster and can handle much larger data sets than existing scale-up methods. For example, CVM with the Gaussian kernel produces superior results on the KDDCUP-99 intrusion detection data, which has about five million training patterns, in only 1.4 seconds on a 3.2GHz Pentium--4 PC.
1,017 citations
TL;DR: It is proved that global finite-time stabilizability of uncertain nonlinear systems that are dominated by a lower-triangular system can be achieved by Holder continuous state feedback.
982 citations
TL;DR: A modification of the exponential time-differencing fourth-order Runge--Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators.
Abstract: A modification of the exponential time-differencing fourth-order Runge--Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made of the performance of this modified exponential time-differencing (ETD) scheme against the competing methods of implicit-explicit differencing, integrating factors, time-splitting, and Fornberg and Driscoll's "sliders" for the KdV, Kuramoto--Sivashinsky, Burgers, and Allen--Cahn equations in one space dimension. Implementation of the method is illustrated by short MATLAB programs for two of the equations. It is found that for these applications with fixed time steps, the modified ETD scheme is the best.
921 citations
TL;DR: The method presented here is a variation of existing methods using empirical Gramians that allows one to compute balancing transformations directly, without separate reduction of the Gramians, and has computational cost similar to that of POD.
Abstract: Many of the tools of dynamical systems and control theory have gone largely unused for fluids, because the governing equations are so dynamically complex, both high-dimensional and nonlinear. Model reduction involves finding low-dimensional models that approximate the full high-dimensional dynamics. This paper compares three different methods of model reduction: proper orthogonal decomposition (POD), balanced truncation, and a method called balanced POD. Balanced truncation produces better reduced-order models than POD, but is not computationally tractable for very large systems. Balanced POD is a tractable method for computing approximate balanced truncations, that has computational cost similar to that of POD. The method presented here is a variation of existing methods using empirical Gramians, and the main contributions of the present paper are a version of the method of snapshots that allows one to compute balancing transformations directly, without separate reduction of the Gramians; and an output p...
916 citations
TL;DR: In this article, a different approach is adopted, and proper orthogonal decomposition is considered, and modes extracted from the decomposition may serve two purposes, namely order reduction by projecting high-dimensional data into a lower-dimensional space and feature extraction by revealing relevant but unexpected structure hidden in the data.
Abstract: Modal analysis is used extensively for understanding the dynamic behavior of structures. However, a major concern for structural dynamicists is that its validity is limited to linear structures. New developments have been proposed in order to examine nonlinear systems, among which the theory based on nonlinear normal modes is indubitably the most appealing. In this paper, a different approach is adopted, and proper orthogonal decomposition is considered. The modes extracted from the decomposition may serve two purposes, namely order reduction by projecting high-dimensional data into a lower-dimensional space and feature extraction by revealing relevant but unexpected structure hidden in the data. The utility of the method for dynamic characterization and order reduction of linear and nonlinear mechanical systems is demonstrated in this study.
TL;DR: In this paper, the critical blow-up exponent for a Keller-Segel-type chemotaxis model was determined, where the chemotactic sensitivity equals some nonlinear function of the particle density.
Book•
01 Jan 2005
TL;DR: In this paper, the authors present a system of equations for evolving pseudomonotone or weakly continuous mappings with set-valued mappings, and a set of auxiliary tools.
Abstract: Preface.- Preface to the 2nd edition.- Notational conventions.- 1 Preliminary general material.- I Steady-state problems.- 2 Pseudomonotone or weakly continuous mappings.- 3 Accretive mappings.- 4 Potential problems: smooth case.- 5 Nonsmooth problems variational inequalities.- 6. Systems of equations: particular examples.- II Evolution problems.- 7 Special auxiliary tools.- 8 Evolution by pseudomonotone or weakly continuous mappings.- 9 Evolution governed by accretive mappings.- 10 Evolution governed by certain set-valued mappings.- 11 Doubly-nonlinear problems.- 12 Systems of equations: particular examples.- References.- Index.
08 Jun 2005
TL;DR: Todorov et al. as discussed by the authors presented an iterative linear-quadratic-Gaussian method for locally-optimal feedback control of nonlinear stochastic systems subject to control constraints.
Abstract: We present an iterative linear-quadratic-Gaussian method for locally-optimal feedback control of nonlinear stochastic systems subject to control constraints. Previously, similar methods have been restricted to deterministic unconstrained problems with quadratic costs. The new method constructs an affine feedback control law, obtained by minimizing a novel quadratic approximation to the optimal cost-to-go function. Global convergence is guaranteed through a Levenberg-Marquardt method; convergence in the vicinity of a local minimum is quadratic. Performance is illustrated on a limited-torque inverted pendulum problem, as well as a complex biomechanical control problem involving a stochastic model of the human arm, with 10 state dimensions and 6 muscle actuators. A Matlab implementation of the new algorithm is availabe at www.cogsci.ucsd.edu//spl sim/todorov.
