An on-line generalised model predictive control (GMPC) strategy is designed and optimised with a novel identification procedure in the presence of different disturbances. The principle of MPC is utilising a discrete-time model of a system to reach the control variables with a prediction over these values, which is followed by computing a cost function for the control aims. Non-inverting buck-boost converter is a non-minimum phase system based on its boost mode, which makes a challenging condition for designing a stable controller. The proposed control technique described in this paper removes the requirement for a system mathematical model adopting a black-box identification method which can decrease the computational burden. Numerous harmful disturbances can affect a DC-DC converter; thus, the GMPC scheme is used along with a novel improved exponential regressive least identification algorithm as an adaptive strategy for the controller to optimise the gains of the controller in an on-line way resulting in better disturbance rejection. A PID controller with particle swarm optimisation algorithm is designed for this converter to be compared with the GMPC controller. Finally, the efficiency of the GMPC is verified in various performances with experimental and simulation results.