Singular value analysis and scaling : applications for multivariable process design and control

Author

Jawaharlal Prasad

Date of Award

12-1984

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Chemical Engineering

Major Professor

Duane D. Bruns

Committee Members

Robert C. Wahler, James J. Downs, S. L., Belle R. Upadhyaya, Charles F. M.

Abstract

Singular Value Analysis (SVA), an analysis structure formalized on Singular Value Decomposition (SVD), has become an important tool in the areas of process design and control. Application has been mainly to the process steady state gain matrix. It decomposes a matrix into three matrices—thus it is the structure of the SVD matrices which provide process analysis and design information.

SVA literature notes that the properties of the SVD matrices depend on the scale or units used for the gains in the process gain matrix. Thus a shortcoming of SVA, its scale dependence, is stated or alluded to by the literature. However, no research has been done on the scaling problem as it relates to SVA for process design and control.

The present research aims to capitalize on the "problem of scaling" and use it as a source of information. The mathematical definition of scaling is used to state the problem and scaling literature is reviewed to initiate the current study. Five two-input two-output models from the multivariable control literature were taken as examples.

SVA is based on the 1₂-norm for the process gain matrix, decomposition matrices, input vectors and output vectors. In terms of SVA, this is represented by the condition number with good conditioning shown by a small condition number. Mathematical literature states that analytically optimal scaling for the 1₂-norm condition number cannot be obtained. Tomlin has introduced the concept of "well-conditioned" matrices. Utilization of Tomlin's scaling procedures suggest that Geometric Mean scaling leads to the best condition number. A numerical search routine, PATERN, has also been employed successfully to search for the optimum condition number. Interaction is another major concern in multivariable process control.

Interaction and process conditioning are two distinct problems. However, it is shown that scaling can alleviate a conditioning problem if the interaction is low and that numerical analysis may be needed to tell the extent to which scaling affects the condition number if interaction is high.

Scaling can be related to a process via sensitivity of transmitters and the range of final control elements. The gain matrix units can be distributed logically on the three SVD matrices U, Σ and V^T. The singular values are dimensionless. U which denotes the out put space is assigned the output units while V which denotes the input space is assigned the input units.

Other areas addressed include structure of U and V, uniqueness of SVD, based on Givens reflections versus rotations, and rules to match the results of analytical SVD and the QR algorithm. The concept of "controller conditioning" versus "process conditioning" has been introduced.

New definitions have been introduced and several theorems were developed to provide research direction and help explain the numerical results. Largely on the basis of these new theorems, additional avenues for SVA, as a formal interaction measure, are suggested.

Recommended Citation

Prasad, Jawaharlal, "Singular value analysis and scaling : applications for multivariable process design and control. " PhD diss., University of Tennessee, 1984.
https://trace.tennessee.edu/utk_graddiss/12944