GLOBAL AND LOCAL CONTROL OF HOMOCLINIC AND HETEROCLINIC BIFURCATIONS

2005 ◽  
Vol 15 (08) ◽  
pp. 2411-2432 ◽  
Author(s):  
HONGJUN CAO ◽  
GUANRONG CHEN
Keyword(s):  
Optimal Control ◽  
Control Method ◽  
Basins Of Attraction ◽  
Stable And Unstable Manifolds ◽  
Optimal Control Method ◽  
Global Optimal ◽  
Unstable Manifolds ◽  
Global And Local ◽  
Theoretical Analyses ◽  
Optimal Control Methods

A comprehensive resonant optimal control method is developed and discussed for suppressing homoclinic and heteroclinic bifurcations of a general one-degree-of-freedom nonlinear oscillator. Based on an adjustable phase shift, the primary resonant optimal control method is presented. By solving an optimization problem for the optimal amplitude coefficients to be used as the control parameters, the force term as the controller can be designed. Three kinds of resonant optimal control methods are compared. The control mechanism of the primary resonant optimal control method is to enlarge to the largest possible degree the control region where homoclinic and/or heteroclinic transversal intersections do not occur, and this is accomplished at lowest cost. It is shown that the primary resonant optimal control method has much better performance than the superharmonic resonant optimal control method, and it works well even when the superharmonic optimal control method fails. In particular, one new global optimal control method is presented, whose central idea is to find a frequency such that the asymmetric homoclinic bifurcations or the multiple homoclinic and heteroclinic bifurcations can attain the same critical values. On the basis of these same critical bifurcation values, chaos resulting from asymmetric homoclinic or multiple homoclinic and heteroclinic bifurcations can be effectively suppressed by the primary resonant optimal control method. This is confirmed by two illustrative examples. The theoretical analyses concerning the suppression of local and global homoclinic and heteroclinic bifurcations are in agreement with the numerical simulations, including the identification of the stable and unstable manifolds and the basins of attraction.

scholarly journals Optimal Control for a Class of Chaotic Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Jianxiong Zhang ◽  
Wansheng Tang
Keyword(s):  
Optimal Control ◽  
Cost Function ◽  
Linear Systems ◽  
Chaotic Systems ◽  
Control Method ◽  
Piecewise Linear ◽  
Robust Optimal Control ◽  
Piecewise Linear Systems ◽  
Optimal Control Method ◽  
Optimal Control Methods

This paper proposes the optimal control methods for a class of chaotic systems via state feedback. By converting the chaotic systems to the form of uncertain piecewise linear systems, we can obtain the optimal controller minimizing the upper bound on cost function by virtue of the robust optimal control method of piecewise linear systems, which is cast as an optimization problem under constraints of bilinear matrix inequalities (BMIs). In addition, the lower bound on cost function can be achieved by solving a semidefinite programming (SDP). Finally, numerical examples are given to illustrate the results.

Stochastic Optimal Control Methods for Investigating the Power of Morphological Computation

2013 ◽  
Vol 19 (1) ◽  
pp. 115-131 ◽  
Author(s):  
Elmar A. Rückert ◽  
Gerhard Neumann
Keyword(s):  
Optimal Control ◽  
Control Method ◽  
Control Methods ◽  
Controlled System ◽  
Control Laws ◽  
Optimal Control Method ◽  
Optimal Controllers ◽  
Morphological Computation ◽  
Link Model ◽  
Optimal Control Methods

One key idea behind morphological computation is that many difficulties of a control problem can be absorbed by the morphology of a robot. The performance of the controlled system naturally depends on the control architecture and on the morphology of the robot. Because of this strong coupling, most of the impressive applications in morphological computation typically apply minimalistic control architectures. Ideally, adapting the morphology of the plant and optimizing the control law interact so that finally, optimal physical properties of the system and optimal control laws emerge. As a first step toward this vision, we apply optimal control methods for investigating the power of morphological computation. We use a probabilistic optimal control method to acquire control laws, given the current morphology. We show that by changing the morphology of our robot, control problems can be simplified, resulting in optimal controllers with reduced complexity and higher performance. This concept is evaluated on a compliant four-link model of a humanoid robot, which has to keep balance in the presence of external pushes.

