State-Feedback Stabilizability Characterization for Switched Positive Linear Systems via Lagrange Duality

A novel state-feedback exponential stabilizability characterization, for discrete-time switched positive linear systems, in terms of a linear mapping of the system modes is presented in this communication. Lagrange duality is used in order to prove that a switched positive linear system is state-feedback exponentially stabilizable if and only if there exists a linear mapping of the system modes whose range contains a Schur matrix. The characterization is specially suitable for state-feedback synthesis, and it further yields to the minimum number of linear functionals required to represent a stabilizing state-feedback mapping. It is furthermore proved that an upper bound for the aforementioned minimum number of linear functionals can be explicitly specified, and computed, in terms of a previously reported state-feedback exponential stabilizability condition. As a result of this constructive prove, a methodology for the synthesis of stabilizing state-feedback mappings (represented by the above mentioned upper bound minimum number of linear functionals) are also obtained.