Microsoft PowerPoint - 107-2_dcs13_zTransform.pptxDCS13-zT-4 Feng-Li Lian © 2019The z-Transform: Example
DCS13-zT-5 Feng-Li Lian © 2019The z-Transform: Table
Astrom & Wittenmark 1997
Astrom & Wittenmark 1997
DCS13-zT-7 Feng-Li Lian © 2019The z-Transform: From State Space to Pulse Transfer Function
DCS13-zT-8 Feng-Li Lian © 2019The z-Transform: From State Space to Pulse Transfer Function
DCS13-zT-9 Feng-Li Lian © 2019The z-Transform: From State Space to Pulse Transfer Function
Pulse Transfer Function
G(z) u[k] y[k]
x[k]
DCS13-zT-10 Feng-Li Lian © 2019The z-Transform: Computation of G(z) from G(s)
DCS13-zT-11 Feng-Li Lian © 2019The z-Transform: Computation of G(z) from G(s)
DCS13-zT-12 Feng-Li Lian © 2019The z-Transform: Computation of H(z) from G(s)
DCS13-zT-13 Feng-Li Lian © 2019The z-Transform: and Shift-Operator Calculus
Example:
But, the solution of the difference equation is:
That is, by using the shift-operator calculus:
Different conclusions for:
DCS13-zT-15 Feng-Li Lian © 2019Poles and Zeros
DCS13-zT-16 Feng-Li Lian © 2019Poles and Zeros
DCS13-zT-17 Feng-Li Lian © 2019Poles and Zeros
DCS13-zT-18 Feng-Li Lian © 2019Poles and Zeros
DCS13-zT-19 Feng-Li Lian © 2019Poles and Zeros
DCS13-zT-20 Feng-Li Lian © 2019Poles and Zeros
DCS13-zT-21 Feng-Li Lian © 2019Poles and Zeros
DCS13-zT-22 Feng-Li Lian © 2019Poles and Zeros
DCS13-zT-23 Feng-Li Lian © 2019Poles and Zeros
DCS13-zT-24 Feng-Li Lian © 2019Poles and Zeros
DCS13-zT-25 Feng-Li Lian © 2019Poles and Zeros
d = #(p) - #(z)
Example:
The zero of the pulse-transfer function:
When h is small:
When h approaches 0:
Laplace transform:
Analysis of discrete systems by the z-transform:
Franklin et al. 2002
The z-transform inversion:
The z-transform
Franklin et al. 2002T: sampling period
DCS13-zT-32 Feng-Li Lian © 2019In Summary
Franklin et al. 2002T: sampling period
DCS13-zT-33 Feng-Li Lian © 2019In Summary
Properties of the z-transform:
Properties of the z-transform:
Oppenheim et al. 1997
Example – Use z-Transform to find system response Consider the difference equation:
DCS13-zT-36 Feng-Li Lian © 2019In Summary
Example – Use z-Transform to find system response If u[k] = unit impulse function:
DCS13-zT-37 Feng-Li Lian © 2019In Summary
Example – Use z-Transform to find system response If u[k] = unit step function:
DCS13-zT-38 Feng-Li Lian © 2019In Summary
Example – Use z-Transform to find system response Step Response:
Impulse Response:
Franklin et al. 2002T: sampling period
DCS13-zT-40 Feng-Li Lian © 2019In Summary
Pole location and response between s and z:
Franklin et al. 2002
s plane z plane
Dynamic Properties between s and z:
Franklin et al. 2002
Franklin et al. 2002
Ogata 1995
Final Value Theorem: