A novel, high-fidelity, low-order model is developed for an air-breathing combustion system designed for supersonic flight with a subsonic combustor. Detailed models for the individual components, intake, combustor, exhaust nozzle, and fuel supply system are interlinked by a low-order global model that captures the physics of the interaction between the intake and the subsonic combustor. Disturbances, both internal to the engine and external, due to atmospheric turbulence, are modeled. The terminal shock location in the intake duct is quantified in terms of the intake backpressure margin, called P(4margin). An innovative controller is designed that meets thrust demands with fuel flow rate and exhaust nozzle throat area as the two inputs, while maintaining a tight control on P(4margin) (intake shock location). Limits on fuel-air ratio, peak combustor temperature, fuel flow rate, and throat area actuator limits are imposed. The controlled combustion system is tested for an accelerated climb mission from a low altitude to a specified cruise Mach number and altitude. An outer guidance loop using dynamic inversion as a rapid prototyping tool is used for this test which, for a reference Mach command profile, provides the engine controller with the expected thrust demand at every instant. A closed-loop simulation is successfully demonstrated that takes the system through the acceleration phase to the desired cruise condition with a smooth switching between the acceleration and cruise segments.