problem_11_21

# problem_11_21 - Problem 11.21 of the Palm textbook Plant TF...

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Problem 11.21 of the Palm textbook Plant TF: GP ( s ) = 4 2 3σ + 3 Design a PID controller that gives a DOMINANT CL pole corresponding to: • a damping ratio of ζ = 0.5 . • a time constant of τ = 1 . SOLUTION PID controller: GC ( s ) = ΚΠ + ΚΙ Κ σ2 + ΚΠσ+ ΚΙ + Κ∆ σ = ∆ σ σ General PID closed-loop characteristic equation: Κ σ2 + ΚΠσ+ ΚΙ 4 0 = 1 + Γ Χ ( σ) Γ Π ( σ) = 1 + ∆ ÷ ÷ 2 σ 3σ + 3 OR DCL ( s ) = σ( 3σ2 + 3) + 4 ( Κ∆ σ2 + ΚΠσ+ ΚΙ ) = 3σ3 + 4 Κ∆ σ2 + ( 3 + 4 ΚΠ ) σ+ 4 ΚΙ = σ3 + 4 4 4 Κ∆ σ2 + 1 + ΚΠ÷ σ+ ΚΙ = 0 3 3 3 (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 ζ = 0.5 ). These dominant poles are governed by the 2 following complex factor: s 2 + 2ζω ν σ+ ω ν . Since we have a third-order CLCE, we need an additional real factor of s + β , where b is, at this point, unknown. Therefore, our total design CLCE is of the form: 2 0 = ( σ2 + 2ζω ν σ+ ω ν ) ( σ+ β) 2 2 = σ3 + ( 2ζω ν + β) σ2 + ( 2βζω ν + ω ν ) σ+ βω ν (2) Comparing coefficients for our general PID CLCE of (1) with those of the design CLCE of (2), we have the following three equations: 4 K D = 2ζω ν + β 3 4 2 1 + ΚΠ = 2βζω ν + ω ν 3 4 2 ΚΙ = βω ν 3 Since we desire ζ = 0.5 and τ = (3) 1 1 1 = 1 , this gives: ω n = = = 2 . Substitution of zwn z 0.5 this into equations (3) gives: 4 K D = 2ζω ν + β = 2 + β 3 4 2 1 + ΚΠ = 2βζω ν + ω ν = 2β + 4 3 4 2 ΚΙ = βω ν = 4 β 3 ⇒ ⇒ 3 ( 2 + β) 4 3 ΚΠ = ( 2β + 3) 4 Κ∆ = ⇒ ΚΙ = 3β 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 > ζω ν = 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_11_21 - Problem 11.21 of the Palm textbook Plant TF...

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

View Full Document
Ask a homework question - tutors are online