| Summary: | The Analytical Triangular Decoupling Internal Model Control (ATDIMC) technique for <inline-formula> <tex-math notation="LaTeX">$2\times 2$ </tex-math></inline-formula> systems is generalized to <inline-formula> <tex-math notation="LaTeX">$n\times n$ </tex-math></inline-formula> systems (<inline-formula> <tex-math notation="LaTeX">$n\ge 2)$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$3\times 3$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$4\times 4$ </tex-math></inline-formula> systems show the effectiveness of these propositions.
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