On the order of simultaneously stabilizing compensators

The simultaneous strong stabilization problem is considered and it is shown that there is no upper bound for the minimal order of a simultaneously strongly stabilizing compensator, in terms of the plant orders. A similar problem was also considered in Smith et al. (1986), where it was shown that such a bound does not exist for the strong stabilization problem of a single plant. But the examples given by Smith et al. were forcing an approximate unstable pole-zero cancellation, or forcing the distance between two distinct unstable zeros to go zero, In this paper it is shown that: 1) if approximate unstable pole-zero cancellation does not occur, and the distances between distinct, unstable zeros are bounded below by a positive constant, then it is possible to find an upper bound for the minimal order of a strongly stabilizing compensator; and 2) for the simultaneous strong stabilization problem (even for the two plant case), such a bound cannot be found