Nonlinear Control of Hydrostatic Thrust Bearing Using Multivariable Optimization
Abstract
:1. Introduction
- The proposed method utilizes an extended state observer to estimate the unknown terms in the state models of the system.
- The combined extended-state-observer-based backstepping controller is an adaptive type compared to the previously reported method of [23].
- The optimal parameters of the proposed control scheme are tuned using the multiobjective particle swarm optimization method, while in the previously reported literature, the criteria for parameters tuning is not given.
- As compared to previously reported methods, the proposed method utilizes fewer sensors to implement the control law.
2. Mathematical Models
2.1. Fluid Viscosity as Control Input
2.2. Active Supply Pressure as Control Input
2.3. Membrane-Controlled Thrust Bearing
3. LESO-Based Controller Design
3.1. Multiobjective Performance Criteria
3.2. Particle Swarm Optimization (PSO)
- . It gives the value of the tuning parameters. For the nth iteration, the ith particle is represented by . The upper and lower bound are defined for the optimizing parameters.
- Velocity : it represents the velocity of particle . For the nth iteration, the ith particle velocity is given by
- Individual best : the cost function value of the particle is compared with the best cost function value. The particle that has the best cost function value is known as the individual best. The individual best is obtained when the ith particle holds the following condition:
- Global best : It is the best outcome among all the individual best values. For the nth iteration, the global best is found when the condition of Equation (43) is satisfied
- Position and velocity are updated with respect to individual best and global best position. The ith particle’s position and velocity is given by
- Termination condition: Termination criteria are often two types. The first is the required number of iterations that have been achieved, while the second is the desired value of the objective function. The current research will try to achieve the “required number of iterations”.
- Step 1: Define a multiobjective function. The simulation is performed online, and variables such as error, settling time, overshoot, and rise time are updated to Equation (38) to find the multiobjective function. The total number iterations and parameters w, c1, c2 are defined.
- Step 2: Stop PSO if the required number of iterations have been reached, or else update the multiobjective function by simulating the model and updating the monitoring variables.
- Step 3: Update the position of individual best according to Equation (41).
- Step 4: Update the position of global best according to Equation (43).
- Step 5: Update position and velocity according to Equation (44). Apply lower and upper boundaries by using Equation (39) if the value goes beyond the desired interval.
- Step 6: go to Step 2.
4. Results
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
Unknown disturbance | |
Fluid film thickness | |
Initial fluid film thickness | |
Supply pressure | |
μ | Fluid viscosity |
Inner and outer radius for bearing | |
Radius for membrane restrictor | |
Recess pressure | |
Unmodelled dynamic | |
Mass of bearing |
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Rehman, W.U.; Khan, W.; Ullah, N.; Chowdhury, M.D.S.; Techato, K.; Haneef, M. Nonlinear Control of Hydrostatic Thrust Bearing Using Multivariable Optimization. Mathematics 2021, 9, 903. https://doi.org/10.3390/math9080903
Rehman WU, Khan W, Ullah N, Chowdhury MDS, Techato K, Haneef M. Nonlinear Control of Hydrostatic Thrust Bearing Using Multivariable Optimization. Mathematics. 2021; 9(8):903. https://doi.org/10.3390/math9080903
Chicago/Turabian StyleRehman, Waheed Ur, Wakeel Khan, Nasim Ullah, M. D. Shahariar Chowdhury, Kuaanan Techato, and Muhammad Haneef. 2021. "Nonlinear Control of Hydrostatic Thrust Bearing Using Multivariable Optimization" Mathematics 9, no. 8: 903. https://doi.org/10.3390/math9080903
APA StyleRehman, W. U., Khan, W., Ullah, N., Chowdhury, M. D. S., Techato, K., & Haneef, M. (2021). Nonlinear Control of Hydrostatic Thrust Bearing Using Multivariable Optimization. Mathematics, 9(8), 903. https://doi.org/10.3390/math9080903