2. دانشگاه صنعتي اميركبير
دانشكده مهندسي پزشكي
استاد درس
دكتر فرزاد توحيدخواه
بهمن 1389
MPC Stability-3
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3. Exponentially Weighted Constrained Control
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4. Recall
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5. weighted variables in the solution of the optimization problem.
As the constraints are specified in terms of the original physical variables, the constraints need to be transformed to correspond with the exponentially
weighted variables in the solution of the optimization problem.
If the constraints are only imposed at the initial sample of the variables within the window, then the constraints
will be used unchanged.
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6. Transformed constraints for future Δu:
Because the constraints on the state variables are imposed one step ahead, the exponential factor needs to be included
Transformed constraints for future Δu:
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7. The constraints on the state variable are:
φ(j) is the data vector used in the design of predictive control.
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8. Constraints on the control variable,
The originally uniformly imposed constant constraints are all transformed into constraints that are functions of the exponential weight factor α−j . The bounds converge to some constants as the future sample index increases
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9. Example 4.9. Suppose that a second-order system with time delay is described by
Design and simulate a predictive control system with unit step input and zero initial conditions.
illustrate the closed-loop performance and numerical condition of the Hessian matrix for α = 1, 1.1, 1.2
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10. We impose the constraints for the first 10 samples.
Solution
We impose the constraints for the first 10 samples.
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11. Predictive control with constraints
Predictive control with constraints. Key: line (1) without constraints; line
(2) constraints on the first sample (α = 1.2); line (3) constraints on the first 10 samples (α = 1.2); line (4) constraints on the first 10 samples (α = 1.1). The plots for case 2, 3, 4 are identical
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12. When transforming the constraints to the forms that embed the exponential weight α, the constrained control system is consistent with respect to the weight factor α.
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13. Additional Benefit
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14. Example 4.10. A mechanical system that is highly
oscillatory and non-minimum-phase,
Poles:
Design and simulate predictive control systems with and without exponential data weighting (α = 1.2 and α = 1); and compare the results.
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15. Open-loop response of mechanical system
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16. Initial state variable condition
Solution.
The LQR control system
Initial state variable condition
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17. LQR control of mechanical system
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18. Because the system is highly oscillatory, it takes about 460 samples for the LQR control trajectory to decay to zero it takes about 20 samples for the system output to decay to zero.
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19. Predictive control without exponential weighting (α = 1)
In order for the MPC to match the LQR system, the prediction horizon needs to be selected as 460 or above.
As for the parameters in the Laguerre function, if a = 0, then N needs to be 400 or above to achieve the LQR desired results.
If a larger a is used, then the parameter N is smaller. Indeed, with the selection of a = 0.9 and N = 60, and predictive control system matches the results obtained from LQR within one optimization window.
In contrast, if a = 0 and N = 199 are selected, then the predictive control signals will match the LQR control up to the sampling instant of 199.
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20. Comparison study: LQR vs predictive control
Comparison study: LQR vs predictive control. Key: line (1) LQR; line (2) predictive control a = 0.9 and N = 60; line (3) predictive control a = 0 and N = 199.
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21. Control increment u continues to oscillate about 500 samples, not shown in Figure. The Hessian matrix is ill-conditioned with either a = 0.9 or a = 0. When a = 0.9, the condition number is × 105, while the condition number is × 104 for the case a = 0.
Why the MPC with Laguerre functions (a = 0.9) has the larger condition number?
This is because the Laguerre functions with a = 0.9 did not decay to zero after 199 samples and the convolution sum φ(m) continued until m reached Np = 480. In comparison, when a = 0, the incremental control is zero after 199 samples.
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22. Predictive control with exponential weighting (α = 1.2)
are transformed into:
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23. Comparison results: LQR vs predictive control
Comparison results: LQR vs predictive control. Key: line (1) LQR; line (2) predictive control a = 0.4 and N = 10 (α = 1.2).
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24. It is seen that the slow oscillation from the original Δu trajectory has disappeared. In addition, the prediction
horizon is much less than 480, instead, 46 is selected. Thus, with Np = 46, N = 10 and a = 0.4, the predictive control system matches the LQR system. The condition number of the Hessian is , which is contrasted with the condition number × 105 when α = 1 (no exponential weighting) and a = 0.9. Also, the feedback gain matrix is:
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25. Constrained control (α = 1.2)
It is seen that the constraint on 0.1 ≤ u was not met at the same time when the constraint on Δu(k) became activated. When both of them are activated, they become conflict constraints. Nevertheless, Hildreth’s quadratic programming method found a compromise solution where the constraint on u was relaxed.
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26. Predictive control with constraints
Predictive control with constraints. Key: line (1) without constraints; line (2) constraints on the first sample (α = 1.2)
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