Modified Z-Transformation

The modified Z-transformation represents an extension of the discrete-time Z-transformation in order to also be able to process signal values between integer sampling times within the framework of discrete-time control technology . The necessary modifications of the Z-transformation were presented in the work of Eliahu Ibrahim jury in 1958. The primary application is the detection of instabilities resulting from the discrete-time signal processing of control systems , for example in the field of power electronics for highly dynamic electrical drive systems.

definition

The modified Z-transformation has the following definition for causal signals with positive and integer k :

{\ displaystyle G (z, m) = \ sum _ {k = 0} ^ {\ infty} f ((k + m) T) z ^ {- k} \}

with the period duration and a "delay factor " that represents part of the period duration and is in the range . ${\ displaystyle T}$ ${\ displaystyle m}$ ${\ displaystyle [0,1)}$

The modified Z-transformation clearly represents a unilateral Z-transformation, which is an additional delay element with the delay before the sampling element to obtain the discrete signal sequence

{\ displaystyle e ^ {- (1-m) Ts}}

having. For time constants , the properties of the Z transformation apply . ${\ displaystyle m}$

Correspondence

Some of the important correspondences of the modified Z-transform are:

Time range f (t)	Spectral range G (z, m)
σ (t)	${\ displaystyle {\ frac {z} {z-1}}}$
${\ displaystyle t}$	${\ displaystyle {\ frac {mT} {z-1}} + {\ frac {T} {(z-1) ^ {2}}}}$
${\ displaystyle e ^ {- at}}$	${\ displaystyle {\ frac {e ^ {- amT}} {ze ^ {- aT}}}}$
${\ displaystyle 1-e ^ {- at}}$	${\ displaystyle {\ frac {1} {z-1}} + {\ frac {e ^ {- - amT}} {ze ^ {- aT}}}}$
${\ displaystyle \ sin (\ omega t)}$	${\ displaystyle {\ frac {z \ sin {(m \ omega T)} + \ sin {((1-m) \ omega T)}} {z ^ {2} -2z \ cos {\ omega T} + 1}}}$

For the correspondences go into the forms of the Z-transformation. The modified Z-transformation is therefore also referred to as the extended Z-transformation . ${\ displaystyle m = 0}$

Individual evidence

↑ Eliahu Ibrahim Jury: Sampled-Data Control Systems . John Wiley & Sons, 1958.

literature

Dierk Schröder: Electric drives - control of drive systems . 3. Edition. Springer, 2009, ISBN 978-3-540-89612-8 , chapter 6.

[jury1-1] Eliahu Ibrahim Jury: Sampled-Data Control Systems . John Wiley & Sons, 1958.