Lecture #08

1. Lecture #08 LTI system stability Time domain analysis 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) signals & systems

3. Continuous-time LTI system Laplace transform Ordinary differential equation Transfer function signals & systems

4. State equation (Simultaneous first-order ODE) Dynamic equation output equation signals & systems

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

8. System response • First order system response • Second order system response • High order system response signals & systems

9. First order signals & systems

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

11. Two characteristic roots are real and distinct. signals & systems

12. Two characteristic roots are equal signals & systems

13. Two characteristic roots are complex numbers Undamped natural frequency Damping ratio signals & systems

14. signals & systems

15. signals & systems

16. signals & systems

17. Higher-order system Dominant root nondominant root signals & systems

18. signals & systems

19. signals & systems

20. signals & systems

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

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