Analytic PID DesignIf G(s) is the loop gain, then the characteristic equation can be written as

We want to find the value of kp, kd, and ki to place a pole of the characteristic equation at a specified location, s1. Inother words, we want to find the value of s1 that satisfies

If we consider the controller

Equation 1

but s1 is also a complex number that can be represented by a magnitude and a phase:

Rewriting equation 1 gives us

Assume we can pick ki based on some other criterion, we can rewrite the equation

Equating real and imaginary parts of this last equation gives us a set of two equations in two unknowns (since ki isknown).

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If s1 and ki are known, they can be used to find M, q, B and b, and the equations can be solved for kp and kd. Doinga little algebra yields the general result that

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2 de 2 12/3/2012 22:33