During the application of Laplace transformProceed from realityFirst, based on the research object, it is planned as a time domainmathematical model Then, with the help of mathematical tools of Laplace transform, it is transformed into a complex domain mathematical model. Finally, if you want the results to be more intuitive, you can use graphics to express them. The graphical representation is based on the transfer function (complex domain mathematical model), so Laplace transform is classicalcontrol theoryInFundamentals of Mathematics。When using Laplace transform to solve mathematical model, it can be regarded as solving alinear equationIn other words, Laplace transform can not only be used to convert simple time-domain signals into complex domain signals, but also be used to solve the differential equations of control systems.Laplace transform is to change the time domain signal into the complex domain signal, on the contrary, Laplace inverse transform is to change the complex domain signal into the time domain signal.[6]
Formula of Laplace transform
Laplace transform [2]Is the continuous time function x (t) whose value is not zero for t ≥ 0
(where - st is the exponent of natural logarithm base e) is transformed into the function X (s) of complex variable s.It is also a function of time x (t)“Complex frequency domain”Representation.
The formula of inverse Laplace transform is: for all t>0, f (t)=mathcal ^ left
=Frac int_ ^ F (s) 'e' ds, c 'is convergentAbscissaValue, which is a real constant and is greater than the real part value of all individual points of F (s) '.
example
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Therefore, in "circuit analysis", the volt ampere relationship of components can beComplex frequency domainIndicated in, that is, resistance element: V=RI,Inductive element：V=sLI,Capacitive element：I=sCV。If resistance R is connected in series with capacitor COutgoing voltageAs output, it is available“partial pressureFormula "shows that the transfer function of the system is H (s)=(1/RC)/(s+(1/RC)), so the Laplace transform Y (s) of the response is equal to the product of the Laplace transform X (s) of the excitation and the transfer function H (s), that is, Y (s)=X (s) H (s)
If definition: f (t) is a function of t, so that when t<0, f (t)=0;S is a complex variable;
Is an operation symbol, which represents the Laplacian integration int_0 ^ infty e 'dt for its object;F (s) is the result of Laplace transform of f (t).
Then the Laplacian transformation of f (t) is given by the following formula:
。
Meaning and function
Operational formula
A function transformation established between real variable function and complex variable function to simplify calculation.To oneReal variableThe function is transformed by Laplace transform, and various operations are performed in the complex number field, and then the operation result is obtained by Laplace inverse transformationReal number fieldThe corresponding result in is often much easier to calculate than the same result directly in the real number field.This operation step of Laplace transform is very important for solvinglinear differential equationEspecially effective, it candifferential equation Make it easy to solvealgebraic equationTo simplify the calculation.stayClassical control theoryThe analysis and synthesis of control systems are based on Laplace transform.One of the main advantages of introducing Laplace transform is thattransfer functionReplace differential equations to describe the characteristics of the system.This is to use an intuitive and simple graphical method to determine the overall characteristics of the control system (seeSignal flow chart、Dynamic structure diagram）Analyze the movement process of the control system (seeNyquist stability criterion、Root locus method）, andIntegrated control systemCorrection device of (seeControl system calibration method）Provides possibilities.Use f (t) to represent a function of the real variable t, and F (s) to represent its Laplace transformation, which is the complex variable s=σ+j ω;A function of, where σ and ω;All are real variables, jtwo=-1。The relationship between F (s) and f (t) is determined by the integrals defined below:
If the real part σ>σcThe above integrals exist for all s values of σ ≤ σcWhen the time integral does not exist, it is called σcIs the convergence coefficient of f (t).For a given real variable function f (t), only when σcWhen it is a finite value, its Laplacian transformation F (s) exists.Traditionally, F (s) is often called the image function of f (t), which is recorded as F (s)=L [f (t)];Called f (t) F (s)Primitive function, recorded as f (t)=L-1[F(s)]。
Function transformation pair and operation transformation property It is easy to establish the transformation pair between the original function f (t) and the image function F (s), and the corresponding relationship between the operation of f (t) in the real number field and the operation of F (s) in the complex number field by using the definition integral.Table 1 and Table 2 respectively list some of the most commonly used function transformation pairs and operation transformation properties.
Existence of Laplacian variation: In order for F (s) to exist, the integral must converge.There are the following theorems:
If the causal function f (t) satisfies: (1) integrable in a finite interval, (2) σ existszeroMake | f (t) | e-σtIf the limit is 0 when t →∞, then for all σ greater than σzeroThe Laplace integral is absolutely and uniformly convergent.
They represent displacement theorems in time domain and complex domain respectively.[3]
Differential property
。[3]
Development history
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French mathematician and astronomerLaplace(1749-1827), main researchCelestial mechanicsAnd physics.He thought mathematics was just a tool to solve problems, but he created and developed many newMathematical method。In 1812, Laplace《Analysis theory of probability》Summarized the wholeprobability theoryThis paper discusses the application of probability in election, judicial investigation, meteorology, etc., and introduces "Laplace transform".Laplace transform led to the later discovery of operation by HeivesideCalculusApplication in electrical theory.[4]
The condition of Fourier transform can be satisfied when it is large enough.
The essence of Laplacian transformation of
Fourier transform of, for
For example, this transformation changes the original function in the forward Fourier transform (the original function times the exponential decay function term) and also changes the integration factor of the inverse Fourier transform（
）, this transformation is
Laplace transform of.Attention should be paid to the
, its discussion scope is no longer just frequency
It is a complex number (including frequency
）Of
。
Application theorem
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In some cases, a real variable functionReal number fieldIt is not easy to perform some operations in, but if the real variable function is Laplacian transformed, and various operations are performed in the complex number field, then the operation results are Laplacian inverse transformed to obtain the corresponding results in the real number field,
In classical control theory, the analysis and synthesis of control systems are based on Laplace transform.One of the main advantages of introducing Laplace transform is thattransfer functionSubstitute constant coefficientdifferential equation To describe the characteristics of the system.This is to determine the overall characteristics of the control system by using an intuitive and simple graphical methodAnalysis control systemAnd provide the possibility of control system adjustment.
By using Laplace transform to solve homogeneous differential equations with constant variables, the differential equation can be transformed intoalgebraic equationTo solve the problem.stayengineeringThe significance of Laplace transform is to transform a signal from time domain to complex frequency domain（S domain）Come up to express;staylinear system, control automation has a wide range of applications.
