Operator calculation according to Mikusiński

The operator calculation according to Mikusiński is an operator calculation of electrical engineering and the system theory of communications engineering , which was worked out in 1950 by Jan Mikusiński . With this he justified the empirical operator calculation according to Heaviside by modern algebraic methods on the basis of the convolution product directly and completely new. His operator calculation does not go the “detour” of the Laplace transformation with separate time and image domains. It is more far-reaching, logically simpler and more generally applicable.

Like all operator calculations, it is primarily used to transform linear differential equations into linear algebraic equations and thus to simplify their solution considerably. In particular when calculating linear networks in electrical engineering , the direct use of the differential operator for inductances and capacitances allows the classic calculation methods of direct current technology to be applied to any signal shape .

From a mathematical-algebraic point of view, the ring of continuous time functions is expanded with the convolution operation as multiplication by forming its quotient field to form the field of the Mikusiński operators , which enables “breaks” of functions as a reverse operation of the convolution.

Advantages and disadvantages

Advantages of the operator calculation according to Mikusiński

• An operator is directly a mathematical model of a signal or system .
• No detour via an image area (frequency area) is necessary, but you always work in the original area ( time area ).
• Convergence studies and the resulting restrictions are not necessary.
• It is not necessary to work with distributions to describe the Dirac impulse (and similar signals).
• No knowledge of complex analysis from function theory is required.

Disadvantages of the operator calculation according to Mikusiński

• The algebraic justification is mathematically very abstract and for the mostly little algebraically trained "practicing engineers" not easy to understand.
• The transition to the “imaginary frequency” often used in practice and thus the spectral representation of signals are not immediately obvious.

For this reason, and because of the limited literature available on operator calculation according to Mikusiński, the Laplace transformation is still the most widely used or learned method of operator calculation, both in engineering practice and in engineering training .

Requirements and notation

Like all operator calculations, the operator calculation according to Mikusiński can only be used for linear time-invariant systems . The signals must vanish for t <0 and be continuous for t ≥ 0 . The constructed operators can, however, also represent signals with a finite number of points of discontinuity .

To a function of time (. For example, x (t)) to be marked as such and lift from its value, and to display the disappearance with a negative time, it is written between curly brackets: . In the literature angle brackets are often used: . Sometimes even the function identifier without "(t)" are written . ${\displaystyle \left\{x(t)\right\}}$ ${\displaystyle \langle x(t)\rangle }$ ${\displaystyle x}$

Construction of the Mikusiński operators

Step 1: the ring of continuous functions

The following operations are defined on the set of functions that are continuous in the interval : ${\displaystyle [0,\infty )}$

The addition assigns the two time functions and the function obtained by "ordinary" addition of the function values: ${\displaystyle \{f(t)\}}$ ${\displaystyle \{g(t)\}}$

${\displaystyle \{f(t)\}+\{g(t)\}=\{f(t)+g(t)\}}$.

The neutral element of addition is therefore the null function . ${\displaystyle \{0\}}$

The multiplication assigns the two functions and their convolution product , so that applies ${\displaystyle \{f(t)\}}$ ${\displaystyle \{g(t)\}}$

${\displaystyle \{f(t)\}\cdot \{g(t)\}=\left\lbrace \int \limits _{0}^{t}f(u)\cdot g(t-u)\cdot du\right\rbrace =\{f(t)\star g(t)\}}$.

In the spelling with curly brackets, the convolution of two functions (also referred to as the product of functions in the literature) is usually symbolized by the simple "painting point" (which can also be omitted), which is not associated with the "normal" multiplication operator (also referred to as the product of values ) may be confused. For example, the following applies to the functional product, i.e. the convolution

${\displaystyle \{t\}\cdot \{e^{t}\}=\{e^{t}-t-1\}\neq \{t\cdot e^{t}\}}$.

There is no neutral element of this multiplication (“one element”), because the delta distribution that would realize this is not a continuous function. This set of continuous functions becomes a commutative ring without a single element, for which applies:

• Titchmarsh theorem : The ring is zero-divisor-free (i.e. the convolution of two functions is only equal if at least one function is). This makes it an area of integrity (in the "broader sense", because of the missing single element). ${\displaystyle \{0\}}$ ${\displaystyle \{0\}}$
• The step function takes on the role of the integration operator. There is no corresponding reverse operator. ${\displaystyle \{1\}}$
• Not every equation has a solution that is a continuous function. The reverse operation of the convolution (“convolution division”) therefore “does not always open”. For example, it can not be solved as a continuous function. ${\displaystyle \{f(t)\}\cdot \{x(t)\}=\{g(t)\}}$ ${\displaystyle \{x(t)\}}$ ${\displaystyle \{t\}\cdot \{x(t)\}=\{e^{t}\}}$

This situation is similar to dividing the whole numbers. There the fractions (which are nothing more than pairs of whole numbers) are introduced as new numbers and the corresponding operations are defined in such a way that the set of whole numbers is embedded in the set of fractions. The “fractions of functions” are constructed in the same way (in the next paragraph).

Step 2: Formation of the quotient field

The body theory shows: Each area of integrity can be clearly assigned a body in which it is isomorphically embedded, the so-called quotient body . The quotient field constructed from the integrity domain of the continuous functions is called the field of the Mikusiński operators . Its elements are called operators . For this purpose, pairs of functions [{x (t)}, {y (t)}] are formed and the following equivalence relation is introduced:

${\displaystyle \left[\{a(t)\},\{b(t)\}\right]=\left[\{c(t)\},\{d(t)\}\right]\Longleftrightarrow \{a(t)\}\cdot \{d(t)\}=\{b(t)\}\cdot \{c(t)\}\Longleftrightarrow a(t)*d(t)=b(t)*c(t)}$

The resulting equivalence classes are the desired operators. They are represented by their representatives, who are usually written with a "fraction line" based on the fraction calculation, but which must not be interpreted as that of the number division:

${\displaystyle \left[\{a(t)\},\{b(t)\}\right]={\frac {\{a(t)\}}{\{b(t)\}}}}$

The addition of two operators is defined as follows (based on the addition of fractions):

${\displaystyle {\frac {\{a(t)\}}{\{b(t)\}}}+{\frac {\{c(t)\}}{\{d(t)\}}}={\frac {\{a(t)\}\cdot \{d(t)\}+\{b(t)\}\cdot \{c(t)\}}{\{b(t)\}\cdot \{d(t)\}}}}$

The multiplication is also defined as separate multiplications (i.e. convolution) of the two "numerators" and "denominators":

${\displaystyle {\frac {\{a(t)\}}{\{b(t)\}}}\cdot {\frac {\{c(t)\}}{\{d(t)\}}}={\frac {\{a(t)\}\cdot \{c(t)\}}{\{b(t)\}\cdot \{d(t)\}}}}$

Finally, the following applies to the division of operators :

${\displaystyle {\frac {\{a(t)\}}{\{b(t)\}}}:{\frac {\{c(t)\}}{\{d(t)\}}}={\frac {\{a(t)\}\cdot \{d(t)\}}{\{b(t)\}\cdot \{c(t)\}}}}$

The function {0} must never appear in the denominator and the “painting point” is always to be understood as a convolution product.

Selected operators and their meanings

The one element

Every operator that has the same functions different from {0} in the numerator and denominator represents the neutral element of the multiplication ("1 element"), the operator 1 (which must not be confused with the jump function {1}):

${\displaystyle 1={\frac {\{x(t)\}}{\{x(t)\}}}}$

By definition, this operator does not change the function itself during multiplication ("convolution"). It thus has the functionality of the delta distribution . Some authors therefore write (not mathematically exact): ${\displaystyle \delta (t)}$

${\displaystyle 1=\{\delta (t)\}}$

The continuous functions

The (original) continuous functions {f (t)} themselves are contained in the operators of the following form as "sub-ring" with isomorphism and can still be written like the original ring elements:

${\displaystyle \{f(t)\}={\frac {\{f(t)\}\cdot \{x(t)\}}{\{x(t)\}}}}$

Their "division" (the reversal of the convolution) can now be carried out indefinitely in the body of the operators:

${\displaystyle \{f(t)\}:\{g(t)\}={\frac {\{f(t)\}}{\{g(t)\}}}}$

This means that the curly brackets can also be viewed as “operator brackets” that turn a time function into an operator.

