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A novel online system‐identification‐based control design for damping inter‐area oscillations in power systems using supplementary wide‐area excitation control is presented. The identification is based on a partly reduced‐order and a partly full‐order model of the grid. Assuming the grid to be divided into coherent areas with phasor measurement units located across the grid, the coherent states of the generators are averaged while the non‐coherent states, such as the internal states of each power system stabilizer, are not. A hybrid state vector consisting of these averaged and non‐averaged states is computed, and used for identifying a hybrid linear time‐invariant state‐space model. A linear quadratic regulator based on this hybrid model is, thereafter, designed. The reduced‐order part of the model is found to save significant amount of online time for both learning and control, while the full‐order part serves to enhance closed‐loop stability. N4SID with randomized singular value decomposition (rSVD) is used for making the identification loop fast. The reduced‐dimensional controller is finally implemented using a broadcast control strategy. The design is extended to non‐linear models of power systems using Carleman bilinearization. Results for linear and bilinear control designs are validated using the IEEE 68‐bus power system model. [ABSTRACT FROM AUTHOR]