Model uncertainty and control consequences: a paper machine study

Figures & data

Figure 1. Functional sketch of PM6, with manipulated inputs u, disturbances d, and controlled outputs y. After [7].

Figure 2. Experimental input (left column) and output (right column) data.

Table 1. Cases of parameter estimation, with parameter set to be estimated, the true parameter values θ, the initial parameter guesses θ⁰, and lower θ _l and upper θ _u bounds for parameters in the estimation procedure

	Parameters	θ	θ^0	θ l	θ u
Case 1	(a11,a21)	(0.9702,0.0018)	(0.9,0.01)	(0.5,0)	(1,0.5)
Case 2	(a11,a12)	(0.9702,0.3283)	(0.9,0.5)	(0.5,0)	(1,1)
Case 3	(a11,a22)	(0.9702,0.9596)	(0.9,0.9)(0.9,0.9)	(0.5,0.5)(0.5,0.5)	(1,1)
Case 4	(a11,a23)	(0.9702,–0.0197)	(0.9,0)(0.9,0)	(0.9,–0.2)	(1,0.2)
Case 5	(a11,a33)	(0.9702,0.8661)	(0.9,0.9)	(0.9,0.8)	(1,1)

Figure 3. Scaled sensivities s_{i ,1} and s_{i ,2} for cases 1–5. The sensitivities are computed using Matlab's lsqnonlin algorithm, see section 3.5.

Figure 4. (Available in colour online). Comparison of real outputs y_t (×, red), initial model outputs (dashed, green), final model outputs (solid, black), as well as scaled model errors (errors for outputs 1 and 2 have been divided by 100).

Table 2. True parameter values θ, estimated parameter values , and boxed confidence regions

	Parameters	θ
Case 1	(a11,a21)	(0.9702,0.0018)	(0.9650,0.0023)	[−4.07,6.00] × [−0.457,0.462]
Case 2	(a11,a12)	(0.9702,0.3283)	(0.9706,0.3239)	[0.881,1.06] × [−1.05,1.70]
Case 3	(a11,a22)	(0.9702,0.9596)	(0.9706,0.9592)	[0.557,1.38] × [0.368,1.55]
Case 4	(a11,a23)	(0.9702,–0.0197)	(0.9703,–0.0198)	[0.955,0.986] × [−0.103,0.0629]
Case 5	(a11,a33)	(0.9702,0.8661)	(0.9703,0.8664)	[0.966,0.975] × [0.811,0.921]

Figure 5. (Available in colour online). Parameter estimates for Case 2. Bootstrap estimates (circles), estimate (pentagram, red), true parameter (square, yellow), Mahalanobis region (solid ellipsoid, green), and contours F_T (θ) = (1 − γ)/2 (dotted, blue) and F_T (θ) = 1 − (1 − γ)/2 (dotted, red).

Figure 6. (Available in colour online). Parameter estimates for Case 5. Bootstrap estimates (circles), estimate (pentagram, red), true parameter (square, yellow), Mahalanobis region (solid ellipsoide, green), and contours F_T (θ) = (1 − γ)/2 (dotted, blue) and F_T (θ) = 1 − (1 − γ)/2 (dotted, red).

Figure 7. Case 2: Open loop operation with bootstrap uncertainty description (left) versus boxed Mahalanobis region description (right). Model outputs y_t (solid) and references r_t (×) are displayed.

Figure 8. Case 5: Open loop operation with bootstrap uncertainty description (left) versus boxed Mahalanobis region description (right). Model outputs y_t (solid) and references r_t (×) are displayed.

Figure 9. Case 5: Closed loop operation with bootstrap uncertainty description (left) versus boxed Mahalanobis region description (right). Model outputs y_t (solid) and references r_t (×) are displayed.

Hauge , T. A. 2003 . Roll-out of Model Based Control with Application to Paper Machines , Norway : Ph.D. thesis, Norwegian University of Science and Technology, and Telemark University College .

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