problem_1121

problem_1121 - of (2), we have the following three...

Info iconThis 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
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background 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

This note was uploaded on 12/23/2011 for the course ME 375 taught by Professor Meckle during the Fall '10 term at Purdue.

Page1 / 2

problem_1121 - of (2), we have the following three...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online