The integration operator

The step function {1} continues to play the role of the integration operator:

${\displaystyle \{1\}={\frac {\{1\}\cdot \{f(t)\}}{\{f(t)\}}}={\frac {\left\lbrace \int \limits _{0}^{t}f(\tau )d\tau \right\rbrace }{\{f(t)\}}}}$

Number operators

The elements of the following form (where k represents any complex number ) are called number operators or constants :

${\displaystyle k={\frac {\{k\cdot x(t)\}}{\{x(t)\}}}}$

These operators are thus a subfield with isomorphism to the complex numbers and take over their function within the operator calculation.

The differential operator

The differential operator is naturally redefined as the inverse of the integration operator and is usually (based on the Laplace transform ) denoted by s (some authors also use p):

${\displaystyle s={\frac {\{f(t)\}}{\{1\}\cdot \{f(t)\}}}={\frac {1}{\{1\}}}}$

If the integration operator is used to derive a differentiable function (the value of which has dropped from 0 when differentiating), then

${\displaystyle \{1\}\cdot \{f'(t)\}=\{f(t)\}-f(0)\cdot \{1\}}$.

If you now replace the integration operator by 1 / s and solve it accordingly, you get the important rule for applying the differential operator to a differentiable function

${\displaystyle s\{f(t)\}=\{f'(t)\}+f(0)\,}$

or for the operator of a differential quotient

${\displaystyle \{f'(t)\}=s\{f(t)\}-f(0)\,}$.

The result of the differential operator is made up of the operator of the derivative of the function and a constant operator. This constant operator represents an initial condition and represents an essential difference to the differential operator of the Heaviside operator calculus , but it agrees with the rules of the Laplace transformation.

The representation of time functions by differential operators

The possibility of representing time functions using expressions with the differential operator s is essential for practical application. In the case of the Mikusiński operators, this gives the exact correspondence to the image functions of the Laplace transform.

• The unit jump as an integration operator is the reciprocal of the differential operator:

${\displaystyle \{1\}={\frac {1}{s}}}$

• The following applies to the ramp function

${\displaystyle \{t\}=\{1\}\cdot \{1\}={\frac {1}{s^{2}}}}$

• For the exponential function we get the property of the differential operator

${\displaystyle s\{e^{\gamma t}\}=\gamma \{e^{\gamma t}\}+1}$

Solving for the exponential function results in its operator representation

${\displaystyle \{e^{\gamma t}\}={\frac {1}{s-\gamma }}}$

• In general the following "correspondence" can be proven:

${\displaystyle \left\lbrace {\frac {t^{n-1}}{(n-1)!}}e^{\gamma t}\right\rbrace ={\frac {1}{(s-\gamma )^{n}}}}$

In practice, the correspondence tables of the Laplace transformation can be used for all of these “conversions” . This also applies to the calculation of the time functions from the operator expressions.

The interpretation of rational operator functions

When working with networks of lumped components and other systems that can be described by ordinary linear differential equations, one obtains a rational operator function (i.e. a fraction of polynomials in s) that describes the system (e.g. as a transfer function ). The interpretation as a function of time (in the Laplace transformation one would call this an inverse transformation) takes place e.g. B. by decomposing partial fractions and interpreting the partial fractions with the help of correspondence tables.

Examples for solving differential equations

An RLC series resonant circuit at the ideal voltage source u (t) can be described by the following (linear) DGL system (in state form ):

${\displaystyle L{\frac {di_{L}}{dt}}=u(t)-u_{C}-Ri_{L}}$

${\displaystyle C{\frac {du_{C}}{dt}}=i_{L}}$

With the introduction of the operator notation one obtains

${\displaystyle L[s\{i_{L}\}-i_{L}(+0)]=\{u(t)\}-\{u_{C}\}-R\{i_{L}\}}$

${\displaystyle C[s\{u_{C}\}-u_{C}(+0)]=\{i_{L}\}}$

Usually the resonance angular frequency , the decay constant and the natural angular frequency are used as abbreviations for resonant circuits . We assume a low damping (periodic case) so that applies . ${\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}}}$ ${\displaystyle \delta ={\frac {R}{2L}}}$ ${\displaystyle \omega ={\sqrt {\omega _{0}^{2}-\delta ^{2}}}}$ ${\displaystyle \omega _{0}>\delta }$

