© 2009 The MathWorks, Inc. ® ® Introduction to PID Controller Design with examples in MATLAB and Simulink Dr. Bora Eryılmaz Engineering Manager Control and Estimation Tools Group The MathWorks, Inc.

2 ® ® What is a PID Controller? A special type of controller C(s) with  Proportional  Integral  Derivative terms acting on the error signal E(s).

3 ® ® What is a PID Controller? (cont.)  Ideal (standard) form:  Series (cascade) form:  Main point is: any second-order controller of the form is a PID controller.

4 ® ® PID Controllers Are Everywhere…  More than 90% of all controllers used in process industries are PID controllers.  A typical chemical plant has 100s or more PID controllers.  PID controllers are widely used in:  Chemical plants  Oil refineries  Pharmaceutical industries  Food industries  Paper mills  Electronic equipments

5 ® ® Some History: Fluid Level Control and at steady state

6 ® ® More History: Flyball Governor in Steam Engines Proportional control  Speed control for engines used proportional control. See the flyball governor by James Watt in 1788.

7 ® ® Many Types of “PID” Controllers…  Proportional (P):  Integral (I):  Proportional + Integral (PI):  Proportional + Derivative (PD):  You might see other combinations with different parameters than Kp, Ki, and Kd.

8 ® ® Low-Order Process Models  Many industrial processes can be modeled using simple stable transfer functions.  First-order process with delay:  Second-order process with delay:  There are many variations of these models, with or without time delays, with transfer functions zeros,...  We can design PI/PID controllers based on these models.

9 ® ® Desirable first-order responses with a tuning parameter K  Remember the open-loop transfer function is given by  Design your PID controller so that L(s) looks like  Then the closed-loop transfer function will look like

10 ® ® Designing a PI controller for a first-order process model  PI controller for a first-order process model  Remember, given K, we want:  Our PI parameters:  Let’s put this in MATLAB and Simulink…

11 ® ® Desirable second-order responses with tuning parameters K and α  Design your PID controller so that L(s) looks like  Then the closed-loop transfer function will look like  K and α are our design parameters.

12 ® ® Designing a PID controller for a second- order process model  PID controller for a second-order process model  Remember, given K and α, we want:   Let’s put this in MATLAB and Simulink…

13 ® ® MATLAB and Simulink Helper  MATLAB commands useful for control design: P = tf(num, den) C = zpk(z, p, K) L = minreal(P*C) K = dcgain(P) T = feedback(L,1) bode(P,L), step(T) sisotool(P)  Simulink blocks useful for control design: Transfer Fcn Zero-Pole Integrator Gain, Sum Transport Delay (time delay) PID Controller

14 ® ® Exam Question (10 points)  How to design PI/PID controllers for higher-order plants?  Key idea is to shape first- or second-order “dominant” plant dynamics. That is, you can ignore fast poles in the model.  Find a first- or second-order model, P(s), that has a similar response as the original model, Po(s). For example, you can use step(P, Po) or bode(P, Po) to compare responses. Similar poles and zeros can be ignored to simplify the model.  Question: Design a PI controller, using the technique of slides 9 & 10, for the plant  Can you get a closed-loop settling time less than 50 sec?