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This article focuses on the stability of uncertain fractional order (FO) polynomials involving integer and FO natural exponential functions. These functions arise from the flexibility property and time delays in the control loop of the system of rigid–flexible coupling space structures. The coefficients of the polynomial are assumed to be complex numbers which are linear functions of uncertain real parameters. An analytical method is presented to check the robust bounded-input bounded-output stability of the control system. Furthermore, a method is developed to determine up to how much the amplitude of the uncertain parameters can grow such that the controller preserves the stability of the system. In the end, an improved optimal FO phase-lead controller is designed for the residual vibration suppression of the FO model of a rigid–flexible coupling space structure. Then, the merit of the presented theoretical results is demonstrated by applying them to the designed control loop.