The speed of your output seems to be pretty slow (ramping up in 1 to 120 seconds, if I understand correctly). So you probably can use the PWM output of a PIC or AVR as your DAC.

So you could set the duty cycle of that PWM output proportional to the desired output voltage. For example, a duty cycle of 0 would correspond to 0 V, of 50% to 5 V, and of 100% to 10 V.

Then, you'd smooth the output waveform with an RC first-order low-pass filter. For example, to get your desired ~7 bits of resolution on a PIC16 with a 20 MHz crystal, you would need to set the PWM frequency to (20 MHz)/(4*128) ~ 40 kHz. So with a filter time constant around 500 ms (for example, R = 510k and C = 1u), the remaining ripple will be very small. A faster (smaller) RC time constant means that the output will respond more quickly to changes in the programmed duty cycle, but it also increases the ripple.

That gives you a voltage that goes between 0 V and the PIC's supply voltage as the PWM duty cycle goes from 0 to 100%. So if that voltage is 5 V, then you can use an amplifier with gain 2 V/V, running from a supply voltage of at least 10 V. You might, for example, use a TLV2371IP, which runs with Vdd up to 16 V, and can usefully deliver perhaps 10 mA of output current.

Then it's just a question of writing the program to set the duty cycle according to the waveforms that you want.