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Where does the Laplace operator apply. TAU. Laplace operator and transfer functions. General curvilinear coordinates and Riemannian spaces

laplacian, is a differential operator defined by the formula

(here - coordinates in), as well as some of its generalizations. L. o. (1) is the simplest elliptic. differential operator of the 2nd order. L. o. plays an important role in mathematical. analysis, math. physics and geometry (see, for example, Laplace equation, Laplace - Beltrami equation, Harmonic function, Harmonic form).

Let Vengeance be an n-dimensional Riemannian space with metric

let be the matrix inverse to the matrix Then L. o. (or the Laplace - Beltrami operator) of the Riemannian metric (2) on M has the form

where - local coordinates on M. Operator (1) differs in sign from L. o. standard Euclidean metric

A generalization of the operator (3) is the L.O. on differential forms. Namely, in the space of all external differential forms on ML. about. has the form

where d - external differentiation operator of the form, d * - operator formally conjugate to d, defined by the following product on smooth finite forms:

where * is the Hodge operator generated by the metric (2) and transforming p-forms into ( etc) -forms. In formula (5), the forms a and b are considered real; on complex forms, the Hermitian continuation of the scalar product (5) must be used. The restriction of operator (4) to O-forms (i.e., functions) is given by formula (3). On p-forms with an arbitrary integer L. o. in local coordinates is written as

Here are the covariant derivatives with respect to

Curvature tensor - Ricci tensor. Let an arbitrary elliptic be given. complex

where E p - real or complex bundles on a manifold M, G ( E p) - spaces of their smooth sections. Introducing in each bundle E p Hermitian metric, as well as an arbitrary setting of the volume element on M, one can define the Hermitian scalar product in spaces of smooth compactly supported sections of bundles E p. Then the operators d *, formally conjugate to operators d. Equation (3) is used to construct an L. o. on each space Г ( E p). If we take the de Rham complex as the complex (6), then with a natural choice of the metric in p-forms and the volume element generated by the metric (2), we obtain the L. o. complex de Rama described above L. o. on forms.

On the complex manifold Mn, along with the de Rham complex, there are elliptic. complexes

where is the space of smooth forms of type ( p, qon M. Introducing a Hermitian structure in the tangent bundle on M, it is possible to build a L. o. (4) the complex of de Rama and L. o. complexes (7), (8):

Each of these operators translates into itself the space If M - is a Kähler manifold, and the Hermitian structure on Mind is induced by the Kähler metric, then

An important fact that determines the role of L. about. elliptic complex, is the existence in the case of a compact manifold of the Hodge Morthogonal decomposition:

In this expansion where - L. o. complex (6), so that is the space of "harmonic" sections of the bundle E p (in the case of the de Rham complex, this is the space of all harmonic forms of degree p). The direct sum of the first two terms on the right-hand side of formula (9) is and the direct sum of the last two terms coincides with In particular, decomposition (9) defines an isomorphism between the cohomology space of complex (6) in the term and the harmonic space. sections

Lit.: Rahm J. de, Differentiable manifolds, trans. from French., M., 1956; Zhen Sheng-shen, Complex manifolds, trans. from English, M., 1961; Wells R., Differential calculus on complex manifolds, trans. from English, M., 1976. M. A. Shubin.

