1. meiling chensignals & systems1 Lecture #2 Introduction to Systems

2. meiling chensignals & systems2 system A system is an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals.

3. meiling chensignals & systems3 Example of system

4. meiling chensignals & systems4 System interconnection

5. meiling chensignals & systems5 System properties Causality Linearity Time invariance Invertibility

6. meiling chensignals & systems6 Causality A system is said to be causal if the present value of the output signal depends only on the present or past values of the input signal.

7. meiling chensignals & systems7 Causal and noncausal system Example: distinguish between causal and noncausal systems in the following: (1) Case I Noncausal system

8. meiling chensignals & systems8 (2) Case II causal system Delay system (3) Case III causal system At present past

9. meiling chensignals & systems9 (4) Case IV noncausal system At presentfuture (5) Case V noncausal system

10. meiling chensignals & systems10

11. meiling chensignals & systems11 Linearity A system is said to be linear in terms of the system input x(t) and the system output y(t) if it satisfies the following two properties of superposition and homogeneity. Superposition ： Homogeneity ：

12. meiling chensignals & systems12 Example 1.19 linear system

13. meiling chensignals & systems13 Example 1.20 Non linear system

14. meiling chensignals & systems14 Properties of linear system : (1) (2)

15. meiling chensignals & systems15 Time invariance A system is said to be time invariant if a time delay or time advance of the input signal leads to an identical time shift in the output signal. Time invariant system

16. meiling chensignals & systems16 Example 1.18 Time varying system

17. meiling chensignals & systems17 Invertibility A system is said to be Invertible if the input of the system can be recovered from the output. HH inv

18. meiling chensignals & systems18 Example 1.15 Inverse system Example 1.16

19. meiling chensignals & systems19 LINEAR TIME-INVARIANT (LTI) SYSTEMS: A basic fact: If we know the response of an LTI system to some inputs, we actually know the response to many inputs System identification

20. meiling chensignals & systems20

21. meiling chensignals & systems21 example The system is governed by a linear ordinary differential equation (ODE) Linear time invariant system linearity

22. meiling chensignals & systems22 LTI System representations 1.Order-N Ordinary Differential equation 2.Transfer function (Laplace transform) 3.State equation (Finite order-1 differential equations) ) 1.Ordinary Difference equation 2.Transfer function (Z transform) 3.State equation (Finite order-1 difference equations) Continuous-time LTI system Discrete-time LTI system

23. meiling chensignals & systems23 constants Order-2 ordinary differential equation Continuous-time LTI system Transfer function Linear system  initial rest

24. meiling chensignals & systems24

25. meiling chensignals & systems25 System response: Output signals due to inputs and ICs. 1. The point of view of Mathematic: 2. The point of view of Engineer: 3. The point of view of control engineer: Homogenous solutionParticular solution ＋ ＋ ＋ Zero-state responseZero-input responseNatural responseForced response Transient responseSteady state response

26. meiling chensignals & systems26 Example: solve the following O.D.E (1) Particular solution:

27. meiling chensignals & systems27 (2) Homogenous solution: have to satisfy I.C.

28. meiling chensignals & systems28 (3) zero-input response: consider the original differential equation with no input. zero-input response

29. meiling chensignals & systems29 (4) zero-state response: consider the original differential equation but set all I.C.=0. zero-state response

30. meiling chensignals & systems30 (5) Laplace Method:

31. meiling chensignals & systems31 Complex response Zero state responseZero input response Forced response (Particular solution) Natural response (Homogeneous solution) Steady state responseTransient response