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Lesson 6

Automatic control 2. Analysis. Lesson 6. (absolute) Stability. Stability. Internal behavior The effect of all characteristic roots. External behavior The effect by cancellation of some transfer function poles. Definition :.

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Lesson 6

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  1. Automatic control 2. Analysis Lesson 6 (absolute) Stability Automatic Cintrol by Meiling CHEN

  2. Stability • Internal behavior • The effect of all characteristic roots. • External behavior • The effect by cancellation of some transfer function poles. Automatic Cintrol by Meiling CHEN

  3. Definition : A system is internal (asymptotic) stable, if the zero-input response decays to zero, as time approaches infinity, for all possible initial conditions. Asymptotic stable =>All the characteristic polynomial roots are located in the LHP (left-half-plan) Automatic Cintrol by Meiling CHEN

  4. Asymptotic stable => BIBO stable BIBO stable=> Asymptotic stable Definition : A system is external (bounded-input, bounded-output) stable, if the zero-state response is bounded, as time approaches infinity, for all bounded inputs.. bounded-input, bounded-output stable =>All the poles of transfer function are located in the LHP (left-half-plan) Automatic Cintrol by Meiling CHEN

  5. System response • First order system response • Second order system response • High order system response Automatic Cintrol by Meiling CHEN

  6. First order Automatic Cintrol by Meiling CHEN

  7. Second order Three cases : • Two characteristic roots are real and distinct. • Two characteristic roots are equal. • Two characteristic roots are complex numbers. Automatic Cintrol by Meiling CHEN

  8. Two characteristic roots are real and distinct. Automatic Cintrol by Meiling CHEN

  9. Two characteristic roots are equal Automatic Cintrol by Meiling CHEN

  10. Two characteristic roots are complex numbers Undamped natural frequency Damping ratio Automatic Cintrol by Meiling CHEN

  11. Automatic Cintrol by Meiling CHEN

  12. Automatic Cintrol by Meiling CHEN

  13. Automatic Cintrol by Meiling CHEN

  14. Higher-order system Dominant root nondominant root Automatic Cintrol by Meiling CHEN

  15. Automatic Cintrol by Meiling CHEN

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  17. Automatic Cintrol by Meiling CHEN

  18. Stability testing • Properties of the polynomial coefficients : • Differing algebraic signs • Zero-valued coefficients • All of the same algebraic sign, non zero At least one RHP root Has imaginary axis roots or RHP roots or both No direct information Automatic Cintrol by Meiling CHEN

  19. Routh-Hurwitz testing The number of RHP roots of P(s) isthe number of algebra sign changes in the elements of the left column of the array. Automatic Cintrol by Meiling CHEN

  20. Example 1 Two roots in the RHP Automatic Cintrol by Meiling CHEN

  21. Example 2 no root in the RHP Automatic Cintrol by Meiling CHEN

  22. Example 3 n 移位次數移至0消失為止 Two roots in the RHP Automatic Cintrol by Meiling CHEN

  23. Example 4 factor Automatic Cintrol by Meiling CHEN

  24. no root in the RHP Automatic Cintrol by Meiling CHEN

  25. + - Example 5 Automatic Cintrol by Meiling CHEN

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