is the integral of motion of a point of constant mass m in the field of the Newton - Coulomb potential L \u003d is the angular momentum - determines the plane of the orbit, and together with the integral of energy - its configuration ...
Encyclopedia of Mathematics
- 1) An integral of the form that performs the Laplace integral transformation of a function f from a real variable t into a function F. of a complex variable p. Was considered by P. Laplace at the end. 18- early. 19th century ....
Encyclopedia of Mathematics
- asymptotic estimates - a method for calculating the asymptotics for l & gt ...
Encyclopedia of Mathematics
- a sequence of congruences in a three-dimensional projective space, in which every two adjacent congruences are formed by tangents to two families of lines of the conjugate network of one surface ...
Encyclopedia of Mathematics
- Laplace transformation, - in a broad sense - Laplace integral of the form where integration is performed along a certain contour L in the plane of the complex variable z, associating the function f ...
Encyclopedia of Mathematics
- established by P. Laplace dependence of capillary pressure Pq on cf. curvature of the surface e of the interface between the adjacent phases and surface tension q: Pq \u003d eq ....
- linear differential operator, to-ry f-tion f assigns in accordance with f-tion Occurs in plural. problems of mat. physics. Ur-tion delta f \u003d 0 called. Laplace's equation ...
Natural science. encyclopedic Dictionary
- one of the main. laws of capillary phenomena. According to L. z., The difference p0 is hydrostatic ...
- linear differential ...
Big Encyclopedic Polytechnic Dictionary
- Primorsky region, South Ussuriysky Territory, on the coast of the North Sea of \u200b\u200bJapan, between the Avseenko and Durynin capes, north of Shkhadgou Bay ...
Encyclopedic Dictionary of Brockhaus and Euphron
- the geodetic azimuth A of the direction to the observed point, obtained from its astronomical azimuth α, corrected taking into account the influence of the deflection of the plumb line at the observation point ...
- the cosmogonic hypothesis of the formation of the solar system - the sun, planets and their satellites from a rotating and contracting gas nebula, expressed by P. Laplace in 1796 in the popular book "Presentation ...
Great Soviet Encyclopedia
- dependence of the hydrostatic pressure drop Δp at the interface between the two phases on the interfacial surface tension σ and the average curvature of the surface ε at the point under consideration: Δр \u003d р1-р2 \u003d εσ, where p1 -...
Great Soviet Encyclopedia
- Laplacian, delta-operator, Δ-operator, linear differential Operator that functions φ in n variables x1, x2, .....
Great Soviet Encyclopedia
- the dependence established by P. Laplace ????? - capillary pressure ?? on the average curvature E of the interface between the adjacent phases and the surface tension? ...
- LAPLASA operator - a linear differential operator, which functions? associates a function Occurs in many problems of mathematical physics. The equation ??? 0 is called the Laplace equation ...
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"LAPLASA OPERATOR" in books

Laplace's resignation

From the book Laplace author

LAPLACE HERITAGE

From the book Laplace author Vorontsov-Velyaminov Boris Nikolaevich

Laplace sugar

From the book of History, old and recent author Arnold Vladimir Igorevich

Sugar Laplace The story of F. Arago: in his youth he was captured by pirates, then ransomed (by some Englishman in Egypt?), When he returned, he became an active scientist, worked with Ampere and in optics. He was nominated to the Academy of Sciences. The candidate (so far) must visit all voters and

Laplace's principle

From the book How Far Until Tomorrow author Moiseev Nikita Nikolaevich

Laplace's Principle Ultimately, I did not become a believer, but I did not become an atheist either. It seemed to me that any categorical statements in this area, which lies on the border of reason and emotion, are inappropriate. Everything is unprovable. No amount of logic will help in solving this eternal question.

Demon Laplace

From the book More Than You Know. An unusual view of the world of finance author Mobussin Michael

Demon Laplace 200 years ago in science dominated by determinism. Inspired by Newton's discoveries, scientists viewed the universe as a clockwork. The French mathematician Pierre Simon Laplace well expressed the essence of determinism in his famous work "The Experience of Philosophy

43. Demon, Laplace

From the book Philosopher at the Edge of the Universe. SF-Philosophy, or Hollywood Comes to the Rescue: Philosophical Problems in Sci-Fi Films by Rowlands Mark

43. Demon, Laplace A hypothetical super-being with comprehensive knowledge of the state of the Universe and is able to accurately predict future changes based on this. Remember at least the breaks from the "Minority Report": if they could see not only the coming

Laplace azimuth

TSB

Laplace hypothesis

From the book Great Soviet Encyclopedia (LA) of the author TSB by Meyers Scott

Rule 52: If you've written the new with placement, write the corresponding delete operator The new and delete operators with placement are not very common in C ++, so it's okay if you are not familiar with them. Remember (rules 16 and 17) that when you write this

1. Operator Select - the basic operator of the structured query language

From the book Databases: lecture notes author author unknown

1. Operator Select - basic operator Structured Query Language The central place in the structured query language SQL is the Select statement, which implements the most popular operation in working with databases - queries.

15.8.2. Placement operator new () and operator delete ()

From the C ++ book for beginners author Lippman Stanley

15.8.2. Placement operator new () and operator delete () The member operator new () can be overloaded, provided that all declarations have different parameter lists. The first parameter must be of type size_t: class Screen (public: void * operator new (size_t); void * operator new (size_t, Screen *); // ...); Other parameters

We considered three main operations of vector analysis: calculating gradtx for a scalar field a and rot a for a vector field a \u003d a (x, y, z). These operations can be recorded in more simple form with the help of the symbolic operator V (“nabla”): The operator V (the Hamilton operator) has both differential and vector properties. Formal multiplication, for example, multiplication ^ by the function u (x, y), will be understood as a partial differentiation: Within the framework of vector algebra, formal operations on the operator V will be carried out as if it were a vector. Using this formalism, we obtain the following basic formulas: 1. If is a scalar differentiable function, then by the rule of multiplication of a vector by a scalar, we obtain where P, Q, R are differentiable functions, then by the formula for finding the scalar product, we obtain the Hamilton operator Second-order differential operations Operator Laplace concept of curvilinear coordinates Spherical coordinates 3. Calculating the vector product, we obtain For a constant function u \u003d с we obtain and for a constant vector с we will have From the distribution property for scalar and vector products we obtain Remark 1. Formulas (5) and (6) can be interpreted tamke as a manifestation of the differential properties of the operator "nabla" (V is a linear differential operator). We agreed that the operator V acts on all values \u200b\u200bwritten behind it. In this sense, for example, is a scalar differential operator. When applying the operator V to the product of some quantities, one must bear in mind the usual rule for differentiating the product. Example 1. Prove that By formula (2), taking into account Remark 1, we obtain or To note the fact that "obl a" does not act on any value included in the complex formula, this value is marked with an index c ("const" ), which is omitted in the final result. Example 2. Let u (xty, z) be a scalar differentiable function and (x, y, z) a vector differentiable function. Prove that 4 Let us rewrite the left-hand side of (8) in symbolic form Taking into account the differential character of the operator V, we obtain. Since ue is a constant scalar, it can be taken out of the scalar product sign, so a (at the last step we dropped the index e). In the expression (V, uac), the operator V acts only on the scalar function and, therefore, As a result, we obtain Remark 2. Using the formalism of action with the operator V as a vector, one must remember that V is not an ordinary vector - it does not length, no direction, so. for example a vector, Where Unable to parse expression (Executable texvc not found; See math / README for configuration help.): H_i \\ - Lame coefficients.

Cylindrical coordinates

In cylindrical coordinates outside the straight line Unable to parse expression (Executable texvc not found; See math / README for configuration help.): \\ R \u003d 0 :

Unable to parse expression (Executable texvc not found; See math / README for setup help.): \\ Delta f \u003d (1 \\ over r) (\\ partial \\ over \\ partial r) \\ left (r (\\ partial f \\ over \\ partial r) \\ right) + ( \\ partial ^ 2f \\ over \\ partial z ^ 2) + (1 \\ over r ^ 2) (\\ partial ^ 2 f \\ over \\ partial \\ varphi ^ 2)

Spherical coordinates

In spherical coordinates outside the origin (in three-dimensional space):

Unable to parse expression (Executable texvc not found; See math / README for setup help.): \\ Delta f \u003d (1 \\ over r ^ 2) (\\ partial \\ over \\ partial r) \\ left (r ^ 2 (\\ partial f \\ over \\ partial r) \\ over r ^ 2 \\ sin ^ 2 \\ theta) (\\ partial ^ 2 f \\ over \\ partial \\ varphi ^ 2) Unable to parse expression (Executable texvc not found; See math / README for setup help.): \\ Delta f \u003d (1 \\ over r) (\\ partial ^ 2 \\ over \\ partial r ^ 2) \\ left (rf \\ right) + (1 \\ over r ^ 2 \\ sin \\ theta) (\\ partial \\ over \\ partial \\ theta) \\ left (\\ sin \\ theta (\\ partial f \\ over \\ partial \\ theta) \\ right) + (1 \\ over r ^ 2 \\ sin ^ 2 \\ theta ) (\\ partial ^ 2 f \\ over \\ partial \\ varphi ^ 2).

If Unable to parse expression (Executable texvc not found; See math / README for configuration help.): \\ F \u003d f (r) at n-dimensional space:

Unable to parse expression (Executable texvc not found; See math / README for setup help.): \\ Delta f \u003d (d ^ 2 f \\ over dr ^ 2) + (n-1 \\ over r) (df \\ over dr).

