{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

problem_1121 - of(2 we have the following three equations 4...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Problem 11.21 of the Palm textbook Plant TF: G P s ( ) = 4 3 s 2 + 3 Design a PID controller that gives a DOMINANT CL pole corresponding to: a damping ratio of ! = 0.5 . a time constant of ! = 1 . SOLUTION PID controller : G C s ( ) = K P + K I s + K D s = K D s 2 + K P s + K I s General PID closed-loop characteristic equation : 0 = 1 + G C s ( ) G P s ( ) = 1 + K D s 2 + K P s + K I s ! " # $ % & 4 3 s 2 + 3 ! " # $ % & OR D CL s ( ) = s 3 s 2 + 3 ( ) + 4 K D s 2 + K P s + K I ( ) = 3 s 3 + 4 K D s 2 + 3 + 4 K P ( ) s + 4 K I = s 3 + 4 3 K D s 2 + 1 + 4 3 K P ! " # $ % & s + 4 3 K I = 0 (1) Design closed-loop characteristic equation : The problem asks us to find values of K P , K I and K D such that the dominant poles are complex (recall that we need ! = 0.5 ). These dominant poles are governed by the following complex factor: s 2 + 2 !" n s + " n 2 . Since we have a third-order CLCE, we need an additional real factor of s + b , where b is, at this point, unknown. Therefore, our total design CLCE is of the form: 0 = s 2 + 2 !" n s + " n 2 ( ) s + b ( ) = s 3 + 2 !" n + b ( ) s 2 + 2 b !" n + " n 2 ( ) s + b " n 2 (2) Comparing coefficients for our general PID CLCE of (1) with those of the design CLCE
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: of (2), we have the following three equations: 4 3 K D = 2 !" n + b 1 + 4 3 K P = 2 b n + " n 2 4 3 K I = b n 2 (3) Since we desire ! = 0.5 and = 1 "# n = 1 , this gives: n = 1 = 1 0.5 = 2 . Substitution of this into equations (3) gives: 4 3 K D = 2 n + b = 2 + b # K D = 3 4 2 + b ( ) 1 + 4 3 K P = 2 b n + n 2 = 2 b + 4 # K P = 3 4 2 b + 3 ( ) 4 3 K I = b n 2 = 4 b # K I = 3 b What should we use for “b”? Recall that b is the third (real) pole of the CLCE. We simply need to choose b such that we are insured that the design poles above are DOMINANT. That is, we need to choose b such that its corresponding time constant is less than our design poles. In other words, we need b > n = 1 ....
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern