JPH08328604A - Arma model identifying method based on step response - Google Patents

Abstract

PURPOSE: To accurately fix the transfer function of an arbitrary degree based on a step response by performing a specified operation. CONSTITUTION: K1 B(s) and A(s) to satisfy the relation of y(s)=K B(s)/A(s)u(s)} are fixed between the step input u(s) for a controlled system and the output y(s) from the controlled system. In this case, the values from P1 to Po are obtained by fixing the degrees m and n of B(s) and A(s), setting the positive infinite matrix W of a degree q to be more than the sum of m, n and 1, setting P(s)=Pq s<q> +Pq-1 s<q-1> +...P1 s+Po to be B(s)/A(s)=1/P(s), setting Po =1, obtaining the value of K from an expression I and successively operating the recurrence formula of an expression II. Between a matrix PP=(P1 , P2 ,..., Pq-1 , Pq )T and a matrix θ=(a1 , a2 ,..., an-1 , an , b1 , b2 ,... bm-1 , bm )T, a determinant III to satisfy PP=Zθ is set and the operation of θ=(ZTWZ)<-1> (ZTWPP) is performed.

Description

Detailed Description of the Invention

[0001]

This invention is based on step response.
Regarding the ARMA model identification method, more specifically,
Based on the step response
It relates to a new method for identification. Where ARMA
What is a model (autoregressive moving average model)?
Auto regression model) and MA model (moving average model)
It is a model that seems to be combined. For example, A (z) y
From (t) = C (z) e (t), output y (t), e
Using (t) as input, the δ-operator {δ = (1-z^-1) / T
s} (where ts is the sampling time),
The AR model is 1 / A (z) = 1 / Ac (δ), and t
If s → 0, 1 / Ac (s), that is, only the denominator.
The MA model has C (z) = Cc (δ) and ts → 0.
Then, Cc (s), that is, only the molecule. ARMA model
Dell has C (z) / A (z) = Cc (δ) / Ac (δ)
If ts → 0, Cc (s) / Ac (s), immediately
It will include the numerator and denominator. However, s is Laplace performance
Arithmetic A, (z) = 1 + a₁z^-1+ ... + a_pz ^-p, C (z)
= 1 + c₁z^-1+ C_qz^-q, (P, q) are autoregressive movement flats
It is the order of the uniform process, and when q = 0, the AR model, p =
When it is 0, it is the MA model.

[0002]

2. Description of the Related Art Conventionally, a step response method and a least squares method have been proposed as methods for identifying a transfer function of a controlled object. The step response method is, as is conventionally known,
The transfer function of the controlled object can be identified based on the step input.

As is well known in the art, the least-squares method can be applied to a controlled object of any order and can identify a transfer function of any order.

[0004]

In the step response method, when a step input of a predetermined height is given, the steady state height of the response and the tangent line of the inflection point are obtained, and the transfer function is obtained from these. Therefore, it can be applied only to a controlled object having an order of 2 or less, and there is an inconvenience that the applicable controlled object has a narrow range.

Since the least squares method obtains a transfer function that minimizes the sum of the squares of the differences, it can be applied to a controlled object of any order. E. FIG. Since it is difficult to satisfy the condition (the condition that n + 1 or more frequencies are required to identify the nth-order transfer function), the identification accuracy of the controlled object deteriorates. Further, in the M-sequence signal (pseudo white signal), the control target generates noise and vibration when the signal is applied, and it is necessary to adjust the parameters for setting the signal. There is an inconvenience that it is necessary.

Therefore, it is impossible to accurately identify a transfer function of an arbitrary order based on the step response, and an identification method capable of accurately identifying a transfer function of an arbitrary order based on the step response. Was longed for.

[0007]

SUMMARY OF THE INVENTION The present invention has been made in view of the above problems, and a novel ARMA capable of accurately identifying a transfer function of an arbitrary order based on a step response.
It aims to provide a model identification method.

