By Edward W Kamen

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**Example text**

We pursue this by beginning with a first-order process. PI C o n t r o l of a F i r s t - O r d e r P r o c e s s Suppose the process has the first-order transfer function Gp(s) = b/(s+a), where a and b are constants. 16) we see that the closed-loop transfer function has a zero at s = -KI/Kp, and it has two poles. 16) that the two poles 55 PI Controller of Gct(S) can be placed wherever desired in the OLHP by choosing the controller gains Kp and K1. In particular, suppose that we want the closed-loop poles to be s = -pl and s = -p2.

Ff a sensor for measuring the output is not available, it is necessary to generate the derivative of the output in order to implement a PID controller. 23) and Gp(s) is the transfer function of the motor. 24) we see that the coefficients of s 2, s, and s o in the denominator of Gcl(S) can be chosen to have any desired values by selecting the controller gains Kp, KI, and KD, and thus all three poles of the closed-loop transfer function can be placed anywhere we wish. 5).

13), when 0 < ~ < 1 the transient part of the step response contains sinusoidal terms that can cause an overshoot in the step response. , when 0 s ~ < 1). In the underdamped case, the two poles of G(s) are complex. 14) Ys~C~ n + COd where COd is the damped natural frequency given by COd = COn~Jl - ~2. 14) that when 0 s ~ < 1, the poles of G(s) are a complex conjugate pair with the values s = -~co, +flOd and s = -~o~, -jcod, where j-- ~/-1. 15) COd where again it is assumed that 0 ~ ~ < 1. 18), we can determine the parameters ~ and o~, from the peak time tp and the settling time ts.