In this paper, we propose an on-chip learning method that can overcome the poor characteristics of pre-developed practical synaptic devices, thereby increasing the accuracy of the neural network based on the neuromorphic system. The fabricated synaptic devices, based on Pr1-xCaxMnO3, LiCoO2, and TiOx, inherently suffer from undesirable characteristics, such as nonlinearity, discontinuities, and asymmetric conductance responses, which degrade the neuromorphic system performance. To address these limitations, we have proposed a conductance-based linear weighted quantization method, which controls conductance changes, and trained a neural network to predict the handwritten digits from the standard database MNIST. Furthermore, we quantitatively considered the non-ideal case, to ensure reliability by limiting the conductance level to that which synaptic devices can practically accept. Based on this proposed learning method, we significantly improved the neuromorphic system, without any hardware modifications to the synaptic devices or neuromorphic systems. Thus, the results emphatically show that, even for devices with poor synaptic characteristics, the neuromorphic system performance can be improved.