D with dash-transform

D with a transform trait is an integral transform associated with continuous and discrete Laplace transforms. The direct D with the transformation transform associates the image of a continuous function with the image of the corresponding discrete function. It is widely used in the sections of control theory related to discrete systems.

Definition

Let be $X_{T}(p)$ ${\ displaystyle X_ {T} (p)}$ $X_{T}(p)$ - Laplace image of some continuous function $x_{T}(t)$ ${\ displaystyle x_ {T} (t)}$ $x_{T}(t)$ , but $X^{*}(q,\varepsilon )$ ${\ displaystyle X ^ {*} (q, \ varepsilon)}$ $X^{*}(q,\varepsilon )$ - image of the corresponding discrete function $x[n,\varepsilon ]=x_{T}((n+\varepsilon )T)$ ${\ displaystyle x [n, \ varepsilon] = x_ {T} ((n + \ varepsilon) T)}$ $x[n,\varepsilon ]=x_{T}((n+\varepsilon )T)$ where T is the sampling period, $n\in \mathbb {N} \cup \{0\}$ ${\ displaystyle n \ in \ mathbb {N} \ cup \ {0 \}}$ $n\in \mathbb {N} \cup \{0\}$ .

We introduce a function $X(q)={\frac {1}{T}}X_{T}\left({\frac {q}{T}}\right)$ ${\ displaystyle X (q) = {\ frac {1} {T}} X_ {T} \ left ({\ frac {q} {T}} \ right)}$ $X(q)={\frac {1}{T}}X_{T}\left({\frac {q}{T}}\right)$ . Then

X^{*}(q,\varepsilon )={\overline {\mathcal {D}}}\{X(q)\}={\frac {1}{2\pi j}}\int \limits _{c-j\infty }^{c+j\infty }X(\eta ){\frac {e^{q}}{e^{q}-e^{\eta }}}e^{\varepsilon \eta }d\eta ,~\mathrm {Re} \,q>c

{\ displaystyle X ^ {*} (q, \ varepsilon) = {\ overline {\ mathcal {D}}} \ {X (q) \} = {\ frac {1} {2 \ pi j}} \ int \ limits _ {cj \ infty} ^ {c + j \ infty} X (\ eta) {\ frac {e ^ {q}} {e ^ {q} -e ^ {\ eta}}} e ^ {\ varepsilon \ eta} d \ eta, ~ \ mathrm {Re} \, q> c}

X^{*}(q,\varepsilon )={\overline {\mathcal {D}}}\{X(q)\}={\frac {1}{2\pi j}}\int \limits _{c-j\infty }^{c+j\infty }X(\eta ){\frac {e^{q}}{e^{q}-e^{\eta }}}e^{\varepsilon \eta }d\eta ,~\mathrm {Re} \,q>c

It can be shown ^[1] that

X^{*}(q,\varepsilon )=\sum _{k=1}^{l}{\underset {\eta =\eta _{k}}{\mathrm {Res} }}{\frac {e^{q}e^{\varepsilon \eta }}{e^{q}-e^{\eta }}}X(\eta ),

{\ displaystyle X ^ {*} (q, \ varepsilon) = \ sum _ {k = 1} ^ {l} {\ underset {\ eta = \ eta _ {k}} {\ mathrm {Res}}} { \ frac {e ^ {q} e ^ {\ varepsilon \ eta}} {e ^ {q} -e ^ {\ eta}} X (\ eta),}

X^{*}(q,\varepsilon )=\sum _{k=1}^{l}{\underset {\eta =\eta _{k}}{\mathrm {Res} }}{\frac {e^{q}e^{\varepsilon \eta }}{e^{q}-e^{\eta }}}X(\eta ),

and the deductions are taken at all poles of the function $X(q)$ ${\ displaystyle X (q)}$ $X(q)$ , So what

X^{*}(q,\varepsilon )=\sum _{r=-\infty }^{+\infty }e^{\varepsilon (q+2\pi jr)}X(q+2\pi jr)

{\ displaystyle X ^ {*} (q, \ varepsilon) = \ sum _ {r = - \ infty} ^ {+ \ infty} e ^ {\ varepsilon (q + 2 \ pi jr)} X (q + 2 \ pi jr)}

X^{*}(q,\varepsilon )=\sum _{r=-\infty }^{+\infty }e^{\varepsilon (q+2\pi jr)}X(q+2\pi jr)

Formula for inverse D with dash-transform:

X(q)={\overline {\mathcal {D}}}^{-1}\{X^{*}(q,\varepsilon )\}=\int \limits _{0}^{1}e^{-q\varepsilon }X^{*}(q,\varepsilon )d\varepsilon

{\ displaystyle X (q) = {\ overline {\ mathcal {D}}} ^ {- 1} \ {X ^ {*} (q, \ varepsilon) \} = \ int \ limits _ {0} ^ { 1} e ^ {- q \ varepsilon} X ^ {*} (q, \ varepsilon) d \ varepsilon}

X(q)={\overline {\mathcal {D}}}^{-1}\{X^{*}(q,\varepsilon )\}=\int \limits _{0}^{1}e^{-q\varepsilon }X^{*}(q,\varepsilon )d\varepsilon

Properties

Linearity: ${\overline {\mathcal {D}}}\left\{\sum _{i=1}^{n}\alpha _{i}X_{i}(q)\right\}=\sum _{i=1}^{n}\alpha _{i}{\overline {\mathcal {D}}}\{X_{i}(q)\}$ ${\ displaystyle {\ overline {\ mathcal {D}}} \ left \ {\ sum _ {i = 1} ^ {n} \ alpha _ {i} X_ {i} (q) \ right \} = \ sum _ {i = 1} ^ {n} \ alpha _ {i} {\ overline {\ mathcal {D}}} \ {X_ {i} (q) \}}$ ${\overline {\mathcal {D}}}\left\{\sum _{i=1}^{n}\alpha _{i}X_{i}(q)\right\}=\sum _{i=1}^{n}\alpha _{i}{\overline {\mathcal {D}}}\{X_{i}(q)\}$
Multiplication by $e^{kq},~k\in \mathbb {Z}$ ${\ displaystyle e ^ {kq}, ~ k \ in \ mathbb {Z}}$ $e^{kq},~k\in \mathbb {Z}$ : ${\overline {\mathcal {D}}}\{e^{kq}X(q)\}=e^{kq}{\overline {\mathcal {D}}}\{X(q)\}$ ${\ displaystyle {\ overline {\ mathcal {D}}} \ {e ^ {kq} X (q) \} = e ^ {kq} {\ overline {\ mathcal {D}}} \ {X (q) \}}$ ${\overline {\mathcal {D}}}\{e^{kq}X(q)\}=e^{kq}{\overline {\mathcal {D}}}\{X(q)\}$
Multiplication by $e^{-\gamma q},~0<\gamma <1$ ${\ displaystyle e ^ {- \ gamma q}, ~ 0 <\ gamma <1}$ $e^{-\gamma q},~0<\gamma <1$ : ${\overline {\mathcal {D}}}\{e^{-\gamma q}X(q)\}={\begin{cases}e^{-q}X^{*}(q,1+\varepsilon -\gamma ),&0\leqslant \varepsilon <\gamma ,\\X^{*}(q,\varepsilon -\gamma ),&\gamma \leqslant \varepsilon <1\end{cases}}$ ${\ displaystyle {\ overline {\ mathcal {D}}} \ {e ^ {- \ gamma q} X (q) \} = {\ begin {cases} e ^ {- q} X ^ {*} (q , 1 + \ varepsilon - \ gamma), & 0 \ leqslant \ varepsilon <\ gamma, \\ X ^ {*} (q, \ varepsilon - \ gamma), & \ gamma \ leqslant \ varepsilon <1 \ end {cases} }}$ ${\overline {\mathcal {D}}}\{e^{-\gamma q}X(q)\}={\begin{cases}e^{-q}X^{*}(q,1+\varepsilon -\gamma ),&0\leqslant \varepsilon <\gamma ,\\X^{*}(q,\varepsilon -\gamma ),&\gamma \leqslant \varepsilon <1\end{cases}}$
Q offset by ± λ: ${\overline {\mathcal {D}}}\{X(q\pm \lambda )\}=e^{\mp \lambda \varepsilon }X^{*}(q\pm \lambda ,\varepsilon )$ ${\ displaystyle {\ overline {\ mathcal {D}}} \ {X (q \ pm \ lambda) \} = e ^ {\ mp \ lambda \ varepsilon} X ^ {*} (q \ pm \ lambda, \ varepsilon)}$
Multiplication by q: ${\overline {\mathcal {D}}}\{qX(q)\}={\frac {\partial }{\partial \varepsilon }}{\overline {\mathcal {D}}}\{X(q)\}$ ${\ displaystyle {\ overline {\ mathcal {D}}} \ {qX (q) \} = {\ frac {\ partial} {\ partial \ varepsilon}} {\ overline {\ mathcal {D}}} \ { X (q) \}}$
Division by q: ${\overline {\mathcal {D}}}\left\{{\frac {X(q)}{q}}\right\}=\int \limits _{0}^{\varepsilon }{\overline {\mathcal {D}}}\{X(q)\}d\varepsilon +{\frac {1}{e^{q}-1}}\int \limits _{0}^{1}{\overline {\mathcal {D}}}\{X(q)\}d\varepsilon$ ${\ displaystyle {\ overline {\ mathcal {D}}} \ left \ {{\ frac {X (q)} {q}} \ right \} = \ int \ limits _ {0} ^ {\ varepsilon} { \ overline {\ mathcal {D}}} \ {X (q) \} d \ varepsilon + {\ frac {1} {e ^ {q} -1}} \ int \ limits _ {0} ^ {1} {\ overline {\ mathcal {D}}} \ {X (q) \} d \ varepsilon}$
Q Differentiation: ${\overline {\mathcal {D}}}\left\{{\frac {d}{dq}}X(q)\right\}={\frac {\partial }{\partial q}}{\overline {\mathcal {D}}}\{X(q)\}-\varepsilon {\overline {\mathcal {D}}}\{X(q)\}$ ${\ displaystyle {\ overline {\ mathcal {D}}} \ left \ {{\ frac {d} {dq}} X (q) \ right \} = {\ frac {\ partial} {\ partial q}} {\ overline {\ mathcal {D}}} \ {X (q) \} - \ varepsilon {\ overline {\ mathcal {D}}} \ {X (q) \}}$

