Multivariable Process Control Using Singular Value Decomposition

Singular value decomposition is a promising tool in the analysis and control of process systems. Singular value decomposition is an established tool in the analysis of linear numerical mathematics and many of the procedures developed there have utility in the study of process systems. The decomposition of the process gain matrix complements the Relative Gain Array in discerning information about interaction and control of a multivariable process. Vector and matrix norms are intimately associated with the decomposition and provide bounds relating process inputs and outputs.

The singular value decomposition provides a framework for a generalized multivariable controller, the singular value controller. This controlled is based on a diagonal matrix of proportional and integral gains and on the two orthogonal matrices obtained from the decomposition. The controller has the property of finding the smallest process input that reduces the error in the output to a minimum. For the square non-singular system, the controller finds the unique input that reduces the error to zero. For systems with more outputs than inputs, the controlled finds the input that reduces the sum of the errors square to its minimum. For systems were more inputs than outputs, the controller maintains the set point by finding the unique minimum input from the infinite set of possible inputs.

Digital simulation provides a means to find the optimal tuning parameters for integral error criteria such as IAE or ITAE. Generally, the singular value controller provided better control for set point changes than did the conventional pairing of single loop PI controllers. The singular value controller was tuned successfully for multivariable processes by adapting a Ziegler Nichols type approach for the singular value controller in cases were conventional Ziegler Nichols tuning failed.