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A criterion is developed for the selection of the best pairing of the control and manipulated variables of a multiloop control system. This criterion is based on the difficulty caused by the interaction terms (the off-diagonal elements) in finding the inverse of the steady state gain matrix. From an analysis based on the proposition that the most desired or best pairing is that one for which the system most closely resembles a set of independent single-loop systems, a quantitative measure of the best pairing is obtained. Although the development of the pairing criterion is based on purely algebraic principles, the validity of the pairing criterion is evident from analogous developments obtained within the control framework. Furthermore, it is shown that the pairing criterion may be used to determine the stability of a multiloop control system, thus enhancing the value of the criterion presented. SCOPEDespite all the attention given by researchers and theoreticians to interaction analysis and variable pairing for multiloop control systems, it continues to represent one of the most difficult control problems facing engineers today. Because of its calculational simplicity, the relative gain (Bristol, 1966) has been by far the most popular pairing technique. This technique has been applied with some success to high-order systems (3 x 3 or higher). However, for high-order systems, cases have appeared in the literature for which Bristol's pairing rule recommends an unstable pairing (McAvoy, 1983), or fails to predict the interaction of the system (Friedly, 1984). This led to the development of extensions to improve the relative gain, such as that presented by McAvoy, which uses Niederlinski's stability theorem, and by Friedly, which analyzes the ratio of the nondiagonal terms over the diagonal terms of the process gain matrix. Also, a new pairing criterion was proposed by Bruns and Smith (1982), which uses the singular value decomposition (SVD) of the process gain matrix. However, the results obtained by this technique are strongly dependent upon the scaling of the process gain matrix. Jerome (1982) and Jensen et al. (1986) concluded that computer-based methods involving interactive graphics are the most reliable methods for pairing. However, the complexity and hardware requirements of these methods have reduced their use and acceptance.In this paper, a simple, reliable, and scale-invariant steady state pairing criterion is developed. The criterion is developed by an analysis of an iterative technique for obtaining the solution to a set of linear equations, or the inverse of a matrix. It is shown that the system interactions can be assessed by analyzing the effect of the off-diagonal elements of the steady state matrix on the difficulty of obtaining the desired solution or matrix inverse by the iterative process. The rate of convergence of the iterative process for the different variable pairings provides the basis for the pairing criterion. It is demonstrated that the algebraic iterative procedure may be represented ...

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