Mixed-signal neuromorphic systems represent a promising solution for solving extreme-edge computing tasks without relying on external computing resources. Their spiking neural network circuits are optimized for processing sensory data on-line in continuous-time. However, their low precision and high variability can severely limit their performance. To address this issue and improve their robustness to inhomogeneities and noise in both their internal state variables and external input signals, we designed on-chip learning circuits with short-term analog dynamics and long-term tristate discretization mechanisms. An additional hysteretic stop-learning mechanism is included to improve stability and automatically disable weight updates when necessary, to enable continuous always-on learning. We designed a spiking neural network with these learning circuits in a prototype chip using a 180 nm CMOS technology. Simulation and silicon measurement results from the prototype chip are presented. These circuits enable the construction of large-scale spiking neural networks with online learning capabilities for real-world edge computing tasks.