Metal-oxide memristors have emerged as promising candidates for hardware implementation of artificial synapses - the key components of high-performance, analog neuromorphic networks - due to their excellent scaling prospects. Since some advanced cognitive tasks require spiking neuromorphic networks, which explicitly model individual neural pulses (spikes) in biological neural systems, it is crucial for memristive synapses to support the spike-time-dependent plasticity (STDP), which is believed to be the primary mechanism of Hebbian adaptation. A major challenge for the STDP implementation is that, in contrast to some simplistic models of the plasticity, the elementary change of a synaptic weight in an artificial hardware synapse depends not only on the pre-synaptic and post-synaptic signals, but also on the initial weight (memristor's conductance) value. Here we experimentally demonstrate, for the first time, STDP protocols that ensure self-adaptation of the average memristor conductance, making the plasticity stable, i.e. insensitive to the initial state of the devices. The experiments have been carried out with 200-nm Al2O3/TiO2-x memristors integrated into 12x12 crossbars. The experimentally observed self-adaptive STDP behavior has been complemented with numerical modeling of weight dynamics in a simple system with a leaky-integrate-and-fire neuron with a random spike-train input, using a compact model of memristor plasticity, fitted for quantitatively correct description of our memristors.