proportional control offset

Question

I understand that with proportional only control you have to have error or the control output goes to zero. The answer to this question Proportional controller was very informative. Mr. Lathrop gives the the example of a car that you desire to go 50mph. Everything makes sense to me until this:

However, the problem now is that you won't ever get to your desired speed. If you were at 50 MPH, then I-S would be 0, and you'd let off the gas completely, which clearly won't maintain 50 MPH. This simple proportional-only controller always requires some error to maintain a non-zero output. For example, let's say to maintain 50 MPH would require a 25% control output (step on the gas 1/4 of the way), and that steady state speed is linear with throttle setting. In that case, the system would asymptotically approach 40 MPH and stay there. K(I - S) is 20, which is the control output required to maintain 40 MPH.

My question is: Why does the system asymptotically approach 40mph and stay there given a 25% controller output? Given C = K(I - S), where I is the control input to the system, S the actual system response, C the control output that drives the system, and K the proportionality constant, .25 = 2(50 - S), so S would approach 49.875mph? I sure I'm missing something real simple here.

Answer 1

Simple answer: if there is no error the output is zero C=K(I-S), if (I-S)=0 then C=0. What is the minimal acheivable error? It is 1/(1+Kp), so increasing Kp you have more and more precise control until the system begins to oscillate. Therefore the P-regultor is tuned to have enough stability margin, so that's why the system approaches asymptoticaly, but if you rise the gain it will be oscillating.