Parabolic coordinates

In parabolic coordinates (in three-dimensional space) outside the origin:

Unable to parse expression (Executable texvc not found; See math / README for setup help.): \\ Delta f \u003d \\ frac (1) (\\ sigma ^ (2) + \\ tau ^ (2)) \\ left [\\ frac (1) (\\ sigma) \\ frac (\\ partial) (\\ partial \\ sigma) \\ left (\\ sigma \\ frac (\\ partial f) (\\ partial \\ sigma) \\ right) + \\ frac (1) (\\ tau) \\ frac (\\ partial) (\\ partial \\ tau) \\ left (\\ tau \\ frac (\\ partial f) (\\ partial \\ tau) \\ right) \\ right] + \\ frac (1) (\\ sigma ^ 2 \\ tau ^ 2) \\ frac (\\ partial ^ 2 f) (\\ partial \\ varphi ^ 2)

Cylindrical parabolic coordinates

In the coordinates of a parabolic cylinder outside the origin:

Unable to parse expression (Executable texvc not found; See math / README for tuning help.): \\ Delta F (u, v, z) \u003d \\ frac (1) (c ^ 2 (u ^ 2 + v ^ 2)) \\ left [\\ frac (\\ partial ^ 2 F) (\\ partial u ^ 2) + \\ frac (\\ partial ^ 2 F) (\\ partial v ^ 2) \\ right] + \\ frac (\\ partial ^ 2 F) (\\ partial z ^ 2).

General curvilinear coordinates and Riemannian spaces

Let on a smooth manifold Unable to parse expression (Executable texvc a local coordinate system is specified and Unable to parse expression (Executable texvc not found; See math / README for setup help.): G_ (ij) is the Riemannian metric tensor on Unable to parse expression (Executable texvc not found; See math / README for configuration help.): X , that is, the metric has the form

Unable to parse expression (Executable texvc not found; See math / README for setup help.): Ds ^ 2 \u003d \\ sum ^ n_ (i, j \u003d 1) g_ (ij) dx ^ idx ^ j .

Let us denote by Unable to parse expression (Executable texvc not found; See math / README for setup help.): G ^ (ij) matrix elements Unable to parse expression (Executable texvc not found; See math / README for setup help.): (G_ (ij)) ^ (- 1) and

Unable to parse expression (Executable texvc not found; See math / README for setup help.): G \u003d \\ operatorname (det) g_ (ij) \u003d (\\ operatorname (det) g ^ (ij)) ^ (- 1) .

Vector field divergence Unable to parse expression (Executable texvc not found; See math / README for configuration help.): F given by coordinates Unable to parse expression (Executable texvc not found; See math / README - setup help.): F ^ i (and representing the first-order differential operator Unable to parse expression (Executable texvc not found; See math / README for tuning help.): \\ Sum_i F ^ i \\ frac (\\ partial) (\\ partial x ^ i)) on the manifold X calculated by the formula

Unable to parse expression (Executable texvc not found; See math / README for setup help.): \\ Operatorname (div) F \u003d \\ frac (1) (\\ sqrt (g)) \\ sum ^ n_ (i \u003d 1) \\ frac (\\ partial) (\\ partial x ^ i) (\\ sqrt (g) F ^ i) ,

and the components of the gradient of the function f - according to the formula

Unable to parse expression (Executable texvc not found; See math / README for setup help.): (\\ Nabla f) ^ j \u003d \\ sum ^ n_ (i \u003d 1) g ^ (ij) \\ frac (\\ partial f) (\\ partial x ^ i).

Operator Laplace - Beltrami on Unable to parse expression (Executable texvc not found; See math / README for configuration help.): X :

Unable to parse expression (Executable texvc not found; See math / README for setup help.): \\ Delta f \u003d \\ operatorname (div) (\\ nabla f) \u003d \\ frac (1) (\\ sqrt (g)) \\ sum ^ n_ (i \u003d 1) \\ frac (\\ partial) (\\ partial x ^ i) \\ Big (\\ sqrt (g) \\ sum ^ n_ (k \u003d 1) g ^ (ik) \\ frac (\\ partial f) (\\ partial x ^ k) \\ Big) ...

Value Unable to parse expression (Executable texvc not found; See math / README - tuning reference.): \\ Delta f is a scalar, that is, it does not change when transforming coordinates.

Application

Through this operator it is convenient to write the Laplace and Poisson equations and the wave equation. In physics, the Laplace operator is applicable in electrostatics and electrodynamics, quantum mechanics, in many equations of the physics of continuous media, as well as in the study of the equilibrium of membranes, films, or interfaces with surface tension (see Laplace pressure), in stationary problems of diffusion and heat conduction, which are reduced, in the continuous limit, to the usual Laplace or Poisson equations or to some of their generalizations.

