ME369 Modeling,Analysis and System Control-A --Lecture 141103 Week 8# Routh Criterion Ch5.7 Nov.3(M) HW3 due Steady State Error Ch5.8 Practical lecture 1-control Nov.5(W) system modeling Quiz 2 Week 9# Nov.10(M) Frequency-Introduction,Response Ch7.1 HW4 due Now.12(W)） Frequency-Nyquist Plot Ch7.3 HW5
ME369 Modeling, Analysis and System Control-A --Lecture_141103 Week 8# Nov.3(M) Routh Criterion Steady State Error Ch5.7 Ch5.8 HW3 due Nov.5(W) Practical lecture 1 – control system modeling Quiz 2 Week 9# Nov.10(M) Frequency-Introduction, Response Ch7.1 Nov.12(W) Frequency-Nyquist Plot Ch7.3 HW4 due HW5
Routh Array a(s)=a,s6+a5s5+…+as+a0=0 06 ao a 43 a 0 S 4 aa4-a6凸 二A asaasa=B 0 as as as Aa3- BasC Aaaoas=D A0-0a=0 0 A A A 2 CB-AD =E Cao-40 S 二do 0 0 C ED E0-C0 S -Cao=F =0 0 0 E F Fao F0-E0 0 0 0 F F The number of sign changes of the first column elements of the Routh Array is equal to the number of unstable roots
Routh Array s 6 a 6 a 4 a2 a 0 s 5 a 5 a 3 a1 0 s 4 a 5 a 4 a 6 a 3 a 5 A a 5 a2 a 6 a1 a 5 B a 5 a 0 a 6 0 a 5 a 0 0 s 3 Aa 3 Ba 5 A C Aa1 a 0 a 5 A D A 0 0 a 5 A 0 0 s 2 CB AD C E Ca 0 A 0 C a 0 0 0 s ED Ca 0 E F E 0 C 0 F 0 00 s 0 Fa 0 E 0 F a 0 F 0 E 0 F 0 00 The number of sign changes of the first column elements of the Routh Array is equal to the number of unstable roots. a ( s ) a 6 s 6 a 5 s 5 a 1 s a 0 0
Example--Verification (s-1)2(s+2)(s+3)=s4+3s3-3s2-7s+6 1 -3 6 3 -7 0 -2 6 3 0 S 20 0 S 6 Examine 1st column,of sign changes= 日96
Example--Verification ( s 1) 2 ( s 2)( s 3) s 4 3 s 3 3 s 2 7 s 6 4 3 2 0 1 36 3 70 s s s s s 2 3 6 20 0 6 0 Examine 1st column, # of sign changes=
Example 10-5,Ogata,pp.541 For 3d-order polynomial: a0s3+a1s2+a2s+a3=0 The array of coefficients becomes s ao 02 a 03 a az -aoa3 0 1 s° a 0 Stable condition: Enough? a az aa;
Example 10-5, Ogata, pp.541 3 0 1 2 2 1 3 a 0 s a s a s a For 3rd-order polynomial: 0 0 3 0 1 1 1 2 0 3 1 3 2 0 2 3 s a a a a a a s s a a s a a The array of coefficients becomes Stable condition: 1 2 0 3 a a a a Enough?
Example--Verification (s2+1)(s+2)(s+3)=6s4+6s3+6s2+6s+5 665 6 6 5 S 5O Examine 1st column,of sign changes= 6
Example--Verification 2 432 ( 1)( 2)( 3) 6 6 6 6 5 s s s ssss 4 3 2 0 665 660 s s s s s 0 5 Examine 1st column, # of sign changes=