The third term in this equation has to do with how fast the error changes: As it's out of spec longer and longer, the I term will continue to increase, eventually overcoming this force or reaching the limits of the motor's capability. This figure is then multiplied by K i and added into u(t).Ĭonsider a situation where a force holds a motor in place and doesn't allow it to return to the setpoint under normal conditions. It's the sum of all errors experienced on the device:Įach time a controller calculates u(t), it adds the instantaneous error to a running tally. The second term in this equation has to do with the combined error over time. Multiply this by K p and you have its contribution to the overall controller output. Here, e(t) is simply the instantaneous error at a point in time-the actual value of a controlled device minus the desired value. As such, the K p value that comes before it is generally larger than the other K values in the equation. The first and most crucial term in this equation is the e(t). We can individually tune each K value for better system performance, which we'll explain further below: Each element has a constant K value in front (K p, K i, and K d), which signifies each element's weight as they form u(t), or the control output at a specific time. And while it never hurts, you don't even have to be able to do calculus.īreaking the first equation down, we produce u(t)-the unitless controller output on the left-hand side of the equation-by adding three mathematical elements on the right-hand side of the equal sign: P, I, and D. The good news is that you don't have to dig out your Modeling and Analysis of Dynamic Systems textbook to understand what's going on here. One advantage of this form is that we can adjust the overall K p constant for the whole equation at one time:Īll of this may look a bit intimidating, perhaps even to someone who graduated with an engineering degree. We can also transpose the equation to extract the K p value and apply it to the entire equation, in what's known as the standard form. This change gives the equation a better relationship to its physical meaning and allows the units to work out properly to a unitless number: We can also replace K i and K d with 1/T i and T d, respectively. K p, K i, and K d are constants that tune how the system reacts to each factor: P, I, and D are represented by the three terms that add together here. We can express PID control mathematically with the following equation. While limit-based control can get you in the ballpark, your system will tend to act somewhat erratically. In this example, they would prevent a car's speed from bouncing from an upper to a lower limit, and we can apply the same concept to a variety of control situations. These more subtle effects are what the I and D terms consider mathematically.

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