A controller design method based on the model and polynomial sum of squares(SOS) programming was put forward for the conventional PI controller of aero-engine featuring poor adaptability and difficulty to guarantee robustness. Combining the robust stability conditions of transfer function model and the weak duality theorem

the robust stability conditions of linear parameter varying(LPV) model depicted by polynomial were proposed. Using the frequency response method

a family of transfer function models was obtained from non-linear model. And then these transfer functions were used to get LPV model by polynomial fitting method. With the proposed theorem and the LPV model

the controller design problem could be transformed into SOS programming problem

which was solved to obtain the linear parameter varying/proportion integration(LPV/PI) controller. Finally the controller was applied to a turbofan engine model. And step simulation results show that the settling time of this high pressure rotor speed control system is less than 3 seconds

and the steady state error is zero.