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I am trying to solve an assignment where the zeros of a PID-controller is given and the polynomial controller-constants need to be found.

The polynomial form of a standard PID-controller should be defined as: $\dfrac{Kd\cdot s^2+Kp\cdot s + Ki}{s}$

And the zeros of the transfer-function are defined as:

  • $z_1 = 1$
  • $z_2 = 3$

I thought, that the correct way would be to solve for the constants using the quadratic equation, but then I end up with 3 unknown parameters in 2 equations. \begin{equation} \begin{split} & \dfrac{-b \pm \sqrt{b^2-4\cdot a\cdot c}}{2\cdot a} = [-1,-3]\\ & \dfrac{-Kp \pm \sqrt{Kp^2-4\cdot Kd\cdot Ki}}{2\cdot Kd} = [-1,-3] \end{split} \end{equation}

I know how to solve it in Matlab with poly([-1;-3]) but I cannot seem to figure it out by hand

Kristian Slot
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  • There's not way to solve it entirely. What you can do is divide the PID controller polynomial by Kd, and solve for Kp/Kd and Ki/Kd. The only way you can know the actual values for Kp, Kd and Ki is if you were given another requirement, such as overshoot, etc. – Carlos Franco Dec 21 '20 at 14:19
  • I have been given no other requirements for the controller. How come MATLAB can find the polynomial coefficients given the roots with poly([root1;root]) if it cannot be done manually? – Kristian Slot Dec 21 '20 at 20:41
  • Most likely it considers the first coefficient as being 1. – Carlos Franco Dec 23 '20 at 13:09

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