Discharge of a series resonant circuit

The special case of discharging the charged capacitor through a short circuit ( ) results ${\displaystyle u_{C}(+0)=U_{C0}}$ ${\displaystyle u(t)=0}$

${\displaystyle sL\{i_{L}\}=-\{u_{C}\}-R\{i_{L}\}}$

${\displaystyle C(s\{u_{C}\}-U_{C0})=\{i_{L}\}}$

By solving the first equation for the capacitor voltage and inserting it into the second equation, we eliminate this for calculating the current in the resonant circuit:

${\displaystyle C[-s(sL\{i_{L}\}+R\{i_{L}\})-U_{C0}]=\{i_{L}\}}$

Rearranging results

${\displaystyle (s^{2}LC+sRC+1)\cdot \{i_{L}\}=-U_{C0}\cdot C}$

With the abbreviations mentioned above and after division by LC, the equation can be rewritten:

${\displaystyle (s^{2}+2\delta s+\omega _{0}^{2})\cdot \{i_{L}\}=-{\frac {U_{C0}}{L}}}$

By solving for the current sought, determining the roots of the denominator and decomposing partial fractions , one gets

${\displaystyle \{i_{L}\}=-{\frac {U_{C0}}{L}}\cdot {\frac {1}{s^{2}+2\delta s+\omega _{0}^{2}}}=-{\frac {U_{C0}}{L}}\cdot {\frac {1}{(s+\delta -j\omega )\cdot (s+\delta +j\omega )}}=-{\frac {U_{C0}}{2j\omega L}}\cdot \left({\frac {1}{s+\delta -j\omega }}-{\frac {1}{s+\delta +j\omega }}\right)}$.

The operator expressions on the right are known as those of the exponential function (see above) and can therefore be written accordingly

${\displaystyle \{i_{L}\}=-{\frac {U_{C0}}{2j\omega L}}\cdot \left(\{e^{-\delta t+j\omega t}\}-\{e^{-\delta t-j\omega t}\}\right)=-{\frac {U_{C0}}{\omega L}}\cdot \left(\{e^{-\delta t}\cdot {\frac {e^{j\omega t}-e^{-j\omega t}}{2j}}\}\right)=-{\frac {U_{C0}}{\omega L}}\cdot \left(\{e^{-\delta t}\cdot \sin {\omega t}\}\right)}$

Finally one writes this operator equation again in "tense":

${\displaystyle i_{L}(t)=-{\frac {U_{C0}}{\omega L}}\cdot \left(e^{-\delta t}\cdot \sin {\omega t}\right)}$

Switching on a series resonant circuit

The special case of charging the uncharged capacitor by switching on a DC voltage results ${\displaystyle \{u(t)\}=\{U_{0}\}}$

${\displaystyle sL\{i_{L}\}=\{U_{0}\}-\{u_{C}\}-R\{i_{L}\}}$

${\displaystyle sC\{u_{C}\}=\{i_{L}\}}$

By solving the first equation for the capacitor voltage and plugging it into the second equation, we eliminate this:

${\displaystyle sC(\{U_{0}\}-sL\{i_{L}\}-R\{i_{L}\})=\{i_{L}\}}$

Rearranging results

${\displaystyle (s^{2}LC+sRC+1)\cdot \{i_{L}\}=\{U_{0}\}\cdot sC}$

With the abbreviations mentioned above and after division by LC, the equation can be rewritten:

${\displaystyle (s^{2}+2\delta s+\omega _{0}^{2})\cdot \{i_{L}\}={\frac {s\{U_{0}\}}{L}}}$

The differential operator s and the integrating switch-on jump cancel each other out to form the constant operator : ${\displaystyle \{U_{0}\}}$ ${\displaystyle U_{0}}$

${\displaystyle (s^{2}+2\delta s+\omega _{0}^{2})\cdot \{i_{L}\}={\frac {U_{0}}{L}}}$.

