Smoothing an LFO Sawtooth Wave
Reason that it's clicking is that you are jumping from 1 to 0 in the time that the audio signal reads the last of the 1 control signal and then the 0 control signal.

You have to slow fast signal transitions over multiple samples, for example 1 to 0 should take 5ms or 10ms which is dozens or 100ds of samples. 

To do that, you can use a lowpass filter at about 300Hz more or less, which can actually be very cheap for a single monophone LFO its less that 1% cpu useage, or you can use a signal smoothing algorithm, for exampel a linear interpolation function or a SmoothStep which is awesome general knowledge for audio programmers.
http://paulbourke.net/miscellaneous/interpolation/

BTW what is the platform? i can't give you code exampels if i dont know what code you use.

the cool thing about smoothstep is how easy it is, its very good knowldedge in synths:
https://en.wikipedia.org/wiki/Smoothstep

https://documentation.apple.com/en/logicstudio/instruments/index.html#chapter=A%26section=3%26tasks=true

check the filtered sawtooth there.

You can also a smoother that says "if difference between last signal and this signal was larger than 0.02, signal - 0.02... and create a secondary smothed signal... there are fairly simple smoothing algo's based on that it's pretty straightforward.

There is also an entire page on the topic.