[0008]

An ARMA model identification method based on a step response according to claim 1 has a y (s) between a step input u (s) for a controlled object and an output y (s) for the controlled object. = K {B (s) / A (s)} u (s) {where B (s) = b _mp s ^mp + b _mp-1 s ^mp-1 + ... + b
₁ s + 1, A (s) = a _np s ^np + a _np-1 s ^np-1 + ...
+ A ₁ s + 1, s is a Laplace operator, A (s) is a stable polynomial, and np ≧ mp, K is a gain}, which satisfies the relation K,
A method for identifying an ARMA model by defining B (s) and A (s), wherein the orders m and n of B (s) and A (s) are defined and the sum of m, n and 1 or more is obtained. A positive definite matrix W of degree q is set, and P (s) = P _qs ^q + P _q-1 s ^q-1 + ... where B (s) / A (s) = 1 / P (s). + P ₁ s + P ₀
And P ₀ = 1 and K
P ₁ by obtaining the value of
To P _q , the matrix PP = (P ₁ , P ₂ , ...
., P _q-1 , P _q ) ^T and the matrix θ = (a ₁ , a ₂ , ...).
a _n-1 , a _n , b ₁ , b ₂ , ..., B _m-1 , b _m ) ^T and PP = Z θ, the number of matrices 3 is set, and θ = (Z ^T
WZ) ⁻¹ (Z ^T WPP).

[0009]

According to the ARMA model identification method based on the step response of claim 1, the step input u to the controlled object is obtained.
Y (s) between (s) and the output y (s) from the controlled object
= K {B (s) / A (s)} u (s) {however, B
(S) = b _mp s ^mp + b _mp-1 s ^{mp -1} + ... + b ₁ s +
1, A (s) = a _np s ^np + a _np-1 s ^np-1 + ... + a ₁
s + 1 and s are Laplace operators, A (s) is a stable polynomial, and np ≧ mp, and K is a gain.
In identifying the ARMA model by defining (s) and A (s), the orders m and n of B (s) and A (s)
And a positive definite matrix W of order q which is equal to or greater than the sum of m, n and 1 is set, and P (B (s) / A (s) = 1 / P (s)) is established.
(S) = By setting the ^{_{P q s q + P q-}} 1 s q-1 + ··· + P 1 s + P 0, it can be equivalently AR model ARMA model. In this case, the output error 0 to q
The steady value of the multiple integral converges to zero. Then, P ₀ = 1 is set, the value of K is obtained by the equation 1, and the values from P 1 to Pq are obtained by sequentially calculating the gradual expression of the equation 2. here,
Since P _i is determined based on P _i-1 to P ₁ , it is not affected by the higher-order terms ignored by the controlled object, as in the step response method. Further, the matrix PP = (P ₁ , P ₂ , ...,
P _q−1 , P _q ) ^T and the matrix θ = (a ₁ , a ₂ , ...).
a _n-1 , a _n , b ₁ , b ₂ , ..., B _m-1 , b _m ) ^T, and the number of matrices 3 that satisfies PP = Zθ is set. here,
The error e and e = Z.theta-PP, an evaluation function J J = e ^T W
If e, W = W ^T > 0, then the minimum value of θ in J is obtained when Eq.

[0010]

[Equation 4]

Therefore, θ = (Z ^T WZ) ⁻¹ (Z ^T WP
By performing the calculation of P), the matrix θ having the parameters of the transfer function as each element can be obtained. In other words,
The parameters a _i and b _i that minimize the steady value of the 0-q multiple integration of the output error can be estimated, and the transfer function of an arbitrary order can be identified with high accuracy.

[0012]

Embodiments of the present invention will now be described in detail with reference to the accompanying drawings showing embodiments. FIG. 1 shows the ARM of the present invention.
It is a flow chart explaining one example of an A model identification method. Between the step input u (s) for the controlled object and the output y (s) from the controlled object, y (s) = K {B (s)
/ A (s)} u (s) {B (s) = b _mp s ^mp +
b _mp-1 s ^mp-1 + ... + b ₁ s + 1, A (s) = a _np s
^np + a _np-1 s ^np-1 + ... + a ₁ s + 1, s is a Laplace operator, A (s) is a stable polynomial, and np ≧ mp, K is a gain}, K and B (s) satisfying the relation , A (s), in order to identify the ARMA model, in step SP1, the order m of B (s) and A (s),
a positive definite matrix W of degree q which is equal to or larger than the sum of m, n and 1
Is set, and P such that B (s) / A (s) = 1 / P (s)
(S) = sets the ^{_{P q s q + P q-}} 1 s q-1 + ··· + P 1 s + P 0, set to P ₀ = 1, yet initialized to i = 0.

