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The cross-Gramian matrixWc0 (T)for linear single-input single-output (SISO) dynamical systems has been related to the controllability and observability Gramians viaW²c0(T)= Wc(T)W₀(T)forT= ∞. The result is now proved for any arbitrary time intervalT, and is extended to symmetric multivariable systems which have symmetric transfer function matrices.
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Fernando et al. (1984) studied this question.
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