Waaay to analytical for the system at hand. The 'control loop' is in the faucet and you are the feedback signal. The only thing that can be controlled is the flow of the hot + cold water at the mixing valve to achieve some desired temperature. You as the controller can control/monitor is the delay to achieve a change in setpoint, e.g. the temp you want. So you, as the controller, have to anticipate the lag in the system and use that to decide when you want to get under the water stream, or enjoy the transient response of cold water...