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      Mills controls systems        
      6   PID controllers (P, PI, PD et PID)        
       6.4   PI controller:        
        - The output of the PI controller is given by the following formula:      
        - The role of the integral action is to cancel the difference between the measurement and the setpoint.
        - The introduction of a PI controller improves accuracy.      
        - So we keep the qualities of proportional correction while significantly reducing the offset.  
        - At the contrary, this type of corrector has some limitations on improving the speed and may even introduce some
          instability of the closed loop system.        
        - Finally, it is more difficult to tune.        
        - Functional diagram of the PI controller:        
        - Step response:        
        - The PI controller transfer function will be:        
        - Ki influence on the system:        
        - We see that the integral term has worked well and that the static error is zero.    
        - We also note that the higher the gain Ki is, more the system converges quickly.    
        - In contrast, greater Ki is, more the system oscillates and overtaking is great.    
        - It is for this that we will integrate the derivative checker.      
       6.5   D controller:        
        - The derivative term reduces the overshoot and oscillations.      
        - Its role is therefore to balance the effects of the dead time (delay) of the process.  
        - The derivative has a stabilizing effect but excessive value can cause instability.    
        - Note that the derivative action can not be used alone.      
        - The output of the corrector is given by the following formula:      
        - Functional diagram:        
        - The D controller transfer function is:        
       6.6   PD controller:        
        - The derivative action can limit the overshoot of the setpoint generated by the PI controller.  
        - When the system approaches the setpoint, this action brakes the system by applying an action in the opposite
          direction, allowing faster stabilization.        
        - The PD controller transfer function is:        
        - We will not insist further on this architecture, because the controller does not work in pure derivative action
          (too unstable).        
       6.7   PID controller:        
        - The PID controller is the combination of three basic actions P, I and D.    
        - Thanks to the term I, it allows the cancellation of the static error while allowing through the action of D speed
          superior performances than a PI controller.        
        - The introduction of a PID controller brings together the various benefits of each action, however, its setting,
          so the weight to be given to each action is rather tricky.      
        - The PID controller control law is given by:        
        - Here below, an example of a functional diagram:      
        - The PID controller transfer function is:        
        - Step response:        
        - Here below is a PID simulator.        
        - Just change the values of Kp, Ki and Kd to see the reactions of the output value.    
        - The setpoint is 80 t/h.        
          Proportional action - value of Kp      
          Integral action - value of Ki      
          Derivative action - value of Kd      
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