Characterization of sign controllability for linear systems with real eigenvalues

A linear time-invariant system of the form ẋ(t) = Ax(t) + Bu{t), or x(t + 1) = Ax(t) + Bu(t) is sign controllable if all linear time-invariant systems whose matrices A and B have the same sign pattern as A and B are controllable. This work characterizes the sign controllability for systems, whose sign pattern of A allows only real eigenvalues. Moreover, we present a combinatorial condition which is necessary for sign controllability and we show that if this condition is satisfied, then in all linear time-invariant systems with that sign pattern, all real eigenvalues of A are controllable. In addition, it is proven that the decision whether a linear time-invariant systems is not sign controllable is NP-complete. We want to emphasize, that our results cover the single and the multi-input case.