TL;DR: In this paper, the dynamics of open flows are considered as a superposition of linear or nonlinear instability waves that behave at each streamwise station as if the flow were homogeneous in the streamwise direction.
Abstract: The objective of this review is to critically assess the different approaches developed in recent years to understand the dynamics of open flows such as mixing layers, jets, wakes, separation bubbles, boundary layers, and so on. These complex flows develop in extended domains in which fluid particles are continuously advected downstream. They behave either as noise amplifiers or as oscillators, both of which exhibit strong nonlinearities (Huerre & Monkewitz 1990). The local approach is inherently weakly nonparallel and it assumes that the basic flow varies on a long length scale compared to the wavelength of the instability waves. The dynamics of the flow is then considered as a superposition of linear or nonlinear instability waves that, at leading order, behave at each streamwise station as if the flow were homogeneous in the streamwise direction. In the fully global context, the basic flow and the instabilities do not have to be characterized by widely separated length scales, and the dynamics is then viewed as the result of the interactions between Global modes living in the entire physical domain with the streamwise direction as an eigendirection. This second approach is more and more resorted to as a result of increased computational capability. The earlier review of Huerre & Monkewitz (1990) emphasized how local linear theory can account for the noise amplifier behavior as well as for the onset of a Global mode. The present survey first adopts the opposite point of view by demonstrating how fully global theory accounts for the noise amplifier behavior of open flows. From such a perspective, there is strong emphasis on the very peculiar nonorthogonality of linear Global modes, which in turn allows a novel interpretation of recent numerical simulations and experimental observations. The nonorthogonality of linear Global modes also imposes severe constraints on the extension of linear global theory to the fully nonlinear regime. When the flow is weakly nonparallel, this limitation is so severe that the linear Global mode theory is of little help. It is then much more appropriate to develop a fully nonlinear formulation involving the presence of a front separating the base state region from the bifurcated state region.
TL;DR: In this article, an up-to-date survey of experimental and theoretical results in spatiotemporal optical solitons can be found, together with a review of the outstanding difficulties and open problems.
Abstract: In the course of the past several years, a new level of understanding has been achieved about conditions for the existence, stability, and generation of spatiotemporal optical solitons, which are nondiffracting and nondispersing wavepackets propagating in nonlinear optical media. Experimentally, effectively two-dimensional (2D) spatiotemporal solitons that overcome diffraction in one transverse spatial dimension have been created in quadratic nonlinear media. With regard to the theory, fundamentally new features of light pulses that self-trap in one or two transverse spatial dimensions and do not spread out in time, when propagating in various optical media, were thoroughly investigated in models with various nonlinearities. Stable vorticity-carrying spatiotemporal solitons have been predicted too, in media with competing nonlinearities (quadratic–cubic or cubic–quintic). This article offers an up-to-date survey of experimental and theoretical results in this field. Both achievements and outstanding difficulties are reviewed, and open problems are highlighted. Also briefly described are recent predictions for stable 2D and 3D solitons in Bose–Einstein condensates supported by full or low-dimensional optical lattices.
TL;DR: In this article, the authors compare several types of nonlinear filters, including: particle filters (PFs), unscented Kalman filters, extended Kalman filter, batch filters and exact recursive filter.
Abstract: Nonlinear filters can provide estimation accuracy that is vastly superior to extended Kalman filters for some important practical applications. We compare several types of nonlinear filters, including: particle filters (PFs), unscented Kalman filters, extended Kalman filters, batch filters and exact recursive filters. The key practical issue in nonlinear filtering is computational complexity, which is often called "the curse of dimensionality". It has been asserted that PFs avoid the curse of dimensionality, but this is generally incorrect. Well-designed PFs with good proposal densities sometimes avoid the curse of dimensionality, but not otherwise. Future research in nonlinear filtering will exploit recent progress in quasi-Monte Carlo algorithms (rather than boring old Monte Carlo methods), as well as ideas borrowed from physics (e.g., dimensional interpolation) and new mesh-free adjoint methods for solving PDEs. This tutorial was written for normal engineers, who do not have nonlinear filters for breakfast.
TL;DR: In this article, a fully nonlinear treatment of surface stress and surface elastic constants is presented, which is illustrated with examples of several crystal faces of some fcc metals modeled with embedded atom potentials.
Abstract: Elastic properties of crystal surfaces are useful in understanding mechanical properties of nanostructures. This paper presents a fully nonlinear treatment of surface stress and surface elastic constants. A method for the determination of surface elastic properties from atomistic simulations is developed. This method is illustrated with examples of several crystal faces of some fcc metals modeled with embedded atom potentials. The key finding in this study is the importance of accounting for the additional relaxations of atoms at the crystal surface due to strain. Although these relaxations do not affect the values of surface stress (as had been determined in previousworks), they have a profound effect on the surface elastic constants.Failure to account for these relaxations can lead to values of elastic constants that are incorrect not only in magnitude but also in sign. A possible method for the experimental determination of the surface elastic constants is outlined.
