This paper focuses on the stabilization problem for rectangular descriptor fractional order systems (FOSs) with 0 < α ≤ 1. Firstly, a fractional order dynamic compensator is constructed to make the closed-loop systems square descriptor FOSs. Secondly, two types of input signals of the compensator are considered, which are state input signal case and output input signal case. Thirdly, a necessary and sufficient condition is proposed for the state input signal case, while a new method of controller design is given for the output input signal case in which the output matrix of the system need not to be full row rank, which reduces the conservativeness of existing methods. Finally, an efficient iterative algorithm for solving the resultant matrix inequalities is proposed, and a numerical example is offered to verify the advantage and feasibility of the results.

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