Derivative Control
    Derivative control quantifies this “need to apply more” correction by linking the amount of accelerator pedal to the “rate of change” of speed. In other words the faster the speed is dropping the more acceleration we apply. A sudden drop in speed requires a large and equally quick depression of the accelerator pedal.  Do not confuse this with the amount of speed drop. It is quite independent. It is also important to realize that on its own derivative control is not sufficient to restore the speed to 30mph. Consider if the change in speed is very slow. For example the speed may be dropping at a rate of 1mph per minute. This would produce an insignificant amount of accelerator pedal depression and even if (after 25 minutes) the speed dropped to 5mph the amount of pedal depression would still be insignificant.  We conclude that we need proportions of both elements to properly control the speed; derivative control to cope with sudden fluctuations and proportional to bring it back from large errors.

   
 

 

 
     
     
     
     
     
     
     
     
                   
   

   We have a very reasonable control system now which can maintain the target speed 30mph within certain limits regardless of flat or hilly roads. What we now need to examine is how close to the target are we capable of controlling the speed. Using the car example in this case is probably a little unfair in that the accuracy of the speedometer and a requirement to travel at an almost exact speed of 30mph are just not sensible. However, lets assume that is exactly what we are trying to achieve.  So, what is wrong with our current accuracy? If I were to estimate what were possible within the current control system I would say that we could hold the speed within the limits of  28 – 32 mph.  So how can we improve that.. Before I answer that lets examine the nature of the speed error.
   If we have a large difference in target and actual speed our proportional control applies a correction. If we have a sudden change in speed the derivative control helps out. However, if we only have a small fixed error the proportional element is so small that it is ineffective and because there is no change in speed the derivative contribution is zero. So the small error persists indefinitely.  What we need here is something that increases in its contribution the longer the error , however small, exists. This is called “Integral Control”.

   
       
       
       
       
         

Next:  Integral Control

     
   
     
   
   
 

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