Tuning digital PID controller

I am using the method of Ziegler-Nichols in the the following form: Kp = 3*Ku/5; Ki = 6*Ku/(5*Pu); Kd = 3*Ku*Pu/40, but it seems it does not work well for object with the following
transfer function: W(z)=(b1*z + b2)/(z^2+a1*z+a2), ts is 2s.
Can someone defie me accurately the equations for Kp, Ki and Kd depending on the transfer function of the objects? Any web links will be very appreciate.
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zlatko wrote:

Z-N is for tuning systems where you _don't_ know the transfer function. If you _do_ have a good nominal transfer function you want to use some other technique, like pole placement, Bode plot, robust, etc.
Z-N tends to result in an underdamped system; you may want to try the Astrom-Hagglund method which is billed as being a significant improvement.
--

Tim Wescott
Wescott Design Services
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If you know a1 and a2 then you know where the two poles are in BOTH the z and the s domain. Set z=1 in your transfer function to calculate the gain. Now you know the two poles and the gain of the plant. Next you define the closed loop transfer function in terms of the plant poles and the PID controller. T(s) = (K(s)*G(s))/(1+K(s)*G(s)). Adding the PID adds a pole so now there are three poles to place. I set the coefficients of s in the characterist equation of the closed loop transfer function equal to the coefficients of the characteristic equation of the desired CLTF and solve the simulataneous equations. This will yield the values of Ki, Kp and Kd that place the poles at the desired location. I usually place the poles on the real axis at the same spot. Once you have the equations you can move the poles where ever you want within reason.
Peter Nachtwey
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