1. Robotics Research Laboratory 1 Chapter 3 Discrete System Analysis - Sampled Data Systems

2. Robotics Research Laboratory 2 Linear Difference Equations If f is linear and has a finite dimension, can be written as is called a linear recurrence equation, or difference equation, or ARMA (Auto Regressive Moving Average) model. Remarks: i)Infinite-impulse response filter or recursive filter ii)Finite-impulse response filter or non-recursive filter iii) Difference equation General description of a discrete-time system is

3. Robotics Research Laboratory 3 A sequence of infinite time sequence Complex plane Time-domain Z-domain (Frequency-domain) If f is a linear time-invariant system, we have a discrete transform function.

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5. 5 Define the z-transform of the sequence {e(k)} as Similarly, can be written as (backward) (forward) or

6. Robotics Research Laboratory 6 Define H(z) is called a pulse(or discrete) transfer function or just a transfer function.

7. Robotics Research Laboratory 7 Derivation ex) unit delay ex) unit discrete pulse (impulse)

8. Robotics Research Laboratory 8 In general Take the z-transform

9. Robotics Research Laboratory 9 H(z)H(z) u(kT)e(kT) E(z)E(z) U(z)U(z) For If

10. Robotics Research Laboratory 10 Discrete Model of Sampled-data Systems D/A output for unit pulse input ( Model of D/A converter) G(s)G(s)A/D y(t)y(t) y(kT) D/A u(kT) G(z)G(z) 2TT 1 0

11. Robotics Research Laboratory 11 Remarks: i)The mathematical model of A/D converter is the same as a sampler (the train of unit impulse) ii)The D/A converter is mathematically modeled as a zero-order hold

12. Robotics Research Laboratory 12 ex)

13. Robotics Research Laboratory 13 Block-Diagram Analysis of Sampled-Data Systems Sampled-Data Systems H(s)G(s)

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15. Robotics Research Laboratory 15 Remark:

16. Robotics Research Laboratory 16 From (1) and (2)

17. Robotics Research Laboratory 17 Remark: G(s)G(s)

18. Robotics Research Laboratory 18 G(s)G(s) H(s)H(s)

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20. Robotics Research Laboratory 20 Digital Controller D/APlant

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23. Robotics Research Laboratory 23 ex)

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25. Robotics Research Laboratory 25 Response between Samples : i)Sub-multiple Sampling ii) Modified z-transform Sub-multiple Sampling G(s)G(s)

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27. Robotics Research Laboratory 27 ex)

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29. Robotics Research Laboratory 29 Modified Z-transform Method – fictitious delay time G(s)G(s) Define

30. Robotics Research Laboratory 30 Complex convolution theorem

31. Robotics Research Laboratory 31 Notes: i) any specific point between two consecutive instants ii) ex)

32. Robotics Research Laboratory 32 G(s)G(s) + H(s)H(s)

33. Robotics Research Laboratory 33 G(s) + e -(1-m)T H(s)

34. Robotics Research Laboratory 34 State Space Form Difference equation Differential equation

35. Robotics Research Laboratory 35 Define Take the z-transform Transfer Function H(z)

36. Robotics Research Laboratory 36 ex)

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38. Robotics Research Laboratory 38 State Space Models for System with Input Delay where l is an integer, m is a positive number less than one.

39. Robotics Research Laboratory 39 Define kT-lTkT-lT+TkT-lT+2T T mTmT 0

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41. Robotics Research Laboratory 41 (n+1) dim

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