An efficient controller design is essential to damp out the rotor speed oscillations for real-life power systems during nominal and faulty conditions. This study presents a decentralised non-linear state observer and controller scheme for power system stabilisers (PSS). Conventional PSS designs are based on linearised models of power system, which is then followed by gain scheduling to cover the complete operating region. Here, the authors propose a derivative-free state-dependent Riccati equation (SDRE) controller and observer scheme for the PSS. Unlike the conventional methods, the proposed scheme is directly based on non-linear models and does not require linearised models. For the proposed SDRE scheme, the non-linear power system model is first converted in state-dependent coefficient form; then the SDRE controller is designed to achieve the desired performance. The SDRE observer is designed to estimate the states of the power systems based on output information; finally, the estimated states are fed back to the SDRE controller. The proposed combined SDRE observer–controller scheme is tested on a 68-bus benchmark system and is compared with the conventional control scheme. Extensive simulations are performed to show the efficacy of the proposed scheme under nominal and faulty conditions.