Neighboring Optimal Feedback Law for Higher-Order Dynamic Systems

2002 ◽  
Vol 124 (3) ◽  
pp. 492-497 ◽  
Author(s):  
Tawiwat Veeraklaew ◽  
Sunil K. Agrawal
Keyword(s):  
Dynamic Systems ◽  
Equations Of Motion ◽  
Optimal Solution ◽  
Higher Order ◽  
Input State ◽  
Linear Quadratic ◽  
First Order ◽  
Optimal Feedback ◽  
Order Form ◽  
Jacobi Equations

In recent years, using tools from linear and nonlinear systems theory, it has been shown that classes of dynamic systems in first-order forms can be alternatively written in higher-order forms, i.e., as sets of higher-order differential equations. Input-state linearization is one of the most popular tools to achieve such a representation. The equations of motion of mechanical systems naturally have a second-order form, arising from the application of Newton’s laws. In the last five years, effective computational tools have been developed by the authors to compute optimal trajectories of such systems, while exploiting the inherent structure of the dynamic equations. In this paper, we address the question of computing the neighboring optimal for systems in higher-order forms. It must be pointed out that the classical solution of the neighboring optimal problem is well known only for systems in the first-order form. The main contributions of this paper are: (i) derivation of the optimal feedback law for higher-order linear quadratic terminal controller using extended Hamilton-Jacobi equations; (ii) application of the feedback law to compute the neighboring optimal solution.

scholarly journals THE CANONICAL STRUCTURE OF THE FIRST-ORDER EINSTEIN–HILBERT ACTION

2010 ◽  
Vol 25 (17) ◽  
pp. 3453-3480 ◽  
Author(s):  
D. G. C. MCKEON
Keyword(s):  
Phase Space ◽  
Gauge Invariance ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Affine Connection ◽  
Canonical Structure ◽  
First Order ◽  
Order Form ◽  
Solving Equations ◽  
Hilbert Action

The Dirac constraint formalism is used to analyze the first-order form of the Einstein–Hilbert action in d > 2 dimensions. Unlike previous treatments, this is done without eliminating fields at the outset by solving equations of motion that are independent of time derivatives when they correspond to first class constraints. As anticipated by the way in which the affine connection transforms under a diffeomorphism, not only primary and secondary but also tertiary first class constraints arise. These leave d(d-3) degrees of freedom in phase space. The gauge invariance of the action is discussed, with special attention being paid to the gauge generators of Henneaux, Teitelboim and Zanelli and of Castellani.

Linear Time-Varying Dynamic Systems Optimization via Higher-Order Method Using Shifted Chebyshev’s Polynomials

1999 ◽  
Vol 121 (2) ◽  
pp. 258-261 ◽  
Author(s):  
Xiaochun Xu ◽  
Sunil K. Agrawal
Keyword(s):  
Differential Equations ◽  
Dynamic Systems ◽  
Linear Time ◽  
Optimal Solution ◽  
Higher Order ◽  
Time Varying ◽  
Varying Coefficients ◽  
Higher Order Method ◽  
Time Varying Coefficients ◽  
Higher Order Differential Equations

For optimization of classes of linear time-varying dynamic systems with n states and m control inputs, a new higher-order procedure was presented by the authors that does not use Lagrange multipliers. In this new procedure, the optimal solution was shown to satisfy m 2p-order differential equations with time-varying coefficients. These differential equations were solved using weighted residual methods. Even though solution of the optimization problem using this procedure was demonstrated to be computation efficient, shifted Chebyshev’s polynomials are used in the paper to solve the higher-order differential equations. This further reduces the computations and makes this algorithm more appropriate for real-time implementation.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Challenging the Multidimensional Conception of Perceived Person-Environment Fit

2020 ◽  
pp. 1-9
Author(s):  
Julian M. Etzel ◽  
Gabriel Nagy
Keyword(s):  
Higher Education ◽  
University Students ◽  
Educational Outcomes ◽  
Factor Model ◽  
Higher Order ◽  
Substantial Proportion ◽  
Unique Variance ◽  
First Order ◽  
Person Environment Fit ◽  
Order Factor

