New regularization embedded nonlinear control designs are proposed for the temperature control of an input-constrained and ill-conditioned thermal process. A classic nonlinear controller applied to such a process is shown to provide good temperature tracking but generates physically unreasonable actuator solutions, i.e. input-constraint-violation. The reason of input-constraint-violating control solutions—ill-conditionedness—is shown by applying singular value decomposition (SVD) on the linear algebraic equivalence of the nonlinear controllers (LAENC). Based on the analogy of LAENC and regularization method for the linear algebraic equations, Tikhonov, truncated singular value decomposition (TSVD) and modified TSVD (MTSVD) methods are embedded in the design of feedback linearizing controllers (FBL) and sliding mode controllers (SMC). These regularization embedded nonlinear controllers (RENLC) provide good temperature tracking and generate physically reasonable and actuator-constraint-satisfying solutions for the ill-conditioned system, in spite of the modeling errors inherent in applying regularization. The optimal Tikhonov parameter is found using an L-curve. Quantitative comparisons of the residuals and standard deviations of the control inputs are used as criteria to select the optimal truncated singular value decomposition (TSVD) parameter.

Issue Section:
Technical Papers
Keywords:
nonlinear control systems, control system synthesis, temperature control, singular value decomposition, linear algebra, feedback, variable structure systems, actuators, optimal control, heat systems
Topics:
Control equipment, Design, Errors, Sliding mode control, Temperature, Modeling, Feedback, Thermal systems, Actuators, Temperature control, Heat
1.
Howell, J. R., 2001, Class Notes for Inverse Design of Thermal Systems with Dominant Radiative Transfer, The University of Texas at Austin.
2.
Erturk, H., Ezekoye, O. A., and Howell, J. R., 2000, “Inverse Solution of Radiative Heat Transfer in Two-Dimensional Irregularly Shaped Enclosures,” Proc. 2000 IMECE, ASME HTD, Vol. 366-1, pp. 109–116.
3.
Erturk, H., Ezekoye, O. A., and Howell, J. R., 2001, “Inverse Transient Boundary Condition Estimation Problem in a Radiating Enclosure,” Proc. 2001 ASME National Heat Transfer Congress.
4.
Erturk, H., Ezekoye, O. A., and Howell, J. R., 2001, “Inverse Design of a Three Dimensional Furnace With Moving Design Environment,” Proc. 2001 IMECE, ASME HTD.
5.
Morales, J. C., 1998, “Radiation Exchange Within Enclosure of Diffuse Gray Surfaces: The Inverse Problem,” Ph.D. dissertation, Department of Mechanical Engineering, The University of Texas at Austin.
6.
Hansen, P. C., 1998, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, Soc. Industrial and Applied Mathematics, Philadelphia.
7.
Hansen
,
P. C.
,
1992
, “
Numerical Tools for Analysis and Solution of Fredholm Integral Equations of the First Kind
,”
Inverse Probl.
,
8
, pp.
849
–
872
.
8.
Breedijk, T., Edgar, T. F., and Trachtenberg, I., 1994, “Model-Based Control of Rapid Thermal Processes,” Proc. of American Control Conference, pp. 887–891.
9.
Norman, S. A., and Boyd, S. P., 1992, “Multivariable Feedback Control of Semiconductor Wafer Temperature,” Proc. of American Control Conference, pp. 811–816.
10.
Huang
,
C. J.
,
Yu
,
C. C.
, and
Shen
,
S. H.
,
2000
, “
Selection of Measurement Locations for the Control of Rapid Thermal Processor
,”
Automatica
,
36
, pp.
705
–
715
.
11.
Cho
,
Y. M.
, and
Gyugyı´
,
P.
,
1997
, “
Control of Rapid Thermal Processing: A System Theoretic Approach
,”
IEEE Trans. Control Syst. Technol.
,
5
, pp.
644
–
653
.
12.
Balakrishinan
,
K. S.
, and
Edgar
,
T. F.
,
2000
, “
Model-Based Control in Rapid Thermal Processing
,”
Thin Solid Films
,
365
, pp.
322
–
333
.
13.
Elia, C. F., 1994, “RTP Multivariable Temperature Controller Development,” Proc. of American Control Conference, pp. 907–912.
14.
Choi
,
J. Y.
, and
Do
,
H. Y.
,
2001
, “
A Learning Approach of Wafer Temperature Control in a Rapid Thermal Processing System
,”
IEEE Trans. Semicond. Manuf.
,
14
, pp.
1
–
10
.
15.
Baker
,
J.
, and
Christofides
,
P. D.
,
1999
, “
Output Feedback Control of Parabolic PDE System With Nonlinear Spatial Differential Operators
,”
Ind. Eng. Chem. Res.
,
38
, pp.
4372
–
4380
.
16.
Park
,
H. M.
,
Yoon
,
T. Y.
, and
Kim
,
O. Y.
,
1999
, “
Optimal Control of Rapid Thermal Processing Systems by Empirical Reduction of Modes
,”
Ind. Eng. Chem. Res.
,
38
, pp.
3964
–
3975
.
17.
Schaper
,
C. D.
,
Kailath
,
T.
, and
Lee
,
Y. J.
,
1999
, “
Decentralized Control of Wafer Temperature for Multizone Rapid Thermal Processing Systems
,”
IEEE Trans. Semicond. Manuf.
,
12
, pp.
193
–
199
.
18.
Aling, H. et al., 1995, “Application of Feedback Linearization to Models of Rapid Thermal Processing (RTP) Reactors,” 3rd Int’l Rapid Thermal Processing Conference-RTP’95, pp. 356–366.
19.
Balakrishnan
,
K. S.
,
Cho
,
W.
, and
Edgar
,
T. F.
,
1994
, “
Comparison of Controller Tuning Methods for Temperature Uniformity Control in a Rapid Thermal Processor
,”
Proc. SPIE
,
2336
, pp.
70
–
80
.
20.
Isidori, A., 1994, Nonlinear Control Systems, 3rd edition, Springer-Verlag.
21.
Slotine, J. J. E., and Li, W., 1991, Applied Nonlinear Control, Prentice–Hall.
22.
Utkin, V., Guldner, J., and Shi, J., 1999, Sliding Mode Control in Electromechanical Systems, Taylor & Francis.
23.
Utkin, V., 1992, Sliding Modes in Control Optimization, Springer-Verlag.
24.
Edwards, C., and Spurgeon, S. K., 1998, Sliding Mode Control: Theory and Applications, Taylor & Francis.
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