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IRREGULAR CASES

Learn how to handle irregular cases in perturbation method with different methods, including augmentation and coefficient listing.

josborn
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IRREGULAR CASES

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  1. IRREGULAR CASES • A first entry zero • Can be handled using different methods • Entire zero row • An important case which yields additional information 5.2 Irregular Cases

  2. FIRST ENTRY ZERO(Perturbation method) + + + - + + s5 1 2 11 s4 2 4 10 s3e 6 s2(4e-12)/e 10 @-12/e 10 s1@6 s0 10 5.2 Irregular Cases

  3. REMARKS • Consider the limit as e approaches zero at the earliest convenience • Simplifies calculations • Avoids ambiguities • It may get tedious. Even a second first entry zero is possible 5.2 Irregular Cases

  4. ANOTHER METHOD(Of limited use) • D(s) and D(1/x) have the same number of RHP roots • Retry by listing coefficients in reversed order x5 10 4 2 + x4 11 2 1 + x3 24/11 12/11 + x2 -3.5 1 - x1132/7 + x0 1 + 2 sign changes, 2 RHP roots 5.2 Irregular Cases

  5. YET ANOTHER METHOD (augmentation) • Multiply D(s) by s+a with a>0 to get Q(s)=(s+a)D(s) • Q(s) & D(s) have the same number of RHP roots • Increases complexity • If it does not work, try another ‘a’ • or even a second order factor 5.2 Irregular Cases

  6. Example(previous) Q(s)=(s5+2s4+2s3+4s2+11s+10)(s+1) =s6+3s5+4s4+6s3+15s2+21s+10 Hence R-H array starts with s6 1 4 15 10 s5 3 6 21 5.2 Irregular Cases

  7. + 1 4 15 10 + 3 6 21 + - + + + s6 / / / 1 2 7 s5 / / / 1 4 5 s4 2 8 10 / / -1 1 s3 -2 2 / / 1 1 s2 5 5 2 sign changes, 2 RHP roots s1 2 s0 1 5.2 Irregular Cases

  8. 2nd irregular caseALL-ZERO ROW D(s)=s5+2s4+2s3+4s2+2s+4 s5 | 1 2 2 s4 | 2 4 4 s3 | 0 0 An all-zero row indicates a factorization of symmetrical roots D(s)=A(s).Q(s) Auxiliary polynomial 5.2 Irregular Cases

  9. Auxilliary polynomial is obtained from the last row before the zero row s5 | 1 2 2 s4 | 2 4 4 s3 | 0 0 A(s)=2s4+4s2+4 has symmetrical roots! -0.4551  j1.0987 0.4551  j1.0987 5.2 Irregular Cases

  10. Symmetrical roots(wrt real&imag axes) OR OR OR OR any combination 5.2 Irregular Cases

  11. How to continue? A’(s): derivative of A(s) Replace next (all-zero) row by the coefficients of A’(s) s5 | 1 2 2 s4 | 2 4 4 s3 | 0 0 A(s)=2s4+4s2+4 A’(s)=8s3+8s 8 8 s2 | 2 4 s1 | -8 s0 | 4 & continue! # of sign changes = # of RHP roots including those of A(s) 5.2 Irregular Cases

  12. ExampleD(s)=s5+s4-s3-2s2-2s s5 | 1 -1 -2 s4 | 1 -2 0 s3 | 1 -2 s2 | 0 0 + + + + - - A(s)=s3-2s A’(s)=3s2-2 3 -2 s1 | -4/3 s0 | -2 1 sign change 1 RHP root 5.2 Irregular Cases

  13. C(s) + K G(s) R(s) - H(s) Feedback application k K 5.2 Irregular Cases

  14. where Characteristic equation zeros Are those of i.e. zeros of 5.2 Irregular Cases

  15. Remarks Use R-H test on Q(s) with K as a parameter & investigate sign changes for different K values Note that: CL poles  Ch. Eq’n zeros Ch. Eq’n. zeros determine system stability 5.2 Irregular Cases

  16. R C + K G(s) - Example Q(s)=s3+3s2+3s+1+K(s+7) = s3+3s2+(3+K)s+1+7K 5.2 Irregular Cases

  17. -1/7 2 0 0 Q(s)=s3+3s2+(3+K)s+1+7K s3 | 1 3+K s2 | 3 1+7K s1 | (8-4K)/3 s0 | 1+7K + + 2-K 1+7K K + + - 2-K - + + 1+7K 1 RHP root No RHP roots 2 RHP roots 5.2 Irregular Cases

  18. For varying K Poles move along a trajectory! called ROOT-LOCUS 5.2 Irregular Cases

  19. End of Chapter Restart section Next chapter Restart chapter i The End General index End show 5.2 Irregular Cases

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