Posted Content•
TL;DR: In this article, an introduction to the nonlinear equations for completely symmetric bosonic higher spin gauge fields in anti de Sitter space of any dimension is provided, and some related issues such as the MacDowell-Mansouri-Stelle-West formulation of gravity, unfolded formulation of dynamical systems in terms of free differential algebras and Young tableaux symmetry properties in terms with Howe dual algesbras are discussed.
Abstract: In this article, an introduction to the nonlinear equations for completely symmetric bosonic higher spin gauge fields in anti de Sitter space of any dimension is provided. To make the presentation self-contained we explain in detail some related issues such as the MacDowell-Mansouri-Stelle-West formulation of gravity, unfolded formulation of dynamical systems in terms of free differential algebras and Young tableaux symmetry properties in terms of Howe dual algebras.
TL;DR: In this paper, a nonlinear adaptive lumped parameter magnetic circuit model is developed to predict the electromagnetic performance of a flux-switching permanent-magnet machine, which enables the air-gap field distribution, the back-electromotive force (back-EMF) waveform, the winding inductances, and the electromagnetic torque to be calculated.
Abstract: A nonlinear adaptive lumped parameter magnetic circuit model is developed to predict the electromagnetic performance of a flux-switching permanent-magnet machine. It enables the air-gap field distribution, the back-electromotive force (back-EMF) waveform, the winding inductances, and the electromagnetic torque to be calculated. Results from the model are compared with finite-element predictions and validated experimentally. The influence of end effects is also investigated, and optimal design parameters, such as the rotor pole width, the stator tooth width, and the ratio of the inner to outer diameter of the stator, are discussed.
TL;DR: In this paper, the mathematical setting of stationary systems modelled by elliptic partial differential equations with stochastic coefficients (random fields) is investigated and stability with respect to stability.
TL;DR: In this paper, a new method is presented to look for exact solutions of nonlinear differential equations by using the general solutions of the simplest nonlinear equations and taking into consideration all possible singularities of equation studied.
Abstract: New method is presented to look for exact solutions of nonlinear differential equations. Two basic ideas are at the heart of our approach. One of them is to use the general solutions of the simplest nonlinear differential equations. Another idea is to take into consideration all possible singularities of equation studied. Application of our approach to search exact solutions of nonlinear differential equations is discussed in detail. The method is used to look for exact solutions of the Kuramoto–Sivashinsky equation and the equation for description of nonlinear waves in a convective fluid. New exact solitary and periodic waves of these equations are given.
TL;DR: A method is proposed for designing an antiwindup gain that maximizes an estimate of the basin of attraction of the closed-loop system that can be modeled by a linear system with a deadzone nonlinearity.
Abstract: This note addresses the design of antiwindup gains for obtaining larger regions of stability for linear systems with saturating inputs. Considering that a linear dynamic output feedback has been designed to stabilize the linear system (without saturation), a method is proposed for designing an antiwindup gain that maximizes an estimate of the basin of attraction of the closed-loop system. It is shown that the closed-loop system obtained from the controller plus the antiwindup gain can be modeled by a linear system with a deadzone nonlinearity. A modified sector condition is then used to obtain stability conditions based on quadratic Lyapunov functions. Differently from previous works these conditions are directly in linear matrix inequality form. Some numerical examples illustrate the effectiveness of the proposed design technique when compared with the previous ones.
TL;DR: The proposed dependence measures provide a natural framework for a limit theory for stationary processes and present limit theorems for partial sums, empirical processes, and kernel density estimates under conditions with quite simple forms.
Abstract: Based on the nonlinear system theory, we introduce previously undescribed dependence measures for stationary causal processes. Our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems. The proposed dependence measures provide a natural framework for a limit theory for stationary processes. In particular, under conditions with quite simple forms, we present limit theorems for partial sums, empirical processes, and kernel density estimates. The conditions are mild and easily verifiable because they are directly related to the data-generating mechanisms.
TL;DR: The dynamic model of a wheeled inverted pendulum (e.g., Segway, Quasimoro, and Joe) is analyzed from a controllability and feedback linearizability point of view and two novel controllers are designed.
