Comparison and Interpretation Methods for Predictive Control of Mechanics
Abstract
:1. Introduction
A key contribution is the design of predictive controllers that are designed using optimization as the very first step, including a formulation of state feedback for robustness in the same optimization while a subsequent step converts the optimal solution from time-parameterization to state-parameterization allowing proportional-derivative gains to be expressed as exact functions of the optimal solution. Thus, feedback errors are de facto expressed exactly in terms of the solutions to the original optimization problem and errors are, thereby, optimally rejected. This notion permits the reader to use only this predictive, optimal feedback controller by itself and also together with the optimal feedforward. Lastly, comparing the optimal feedforward to the predictive optimal feedback control permits expression of a proposed controller called “2DOF” to imply the twice-invocation of the original optimization problem. This proposed 2DOF topology achieves near-machine precision target tracking errors while using near-minimal costs.
2. Materials and Methods
2.1. Open Loop Optimum Controller
- Write the control Hamiltonian.
- Implement the Hamiltonian Minimization Condition for the static problem of Equation (8).This is a constrained minimization problem, so use Equations (9) and (10) where is the Lagrange multiplier associated with the co-state.Confirm optimality by verifying the convexity condition in Equation (11).The result: once we find the co-state, we will have optimum control. Notice Karush-Kuhn-Tucker conditions often used with inequality constraints are not necessary here.
- Apply the Adjoint Equations per Equations (12) and (14), which result in Equations (13) and (15), respectively, and are plotted in Figure 1b.Integrating (14)
- Rewrite the Hamiltonian Minimization Condition in Equation (8) by substituting Equation (9).
- Substituting Equation (15) into Equation (16) produces Equation (17).
ASIDE: We’ll see in the next section how to implement the more general optimum control and solve for constants a and b as time, t progresses. That controller is referred to as the continuous predictive optimal closed-loop controller.
2.2. Continuous Predictive Closed Loop Optimum Controller
- Recall Equations (18)–(20) where yields:
- Set up a matrix equation in the form simulated in Figure 3.
2.3. Sampled-Data Predictive Optimum Controller
2.4. Proportional Plus Derivative (PD) Controller Derived Foremost from an Optimization Problem
Definingand, the optimum controlas a function of the time may be written as a function of states:where the K’s are feedback gains that are functions of the θ and ω error.
2.5. Feedforward/Feedback PD Controller
2.6. Two-DOF Controller: Optimal Control Augmented with Feedback Errors Calcuated with Optimal States
3. Results
3.1. Monte Carlo Analysis on a Deterministic Plant with Noise
3.2. Open Loop Optimal Controller
3.3. Continuous-Update Optimal Controller
3.4. Sample-Predictive Optimum Controller
3.5. PD Control Derived Foremost from an Optimization Problem
3.6. Feedforward/Feedback PD Controller
3.7. Two-DOF Controller: Optimal Control Augmented with Feedback Errors Calcuated with Optimal States
3.8. Monte Carlo Analysis on a Mismodeled Plant with Noise
4. Discussion
Proposed 2DOF control designed foremost as an optimization problem: The open-loop optimal control is used as a feedforward, while the optimal states derived from a time-parameterized optimal control are compared to the feedback signal to generate the error fed to the feedback controller whose gains are a reparameterization of the optimal solution.