Abstract. In the current study, we examined the viability of a multidimensional conception of perceived person-environment (P-E) fit in higher education. We introduce an optimized 12-item measure that distinguishes between four content dimensions of perceived P-E fit: interest-contents (I-C) fit, needs-supplies (N-S) fit, demands-abilities (D-A) fit, and values-culture (V-C) fit. The central aim of our study was to examine whether the relationships between different P-E fit dimensions and educational outcomes can be accounted for by a higher-order factor that captures the shared features of the four fit dimensions. Relying on a large sample of university students in Germany, we found that students distinguish between the proposed fit dimensions. The respective first-order factors shared a substantial proportion of variance and conformed to a higher-order factor model. Using a newly developed factor extension procedure, we found that the relationships between the first-order factors and most outcomes were not fully accounted for by the higher-order factor. Rather, with the exception of V-C fit, all specific P-E fit factors that represent the first-order factors’ unique variance showed reliable and theoretically plausible relationships with different outcomes. These findings support the viability of a multidimensional conceptualization of P-E fit and the validity of our adapted instrument.

Computational Models for Multilayered Composite Shells with Application to Tires

1996 ◽  
Vol 24 (1) ◽  
pp. 11-38 ◽  
Author(s):  
G. M. Kulikov
Keyword(s):  
Type Theory ◽  
Computational Models ◽  
Geometrical Nonlinearity ◽  
Higher Order ◽  
Stress Strain ◽  
Strain Fields ◽  
First Order ◽  
Timoshenko Type ◽  
Joint Influence ◽  
Discrete Layer

Abstract This paper focuses on four tire computational models based on two-dimensional shear deformation theories, namely, the first-order Timoshenko-type theory, the higher-order Timoshenko-type theory, the first-order discrete-layer theory, and the higher-order discrete-layer theory. The joint influence of anisotropy, geometrical nonlinearity, and laminated material response on the tire stress-strain fields is examined. The comparative analysis of stresses and strains of the cord-rubber tire on the basis of these four shell computational models is given. Results show that neglecting the effect of anisotropy leads to an incorrect description of the stress-strain fields even in bias-ply tires.

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

scholarly journals Higher order Hamiltonian Monte Carlo sampling for cosmological large-scale structure analysis

2021 ◽  
Vol 502 (3) ◽  
pp. 3976-3992
Author(s):  
Mónica Hernández-Sánchez ◽  
Francisco-Shu Kitaura ◽  
Metin Ata ◽  
Claudio Dalla Vecchia
Keyword(s):  
Fourth Order ◽  
Large Scale ◽  
Equations Of Motion ◽  
Large Scale Structure ◽  
Real Space ◽  
Higher Order ◽  
Second Order ◽  
Scale Structure ◽  
Integration Schemes ◽  
Reconstruction Methods

ABSTRACT We investigate higher order symplectic integration strategies within Bayesian cosmic density field reconstruction methods. In particular, we study the fourth-order discretization of Hamiltonian equations of motion (EoM). This is achieved by recursively applying the basic second-order leap-frog scheme (considering the single evaluation of the EoM) in a combination of even numbers of forward time integration steps with a single intermediate backward step. This largely reduces the number of evaluations and random gradient computations, as required in the usual second-order case for high-dimensional cases. We restrict this study to the lognormal-Poisson model, applied to a full volume halo catalogue in real space on a cubical mesh of 1250 h−1 Mpc side and 2563 cells. Hence, we neglect selection effects, redshift space distortions, and displacements. We note that those observational and cosmic evolution effects can be accounted for in subsequent Gibbs-sampling steps within the COSMIC BIRTH algorithm. We find that going from the usual second to fourth order in the leap-frog scheme shortens the burn-in phase by a factor of at least ∼30. This implies that 75–90 independent samples are obtained while the fastest second-order method converges. After convergence, the correlation lengths indicate an improvement factor of about 3.0 fewer gradient computations for meshes of 2563 cells. In the considered cosmological scenario, the traditional leap-frog scheme turns out to outperform higher order integration schemes only when considering lower dimensional problems, e.g. meshes with 643 cells. This gain in computational efficiency can help to go towards a full Bayesian analysis of the cosmological large-scale structure for upcoming galaxy surveys.

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