a) providing a NOx purifying device in said exhaust system for purifying NOx in exhaust gases from said engine under a condition where a concentration of oxygen in the exhaust gases is higher than a concentration of reducing components in the exhaust gases, said NOx purifying device generating ammonia and retaining the generated ammonia when the concentration of reducing components in the inflowing exhaust gases is higher than the concentration of oxygen, and said NOx purifying device purifying NOx with the retained ammonia when the concentration of reducing components in the inflowing exhaust gases is lower than the concentration of oxygen, said NOx purifying device adsorbing NOx when the concentration of oxygen is higher than the concentration of reducing components, and said NOx purifying device generating ammonia from the adsorbed NOx when the concentration of reducing components is higher than the concentration of oxygen;

b) estimating a residual amount of ammonia retained in said NOx purifying device, wherein the residual amount of ammonia is estimated according to a temperature of said NOx purifying device;

c) switching the concentration of reducing components in the exhaust gases flowing into said NOx purifying device from a lower side to a higher side with respect to the concentration of oxygen, and from the higher side to the lower side, according to the estimated residual amount of ammonia;

d) detecting a rotational speed of said engine; and

e) detecting an operation amount of an accelerator pedal provided in a vehicle driven by said engine, wherein the residual amount of ammonia is estimated according to the detected engine rotational speed and the detected accelerator pedal operation amount, and

wherein said step b) of estimating the residual amount of ammonia includes the steps of:

i) calculating a basic generation amount of ammonia according to the detected engine rotational speed and the detected accelerator pedal operation amount, when the concentration of reducing components is higher than the concentration of oxygen;

ii) calculating a first temperature correction coefficient according the temperature of said NOx purifying device;

iii) calculating a generation amount of ammonia by correcting the basic generation amount with the first temperature correction coefficient;

iv) calculating a basic consumption amount of ammonia according to the detected engine rotational speed and the detected accelerator pedal operation amount, when the concentration of reducing components is lower than the concentration of oxygen;

v) calculating a second temperature correction coefficient according the temperature of said NOx purifying device; and

vi) calculating a consumption amount of ammonia by correcting the basic consumption amount with the second temperature correction coefficient, wherein the residual amount is calculated using the generation amount and the consumption amount of ammonia.