Automatic Cintrol by Meiling CHEN1 Lesson 6 (absolute) Stability Automatic control 2. Analysis

Automatic Cintrol by Meiling CHEN2 Stability Internal behavior –The effect of all characteristic roots. External behavior –The effect by cancellation of some transfer function poles.

Automatic Cintrol by Meiling CHEN3 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 CHEN4 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) Asymptotic stable => BIBO stable BIBO stable=> Asymptotic stable

Automatic Cintrol by Meiling CHEN5 System response (i)First order system response (ii)Second order system response (iii)High order system response

Automatic Cintrol by Meiling CHEN6 First order

Automatic Cintrol by Meiling CHEN7 Second order (a)Two characteristic roots are real and distinct. (b)Two characteristic roots are equal. (c)Two characteristic roots are complex numbers. Three cases :

Automatic Cintrol by Meiling CHEN8 Two characteristic roots are real and distinct.

Automatic Cintrol by Meiling CHEN9 Two characteristic roots are equal

Automatic Cintrol by Meiling CHEN10 Two characteristic roots are complex numbers Undamped natural frequency Damping ratio

Automatic Cintrol by Meiling CHEN11

Automatic Cintrol by Meiling CHEN12

Automatic Cintrol by Meiling CHEN13

Automatic Cintrol by Meiling CHEN14 Higher-order system Dominant rootnondominant root

Automatic Cintrol by Meiling CHEN15

Automatic Cintrol by Meiling CHEN16

Automatic Cintrol by Meiling CHEN17

Automatic Cintrol by Meiling CHEN18 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 CHEN19 Routh-Hurwitz testing The number of RHP roots of P(s) is the number of algebra sign changes in the elements of the left column of the array.

Automatic Cintrol by Meiling CHEN20 Example 1 Two roots in the RHP

Automatic Cintrol by Meiling CHEN21 Example 2 no root in the RHP

Automatic Cintrol by Meiling CHEN22 Example 3 Two roots in the RHP n 移位次數移 至 0 消失為止

Automatic Cintrol by Meiling CHEN23 Example 4 factor

Automatic Cintrol by Meiling CHEN24 no root in the RHP

Automatic Cintrol by Meiling CHEN25 Example 5 + -