I'm trying to understand PIDs and there is just one thing in the general equation that I can't figure out, that is what does the Tau in the integration mean? in the pdf equation 10.1 there is ki * integral[ e(tau) dtau ] but it doesn't say that it means by tau
http://www.cds.caltech.edu/~murray/books/AM08/pdf/am06-pid_16Sep06.pdf
Tau is just time. You're integrating that shaded area of the curve from tau = 0 to tau = t. You might see this written integral[e(t) dt]. But you can't write t here, because t is already in use in the equation. So tau is just a substitute for t. Doesn't matter what variable names you use. You're integrating over time.