Variations and generalizations

The d'Alembert operator is a generalization of the Laplace operator for hyperbolic equations. Includes the second time derivative.
The vector Laplace operator is a generalization of the Laplace operator to the case of a vector argument.

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For some reason, the usual high-speed Stellino "self-awareness" did not work this time ... Apparently, it was too painful to lose her dear friends, especially knowing that no matter how much she missed them later, she would not see them anywhere else and never ... It was not an ordinary bodily death, when we all get a great chance - to incarnate again. It was their soul that died ... And Stella knew that neither the brave girl Maria, nor the "eternal warrior" Luminary, nor even the ugly, kind Dean, would never incarnate, sacrificing their eternal life for others, perhaps very good, but completely strangers to them ...
Just like Stella, my soul ached very much, because this was the first time when I saw in reality how, of their own free will, brave and very kind people left for eternity ... my friends. And it seemed that sadness had settled in my wounded childish heart forever ... But I also already understood that no matter how I suffered, and no matter how I wished, nothing would bring them back ... Stella was right - you can't it was to win at such a price ... But it was their own choice, and we had no right to refuse them this. And to try to convince us - we simply did not have enough time for this ... But the living had to live, otherwise all this irreplaceable sacrifice would have been in vain. But it was precisely this that could not be allowed.
- What are we going to do with them? - Sighing convulsively, pointed to the huddled kids, Stella. - You can't leave here.
Before I could answer, a calm and very sad voice sounded:
- I will stay with them, if, of course, you will allow me.
We jumped up together and turned around - this was the person saved by Maria spoke ... But we somehow completely forgot about him.
- How are you feeling? I asked as kindly as possible.
I honestly did not wish harm to this unfortunate, rescued stranger at such a high price. It was not his fault, and Stella and I knew it perfectly well. But the terrible bitterness of the loss still covered my eyes with anger, and although I knew that it was very, very unfair to him, I could not pull myself together and push this terrible pain out of me, leaving it "for later" when I all alone, and, closing myself “in my corner”, I can give vent to bitter and very heavy tears ... And I was also very afraid that the stranger would somehow feel my “rejection”, and thus his release would lose that importance and beauty victory over evil, in the name of which my friends died ... Therefore, I tried with all my strength to gather myself and, smiling as sincerely as possible, waited for an answer to my question.
The man looked around sadly, apparently not quite understanding what had happened here, and what had happened to him all this time ...
- Well, where am I? .. - in a voice hoarse with excitement, he asked quietly. - What is this place, so terrible? It doesn't sound like what I remember ... Who are you?
- We are friends. And you are absolutely right - this is not a very pleasant place ... And a little further the places are generally scary to savagery. Our friend lived here, he died ...
- I'm sorry, little ones. How did your friend die?
“You killed him,” Stella whispered sadly.
I froze, staring at my girlfriend ... It was not the "sunny" Stella who was familiar to me, who "without fail" felt sorry for everyone, and would never make anyone suffer! .. But, apparently, the pain of loss, like me, it caused her an unconscious feeling of anger “at everyone and everything,” and the baby was not yet able to control this in herself.
- Me?! .. - exclaimed the stranger. - But it can't be true! I've never killed anyone! ..
We felt that he was telling the pure truth, and knew that we had no right to shift the blame on him. Therefore, without even saying a word, we smiled together and immediately tried to quickly explain what really happened here.
For a long time, the person was in a state of absolute shock ... Apparently, everything he heard sounded wild to him, and certainly did not coincide with what he really was, and how he treated such a terrible evil that did not fit into normal human frames ...
- How can I compensate for all this?! .. I just can't? And how to live with this ?! .. - he grabbed his head ... - How many have I killed, tell me! .. Can anyone say this? And your friends? Why did they do this? But why?!!!..
- So that you can live as you should ... As you wanted ... And not as someone wanted ... To kill the Evil that killed others. Because, probably ... - Stella said sadly.
- Forgive me, dear ... Forgive ... If you can ... - the man looked completely killed, and I was suddenly "pricked" by a very bad feeling ...
- Well, I do not! - I exclaimed indignantly. - Now you must live! Do you want to nullify all their sacrifice ?! Don't even think about it! Now you will do good instead of them! It will be right. And “leaving” is the easiest thing. And now you no longer have such a right.
The stranger stared at me in a daze, apparently not expecting such a stormy surge of "righteous" indignation. And then he smiled sadly and said quietly:
- How did you love them! .. Who are you, girl?
My throat was very tight and for a while I could not squeeze out a word. It was very painful because of such a heavy loss, and, at the same time, it was sad for this "restless" person, who will, oh, how difficult it is to exist with such a burden ...
- I am Svetlana. And this is Stella. We're just walking around here. We visit friends or help someone when we can. True, there are no friends left now ...
- Forgive me, Svetlana. Although for sure it will not change anything if I ask you for forgiveness every time ... What happened happened, and I cannot change anything. But I can change what will happen, right? - the man glared at me with his eyes blue like the sky and, smiling, a woeful smile, said: - And yet ... You say, I am free in my choice? .. But it turns out - not so free, dear .. Rather, it looks like an atonement ... With which I agree, of course. But this is your choice that I am obliged to live for your friends. Due to the fact that they gave their lives for me .... But I did not ask for this, did I? .. Therefore, it is not my choice ...
I looked at him, completely dumbfounded, and instead of "proud indignation", ready to immediately escape my lips, I gradually began to understand what he was talking about ... No matter how strange or offensive it may sound, but all this was sincerely true! Even if I didn't like it at all ...

