دانشگاه صنعتي اميركبير دانشكده مهندسي پزشكي استاد درس دكتر فرزاد توحيدخواه بهمن 1389 MPC Stability-3 کنترل پيش بين-دکتر توحيدخواه 2
Exponentially Weighted Constrained Control کنترل پيش بين-دکتر توحيدخواه
Recall کنترل پيش بين-دکتر توحيدخواه
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. کنترل پيش بين-دکتر توحيدخواه
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: کنترل پيش بين-دکتر توحيدخواه
The constraints on the state variable are: φ(j) is the data vector used in the design of predictive control. کنترل پيش بين-دکتر توحيدخواه
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 کنترل پيش بين-دکتر توحيدخواه
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 کنترل پيش بين-دکتر توحيدخواه
We impose the constraints for the first 10 samples. Solution We impose the constraints for the first 10 samples. کنترل پيش بين-دکتر توحيدخواه
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 کنترل پيش بين-دکتر توحيدخواه
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. کنترل پيش بين-دکتر توحيدخواه
Additional Benefit کنترل پيش بين-دکتر توحيدخواه
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. کنترل پيش بين-دکتر توحيدخواه
Open-loop response of mechanical system کنترل پيش بين-دکتر توحيدخواه
Initial state variable condition Solution. The LQR control system Initial state variable condition کنترل پيش بين-دکتر توحيدخواه
LQR control of mechanical system کنترل پيش بين-دکتر توحيدخواه
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. کنترل پيش بين-دکتر توحيدخواه
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. کنترل پيش بين-دکتر توحيدخواه
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. کنترل پيش بين-دکتر توحيدخواه
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 4.6595 × 105, while the condition number is 1.7418 × 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. کنترل پيش بين-دکتر توحيدخواه
Predictive control with exponential weighting (α = 1.2) are transformed into: کنترل پيش بين-دکتر توحيدخواه
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). کنترل پيش بين-دکتر توحيدخواه
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 59.2435, which is contrasted with the condition number 4.6595× 105 when α = 1 (no exponential weighting) and a = 0.9. Also, the feedback gain matrix is: کنترل پيش بين-دکتر توحيدخواه
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. کنترل پيش بين-دکتر توحيدخواه
Predictive control with constraints Predictive control with constraints. Key: line (1) without constraints; line (2) constraints on the first sample (α = 1.2) کنترل پيش بين-دکتر توحيدخواه
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