5. Future Works
Funding
Acknowledgments
Conflicts of Interest
Appendix A
- Optimum Open Loop no noise (Figure A1)
- Optimum Open Loop with noise (Figure A2)
- PD Controller no noise (Figure A3)
- PD Controller with noise (Figure A4)
- Continuous Predictive Optimum no noise (Figure A5)
- Continuous Predictive Optimum with noise (Figure A6)
- Feedforward/Feedback PD no noise (Figure A7)
- Feedforward/Feedback PD with noise (Figure A8)
- 2DOF PD Controller no noise (Figure A9)
- 2DOF PD Controller with noise (Figure A10)
- Sampled Predictive Controller without Noise (Figure A11)
- Sampled Predictive Controller with Noise (Figure A12)
- Mis-modeled Plant (Figure A13)
- Error Norms (Figure A14)
Appendix A.1. Optimum Open Loop Controller-No Noise
Appendix A.2. Optimum Open Loop Controller-with Noise
Appendix A.3. PD Controller No Noise
Appendix A.4. PD Controller with Noise
Appendix A.5. Continuouse Predictive Optimum Controller-No Noise
Appendix A.6. Continuouse Predictive Optimum Controlle-with Noise
Appendix A.7. Feedforward/Feedback PD Controller-No Noise
Appendix A.8. Feedforward/Feedback PD Controlle-with Noise
Appendix A.9. 2DOF PD Controller-No Noise
Appendix A.10. 2DOF PD Controller-with Noise
Appendix A.11. Sampled Predictive Controller-without Noise
Appendix A.12. Sampled Predictive Controller-with Noise
Appendix A.13. Mismodeled Plant
Appendix A.14. Error Norms
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Controller | Deviation, | Mean Error, | Deviation, | Mean Error, | Mean Cost, J |
---|---|---|---|---|---|
Optimal open loop | 0.0328 | 0.0439 | 0.03 | 0.045 | 5.9787 |
Continuous predictive | 0.0015 | 0.002 | 0.0192 | 0.0251 | 6.0117 |
Sampled predictive | 0.0031 | 0.0074 | 0.0249 | 0.0321 | 6.1137 |
Proportional-derivative (PD) | 9.66 × 10−4 | 1.41 × 10−2 | 3.14 × 10−16 | 1.02 × 10−15 | 59.6885 |
Feedforward + feedback | 7.86 × 10−16 | 9.83 × 10−16 | 0.0058 | 0.0087 | 8.4507 |
2DOF | 7.07 × 10−16 | 1.02 × 10−15 | 1.89 × 10−15 | 2.51 × 10−15 | 8.619 |
Controller | Deviation, | Mean Error, | Deviation, | Mean Error, | Mean Cost, J |
---|---|---|---|---|---|
Optimal open loop | 0.034 | 0.044 | 0.0283 | 0.0484 | 6.004 |
Continuous predictive | 0.0015 | 0.0019 | 0.02 | 0.025 | 6.0137 |
Sampled predictive | 0.0029 | 0.007 | 0.0222 | 0.0307 | 6.0183 |
Proportional-derivative (PD) | 9.89 × 10−4 | 0.014 | 3.12 × 10−16 | 1.02 × 10−15 | 59.3244 |
Feedforward + feedback | 7.36 × 10−16 | 9.69 × 10−16 | 0.0058 | 0.0086 | 8.9637 |
2DOF | 6.42 × 10−16 | 9.02 × 10−16 | 1.90 × 10−15 | 2.56 × 10−15 | 7.8284 |
Controller | Benefits | Weakness |
---|---|---|
Optimal open loop | Establishes optimal case | Not realistically implementable |
Continuous predictive * | Good cost and position | High rate error and deviation |
Sampled predictive * | Good cost and position | High rate error and deviation |
Proportional-derivative (PD) * | Best rate control | Worst cost |
Feedforward + feedback | Best position control | Slight rate error and deviation |
2DOF * | All-around good | Slightly higher cost than optimal |
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Sands, T. Comparison and Interpretation Methods for Predictive Control of Mechanics. Algorithms 2019, 12, 232. https://doi.org/10.3390/a12110232
Sands T. Comparison and Interpretation Methods for Predictive Control of Mechanics. Algorithms. 2019; 12(11):232. https://doi.org/10.3390/a12110232
Chicago/Turabian StyleSands, Timothy. 2019. "Comparison and Interpretation Methods for Predictive Control of Mechanics" Algorithms 12, no. 11: 232. https://doi.org/10.3390/a12110232
APA StyleSands, T. (2019). Comparison and Interpretation Methods for Predictive Control of Mechanics. Algorithms, 12(11), 232. https://doi.org/10.3390/a12110232