Control system synthesis using linear transformations

A new approach to the synthesis of linear multivariable control systems is presented. The main result demonstrates how a multivariable cross-coupled process within a closed loop may be transformed such that single-variable synthesis tools may be used to design the controller. Cross-coupled processes may be represented by a matrix equation. The new approach uses elementary transformations to diagonalize the polynomial matrix representing the open-loop process. This yields
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uncoupled variables which describe the process. Closed-loop diagonalization is obtained by using a pro-multiplier matrix to relate the uncoupled variables to the physical variables of the process. This premultiplier matrix is a precontroller in the physical system. It allows synthesis to be carried out in the uncoupled coordinate system to get the physical coordinate system controller. The complete controller is designed using time or frequency single-variable synthesis techniques, and the uncoupled variables are handled one at a time during this phase. A second result follows from the main result. Disturbance inputs to the multivariable process are minimized just as they are in the single-variable case.