A gas turbine engine has, in flow series, a compressor section, a combustor, and a turbine section. The engine further has a cooling system which diverts a cooling air flow received from the compressor section to a heat exchanger and then to the turbine section to cool a component thereof. The cooling air flow by-passes the combustor and is cooled in the heat exchanger. The cooling system has a valve arrangement which regulates the cooling air flow. The engine further has a closed-loop controller which estimates and/or measures one or more temperatures of the cooled component, compares values derived from the estimated and/or measured temperatures with one or more corresponding targets, and issues a demand signal to the valve arrangement based on the comparison and a value of the demand signal at a previous time step.