Robust MIMO PID controllers tuning based on complex/real ratio of the characteristic matrix eigenvalues

Robust MIMO PID controllers tuning based on complex/real ratio of the characteristic matrix eigenvalues is proposed. It is showed that this tuning criterion is equivalent to H∞H∞ optimal control. Under the proposed criterion, the tuning problem is stated as an optimization problem, in which the complex/real ratio of the characteristic matrix eigenvalues, a Lyapunov quadratic index, and the spectral abscissa were simultaneously minimized. Proposed criterion was applied to the multivariate controls of a distillation column, and a non-linear chemical reactor, both reported in the literature.