1. Digital Control Systems The z-Transform

2. The z Transform Definition of z-Transform The z transform method is an operational method that is very powerful when working with discrete time systems. In considering the z-transform of a time function x(t), we consider only the sampled values of x(t), that is: x(0), x(T),x(2T),x(3T)…. The z-Transform of a time function x(t), where t is nonnegative, or of a sequence of values x(kT), where k takes zero or positive integers and T is sampling period is defined by the following equation: For a sequence of numbers x(k), the z-transform is defined by In the one sided z transform we assume x(t)=0 for t<0 or x(k)=0 for k<0. Note that z is a complex variable

3. The z Transform In the two sided z transform, the time function x(t) is assumed to be nonzero values for k<0 we assume x(t)=0 for t<0 or x(k)=0 for k<0. Note that z is a complex variable

4. Z Transforms of Elementary Functions

5. Decaying Exponential Function Damped Cosine Wave

6. Z Transforms of Elementary Functions Unit Ramp Function

7. Z Transforms of Elementary Functions Polynomial Function a k

8. Table of Z Transforms

9. Z-Transform Examples *

10. Z Transform Examples ****

11. Properties of the z-Transform Linearity Time Shift Multiplication by an Exponential Sequence Differentiation in z-Domain

12. Properties of the z-Transform Time Reversal Initial Value Theorem Convolution

13. Properties of the z-Transform Example:

14. Properties of the z-Transform Example Find the Z-transform of the causal sine sequence.

15. Properties of the z-Transform

16. Inverse z-Transform Other than referring to z transform tables, four methods for obtaining the inverse z transform are available: Direct Division Method Computational Method Partial Fraction Expansion Method Inversion Integral Method The inverse Z-transform can yield the corresponding time sequence f(kt) uniquely. However, it says nothing about f(t). There might be numerous f(t) for a given f(kT).

17. Inverse z-Transform Direct Division Method Express X(z) in powers of z −1

18. Inverse z-Transform Computational Method-Difference Equation Approach Example:

19. Inverse z-Transform Computational Method-Difference Equation Approach Example:

20. Inverse z-Transform Computational Method-Difference Equation Approach Example(cntd.): By substituting k=-2 into eqn., we find y(0)=0 By substituting k=-1 into eqn., we find y(1)=0.4673 By substituting k=0 into eqn., we find y(2)=0.3769. ………………………………. ……… Finding difference equation from TF.

21. Inverse z-Transform Computational Method-Matlab Approach Example:

22. Inverse z-Transform Computational Method-Matlab Approach Example (cntd.):

23. Inverse z-Transform Partial Fraction Expansion

24. Inverse z-Transform Example:

25. z-Transform Method for Solving Difference Equation Time shift Property of Z-transform

26. z-Transform Method for Solving Difference Equation Example:

27. z-Transform Method for Solving Difference Equation Example: