Download presentation
Presentation is loading. Please wait.
1
LTI system stability Time domain analysis
Lecture #08 LTI system stability Time domain analysis meiling chen signals & systems
2
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
3
Continuous-time LTI system
Laplace transform Ordinary differential equation Transfer function meiling chen signals & systems
4
State equation (Simultaneous first-order ODE) Dynamic equation
output equation meiling chen signals & systems
5
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
6
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
7
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
8
System response First order system response
Second order system response High order system response meiling chen signals & systems
9
First order meiling chen signals & systems
10
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
11
Two characteristic roots are real and distinct.
meiling chen signals & systems
12
Two characteristic roots are equal
meiling chen signals & systems
13
Two characteristic roots are complex numbers
Undamped natural frequency Damping ratio meiling chen signals & systems
14
meiling chen signals & systems
15
meiling chen signals & systems
16
meiling chen signals & systems
17
Higher-order system Dominant root nondominant root meiling chen
signals & systems
18
meiling chen signals & systems
19
meiling chen signals & systems
20
meiling chen signals & systems
21
= natural response + forced response
Part I : Characteristic roots Part II : Input function Steady state error Sensitivity …etc. meiling chen signals & systems
22
= transient response + steady state response
rising time delay time maximum overshoot settling time …etc. Steady state error meiling chen signals & systems
Similar presentations
