Pid Controller

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Introduction
A proportional-integral-derivative (PID) controller is one of the most common algorithms used for control systems. It is widely used because the algorithm does not involve higher order mathematics, but still contains many variables. The amount of variables that are used allows the user to easily adjust the system to the desired settings. The algorithm for the PID uses a feedback loop to correct the difference between some measured value and the setpoint. It does this by calculating and outputting some action that will correct this error in the system. A PID controller has a proportional, integral and a derivative control which handles the current, past and predicted future of the signal error. For more information about PID, please refer to PID Intro. The PID controller can operate systems that run in a linear or nonlinear fashion. Tuning processes are done to the controller to tackle the possible nonlinear system. Limitations arise within the system because tuning is limited to only three different parameters (proportional, integral, and derivative controls). Additional information on tuning of PID can be found at [1] or [2]. The most common limitations that occur within the PID control specifically involve the integral control. The following article addresses some of the common limitations faced by each control type, with an emphasis on the integral control, and some solutions to overcome each of these limitations. Proportional Control

The main purpose of the proportional control is minimize the fluctuations that occur within the system. Limitations
The P-controller usually has steady-state errors (the difference in set point and actual outcome) unless the control gain is large. As the control gain becomes larger, issues arise with the stability of the feedback loop. For instance, reducing the rise time implies a high proportional gain, and reducing overshoot and oscillations implies a small proportional gain. This is not possible to achieve in all systems. Below is a general process outcome diagram showing the terminology used above.

Below is a sample diagram of process output of proportional control.

Solutions
The way to eliminate these steady-state errors is by adding an integral action. The integral term in the equation drives the error to zero. Higher Integral constant (1 / Tt) drives the error to zero sooner but also invites oscillations and instability. Read on the integral control section below to know more about limitations associated with this integral term. Below is a sample process output diagram when integral control is added.

The above picture shows the reduction of overshoots and oscillations compared to the picture before adding the integral action. Integral Control
The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error. Summing the instantaneous error over time (integrating the error) gives the accumulated offset that should have been corrected previously. The accumulated error is then multiplied by the integral gain and added to the controller output. The magnitude of the contribution of the integral term to the overall control action is determined by the integral gain, Ki. The integral term is given by:

Where
* Dout: Derivative output
* Kd: Derivative Gain, a tuning parameter
* e: Error = SP − PV
* t: Time or instantaneous time (the present)

Limitations
Windup
A basic knowledge of the concept of windup is useful before describing a specific type. Windup is defined as the situation when the feedback controller surpasses the saturation (i.e. maximum) limits of the system actuator and is not capable of instantly responding to the changes in the control error. The concept of the control variable reaching the actuator’s operation limits is reasonable considering the wide variety of operating conditions that are possible. When windup occurs the actuator constantly runs at its saturation limit...
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