An air-fuel-ratio of each cylinder is estimated based on a detection value of an air-fuel-ratio sensor. An air-fuel-ratio detection timing determination is executed to determine whether a deviation in air-fuel-ratio detection timing exists based on the estimated air-fuel-ratio while the cylinder-by-cylinder air-fuel-ratio control is performed. An observation residual is computed based on the detection value of the air-fuel-ratio sensor and the estimated air-fuel-ratio. When the observation residual is greater than or equal to a specified threshold value, the air-fuel-ratio detection timing determination is prohibited. Therefore, when the observation residual is still large before the estimated air-fuel-ratio of each cylinder is converged, or when the observation residual is still large due to a temporal deterioration in estimating accuracy of the estimated air-fuel-ratio, the air-fuel-ratio detection timing determination can be prohibited.