Extended Approach to Analytical Triangular Decoupling Internal Model Control of Square Stable Multivariable Systems With Delays and Right-Half-Plane Zeros

The Analytical Triangular Decoupling Internal Model Control (ATDIMC) technique for $2\times 2$ systems is generalized to $n\times n$ systems ( $n\ge 2)$ with delays and right-half-plane (RHP) transmission zeros. The formulation is done by first creating a triangular closed-loop transfer function matrix corresponding to the achievement of the triangular decoupling objective of restraining inverse-response and control-loop-interaction characteristics to a single plant output. Subsequently, the corresponding multivariable internal model controller is calculated, with transfer-function approximations made using an optimization algorithm that minimizes the Integral Time-Weighted Absolute Error (ITAE) of the difference between the step responses of the original and reduced expressions. It is shown that $n$ ATDIMC designs emerge that achieve the shifting of inverse responses and interactions to a least-desired output, with delays retained for all outputs and asymptotic tracking of setpoints achieved for all $n$ outputs of each design. To mitigate the possible effect of severe interaction on the least-desired output, a modification of this formulation is performed to spread inverse-response behavior to a second output, while minimizing the interaction of that output with the initial least-desired output. Simulation results for selected $3\times 3$ and $4\times 4$ systems show the effectiveness of these propositions.