Chapter 2 Mathematical Description of Systems

1. Chapter 2 Mathematical Description of Systems • To introduce some important concepts of systems, and then develop their associated mathematical descriptions. 長庚大學電機系

2. Outline • Some important concept of systems • Linear systems • Linear time-invariant systems • Linearization • Examples • Discrete-time systems 長庚大學電機系

3. System • A system is generally described in the Figure below: where we assume that an input results in a unique output. • SISO, MIMO, SIMO, Continuous-time system, discrete-time system. 長庚大學電機系

4. Memoryless system:output y(t)at depends only on the input applied at (independent of past and future input) • Causal or non-anticipatory system: current output depends on past and current input but not on future input. • The state of a system at time is the information at t0 that, together with the input for determines uniquely the output for all • If the state at is known, the output is completely determined by and the input after . This implies that the state summarizes the effect of past input on future output. 長庚大學電機系

5. A system is said to belumpedif its number of state variable is finite. Otherwise it is called adistributed system. Example: Consider the spring system If we know and the input u(t) for Then output is uniquely determined. The state at is Lumped system 長庚大學電機系

6. Example:Consider the unit-time delay system We need the information to determine Initial state(state at ) is , which has infinite many point distributed system. 長庚大學電機系

7. A system is said to be linear if • This is called thesuperposition property 長庚大學電機系

8. is calledzero input response is calledzero state response (superposition principle) Output due to= output due to + output due to i.e.,response = zero-input response + zero-state response 長庚大學電機系

9. Input-Output Description of linear system Define : Then 長庚大學電機系

10. Consider the input-output pairs • is calledimpulseresponsewhich denotes the output observed at time t due to impulse input applied at time . 長庚大學電機系

11. Causal for t< • A system is calledrelaxedat if x( )=0 • A linear system that is causal and relaxed at can be described by causal Linear relaxed at 長庚大學電機系

12. MIMO System • A p input and q output linear system that is also causal and relaxed at can be described by where is calledimpulse response. • is the impulse response at time t at the ith output terminal due to an impulse applied at time at the jth input terminal. 長庚大學電機系

13. State-space Description Every linear lumped system can be described by Implementation by Op-Amp circuits in pp. 16-17 長庚大學電機系

14. Linear Time-invariant Systems • A system is said to be time-invariant if Otherwise, the system is time-varying • Most of system are time-varying, why study LTI system? 長庚大學電機系

15. Example: Two time-varying examples: • In state space representation, how to determine a system is linear? time invariant? 長庚大學電機系

16. Input-output Description • Time invariant linear, time-invariant, causal and relax at t=0 can be described as convolution integral • output at time t due to an impulse input applied at time 0. 長庚大學電機系

17. Unit delay Example: Consider the unit delay system • The system is Linear, time-invariant and causal, but not lumped 長庚大學電機系

18. Example:Consider the unity feedback system • If • Let r(t) be any input with 長庚大學電機系

19. Transfer-function matrix • is called the Laplace transform of y(t) • Forlinear, causal, relax at t=0 and time-invariant system, we have • causal 長庚大學電機系

20. is called the transfer function of the system 長庚大學電機系

21. ForMIMO system where is the transfer function from the jth input to the ith output. • is called transfer-function matrix or transfer matrix. 長庚大學電機系

22. For unit-delay system, • For unity-feedback system, 長庚大學電機系

23. Rational Transfer Function • For linear lumped system, is a rational function. We define that isproper is strictly proper isbiproper isimproper 長庚大學電機系

24. Improper rational transfer function will amplify high-frequency noise rarely arise in practice • is called - a pole of - a zero of • D(s) and N(s) are calledcoprime if they have no common factor. 長庚大學電機系

25. State-Space Equation • Every linear, time-invariant, lumped system can be described by • If 長庚大學電機系

26. Linearization (Taylor’s Series Expansion) • If is differentiable at , then for small, where is the Jacobian matrix. • Let =f(x), is the 1st order (or linear) approximation of y=f(x) at 長庚大學電機系

27. Example: consider the pendulum equation. - by letting with - The Jacobian matrix is - If - If 長庚大學電機系

28. Examples • Consider the mechanical system. Suppose that - input u is the external force - output y is the displacement from the equilibrium - friction ~ - spring force = Then the input-output description has the form 長庚大學電機系

29. - Transfer function (assume that y(0)=0) - Let the state-space equation is 長庚大學電機系

30. Consider - Input-output description - Transfer matrix • where 長庚大學電機系

31. State-space description: Let 長庚大學電機系

32. Consider the RLC network - Apply KCL and KVL we have the state space description 長庚大學電機系

33. Discrete-Time System • Suppose that the input and output have the same sampling period • If superposition property holds linear system response=zero-state response + zero-input response 長庚大學電機系

34. A discrete-time system is causal if current output depends on current and past input. • is the state of the system at time . The entries of x are called state variables. • Lumped system: number of state variables is finite. Otherwise it is called a distributed system. 長庚大學電機系

35. Input-output Description • Define : impulse sequence • Consider the input-output pairs • is called the impulse response sequencewhich is the output at time instance k excited by impulse sequence at time instance m. 長庚大學電機系

36. Linear, causal and relaxed at k0 causal linearrelaxed at • Linear, causal, relaxed at and time-invariant discrete-convolution 長庚大學電機系

37. Discrete Transfer Function • is called the z-transformof • For a linear, causal, relaxed at 0 and time-invariantsystem • is called discrete transfer function 長庚大學電機系

38. Examples • Consider the unit sampling-time delay system Then • Consider the discrete-time feedback system. Then 長庚大學電機系

39. State-space Equation • Every linear, lumped and time-invariant discrete-time system can be described as • If • Discrete transfer matrix 長庚大學電機系

40. Example:(A money market account) • Interest rate r depends on the amount of money in the account nonlinear system. • r is constant linear system. • r is changing with time time-varying system. Otherwise it is a time-invariant system. • Consider LTI case with r=0.015% per day and compound daily. Then • If 長庚大學電機系

41. To develop state-equation, define • Why not choose ? 長庚大學電機系