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The mathematical model of the system differential equation and transmission operator

does not involve any mathematical transformation, but directly analyzes the system in the time variable domain, which is called the time domain analysis of the system. There are two methods: time-domain classical method and time-domain convolution method

time domain classical method is a method to directly solve the system differential equations. The advantages of this method are intuitionistic and clear physical concepts. The disadvantages are that the solving process is tedious and there are limitations in application. Therefore, before the 1950s, people generally liked to use transform domain analysis methods (such as Laplace transform), and less used time-domain classical methods. Since the 1950s, due to the widespread application of (T) function and computer, the time-domain convolution method has developed rapidly, and is becoming more and more mature and perfect. It is only about the appearance of words to determine the order of documents, which has become one of the important methods of system analysis. Time domain analysis is the basis of various transform domain analysis methods

in this chapter, firstly, the mathematical model differential equation of the system is established, then the zero input response of the system is obtained by the classical method, and the zero state response of the system is obtained by the time-domain convolution method, and then the zero input response is added to the special strong zero state response to obtain the full response of the system. The idea and procedure are as follows:

secondly, it will be introduced that the system is equivalent to a differential equation; The system is equivalent to a transmission operator H (P); The system is equivalent to a signal impulse response H (T). The analysis of the system is to study the relationship between the excitation signal f (T) and the impulse response signal H (T), which is called convolution integral

can be divided into two types of

research systems: manual grinding and mechanical grinding. To immediately reverse the proportional valve to supply oil to the other chamber of the oil cylinder, first establish the mathematical model differential equation of the system. The basis for establishing the differential equation of circuit system is two kinds of constraints of circuit: topological constraints (KCl, KVL) and component constraints (time domain volt ampere relationship of components). In order to make it easy for readers to understand and accept, we adopt the method from special to general

figure (a) shows a two hole circuit with three independent dynamic elements, where is excitation and is response. For two hole circuits, KVL equation can be listed as

according to equation (), the circuit model in operator form can be drawn, as shown in figure (b). Comparing figure (a) with figure (b),

can easily draw figure (b) according to figure (a), that is, l is rewritten into LP, C is rewritten into,

everything else remains the same. When the operator circuit model is drawn, it is easy to write the formula () according to the operator circuit model in figure (b)

note that during the calculation of the above formula, the common factor P in the numerator and denominator is eliminated. This is because the circuit studied is of third order,

so the differential equation of the circuit should also be of third order. However, it should be noted that the common factor in the numerator and denominator may not be eliminated in any case

some cases can be eliminated, while others cannot, depending on the specific situation. Therefore, the above formula of

is a third-order linear non-homogeneous ordinary differential equation with constant coefficients whose variable to be solved is I1 (T)

The left end of the equal sign of theequation is the linear combination of response I1 (T) and its derivatives, and the right end of the equal sign of

is the linear combination of excitation f (T) and its derivatives

generalization, for n-order systems, if y (T) is the response variable and f (T) is the excitation, as shown in the figure, the general form of the system differential equation is

h (P), which is called the transfer operator or transfer operator of response y (T) to excitation f (t), which is the ratio of two real coefficient rational polynomials of P, and the denominator of

is the characteristic polynomial D (P) of the differential equation. H (P) describes the characteristics of the system itself, which is independent of the excitation and response of the system

here is a point: the letter P is essentially a differential operator, but from the perspective of mathematical form, it can be artificially regarded as a variable (usually a complex number). Thus, the transfer operator H (P) is the ratio of two real coefficient rational polynomials of P

example circuit shown in figure (a). Find the differential equations of the transfer operator and U1 (T) and U2 (T) to I (T) in response to the excitation of U1 (T) and U2 (T)

the circuit of solving its operator form is shown in figure (b). For nodes ① and ②, the KCl equation in the form of column operators is

. It can be seen that for different responses U1 (T) and U2 (T), the characteristic polynomials are the same,

which is the invariance and similarity of the characteristic polynomials of the system

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