Difference between revisions of "PID Control"
(→Chapter Summary) 

Line 27:  Line 27:  
== Chapter Summary ==  == Chapter Summary ==  
+  This chapter describes the design and use of PID (proportionalintegralderivative) control:  
+  <ol>  
+  <li><p> The basic PID controller as the form  
+  <center><amsmath>  
+  u(t) = k_p e(t) + k_i \int_0^t e(\tau)\,d\tau + k_d \frac{de}{dt},  
+  </amsmath></center>  
+  where <math>u</math> is the control signal and <math>e</math> 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.  
+  <center>[[Image:pid.png]]</center>  
+  </p></li>  
+  
+  <li><p> ''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:  
+  <center><amsmath>  
+  u=k_pe+\frac{1}{1+sT_i}u,  
+  </amsmath></center>  
+  </p></li>  
+  
+  <li><p> ''Derivative action'' provides a method for predictive action. The inputoutput relation of a controller with proportional and derivative action is  
+  <center><amsmath>  
+  u=k_pe+k_d\frac{de}{dt}=k\bigl(e+T_d\frac{de}{dt}\bigr),  
+  </amsmath></center>  
+  where <math>T_d=k_d/k_p</math> 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 <math>T_d</math> time units into the future using the tangent to the error curve.  
+  </p></li>  
+  
+  <li><p>  
+  </p></li>  
+  
+  </ol>  
== Exercises ==  == Exercises == 
Revision as of 15:30, 8 January 2007
Prev: Loop Analysis  Chapter 10  PID Control  Next: Loop Shaping 
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.
Textbook Contents

Lecture MaterialsSupplemental Information

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
Frequently Asked Questions