PID Control
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This chapter describes the use of proportional integral derivative (PID) feedback for control systems design. We discuss the basic concepts behind PID control and the methods for choosing the PID gains.
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Chapter Summary
This chapter describes the design and use of PID (proportionalintegralderivative) control:
The basic PID controller as the form
where is the control signal and is the control error. The control signal is thus a sum of three terms: a proportional term that is proportional to the error, an integral term that is proportional to the integral of the error, and a derivative term that is proportional to the derivative of the error.
Integral action guarantees that the process output agrees with the reference in steady state and provides an alternative to including a feedforward term for tracking a constant reference input. Integral action can be implemented using automatic reset, where the output of a proportional controller is fed back to its input through a low pass filter:
Derivative action provides a method for predictive action. The inputoutput relation of a controller with proportional and derivative action is
where is the derivative time constant. The action of a controller with proportional and derivative action can be interpreted as if the control is made proportional to the \emph{predicted} process output, where the prediction is made by extrapolating the error time units into the future using the tangent to the error curve.
Exercises
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Errata

 Errata: Caption for Figure 10.3b should be "Derivative action"
 Errata: After equation (10.5), low frequency limit bound be 1/T d instead of T d
 Errata: Closed loop time constant after equation (10.6) has an extra factor of 2
 Errata: The b k i term in the characteristic polynomial has an extra factor of s
 Errata: Toward end of Section 10.4, k i should be k t
 Errata: Figure 10.11 is missing filter on derivative term
 Errata: In equation (10.4), upper limit of the integral should be t, not infinity
 Errata: In Figure 10.13, the indices on the resistors and capacitors are incorrect
 Errata: The expressions for the PID parameters for the op amp implementation are incorrect
 Errata: The PID integral time constant T i for the op amp implementation should be equal to R 2 C 2
 Errata: Missing ydot after equation (10.16)
 Errata: In Exercise 10.1, the second term in the denominator should be kp not kd
 Errata: In Exercise 10.11, the dynamics for x 2 should test for e less than e 0, not e less than 1