Investigation of Optimal Control With a Minimum-Fuel Consumption Criterion for a Fourth-Order Plant With Two Control Inputs; Synthesis of an Efficient Suboptimal Control

1965 ◽  
Vol 87 (1) ◽  
pp. 39-57 ◽  
Author(s):  
A. J. Craig ◽  
I. Flu¨gge-Lotz
Keyword(s):  
Feedback Control ◽  
Fuel Consumption ◽  
Fourth Order ◽  
Attitude Control ◽  
Equations Of Motion ◽  
Suboptimal Control ◽  
Optimal Solutions ◽  
State Variables ◽  
Time Control ◽  
Heuristic Analysis

Application of Pontryagin’s optimal principle to control system problems eventually requires that a two-point boundary-value problem be solved. For plants whose equations of motion are greater than second order this represents a formidable barrier in realizing a practical feedback control. When the criterion for optimization is minimum-fuel consumption, choice of time for solution may be used as a free parameter, and this plus consideration of the efficiency of application of control leads to an approximate method which avoids this difficulty. A heuristic analysis of the geometry of state-space trajectories, using true optimal solutions as a guide, provides laws for constructing a feedback control from the state variables. It further provides a knowledge of the bounds of performance of the mechanized suboptimal control and an estimate on the performance of a minimum-time control for comparison purposes. As an example, the method is applied to the problem of minimum-fuel attitude control of an earth-orbiting satellite, a fourth-order plant with two controls.

The Near-Minimum-Time Control Of Open-Loop Articulated Kinematic Chains

1971 ◽  
Vol 93 (3) ◽  
pp. 164-172 ◽  
Author(s):  
M. E. Kahn ◽  
B. Roth
Keyword(s):  
Optimal Control ◽  
Feedback Control ◽  
Response Times ◽  
Equations Of Motion ◽  
Open Loop ◽  
Rigid Bodies ◽  
Minimum Time ◽  
Suboptimal Control ◽  
Dynamical Equations ◽  
Highly Nonlinear

The time-optimal control of a system of rigid bodies connected in series by single-degree-of-freedom joints is studied. The dynamical equations of the system are highly nonlinear, and a closed-form representation of the minimum-time feedback control is not possible. However, a suboptimal feedback control, which provides a close approximation to the optimal control, is developed. The suboptimal control is expressed in terms of switching curves for each of the system controls. These curves are obtained from the linearized equations of motion for the system. Approximations are made for the effects of gravity loads and angular velocity terms in the nonlinear equations of motion. Digital simulation is used to obtain a comparison of response times of the optimal and suboptimal controls. The speed of response of the suboptimal control is found to compare quite favorably with the response speed of the optimal control.

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.

A novel PID control parameters tuning approach for robot manipulators mounted on oscillatory bases

Robotica ◽  
2007 ◽  
Vol 25 (4) ◽  
pp. 467-477 ◽  
Author(s):  
J. Lin ◽  
Z.-Z. Huang
Keyword(s):  
Pid Control ◽  
Pid Controller ◽  
Low Cost ◽  
Equations Of Motion ◽  
Grey Theory ◽  
Control Parameters ◽  
State Variables ◽  
Relational Analysis ◽  
Parameters Tuning ◽  
Grey Relational

SUMMARYThis research focuses on the issue of dynamic modeling and controlling a robotic manipulator attached to a compliant base. Such a system is known under the name macro–micro system, characterized by the number of control actuators being less than the number of state variables. The equations of motion for a two-link planar elbow arm mounted on an oscillatory base has been presented in this investigation. In order to study the sensitivity of tuning the PID parameters to achieve the desired performance, the Grey relational analysis has first been proposed. Therefore, the aim of this work is to apply Grey theory to optimize parameters for partial states feedback of a PID controller for such a structure. The experimental results of the proposed methodology also show that it is technically and economically feasible to develop a low-cost, reliable, automatic, less time-consuming controller for robotics mounted on oscillatory bases.

Trajectory Tracking of Underactuated Multibody Systems

2011 ◽  
Author(s):  
Stefan Reichl ◽  
Wolfgang Steiner
Keyword(s):  
Trajectory Tracking ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
State Variables ◽  
Algebraic Equations

This work presents three different approaches in inverse dynamics for the solution of trajectory tracking problems in underactuated multibody systems. Such systems are characterized by less control inputs than degrees of freedom. The first approach uses an extension of the equations of motion by geometric and control constraints. This results in index-five differential-algebraic equations. A projection method is used to reduce the systems index and the resulting equations are solved numerically. The second method is a flatness-based feedforward control design. Input and state variables can be parameterized by the flat outputs and their time derivatives up to a certain order. The third approach uses an optimal control algorithm which is based on the minimization of a cost functional including system outputs and desired trajectory. It has to be distinguished between direct and indirect methods. These specific methods are applied to an underactuated planar crane and a three-dimensional rotary crane.

