Figure 18 The simulation show how the control-signal fail to follow the optimum if the optimum varies fast. Two different sets of u and t was used.
The control-algorithm was based on the assumption that the change u of the control-variable u was the only change affecting the controlled variable y. Thus the controller neglected any influence of disturbances. This meant that any effects on y due to disturbances (in this case a changing ) was treated as if the change was caused by the change in u . If the disturbances caused a larger change in y than the change u did, then the behaviour of the controller could become somewhat unintuitive. The simulations shown in Figure 19 illustrates this. If was changed faster than u could grow (as in the left plot in Figure 19), the change in y depended more on then on u. At first, u was constantly changed in same direction since y was increasing. u was actually changed in the wrong direction but since affected y more than u , y increased regardless. The controller acted as if the change in y was due to the change in u and thus kept on changing u in the same faulty direction (Note that the control algorithm could just as well have changed u in the correct direction if it would have started with a negative u.). When the rapidly changing optimum overtook u the behaviour changed. y started to decrease no matter which direction u was changed. This caused the controller to switch sign of u every iteration, effectively “locking” u since it would only alternate between two values. In the other simulation (the right plot in Figure 19) the change in was slower than the maximum growth-rate of u and thus the problems with u growing in the wrong direction or becoming locked did not occur. Another way to look at the scenario analysed above was that y changed independent of u . If y was independent of u and increased, u would constantly be changed in one direction. On the other hand, if y was independently decreased, u would get locked between two values.
Published on Nov 9, 2011