Any part of the control system, be it a regulator, an object or a sensor, has an input and an output. With the help of inputs and outputs, they interact with other elements of the system and with the external environment. When an input signal acts on a system element, some internal state changes occur in this element, which lead to a change in the output signal. That is, an element of the system is some function of the dependence of y on x. This can be depicted in Figure 1.

Figure 1 - control system element with input and output

The definition of the function F (x) is, in fact, the main problem solved within the framework of the theory of automatic control. Knowing the F (x) of the object will help to compose correct algorithm control, F (x) of the sensor will determine the character feedback, and the synthesis of F (x) will make the system truly operational. F itself is also sometimes referred to as an operator, as it operates on an input signal.

Integration and differentiation are basic operations in TAU. Suppose the signal grows for some time, which is often very typical for signals in control systems, then to describe this process it should be "collected" by an integral over the entire time interval:

Differentiation is also extremely useful in automatic control theory. The differentiation operator, as opposed to the integration operator, takes the derivative from the input signal, that is:

Here a very important concept in TAU arises - the Laplace operator p, which is intended to replace the notation d / dt, in other words

Also, in some sources, this operator is represented by the product of the imaginary unit and the angular frequency, that is, p \u003d jω. But we will not touch the frequency range for now, because this is a broad topic and just remember two simple rules:

What does signal integration and differentiation look like? The integration of the hopping waveform is shown in Figure 2a. Everything is simple here, the signal will be incremented at each integration step until it reaches the initial set value in time t1. What if we differentiate such a signal? In no case! This is a threat to the safety of the Universe, such a signal will pierce the firmament and rush to infinity towards the stars (Figure 2b)! In short, mathematics says that the derivative of an instantaneously changed signal is equal to infinity, and since infinity is an ideal and unattainable value, such an operation does not make sense in the real world. Otherwise, they say that such an operation is not physically realizable. In general, p is not used in its pure form, but is used only in more complex expressions, where this p will be somehow compensated.

Figure 2 - signal integration and differentiation

Now that we know about the ratio of the output signal to the input signal and about the Laplace operator, we can move on to such a concept as a transfer function. Basically, the transfer function, written as W (p), is the output / input ratio. The system written through transfer functions is more visual, and more or less simple methods of analysis and synthesis can be applied to it. But more about them later, and now let's look at a simple example of how such functions are obtained.

Suppose we have a link, the processes occurring in which are described by the following equation:

On the left is the output value (and its derivative), on the right is the input value (in complex expressions there may also be derivatives). T is some kind of time constant, K is some kind of coefficient. Now we replace it with the Laplace operator:

As noted above, the transfer function is equal to the output / input ratio:

This is how we got the first-order inertial link transfer function. In TAU there are several typical links (including this), from which you can make up any system, any link of any complexity. For now, just note that the transfer functions, depending on the orders of the numerator and denominator, can be correct or incorrect. The above function is correct, it is also said to be strictly correct, because the order of the denominator is greater than the order of the numerator. And that's good, it is realizable. There are a couple more functions below.

Function type 1 is also correct, but not strictly. The degree of the numerator is equal to the degree of the denominator, but that's okay, it is also realizable. But a function like 2 is not realizable due to the presence of a square in the numerator and the absence of a square or more high degree in the denominator, that is, in this case there will be some kind of uncompensated derivative. Thus, the order in the transfer functions must be strictly observed!

Type