RU2166789C2 - Method for determining transfer function coefficients of linear dynamic entities - Google Patents

Abstract

FIELD: automatic control and measurement technology. SUBSTANCE: transient process time T is measured at instant of transient process completion, integral estimates of quality are calculated, and factors a_i and b_j of transfer-function numerator and denominator polynomials are calculated using data obtained together with interrelations between integral estimates of quality and factors of transfer function expansion into a power series. EFFECT: improved accuracy and speed. 3 dwg

Description

 The invention relates to the field of automation, namely to measuring equipment and automatic regulation, and can be used to determine the transfer function coefficients of dynamic objects of complex structures, for example, when designing and tuning aircraft control systems, a gas turbine engine, etc.

 Known methods for identifying linear dynamic objects based on the formation of the spectral density of the input signal of the studied object and the mutual spectral density of the input and output signals of the studied object [Pugachev VS The theory of random functions and its application to automatic control problems, M. 1957, p. 395-403, 592-597].

 The main drawback of the known spectral identification methods is the incorrectness of their application for the identification of dynamic objects in those cases where the observed signals along with random components contain nonrandom harmonic components, which are usually characterized by signals recorded in normal operation. In this case, the signals of spectral densities are characterized by the presence of narrow peaks and zero dips, and obtaining statistical estimates of spectral densities from estimates of the signal of the correlation function on the finite observation interval by directly applying the Fourier transform does not give the desired results, since in this case the variances of the estimates of the signals of spectral densities have the same order as the spectral densities themselves, which leads to large errors [Pugachev VS The theory of random functions and its application to automatic control problems, M. 1957, p. 395-403, 592-597]. In addition, these methods are very critical to interference present simultaneously in the input and output signals.

 There is also a known method [USSR Author's Certificate N 696416, cl. G 05 B 23/00, 1979] determining the transfer function coefficients of dynamic objects by applying time-varying test signals to the object input as a sequence of single pulses. Using this method, it is possible to determine the coefficients of the transfer functions of a higher order.

However, the implementation of this method requires significant time-consuming, which is associated with the experimental determination of all coefficients α _{k of time-} varying test signals α _BX = α _k t ^k and with the need for sequential tuning of the object model. In addition, the model settings affect the accuracy of determining the transfer function coefficients.

и коэффициента передачи линейного динамического объекта

Этот способ не пригоден для определения коэффициентов передаточных функций объектов второго и более высоких порядков.Closest to the proposed invention is a method [Copyright certificate of the USSR N 661511, class. G 05 B 23/00, 1979 (prototype)] determining the transfer function coefficients of linear dynamic objects by applying a test signal tending to a steady value to the input of a linear dynamic object and integrating its output signal, measuring time intervals t ₁ and t ₂ from the moment of supplying the test signal of value A to the moment of reaching levels J ₁ and J _2, respectively, and determining the time constant

and transmission coefficient of a linear dynamic object

This method is not suitable for determining the transfer function coefficients of objects of the second and higher orders.

 The task to which the invention is directed is to increase the accuracy and speed when determining the transfer function coefficients, by calculating the integral quality estimates and using their relationship with the transfer function coefficients of linear, asymptotically stable dynamic objects.

вычисляют линейные интегральные оценки качества, которые можно записать в виде

где k=n+m-l
m - порядок числителя;
n - порядок знаменителя;
ε_СВ (t)=y_уст-y(t),
и по ним, используя взаимосвязь интегральных оценок качества и коэффициентов разложения

передаточной функции в степенной ряд

,
вычисляют коэффициенты d_i полинома числителя и a_j полинома знаменателя передаточной функции

где i=m-l,j=n,
путем решения полученной системы линейных алгебраических уравнений

а коэффициенты b_i вычисляются по формуле

При b_i= 0 процесс вычислений завершается. При этом теоретически не существует ограничения на порядок числителя и знаменателя исследуемых объектов. Кроме того, время эксперимента равно T, что позволяет сэкономить время определения коэффициентов для объектов с высоким порядком.The problem is solved in that in the method for determining the transfer function coefficients of linear dynamic objects by supplying a step test signal to the input of the test object, the duration T of the transient process at the output of the test object is measured, the completion time of which is determined by the attribute
| y _mouth -y (t) | ≅ Δ _d ,
where y (t) is the value of the output signal of the studied object, y _mouth is the steady-state value of the output signal, Δ _d is the permissible error in completing free oscillations, then for objects with a transfer function

calculate linear integral quality estimates that can be written as

where k = n + ml
m is the order of the numerator;
n is the order of the celebrity;
ε _CB (t) = y _mouth -y (t),
and on them, using the relationship of integrated quality estimates and decomposition coefficients

transfer function in a power series

,
the coefficients d _{i of} the numerator polynomial and a _{j of} the denominator polynomial of the transfer function are calculated

where i = ml, j = n,
by solving the resulting system of linear algebraic equations

and the coefficients b _i are calculated by the formula

When b _i = 0, the calculation process ends. In this case, theoretically there is no restriction on the order of the numerator and denominator of the studied objects. In addition, the experiment time is T, which saves the time of determining the coefficients for objects with a high order.

 In FIG. 1 shows a structural diagram of a device that implements the proposed method; in FIG. 2 - illustration of the processes for determining the coefficients; in FIG. 3 is a block diagram of a program for calculating coefficients implemented by a computer.

 The device comprises an input signal shaper 1, an object under investigation 2, an analog-to-digital converter 3, and a computer 4.

Затем по формуле (5) и системе уравнений (4) вычисляют коэффициенты b_i и a_j.As seen in FIG. 2 and FIG. 3, when determining the coefficients in the time interval, the transient process completion time T is determined and, at equal time intervals Δ t, depending on the quantization frequency, the response values of the object y (t) to the action of a single step signal are recorded in the computer memory. Next, the integration operation is carried out, the integral is calculated and the integral quality assessment is determined

Then, according to the formula (5) and the system of equations (4), the coefficients b _i and a _j are calculated.

где τ - постоянная времени;
ξ - коэффициент демпфирования.Consider the implementation of this method on the example of an object with a transfer function

where τ is the time constant;
ξ is the damping coefficient.

a₁-b₁=J₀a₀=1,8

a₁-b₁=1,8
a₂-b₂=-2,04+1,8a₁
2,04a₁-1,8a₂=-1,488
1,488a₁+2,04a₂=12,339
a₁=2,803
a₂=4,004
b₁=1,003
b²=0,9986.Using the transition function, linear integral quality estimates J _о = 1.8 were calculated; J ₁ = 2.04; J ₂ = 2.976; J ₃ = -74.037; J ₄ = -690.834 when t varies from 0 to 30 s. Define the image of the free movement of the object

a ₁ -b ₁ = J ₀ a ₀ = 1.8

a ₁ -b ₁ = 1.8
a ₂ -b ₂ = -2.04 + 1.8a ₁
2.04a ₁ -1.8a ₂ = -1.488
1,488a ₁ + 2,04a ₂ = 12,339
a ₁ = 2.803
a ₂ = 4.004
b ₁ = 1.003
b ² = 0.9986.

.Simulated object with transfer function

.

 The relative error in determining the coefficients of the transfer function is small.

 Thus, the proposed method allows you to calculate its coefficients with greater accuracy and speed.

Claims (1)

A method for determining the transfer function coefficients of linear dynamic objects by applying a step test signal to the input of the research object, characterized in that they measure the duration T of the transient at the output of the test object, the completion time of which is determined by
| y _mouth - y (t) | ≅ Δ _d
where y (t) is the value of the output signal of the investigated object;
y _mouth - steady-state value of the output signal;
Δ _d is the permissible error of completion of free vibrations,
then, for objects with transfer function

calculate linear integral quality estimates, which can be written as:

where k = n + m - 1;
m is the order of the numerator;
n is the order of the denominator;
ε _sv (t) = y _mouth - y (t),
and on them, using the relationship of integrated quality estimates and decomposition coefficients

transfer function in a power series

the coefficients d _{i of} the numerator polynomial and a _{j of} the denominator polynomial of the transfer function are calculated

where i = m - I, j = n,
by solving the resulting system of linear algebraic equations:

and the coefficients b _i are calculated by the formula:

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