LTI system stability Time domain analysis Lecture #08 LTI system stability Time domain analysis meiling chen signals & systems

System representations Continuous-time LTI system Ordinary differential equation Transfer function (Laplace transform) Dynamic equation (Simultaneous first-order ODE) Discrete-time LTI system Ordinary difference equation Transfer function (Z-transform) Dynamic equation (Simultaneous first-order ordinary difference equation) meiling chen signals & systems

Continuous-time LTI system Laplace transform Ordinary differential equation Transfer function meiling chen signals & systems

State equation (Simultaneous first-order ODE) Dynamic equation output equation meiling chen signals & systems

Stability Internal behavior External behavior The effect of all characteristic roots. External behavior The effect by cancellation of some transfer function poles. meiling chen signals & systems

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-plane) meiling chen signals & systems

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-plane) Asymptotic stable => BIBO stable BIBO stable=> Asymptotic stable meiling chen signals & systems

System response First order system response Second order system response High order system response meiling chen signals & systems

First order meiling chen signals & systems

Second order Two characteristic roots are real and distinct. Three cases : Two characteristic roots are real and distinct. Two characteristic roots are equal. Two characteristic roots are complex numbers. meiling chen signals & systems

Two characteristic roots are real and distinct. meiling chen signals & systems

Two characteristic roots are equal meiling chen signals & systems

Two characteristic roots are complex numbers Undamped natural frequency Damping ratio meiling chen signals & systems

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Higher-order system Dominant root nondominant root meiling chen signals & systems

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= natural response + forced response Part I : Characteristic roots Part II : Input function Steady state error Sensitivity …etc. meiling chen signals & systems

= transient response + steady state response rising time delay time maximum overshoot settling time …etc. Steady state error meiling chen signals & systems