Some Transformation

$X_{T}(s)={\mathcal {L}}\{x_{T}(t)\}$ ${\ displaystyle X_ {T} (s) = {\ mathcal {L}} \ {x_ {T} (t) \}}$	$X(q)={\frac {1}{T}}X_{T}\left({\frac {q}{T}}\right)$ ${\ displaystyle X (q) = {\ frac {1} {T}} X_ {T} \ left ({\ frac {q} {T}} \ right)}$	$X^{}(q,\varepsilon )={\mathcal {D}}\{x[n,\varepsilon ]\}={\overline {\mathcal {D}}}\{X(q)\}$ ${\ displaystyle X ^ {} (q, \ varepsilon) = {\ mathcal {D}} \ {x [n, \ varepsilon] \} = {\ overline {\ mathcal {D}}} \ {X (q ) \}}$
${\frac {1}{s}}$ ${\ displaystyle {\ frac {1} {s}}}$	${\frac {1}{q}}$ ${\ displaystyle {\ frac {1} {q}}}$	${\frac {e^{q}}{e^{q}-1}}$ ${\ displaystyle {\ frac {e ^ {q}} {e ^ {q} -1}}}$
${\frac {T}{Ts+\beta }}$ ${\ displaystyle {\ frac {T} {Ts + \ beta}}}$	${\frac {1}{q+\beta }}$ ${\ displaystyle {\ frac {1} {q + \ beta}}}$	${\frac {e^{q}e^{-\beta \varepsilon }}{e^{q}-e^{-\beta }}}$ ${\ displaystyle {\ frac {e ^ {q} e ^ {- \ beta \ varepsilon}} {e ^ {q} -e ^ {- \ beta}}}}$
${\frac {T}{T^{2}s^{2}+2\zeta Ts\beta +\beta ^{2}}}$ ${\ displaystyle {\ frac {T} {T ^ {2} s ^ {2} +2 \ zeta Ts \ beta + \ beta ^ {2}}}$	${\frac {1}{q^{2}+2\zeta q\beta +\beta ^{2}}}$ ${\ displaystyle {\ frac {1} {q ^ {2} +2 \ zeta q \ beta + \ beta ^ {2}}}$	${\frac {e^{q}e^{-\zeta \beta \varepsilon }(e^{q}\sin \beta \varepsilon {\sqrt {1-\zeta ^{2}}}+e^{-\zeta \beta }\sin \beta (1-\varepsilon ){\sqrt {1-\zeta ^{2}}})}{\beta {\sqrt {1-\zeta ^{2}}}(e^{2q}-2e^{q}e^{-\zeta \beta }\cos \beta {\sqrt {1-\zeta ^{2}}}+e^{-2\zeta \beta })}}$ ${\ displaystyle {\ frac {e ^ {q} e ^ {- \ zeta \ beta \ varepsilon} (e ^ {q} \ sin \ beta \ varepsilon {\ sqrt {1- \ zeta ^ {2}}} + e ^ {- \ zeta \ beta} \ sin \ beta (1- \ varepsilon) {\ sqrt {1- \ zeta ^ {2}})} {\ beta {\ sqrt {1- \ zeta ^ {2} }} (e ^ {2q} -2e ^ {q} e ^ {- \ zeta \ beta} \ cos \ beta {\ sqrt {1- \ zeta ^ {2}}} + e ^ {- 2 \ zeta \ beta})}}}$

Notes

↑ Golovanov M. A., Ivanov V. A. Lecture notes for the course “Theory of digital automatic control systems”: Part 1. - Moscow: MGTU publishing house, 1990. - P. 44−46.

[1] Golovanov M. A., Ivanov V. A. Lecture notes for the course “Theory of digital automatic control systems”: Part 1. - Moscow: MGTU publishing house, 1990. - P. 44−46.

D with dash-transform

Definition

Properties

Some Transformation

Notes

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