Consider the simplest case, a body is acted on only by the force that you control. You want to reach certain speed; how should you control the force. If we elect to use the proportion gain control, i.e. the force is proportion to the difference between the desired speed and the current speed. We would have an exponential speed profile, that approaches the desired speed asymptotically with a time constant of m/K, which m is the mass and K is the proportional constant.
If there is a constant force also acting on the body, such as the gravity, and we apply the proportional control, the speed profile is again an exponential with the same time constant. But now our final asymptotic speed is not the desired speed, but offseted by mg/K. So the proportional control results in a steady state error.
The common cure is to add another force that is proportional to the accumulated or integrated velocity error. Now we have a second order system. If the integrated gain is sufficiently high, we would have an oscillatory response.
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