Next, in step SP2, the value of K is obtained by the equation 1. Then, in step SP3, the value of i is incremented, and in step SP4, the gradual expression of Expression 5 is calculated to obtain the value of Pi. afterwards,
In step SP5, it is determined whether i is greater than or equal to q. If i is less than q, the processing in step SP3 is performed again. On the contrary, when i is determined to be equal to or greater than q, the matrix PP = (P ₁ , P ₂ ,
..., P _q-1 , P _q ) ^T, and the matrix θ = (a ₁ , a ₂ , ...
, A _n-1 , a _n , b ₁ , b ₂ , ..., B _m-1 , b _m ) ^T and the number of matrices 3 satisfying PP = Zθ is set, and in step SP7, The unknown parameter of the transfer function is estimated by performing the calculation of θ = (Z ^T WZ) ⁻¹ (Z ^T WPP).

After that, an ARMA model is constructed based on the estimated parameters and gains, and the input / output relationship is confirmed. If there is no deviation as a result of this confirmation, the transfer function of the controlled object is determined based on the estimated parameter and gain. On the contrary, if there is a deviation, the order of A (s), B (s) and W is changed and the above series of processing is performed again.

It should be noted that although the above flow chart does not mention the calculation of the dead time, it can be easily and accurately obtained from the relationship between the step input and the output.

[0016]

(Equation 5)

[0017]

[Specific example 1] As a control target, the pump pressure is 70 kgf /
Using a hydraulic cylinder of cm2, giving a step input, 4
Sampling was performed at msec intervals, the transfer function was identified based on the input / output signal, the output of the control target was calculated, and the output was compared with the actual output (see the solid line). The results of this comparison are shown in FIGS. Here, as the identification method of the transfer function, the method according to the above-described embodiment, the least squares method, and the step response method are adopted. The outputs obtained by adopting these methods are shown by a dotted line, a broken line, and an alternate long and short dash line, respectively. Further, in the method according to the above embodiment, m = 3, n = 4, q = 8, and W are set in the unit matrix. In the least-squares method, a conventionally known ridge constant is introduced, and the numerator order is set to the third order and the denominator order is set to the fourth order. In the step response method, the transfer function is K / (TS +
It was set to 1).

2 and 4, the horizontal axis represents time (seconds).
And the vertical axis is the operating speed (mm / sec) of the controlled object. 3 and 5, the horizontal axis represents time (second) and the vertical axis represents the error (mm / se) with respect to the actual operating speed of the controlled object.
c). Since the velocity error corresponds to the position error, the position error and the like can be confirmed from these figures.

As is clear from these figures, when the identification method according to the above-mentioned embodiment is adopted, an output which can be approximated to the actual output with high accuracy is obtained not only at the time of rising but also at the time of almost steady state. . In other words, it can be seen that the transfer function of the controlled object can be identified with high accuracy.

[0020]

[Specific Example 2] A pump pressure of 20 kgf /
Using a hydraulic cylinder of cm2, giving a step input, 4
Sampling was performed at msec intervals, the transfer function was identified based on the input / output signal, the output of the control target was calculated, and the output was compared with the actual output (see the solid line). The results of this comparison are shown in FIGS. 6 and 7. Here, as the identification method of the transfer function,
The method according to the above-mentioned embodiment, the least squares method was adopted. Outputs obtained by adopting these methods are shown by broken lines and dotted lines in FIG. 6, respectively, and by solid lines and broken lines in FIG. 7, respectively. In the method according to the above embodiment, m = 1 and n =
2, q = 10, and W is a diagonal matrix {= diag (2 ¹⁰ ,
2 ⁹ , ..., 21 ¹ )}. In the least-squares method, the numerator order is set to the first order and the denominator order is set to the second order.