Super-twisting disturbance observer-based finite- time attitude stabilization of flexible spacecraft subject to complex disturbances

2018 ◽  
Vol 25 (5) ◽  
pp. 1008-1018 ◽  
Author(s):  
Ruidong Yan ◽  
Zhong Wu
Keyword(s):  
Attitude Control ◽  
Disturbance Observer ◽  
Sliding Mode ◽  
State Variables ◽  
Attitude Dynamics ◽  
Attitude Control System ◽  
Terminal Sliding Mode ◽  
Nonlinear Disturbance Observer ◽  
Attitude Stabilization ◽  
Nonsingular Terminal Sliding Mode

There exist complex disturbances in the attitude control system of flexible spacecrafts, such as space environmental disturbances, flexible vibrations, inertia uncertainties, payload motions, etc. To suppress the effects of these disturbances on the performance of attitude stabilization, a super-twisting disturbance observer (STDO)-based nonsingular terminal sliding mode controller (NTSMC) is proposed in this paper. First, STDO is designed for a second-order dynamical system constructed by applying the lumped disturbance and its integral as state variables, and applying the integral as virtual measurement. Since the virtual measurement is obtained by integrating the inverse attitude dynamics, STDO not only avoids the differential operation of angular velocity, but also fully utilizes the information of a nonlinear model. By combining STDO with NTSMC, a composite controller is designed to achieve high-accuracy spacecraft attitude stabilization. Since most of the disturbances are compensated for by a STDO-based feedforward compensator, only a small switching gain is required to deal with the residual disturbances and uncertainties. Thus, the chattering phenomenon of the controller can be alleviated to a great extent. Finally, numerical simulations for the comparison between STDO-based NTSMC and nonlinear disturbance observer-based NTSMC are carried out in the presence of complex disturbances to verify the effectiveness of the proposed approach.

Fourth‐order finite‐difference P-SV seismograms

Geophysics ◽  
1988 ◽  
Vol 53 (11) ◽  
pp. 1425-1436 ◽  
Author(s):  
Alan R. Levander
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Fourth Order ◽  
Equations Of Motion ◽  
Dispersion Analysis ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Order System ◽  
Surface Boundary ◽  
Lamb’S Problem

I describe the properties of a fourth‐order accurate space, second‐order accurate time, two‐dimensional P-SV finite‐difference scheme based on the Madariaga‐Virieux staggered‐grid formulation. The numerical scheme is developed from the first‐order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga‐Virieux staggered‐grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic‐elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free‐surface or within a layer and to satisfy free‐surface boundary conditions. Benchmark comparisons of finite‐difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite‐difference and reflectivity solutions for elastic‐elastic and acoustic‐elastic layered models.

Multi-Parameter Optimization of Damped Linear Continuous Systems

1972 ◽  
Vol 94 (1) ◽  
pp. 1-7 ◽  
Author(s):  
O. B. Dale ◽  
R. Cohen
Keyword(s):  
Optimal Parameter ◽  
Equations Of Motion ◽  
Analytic Solutions ◽  
System Response ◽  
Continuous Systems ◽  
Frequency Interval ◽  
State Variables ◽  
Initial Value ◽  
Unknown Parameters ◽  
One Dimensional

A method is presented for obtaining and optimizing the frequency response of one-dimensional damped linear continuous systems. The systems considered are assumed to contain unknown constant parameters in the boundary conditions and equations of motion which the designer can vary to obtain a minimum resonant response in some selected frequency interval. The unknown parameters need not be strictly dissipative nor unconstrained. No analytic solutions, either exact or approximate, are required for the system response and only initial value numerical integrations of the state and adjoint differential equations are required to obtain the optimal parameter set. The combinations of state variables comprising the response and the response locations are arbitrary.

Simulation of Planar Dynamic Mechanical Systems With Changing Topologies: Part I — Characterization and Prediction of the Kinematic Constraint Changes

1987 ◽  
Author(s):  
B. J. Gilmore ◽  
R. J. Cipra
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
State Variables ◽  
Kinematic Constraint ◽  
Kinematic Constraints ◽  
Force Closure ◽  
Simulation Strategy ◽  
Reaction Forces ◽  
Discontinuous Equations

Abstract Due to changes in the kinematic constraints, many mechanical systems are described by discontinuous equations of motion. This paper addresses those changes in the kinematic constraints which are caused by planar bodies contacting and separating. A strategy to automatically predict and detect the kinematic constraint changes, which are functions of the system dynamics, is presented in Part I. The strategy employs the concepts of point to line contact kinematic constraints, force closure, and ray firing together with the information provided by the rigid bodies’ boundary descriptions, state variables, and reaction forces to characterize the kinematic constraint changes. Since the strategy automatically predicts and detects constraint changes, it is capable of simulating mechanical systems with unpredictable or unforeseen changes in topology. Part II presents the implementation of the characterizations into a simulation strategy and presents examples.

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