Laplace Transform is an integral transform commonly used in engineering mathematics.
If defined:
F(t) is a function of t, so t < when t
S is a complex variable;
Mathcal is an operation symbol, which represents Laplace integral int _ 0 infity e, dt; F(s) is the laplace transformation result of f(t).
Then the Laplace transform of f(t) is given by the following equation:
F(s)，=mathcal left =int_ ^infty f(t),e^，dt
Laplace inverse transformation is the process of solving f(t) when F(s) is known. Represented by mathematical symbols.
The formula of Laplace inverse transformation is:
For all t>0,;
f(t)
= Math Left
= f (s), e interest, ds
The abscissa value c of the convergence interval is a real constant and is greater than the real part value of a single point of all F(s).
Function transformation between real variable function and complex variable function established to simplify calculation. It is much easier to calculate a real variable function with Laplace transform, then do various operations in complex number domain, and then get the corresponding results in real number domain with Laplace inverse transform than to get the same results directly in real number domain. This operation step of Laplace transform is particularly effective for solving linear differential equations, which can be transformed into algebraic equations that are easy to solve, thus simplifying the calculation. In classical control theory, the analysis and synthesis of control system is based on Laplace transform. One of the main advantages of introducing Laplace transform is that it can use transfer function instead of differential equation to describe the characteristics of the system. This makes it possible to determine the overall characteristics of the control system (see signal flow chart and dynamic structure chart), analyze the motion process of the control system (see Nyquist stability criterion and root locus method) and the correction device of the integrated control system (see control system correction method) by intuitive and simple graphic methods.
F(t) represents a function of the real variable T, and F(s) represents its Laplace transform, which is a complex variable S =σ+J &；; The function of σ and&in owega; Owega are both real variables, J2 =- 1. The relationship between F(s) and f(t) is determined by integral and is defined as follows:
If the above integral exists for all S values of the real part σ > σc, but does not exist for σ ≤σc, σc is called the convergence coefficient of f(t). For a given real variable function f(t), Laplace transform F(s) only exists when σc is finite. Traditionally, F(s) is often called the image function of f(t), which is denoted as f (s) = l [f (t)]; Let f(t) be the original function of F(s) and let FT = L- 1 [f (s)].
Transformation properties of function transformation pair and operation By using the definition integral, 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. Table 1 and Table 2 respectively list some commonly used function transformation pairs and operational transformation properties.