In FIG. 6, the horizontal axis represents time (seconds) and the vertical axis represents the operating speed (mm / sec) of the controlled object. Figure 7
The horizontal axis represents time (seconds) and the vertical axis represents the error (mm / sec) with respect to the actual operating speed of the controlled object. As is clear from these figures, when the identification method according to the above-described embodiment is adopted, an output that can be approximated to the actual output with high accuracy is obtained not only at the time of rising but also at the time of almost steady state. In other words, it can be seen that the transfer function of the controlled object can be identified with high accuracy.

Further, FIGS. 8 to 12 show changes with time in the 1 to 5 multiple integral values of the errors, respectively. The solid line shows the case of the method of the above-mentioned embodiment, and the broken line shows the case of the least squares method. As is clear from these figures, in the case of the method of the above-mentioned embodiment, the absolute value of the integrated value of the error becomes smaller as the multiple integration is performed, whereas in the case of the least squares method, the local value becomes smaller. Approximate accuracy is improved. However, the absolute value of the integral value of the error increases with the passage of time, and even if the multiple integral is applied to the conventional least-squares method, the approximation accuracy cannot be improved over a wide range.

[0023]

The invention according to claim 1 has a peculiar effect that the identification of the transfer function of an arbitrary order can be achieved with high accuracy based on the step response.

[Brief description of drawings]

FIG. 1 is a flowchart illustrating an embodiment of an ARMA model identification method of the present invention.

FIG. 2 is a diagram showing a change over time in output speed based on the transfer function identified by the method of FIG. 1 and the transfer function identified by the conventional method.

FIG. 3 is a diagram showing a temporal change of an output speed error based on a transfer function identified by the method of FIG. 1 and a transfer function identified by a conventional method.

FIG. 4 is a diagram showing a change over time in output speed based on the transfer function identified by the method of FIG. 1 and the transfer function identified by the conventional method.

5 is a diagram showing a temporal change of an error of an output speed based on a transfer function identified by the method of FIG. 1 and a transfer function identified by a conventional method.

FIG. 6 is a diagram showing a change over time in output speed based on the transfer function identified by the method of FIG. 1 and the transfer function identified by the conventional method.

FIG. 7 is a diagram showing a temporal change of an output speed error based on the transfer function identified by the method of FIG. 1 and the transfer function identified by the conventional method.

FIG. 8 is a diagram showing a temporal change of a single integral value of an error.

FIG. 9 is a diagram showing a temporal change of a double integral value of an error.

FIG. 10 is a diagram showing a change with time of a triple integral value of an error.

FIG. 11 is a diagram showing a time change of a quadruple integral value of an error.

FIG. 12 is a diagram showing a temporal change of a quintuple integral value of an error.

Claims (1)

[Claims] 1. A step input u (s) for a controlled object
And the output y (s) from the controlled object, y (s) = K
{B (s) / A (s)} u (s) {where B (s) =
b _mp s ^mp + b _mp-1 s ^mp-1 + ... + b ₁ s + 1, A
(S) = a _np s ^np + a _np-1 s ^np-1 + ... + a ₁ s +
1, s is a Laplace operator, A (s) is a stable polynomial, and np
≧ mp, K is a gain}, K, B (s),
A method for identifying an ARMA model by defining A (s), wherein the orders m and n of B (s) and A (s) are defined, and m, n and 1
A positive definite matrix W of degree q which is equal to or larger than the sum of and is set, and P (s) = P where B (s) / A (s) = 1 / P (s)
Set _q s ^q + P _q-1 s ^q-1 + ... + P ₁ s + P ₀ and set P ₀ = 1 to obtain the value of K by the equation 1 and sequentially calculate the gradual expression of the equation 2 By obtaining the values from P1 to Pq, the matrix PP = (P ₁ , P ₂ , ..., P _q-1 , P _q ) ^T and the matrix θ = (a ₁ , a ₂ , ... , A _n-1 , a _n , b ₁ , b ₂ , ...
.., b _m−1 , b _m ) ^T and the number of matrices 3 that satisfies PP = Zθ are set, and θ = (Z ^T WZ) ⁻¹ (Z ^T WPP) is calculated. An ARMA model identification method based on the step response. [Equation 1] [Equation 2] (Equation 3)