Transitivity in simultaneous stabilization for a family of time-varying systems

On the simultaneous stabilization for a family of time‐varying linear systems under the transmission condition

“…This paper has dealt with the simultaneous stabilization problems of a finite family of discrete time-varying systems from the transmission version. These simultaneous stabilizability criteria extend the results in [15] and [16] by only requiring the strong representation of one controller in the transmission condition.…”

“…An equivalent condition is derived that is not depending on the strong representations of L 0 or L 2 [14]. As an improvement of the previous work, the authors considered the simultaneous of n + 1 plants under the transmission condition, a necessary and sufficient condition is established in terms of strong representations for only one controller and one plant in the transmission condition [15]. Recently, a sufficient condition for simultaneous stabilizability of n + 1 plants is obtained in [16], this criteria relies on the strong representations of n controllers, instead of the given n + 1 plants.…”

Necessary and sufficient conditions to the transitivity in simultaneous stabilisation of time‐varying systems

“…It was also shown in [10] that the simultaneous stabilisation of n + 1 multiple‐input multiple‐output(MIMO) plants is equivalent to the strong simultaneous stabilisation of n MIMO plants. Lu and Xu extended the simplified version of the necessary and sufficient condition for simultaneous stabilisation of n + 1 discrete linear time‐varying plants in [11]. For the case n > 2, some sufficient conditions are found for the simultaneous stabilisation of n plants (see [12, 13]).…”

A new necessary and sufficient condition for the simultaneous stabilization of three time-varying linear systems

“…14-those conditions do not employ strong representations of L 0 and L 2 . As an improvement of the previous work, an equivalent condition not depending on the strong representations of L 0 or L 2 will be derived 15 . Up to the present, some criteria for simultaneous stabilizability of three plants have been given within framework of nest algebra.…”

“…12, 14, one of main contributions here is establishing a necessary and sufficient condition to the transitivity of timevarying linear systems. The early equivalent conditions to the simultaneous stability depend on at least one strong representation of the considering plants and one of the controllers 7,13,15 , while the simultaneous stability criteria depend only on the single strong representation of one controller, which avoids any information about strong representations of each given plant. Additionally, a parametrization for all simultaneous stabilizers is derived under the restrictive transmission condition.…”