Preliminary Design of a Receding Horizon Controller Supported by Adaptive Feedback
Abstract
:1. Introduction
2. Formulation of the Optimization Problem without Using Constraint Terms
- It is given the horizon for design in the discrete time steps containing the state variables as , and , the exerted control forces , and the second coordinate time derivatives as ;
- in the discrete time representation the above data contain certain redundancy because the relationship must be valid for , and for ;
- the dynamic model of the system determines ; and
- the initial conditions are determined by redundantly with .
3. Solution for the Minimization
- The minimization was commenced from zero force components, and started with a big step using the Newton–Raphson algorithm. A parameter was introduced in the first step.
- Until the Newton–Raphson algorithm with the given step-length yielded a smaller cost value than the previously visited point, it proceeded forward with this big step length;
- When in the suggested next point the function value found was bigger than that in the starting point, has been halved, and times the step-length of the Newton–Raphson algorithm was applied. If it yielded a better point, the calculated step-length was used; otherwise was kept halved until it either provided a better next point, or it achieved a preset minimum value at which point the algorithm stopped.
4. Utilization of the Result of Cost Minimization
5. Simulation Results
5.1. Simulations without Force Limitation
5.1.1. Investigations for Different Time Resolutions
5.1.2. Computations for Less Precise Minimum Seeking
5.2. With Force Limitation
6. Discussion
- In general it seems to be an interesting research area to consider the problem of adaptive optimal control design for a wider set that is exempt of the limitations of linear time-invariant dynamic system models and quadratic cost function contributions.
- The prevailing general approach in this field is linear programming that tackles the problem in discrete time grid approach and the use of Lagrange’s reduced gradient algorithm that is professionally implemented e.g., in the EXCEL’s Solver package.
- It is evident that the direct problem formulation applied for evading the use of the constraint-based formalism leads to considerable reduction in complexity and computational needs. The number of the independent variables of the original approach is considerably decreased. Instead of the variables , , only the variables remain the independent ones, and there is no need to calculate the Lagrange multipliers .
- The problem of small steps in the case of the suggested solution can be tackled by a modification of the Newton–Raphson algorithm (generally it is an issue in using the reduced gradient method, too).
- It was found that for keeping the computational time low, it is expedient to use not very precise minimum seeking.
- The scattering of the force values and the related effects rather can be tackled by a simple noise-filtering approach applied for the optimized trajectory to be adaptively tracked.
- The suggested adaptive controller can well track the smoothed signal.
- The mathematical frameworks of optimization and adaptive tracking can be separated from each other in a simple manner.
- The basic concept was that the force-limited optimization can be executed by the use of a heavy dynamic model. Therefore, the force limitation issues can be tackled in the optimization phase. It can be expected that the trajectory that was optimized for the heavy model can be tracked by an easier mechanical construction for the acceleration of the components of which smaller force or torque components can be expected. For tracking the optimized trajectory, a simple CTC type or an improved adaptive CTC type control strategy can be used that is free of the burden of the force-limitation issues.
- The necessary time grid resolution depends on various factors as the dynamics of the nominal trajectory to be tracked, the structure of the cost function applied, the parameters used in smoothing the optimized trajectory, and that of the adaptive controller that tracks the smoothed trajectory.
- In general all the above factors can be clarified via making numerical simulations for a given problem or problem class.
- In the given investigations, the execution time was measured by the use of the given hardware that was a laptop with a single CPU and a multitasking operating system. Consequently, the measured data also contain the time sections during which the actual task was interrupted and the processor was working on other tasks. However, because during the calculations no heavy software application was running, these data provide approximate and reliable information. (Parallel or simultaneous’use for instance of a video player drastically modifies these observed data.)
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Parameter | Exact Model for Simulation | Heavy Model for Optimization | Approximate Model for Adaptive Control |
---|---|---|---|
component’s mass | 10.0 | 15.0 | 12.0 |
component’s mass | 20.0 | 25.0 | 21.0 |
component’s mass | 10.0 | 13.0 | 12.0 |
load’s mass | 50.0 | 55.0 | 52.0 |
grav. accel. | 9.81 | 10.0 | 9.0 |
load’s inertia moment | 45.0 | 50.0 | 42.0 |
arm length | 2.0 | 2.0 | 2.0 |
arm length | 1.0 | 1.0 | 1.0 |
Parameter | Meaning | Value |
---|---|---|
Discrete time resolution | ||
Trajectory tracking exponential coeff. | ||
Trajectory smoothing exponential coeff. | ||
Cost contribution coeffs. | ||
Cost contribution coeff. | ||
Cost parameter 1 | or | |
Cost parameter 2 | ||
Force cost parameter 1 | ||
p | Cost parameter 3 | |
Force cost parameter 2 | varying or | |
Force cost parameter 3 | ||
Augmented arrays’ Frobenius norm | or | |
H | Discrete horizon length | 12 |
Moderating factor in adaptive control | or | |
Moderating factor in kinematic block | or | |
Stopping limit in minimum seeking | ||
Adaptive interpolation factor |
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Issa, H.; Tar, J.K. Preliminary Design of a Receding Horizon Controller Supported by Adaptive Feedback. Electronics 2022, 11, 1243. https://doi.org/10.3390/electronics11081243
Issa H, Tar JK. Preliminary Design of a Receding Horizon Controller Supported by Adaptive Feedback. Electronics. 2022; 11(8):1243. https://doi.org/10.3390/electronics11081243
Chicago/Turabian StyleIssa, Hazem, and József K. Tar. 2022. "Preliminary Design of a Receding Horizon Controller Supported by Adaptive Feedback" Electronics 11, no. 8: 1243. https://doi.org/10.3390/electronics11081243
APA StyleIssa, H., & Tar, J. K. (2022). Preliminary Design of a Receding Horizon Controller Supported by Adaptive Feedback. Electronics, 11(8), 1243. https://doi.org/10.3390/electronics11081243