scholarly journals Feedback Control for a Diffusive and Delayed Brusselator Model: Semi-Analytical Solutions

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 725
Author(s):  
Hassan Yahya Alfifi
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Chemical Species ◽  
Periodic Oscillation ◽  
Delayed Feedback Control ◽  
Hopf Bifurcation Point ◽  
Analytical Technique ◽  
Bifurcation Points ◽  
Diffusion Parameters ◽  
The Stability

This paper describes the stability and Hopf bifurcation analysis of the Brusselator system with delayed feedback control in the single domain of a reaction–diffusion cell. The Galerkin analytical technique is used to present a system equation composed of ordinary differential equations. The condition able to determine the Hopf bifurcation point is found. Full maps of the Hopf bifurcation regions for the interacting chemical species are shown and discussed, indicating that the time delay, feedback control, and diffusion parameters can play a significant and important role in the stability dynamics of the two concentration reactants in the system. As a result, these parameters can be changed to destabilize the model. The results show that the Hopf bifurcation points for chemical control increase as the feedback parameters increase, whereas the Hopf bifurcation points decrease when the diffusion parameters increase. Bifurcation diagrams with examples of periodic oscillation and phase-plane maps are provided to confirm all the outcomes calculated in the model. The benefits and accuracy of this work show that there is excellent agreement between the analytical results and numerical simulation scheme for all the figures and examples that are illustrated.

Time-Delayed Sampled-Data Feedback Control of Differential Systems Undergoing Hopf Bifurcation

2021 ◽  
Vol 31 (01) ◽  
pp. 2150004
Author(s):  
Huan Su ◽  
Jing Xu
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Blood Transfusion ◽  
Sampling Period ◽  
Delayed Feedback Control ◽  
Control Technique ◽  
Blood Collection ◽  
Sampled Data ◽  
Differential Systems ◽  
Data Feedback

In this paper, time-delayed sampled-data feedback control technique is used to asymptotically stabilize a class of unstable delayed differential systems. Through the analysis for the distribution change of eigenvalues, an effective interval of the control parameter is obtained for a given sampling period. Here an indirect strategy is taken. Specifically, the system of continuous-time delayed feedback control is studied first by Hopf bifurcation theory. And then, the result and implicit function theorem are used to analyze the system of time-delayed sampled-data feedback control with a sufficiently small sampling period. Considering the practical criterion for the size of sampling period, the upper bound of sampling period is estimated. Finally, an application example, an unstable Mackey–Glass model, is asymptotically stabilized by introducing a blood transfusion item with time-delayed sampled-data feedback control. The blood transfusion speed and blood collection test period are derived from the main results. Some simulations and comparisons show the correctness and advantages of the main theoretical results.

An Efficient Approach for Stability Analysis and Parameter Tuning in Delayed Feedback Control of a Flying Robot Carrying a Suspended Load

2019 ◽  
Vol 141 (8) ◽  
Author(s):  
Wei Dong ◽  
Ye Ding ◽  
Luo Yang ◽  
Xinjun Sheng ◽  
Xiangyang Zhu
Keyword(s):  
Time Delay ◽  
Feedback Control ◽  
Spectral Radius ◽  
Stability Region ◽  
Parameter Tuning ◽  
Suspended Load ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
The Stability ◽  
Flying Robot