An Optimal Control Approach to the Path Tracking Problem for an Automobile

1986 ◽  
Vol 10 (4) ◽  
pp. 233-241 ◽  
Author(s):  
H. Hatwal ◽  
E.C. Mikulcik
Keyword(s):  
Optimal Control ◽  
Control Method ◽  
Lane Change ◽  
Control Methods ◽  
Tracking Problem ◽  
Control Approach ◽  
Optimal Control Method ◽  
Linear And Nonlinear ◽  
Optimal Control Methods ◽  
Optimal Control Approach

The problem of determining the steering required for an automobile to follow a specified path is investigated using exact inverse and optimal control methods applied to simple linear and nonlinear vehicle models in lane-change maneuvers. The optimal control method results in solutions which provide for reasonably close tracking of the specified trajectories, but with reduced steering activity in comparison with what would be required for exact tracking, and also with less severe excursions in the slip angles, yaw rates and lateral accelerations. The results appear to be qualitatively consistent with the manner in which a real driver would react and the method could find applicability as a tool to compare the handling properties of different vehicles in a manner analogous to that of test drivers following a specified test course.

The human-like trajectory planning for autonomous vehicles based on optimal control in a test track environment

2021 ◽  
pp. 095440702110090
Author(s):  
Xing Xu ◽  
Minglei Li ◽  
Feng Wang ◽  
Ju Xie ◽  
Xiaohan Wu ◽  
...  
Keyword(s):  
Optimal Control ◽  
Autonomous Vehicles ◽  
Optimal Trajectory ◽  
Control Method ◽  
Autonomous Vehicle ◽  
Trajectory Data ◽  
Test Track ◽  
Optimal Control Method ◽  
The Future ◽  
On The Road

A human-like trajectory could give a safe and comfortable feeling for the occupants in an autonomous vehicle especially in corners. The research of this paper focuses on planning a human-like trajectory along a section road on a test track using optimal control method that could reflect natural driving behaviour considering the sense of natural and comfortable for the passengers, which could improve the acceptability of driverless vehicles in the future. A mass point vehicle dynamic model is modelled in the curvilinear coordinate system, then an optimal trajectory is generated by using an optimal control method. The optimal control problem is formulated and then solved by using the Matlab tool GPOPS-II. Trials are carried out on a test track, and the tested data are collected and processed, then the trajectory data in different corners are obtained. Different TLCs calculations are derived and applied to different track sections. After that, the human driver’s trajectories and the optimal line are compared to see the correlation using TLC methods. The results show that the optimal trajectory shows a similar trend with human’s trajectories to some extent when driving through a corner although it is not so perfectly aligned with the tested trajectories, which could conform with people’s driving intuition and improve the occupants’ comfort when driving in a corner. This could improve the acceptability of AVs in the automotive market in the future. The driver tends to move to the outside of the lane gradually after passing the apex when driving in corners on the road with hard-lines on both sides.

scholarly journals Reconfiguring Smart Structures Using Approximate Heteroclinic Connections in a Spring-Mass Model

2015 ◽  
Author(s):  
Jiaying Zhang ◽  
Colin R. McInnes
Keyword(s):  
Optimal Control ◽  
Control Method ◽  
Smart Structures ◽  
Polynomial Method ◽  
Reference Trajectory ◽  
Sensors And Actuators ◽  
Embedded Computing ◽  
Equal Energy ◽  
Optimal Control Method ◽  
Heteroclinic Connection

Several new methods are proposed to reconfigure smart structures with embedded computing, sensors and actuators. These methods are based on heteroclinic connections between equal-energy unstable equilibria in an idealised spring-mass smart structure model. Transitions between equal-energy unstable (but actively controlled) equilibria are considered since in an ideal model zero net energy input is required, compared to transitions between stable equilibria across a potential barrier. Dynamical system theory is used firstly to identify sets of equal-energy unstable configurations in the model, and then to connect them through heteroclinic connection in the phase space numerically. However, it is difficult to obtain such heteroclinic connections numerically in complex dynamical systems, so an optimal control method is investigated to seek transitions between unstable equilibria, which approximate the ideal heteroclinic connection. The optimal control method is verified to be effective through comparison with the results of the exact heteroclinic connection. In addition, we explore the use of polynomials of varying order to approximate the heteroclinic connection, and then develop an inverse method to control the dynamics of the system to track the polynomial reference trajectory. It is found that high order polynomials can provide a good approximation to true heteroclinic connections and provide an efficient means of generating such trajectories. The polynomial method is envisaged as being computationally efficient to form the basis for real-time reconfiguration of real, complex smart structures with embedded computing, sensors and actuators.

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