Simultaneous Stabilization of Three or More Plants: Conditions on the Positive Real Axis Do Not Suffice

The problem of the simultaneous stabilizability of a finite family of single-input, single-output time-invariant systems by a time- invariant controller is studied. The link between stabilization and avoidance is shown and is used to derive necessary conditions for the simultaneous stabilization of k plants. These necessary conditions are proved to be, in general, not sufficient. This result also disproves a long- standing conjecture on the stabilizability condition of a single plant with a stable minimum phase controller. The main result is to show that, unlike the case of two plants, the existence of a simultaneous stabilizing controller for more than two plants is not guaranteed by the existence of a controller such that the closed loops have no real unstable poles.

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