ABSTRACT A novel convex necessary and sufficient condition for state-feedback exponential stabilisability in discrete-time switched linear systems, whose modes are described by rank-one matrices, is reported and proved in the present communication. A switched linear system, of this class, is shown to be state-feedback exponentially stabilisable if and only if a set of linear matrix inequalities (LMIs) associated to the system is feasible. And the solvability of this set of LMIs associated to the system is shown to be equivalent to that of a set of (standard) linear inequalities associated to the system. It is also proved that each solution to this set of LMIs (associated to the system) yields, through explicit formulas, to an exponentially stabilising state-feedback mapping, and also to a Lyapunov function for the exponential stability of the trivial solution of the corresponding closed-loop system (obtained by means of that feedback mapping). And such a Lyapunov function is always represented by a number of quadratic functionals that equals the number of modes composing the switched system.

中文翻译：

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一阶模式切换线性系统的凸充要稳定条件

摘要 本文报道并证明了离散时间切换线性系统中状态反馈指数稳定性的新凸充要条件，其模式由秩一矩阵描述。当且仅当与系统相关联的一组线性矩阵不等式 (LMI) 可行时，此类切换线性系统才被证明是状态反馈指数稳定的。并且这组与系统相关的 LMI 的可解性被证明等效于与系统相关的一组（标准）线性不等式的可解性。还证明了这组 LMI（与系统相关）的每个解决方案都通过显式公式产生了指数稳定的状态反馈映射，以及用于对应闭环系统的平凡解的指数稳定性的 Lyapunov 函数（通过该反馈映射获得）。并且这样的李雅普诺夫函数总是由许多二次泛函表示，这些二次泛函等于组成切换系统的模式数。