In recent years the use of apparatus for controlling various processes such as chemical processes, petrochemical processes and processes for the distillation, extraction and refining of petroleum and the like have been developed.  With the help of these apparatus, certain variables of the process may be measured and, in response, certain inputs controlled to enable the process to be operated in the most economical manner consistent with safe operation. 
For example, in furnaces for heating process fluids, the temperature of the heated fluid leaving the furnace is measured and the amount of fuel is automatically regulated to maintain the heated fluid at the desired temperature.  Under given furnace, fuel and atmospheric conditions, it takes a specific volume of combustion air to completely burn the fuel.  An insufficient supply of combustion air (oxygen) leaves unburned fuel in the combustion zone--which is very inefficient and potentially dangerous.  On the other hand, if there is an excess of combustion air, extra fuel is required to heat it, and the heated excess air is then usually passed uselessly out of the furnace stack--an inefficient mode of operation.  Thus, there is a need for controlling the supply of combustion air to furnaces to minimize periods of operation under conditions of excess air or excess fuel. 
On many furnaces, especially natural-draft furnaces, the air required for combustion is controlled manually, such as by a damper arrangement in the incoming air stream or in the furnace stack.  Normally, too much air is supplied to the furnace because, although inefficient, this represents safe operation and requires minimal operator attention. 
One type of existing controller maintains a preset air-to-fuel ratio by varying the flow of air responsive to changes in the flow of fuel.  Another type maintains a predetermined level of oxygen in the flue gas by using an oxygen analyzer. 
A more advanced system, described in U.S. Pat.  No.  3,184,686, describes an apparatus which controls the operation of a furnace by slowly reducing excess air until an optimum is reached, and then oscillating the amount of air about the optimum.  Thus, the combustion zone is operated part of the time under fuel-rich conditions and part of the time under oxygen-rich conditions. 
Yet another control system, described in an article entitled "Improving the Efficiency of Industrial Boilers by Microprocessor Control" by Laszlo Takacs in Power 121, 11, 80-83 (1977), uses a microprocessor to optimize the air-fuel ratio of a boiler based on feedback signals from stack-gas oxygen and combustible-materials analyzers, with the use of a CO analyzer being discussed. 
A need still exists, however, for an optimizing controller and method which will allow a combustion zone to be operated so that maximum efficiency can be achieved safely even under varying process and atmospheric conditions and fuel composition.  Particularly with respect to fired furnaces, a need exists for a method and apparatus which will control the supply of combustion air at a minimum without creating fuel-rich conditions and minimize the production of pollutants such as NO.sub.x in the stack gas.