So we have practically the same equation as for the switch-off process and thus the same solution (with the opposite sign )

${\displaystyle i_{L}(t)={\frac {U_{0}}{\omega L}}\cdot \left(e^{-\delta t}\cdot \sin {\omega t}\right)}$

Operator series and operator functions

limit

For necessary border crossings when working with operator series, partial differential equations and non-rational operators, an extended concept of the convergence of operator sequences is introduced: If a function in the classical sense converges to, then the sequence of operators also converges to the operator , where g is any operator . For example, it follows from the known convergence of ${\displaystyle \{f_{n}(t)\}}$ ${\displaystyle \{f(t)\}}$ ${\displaystyle g\cdot \{f_{n}(t)\}}$ ${\displaystyle g\cdot \{f(t)\}}$

${\displaystyle \lim _{\omega \to \infty }\left\lbrace {\frac {\sin \omega t}{\omega }}\right\rbrace =\{0\}}$

the convergence in the sense of the operator calculation

${\displaystyle \lim _{\omega \to \infty }\{\cos \omega t\}=\lim _{\omega \to \infty }s\cdot \left\lbrace {\frac {\sin \omega t}{\omega }}\right\rbrace =s\cdot \{0\}=\{0\}}$.

Operator series

With this concept of convergence, the regularities and sum formulas of the "classical" series theory can be adopted on operator series. A typical example is the binomial series :

${\displaystyle \sum _{n=0}^{\infty }{\nu \choose n}{\{f(t)\}}^{n}=(1+\{f(t)\})^{\nu }}$

Operator functions

An operator that depends on a real number from a certain interval is called an operator function . In particular, parametric operator functions are defined for every real number. Operator functions can be continuous and differentiable with respect to the parameter. Then the derivative of the operator function is defined as the operator of the partial derivative of the function defining the operator. So is for example , then applies . ${\displaystyle u(x)=\{\sin xt\}}$ ${\displaystyle u'(x)=\left\lbrace {\frac {\partial }{\partial x}}\sin xt\right\rbrace }$

The displacement operator

The shift operator is an operator function dependent on the parameter T, which shifts the function to which it is applied by the time T "to the right": ${\displaystyle h^{T}}$

${\displaystyle h^{T}\{f(t)\}={\begin{cases}0&:t<T\\f(t-T)&:t\geq T\end{cases}}}$

On the basis of the calculation rules for the "usual" power calculus that can be derived from the definition and some other definitions for the displacement operator , this can advantageously be defined as an exponential function in relation to the differential operator s:

${\displaystyle h^{T}=e^{-sT}}$.

With the help of the shift operator, discontinuous functions (e.g. the step function ) and periodic functions can now also be represented as operators. This ensures compatibility with the complex alternating current calculation and the Laplace transformation .

annotation

In the older literature (e.g. Berg ) it is not the convolution that is used as the functional product, but the time derivative of the convolution product. Although this is only a formal difference, it disrupts the compatibility with the “modern” Laplace transformation , but is compatible with the so-called Laplace-Carson transformation .

See also

• Complex AC bill
• Extended symbolic method of alternating current engineering
• Operator calculation according to Heaviside
• Systems theory (engineering sciences)

literature

• Jan Mikusiński : Operator calculation . German Science Publishers, Berlin 1957.
• FH Lange: Signals and Systems - Volume 1: Spectral Representation . Verlag Technik, Berlin 1965.
• Lothar Berg : Introduction to operator calculus . German Science Publishing House, Berlin 1965.
• Manfred Peschel : Modern applications of algebraic methods . Verlag Technik, Berlin 1971.
• Peter Vielhauer : Theory of transmission on electrical lines . Verlag Technik, Berlin 1970.
• Peter Vielhauer: Linear Networks . Verlag Technik, Berlin 1982.
• Gerhard Wunsch : Systems Theory . Academic publishing company Geest & Portig K.-G., Leipzig 1975.
• Gerhard Wunsch: History of Systems Theory . Akademie-Verlag, Leipzig 1985.

Individual evidence

1. ^ Gerhard Wunsch : Algebraic Basic Concepts . Verlag Technik, Berlin 1970, DNB  458706388 .