How the RGA Came into Being and Where it Has Gone

In the early '60's, Mike Masarivic of Case Western Reserve published a book on Multivariable Control based on his MIT PhD thesis which included measures of interaction and posed examples where decoupling control was not always optimal. At Foxboro a friend, John Chang, and I were discussing this work and decided that, as bright young engineers, we ought to make improvements in the way that multivariable processes were controlled. John had a chance to go into a power plant and take step response data for three different operating points, thus capturing both dynamics and nonlinearity. Out of this came a complete internal report. John left to the University of Delaware to get a PhD.

And I got the job of making sense of the report. Perversely in a young smart-ass attempt to improve control, I decided to try and define the motivation of the existing designs. I went through a series of normalizations of the gain matrices of John's data to try and see if anything stood out. I sought obvious dominant diagonal behavior, suggesting the basic single loop control. None worked completely. Considering Masaravic's interaction measures, it was clear to me that any "good" measure of interaction must be dimensionless, since single loop control could arbitrarily alter loop scaling.

Leaving natural and theory-arbitrary process based scaling for more mathematics based "scaling", I played with cofactors and determinants. The determinant was the "absolute value" of the matrix, but dividing the matrix by the determinant did not arrive at dimensionlessness. But multiplying each element by its cofactor got teh desired result and the existing formulation. What did this mean?

The future of the RGA was settled when Greg Shinskey visited a plant where they were having trouble controling a process and calculated an RGA of 27. I can almost forgive him his coinage of a so dramatic a name (RGA). As a result, successive efforts to recast the measure as a "Bristol Measure" have largely failed! I always felt that the "Root Locus" terminaolgy represented a lost opportunity. Had a more neutral term been selected everone would know that Evans invented the "Evans Graph"!

The RGA notion has been extended in a number of ways, initially as a fully transfer function generalization, but this doesn't seem to have theoretical support. Then the variations on the Relative Disturbance Gain Array, which creates a useful different categorization of interaction effects, and the Block Relative Gain, with substantial mathematical connections. In one sense the original RGA captures three levels of cofactor, adequate to represent matrices up to 3x3, but higher dimentional matrices will have more complex interactions. On the other hand the RGA therefore captures the interaction of matrices whose 3x3 or lower dimension sub-matrices dominate.

RGA Bibliography

The Chemical Engineering Literature contains a large number of papers, theses, and books, starting with Tom McAvoys original text and Greg Shinskey's process control books and papers. With a couple of exceptions, this bibliography includes only pure RGA papers by other authors and links to more complete web sites. The author's papers, multivariable or otherwise, are covered elsewhere. Much of this will undoubtedly become dated, but googling Relative Gain Analysis (RGA gets other stuff) is always fruitful (The most recent effort found a Google book: Deep Fat Frying: Fundamentals and Applications, as an illustration of the current reach of the RGA. This time there were over a million references of which a quick look at page 13 still had valid RGA references among the now prevalent junk.).

  1. E.H. Bristol, "On a New Measure of Interaction for Multivariable Control", IEEE PTGAC, Vol. AC-11, No. 1, January 1966.
  2. T.J McAvoy, Interaction Analysis, ISA, Research Triangle Park, NC, 1983. [Sour Grapes: this reference has been cited by several references to the RGA without citing my original paper. The same happened with Greg Shinskey's Process Control book.]
  3. C.N. Nett and V. Manousiouthakis, "Euclidian condition and block relative gain: connections, conjectures, and clarifications", IEEE PTGAC, Vol. AC-32, pp. 405-7, 1987.
  4. b To be filled in
  5. Charles R. Johnson and Helene M. Shapiro "Mathematical Aspects of the Relative Gain Array (A⊗A-T)", Siam J. Alg. Disc. Meth., Vol. 7, No. 4, Oct. 1986. As an exception to the Chem E references this is from the mathematics literature.
  6. Roger A. Horn "The Hadamard Product", Matrix Theory and Applications , Charles R. Johnson, editor, Proceedings of Symposia in Applied Mathematics, Vol. 40, Phoenix, Jan. 1989.
  7. Roger A. Horn and Charles R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991 (Sequel to Matrix Analysis, 1985).