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

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