This paper presents an accurate and computationally efficient time-domain design method for the stability region determination and optimal parameter tuning of delayed feedback control of a flying robot carrying a suspended load. This work first utilizes a first-order time-delay (FOTD) equation to describe the performance of the flying robot, and the suspended load is treated as a flying pendulum. Thereafter, a typical delayed feedback controller is implemented, and the state-space equation of the whole system is derived and described as a periodic time-delay system. On this basis, the differential quadrature method is adopted to estimate the time-derivative of the state vector at concerned sampling grid point. In such a case, the transition matrix between adjacent time-delay duration can be obtained explicitly. The stability region of the feedback system is thereby within the unit circle of spectral radius of this transition matrix, and the minimum spectral radius within the unit circle guarantees fast tracking error decay. The proposed approach is also further illustrated to be able to be applied to some more sophisticated delayed feedback system, such as the input shaping with feedback control. To enhance the efficiency and robustness of parameter optimization, the derivatives of the spectral radius regarding the parameters are also presented explicitly. Finally, extensive numeric simulations and experiments are conducted to verify the effectiveness of the proposed method, and the results show that the proposed strategy efficiently estimates the optimal control parameters as well as the stability region. On this basis, the suspended load can effectively track the pre-assigned trajectory without large oscillations.

scholarly journals Adaptive Control of Electromagnetic Suspension System by HOPF Bifurcation

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Aming Hao ◽  
Xiaolong Li ◽  
Longhua She
Keyword(s):  
Adaptive Control ◽  
Hopf Bifurcation ◽  
Large Scale ◽  
Stability Range ◽  
Hopf Bifurcation Point ◽  
Maglev System ◽  
Large Scale Disturbance ◽  
Excited Vibration ◽  
Parameter Variance ◽  
The Stability

EMS-type maglev system is essentially nonlinear and unstable. It is complicated to design a stable controller for maglev system which is under large-scale disturbance and parameter variance. Theory analysis expresses that this phenomenon corresponds to a HOPF bifurcation in mathematical model. An adaptive control law which adjusts the PID control parameters is given in this paper according to HOPF bifurcation theory. Through identification of the levitated mass, the controller adjusts the feedback coefficient to make the system far from the HOPF bifurcation point and maintain the stability of the maglev system. Simulation result indicates that adjusting proportion gain parameter using this method can extend the state stability range of maglev system and avoid the self-excited vibration efficiently.

Bifurcation Analysis of a Railway Wheelset With Nonlinear Wheel-rail Contact

2021 ◽  
Author(s):  
Jinying Guo ◽  
Huailong Shi ◽  
Ren Luo ◽  
Jing Zeng
Keyword(s):  
Contact Angle ◽  
Hopf Bifurcation ◽  
Bifurcation Point ◽  
Critical Speed ◽  
Nonlinear Term ◽  
Linear Term ◽  
Railway Vehicle ◽  
Hopf Bifurcation Point ◽  
Quintic Polynomial ◽  
The Stability

Abstract Stability is a key factor for the operation safety of railway vehicles, while current work employs linearized and simplified wheel/rail contact to study the bifurcation mechanism and assess the stability. To study the stability and bifurcation characters under real nonlinear wheel/rail contact, a fully parameterized nonlinear railway vehicle wheelset model is built. In modeling, the geometry nonlinearities of wheel and rail profiles come from field measurements, including the rolling radius, contact angle, and curvatures, etc. Firstly, four flange force models and their effects on the stability bifurcations are compared. It shows that an exponent fitting is more proper than a quintic polynomial one to simulate the flange, and works well without changing the Hopf bifurcation type. Then the effects of each term of the nonlinear geometry of wheel/rail contact on the Hopf bifurcation and Limit Circle bifurcation are discussed. Both the linear term and nonlinear term of rolling radius have a significant influence on Hopf bifurcation and Limit Point of Circle (LPC) bifurcation. The linear critical speed (Hopf bifurcation point) and the nonlinear critical speed (LPC bifurcation point) changes times while within the calculated range of the linear term of the rolling radius. Its nonlinear term changes the bifurcation type and the nonlinear critical speed almost by half. The linear term of contact angle, the radius of curvature of wheel, and rail profile should be taken into consideration since they can change both the bifurcation point and type, while the cubic term can be ignored. Furtherly, the field measured wheel profiles for several running mileages are employed to examine the real geometry nonlinearities and the according Hopf bifurcation behavior. The result shows that a larger suspension stiffness would increase the running stability under wheel wear.

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