Abstract: In this paper, the dynamic model of a wheeled inverted pendulum (eg, Segway, Quasimoro, and Joe) is analyzed from a controllability and feedback linearizability point of view First, a dynamic model of this underactuated system is derived with respect to the wheel motor torques as inputs while taking the nonholonomic no-slip constraints into considerations This model is compared with the previous models derived for similar systems The strong accessibility condition is checked and the maximum relative degree of the system is found Based on this result, a partial feedback linearization of the system is obtained and the internal dynamics equations are isolated The resulting equations are then used to design two novel controllers The first one is a two-level velocity controller for tracking vehicle orientation and heading speed set-points, while controlling the vehicle pitch (pendulum angle from the vertical) within a specified range The second controller is also a two-level controller which stabilizes the vehicle's position to the desired point, while again keeping the pitch bounded between specified limits Simulation results are provided to show the efficacy of the controllers using realistic data
TL;DR: A solution to the reactive power dispatch problem with a novel particle swarm optimization approach based on multiagent systems (MAPSO) is presented and it is shown that the proposed approach converges to better solutions much faster than the earlier reported approaches.
Abstract: Reactive power dispatch in power systems is a complex combinatorial optimization problem involving nonlinear functions having multiple local minima and nonlinear and discontinuous constraints. In this paper, a solution to the reactive power dispatch problem with a novel particle swarm optimization approach based on multiagent systems (MAPSO) is presented. This method integrates the multiagent system (MAS) and the particle swarm optimization (PSO) algorithm. An agent in MAPSO represents a particle to PSO and a candidate solution to the optimization problem. All agents live in a lattice-like environment, with each agent fixed on a lattice point. In order to obtain optimal solution quickly, each agent competes and cooperates with its neighbors, and it can also learn by using its knowledge. Making use of these agent-agent interactions and evolution mechanism of PSO, MAPSO realizes the purpose of optimizing the value of objective function. MAPSO applied to optimal reactive power dispatch is evaluated on an IEEE 30-bus power system and a practical 118-bus power system. Simulation results show that the proposed approach converges to better solutions much faster than the earlier reported approaches. The optimization strategy is general and can be used to solve other power system optimization problems as well.
Proceedings Article•
05 Dec 2005TL;DR: This paper marginalize out the model parameters in closed-form, using Gaussian Process (GP) priors for both the dynamics and the observation mappings, resulting in a nonparametric model for dynamical systems that accounts for uncertainty in the model.
Abstract: This paper introduces Gaussian Process Dynamical Models (GPDM) for nonlinear time series analysis. A GPDM comprises a low-dimensional latent space with associated dynamics, and a map from the latent space to an observation space. We marginalize out the model parameters in closed-form, using Gaussian Process (GP) priors for both the dynamics and the observation mappings. This results in a nonparametric model for dynamical systems that accounts for uncertainty in the model. We demonstrate the approach on human motion capture data in which each pose is 62-dimensional. Despite the use of small data sets, the GPDM learns an effective representation of the nonlinear dynamics in these spaces. Webpage: http://www.dgp.toronto.edu/~jmwang/gpdm/
TL;DR: This paper is concerned with sliding mode control for uncertain stochastic systems with time-varying delay, and an integral sliding surface is first constructed, and a sufficient condition is derived to guarantee the global Stochastic stability of the stoChastic dynamics in the specified switching surface for all admissible uncertainties.
Book•
01 Jan 2005
TL;DR: In this article, a sampling of design methodologies Linear and nonlinear potential shaping for stabilization and tracking for fully actuated systems Stabilization and tracking using oscillatory controls Motion planning for underactuated systems Appendices Timedependent vector fields Some proofs.
Abstract: Part I: Modeling of mechanical systems Introductory examples and problems Linear and multilinear algebra Differential geometry Simple mechanical control systems Lie groups, systems on groups, and symmetries.- Part II: Analysis of mechanical control systems Stability Controllability Low-order controllability and kinematic reduction Perturbation analysis.- Part III: A sampling of design methodologies Linear and nonlinear potential shaping for stabilization Stabilization and tracking for fully actuated systems Stabilization and tracking using oscillatory controls Motion planning for underactuated systems Appendices Time-dependent vector fields Some proofs.
TL;DR: This article outlines conditions that lead to the formation of multiple optima in the estimator for systems tending to a steady state and proposes tests that determine when these conditions hold for chemical reaction networks.
Abstract: The goal of state estimation is to reconstruct the state of a system from process measurements and a model. State estimators for most physical processes often must address many different challenges, including nonlinear dynamics, states subject to hard constraints (e.g. nonnegative concentrations), and local optima. In this article, we compare the performance of two such estimators: the extended Kalman filter (EKF) and moving horizon estimation (MHE). We illustrate conditions that lead to estimation failure in the EKF when there is no plant-model mismatch and demonstrate such failure via several simple examples. We then examine the role that constraints, the arrival cost, and the type of optimization (global versus local) play in determining how MHE performs on these examples. In each example, the two estimators are given exactly the same information, namely tuning parameters, model, and measurements; yet MHE consistently provides improved state estimation and greater robustness to both poor guesses of the initial state and tuning parameters in comparison to the EKF.