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

Solution to the Generalized Champagne Problem on simultaneous stabilization of linear systems

“…The well-known "Generalized Champagne Problem" on simultaneous stabilization of linear systems is solved by using complex analysis [1,11,13,18,22] and Blondel's technique [5,6,8].…”

“…We give a complete answer to the open problem proposed by Patel et al [20,21], which automatically includes the solution to the original "Champagne Problem" [6,8,9,17,20,21]. Based on the recent development in automated inequality-type theorem proving [31,32,33,35,36], a new stabilizing controller design method is established.…”

“…For k ≥ 3, simultaneous stabilization problem is much more complicated than one expected [5,6,7,8]. Vidyasagar and Viswanadham [26] stated that it is possible to transform simultaneous stabilization of k systems to strong stabilization of corresponding k − 1 systems [5,15,25].…”

“…Vidyasagar and Viswanadham [26] stated that it is possible to transform simultaneous stabilization of k systems to strong stabilization of corresponding k − 1 systems [5,15,25]. Blondel et al [8] proved that simultaneous stabilization of k systems is equivalent to bistable stabilization of associated k − 2 systems. Bistable stabilization means stabilization with a stable and inverse-stable controller.…”

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Abstract:The well-known "Generalized Champagne Problem" on simultaneous stabilization of linear systems is solved by using complex analysis [1,11,13,18,22] and Blondel's technique [5,6,8].We give a complete answer to the open problem proposed by Patel et al [20,21], which automatically includes the solution to the original "Champagne Problem" [6,8,9,17,20,21]. Based on the recent development in automated inequality-type theorem proving [31,32,33,35,36], a new stabilizing controller design method is established.

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