JP2987639B2 - Control system frequency response analysis method - Google Patents

Description

Description: TECHNICAL FIELD The present invention relates to a control system analysis method, and more particularly to a control suitable for analyzing a frequency response of a control system for controlling a vibration system having a large degree of freedom and a large scale. The present invention relates to a frequency response analysis method for a system.

[Conventional technology]

Conventionally, the state equation of the entire control system for controlling a multi-degree-of-freedom vibration system is described in Transactions of the Japan Society of Mechanical Engineers (C) 52, 475 (Showa 61).
-3) Pages 1030 to 1036 discuss the subject of "high-speed positioning control of a multi-degree-of-freedom vibration system".

On the other hand, pages 190 to 1 of "Control System Design by Numerical Analysis" published by the Society of Instrument and Control Engineers (first edition on October 30, 1986)
On page 94, there is described a method of obtaining a frequency response from a state equation by solving a simultaneous linear equation of a coefficient matrix having a complex number as an element by performing a Laplace transform on the state equation and s = jω = 2πjfω: angular frequency.

[Problems to be solved by the invention]

The method of expressing the state equation described in "High-speed positioning control of multi-degree-of-freedom vibration systems" in the Transactions of the Japan Society of Mechanical Engineers, which is the above-mentioned prior art, considers the control system that controls the vibration system with a large degree of freedom and scale. However, as described therein, it takes a long time to perform the diagonal ratio of the equation of state without reducing the degree of freedom of the vibration system.

On the other hand, the Fadeev method described in “Control System Design by Numerical Analysis Method” issued by the Society of Instrument and Control Engineers
This algorithm does not take into account the application to multi-degree-of-freedom systems beyond the following, and there is a problem in performing frequency response analysis of multi-degree-of-freedom vibration systems.

In addition, a method of obtaining a frequency response from a state equation by Laplace transforming a state equation described in the same book and solving a simultaneous linear equation of a coefficient matrix having a complex number as s = 2πjf is used as a complex eigenvalue λ _{i of the} state equation. Inside the real part Re
(Λ _i ) is close to 0 and l _m (λ _i ) = 2π
No consideration is given to the case of f, and there is a problem that the determinant becomes singular and the inverse matrix cannot be obtained.

An object of the present invention is to provide a method for stably and quickly analyzing a frequency response of a control system for controlling a vibration system having a large degree of freedom and a large scale.

[Means for solving the problem]

The object described above comprises an N-degree-of-freedom vibration system having N degrees of freedom, a control system for controlling the N-degree-of-freedom vibration system, and a feedback system for feeding back the characteristics of the N-degree-of-freedom vibration system and the control system. In a frequency response analysis method of a control system in which a control input is given to a system to be controlled to perform a frequency response analysis, wherein a step of leading a vibration system given as a control target to an equation of a vibration system having N degrees of freedom exists in a control band. Calculating P eigenvectors smaller than N, and performing coordinate transformation using the eigenvectors to change the degree of freedom from N to P
Deriving a state equation of the vibration system with P degrees of freedom reduced to: a step of deriving a state equation of the control system and a state equation of the feedback system; a state equation of the P degree of freedom vibration system; Combining the equation and the feedback system state equation to derive a state equation of the entire system; converting the state equation of the entire system to a standard form; transforming the state equation of state into a Laplace transform; Solving a simultaneous linear equation of an equation obtained by substituting s = 2πjf into the Laplace-transformed standard form equation, and providing a frequency f _n as the control input and obtaining a frequency response thereof.

The above object is achieved by the similarity transformation in which an eigenvector of a state equation of the entire system is calculated and coordinate transformation is performed using orthogonality of the eigenvector to derive a diagonal state equation.

The above object is attained by the above-mentioned similarity transformation performing a Householder transformation on the state equation of the entire system to derive a Hessenberg-type state equation.

[Action]

According to the above configuration, a vibration system given as a control object is derived into an equation of an N-degree-of-freedom vibration system, and coordinate transformation is performed using P eigenvectors to reduce the degree of freedom from N to P. By deriving the state equation, combining the state equation of the control system and the state equation of the feedback system to derive the state equation of the entire system, the order of the state equation of the entire system can be reduced, and the subsequent calculation time can be reduced. .

In addition, the state equation of the entire system is similarly transformed into a standard form.
By performing a similarity transformation and obtaining a diagonal state equation,
Since the determinant becomes singular and the problem that the inverse matrix cannot be obtained is reduced, stable calculation can be performed.

Furthermore, by obtaining a Hessenberg-type state equation by performing a similarity transformation on the state equation of the entire system, the inverse matrix of a simple determinant is solved, and the time required for the calculation is reduced. Is shortened.

〔Example〕

 Hereinafter, an embodiment of the present invention will be described with reference to the drawings.

As shown in FIG. 1, this embodiment includes the following components.

a. an element for modeling the vibration system into an equation of an N-degree-of-freedom vibration system; and b. a primary to P-th order (P
An element for calculating an eigenvalue and an eigenmode of N); c. An element for performing coordinate transformation to derive an equation of P degrees of freedom using the first to Pth eigenmodes obtained in b above; E. An element for obtaining the state equation of the system and the feedback system; e. An element for connecting the equation of the P degree of freedom system obtained in c above and the state equation of the control system and the feedback system obtained in d above; G. An element that defines the state equation obtained in the above as a state equation of the whole system, g. An element that performs a Laplace transform on the equation obtained by performing similarity transformation on the state equation of the whole system obtained by f, and h. It consists of an element for obtaining the frequency response at the frequency fn from the equation obtained by the above g.

The features of this embodiment are as follows: (1) Rather than directly coupling the equation of the N-degree-of-freedom vibration system to the equation of the control system, coordinate transformation is performed before that to reduce the degree of freedom from N to P. Point (2) The point that the state equation obtained by the combination is further subjected to coordinate transformation to stabilize the operation and reduce the amount of operation.

 Hereinafter, the calculation procedure will be described first with respect to the point (1).

A method equation when a multi-degree-of-freedom vibration system is modeled as an N-degree-of-freedom system can be expressed by the following equation.

M + C + KZ = Du (1.1) where M: mass matrix C: damping matrix K: stiffness matrix Z: displacement vector u: control force vector D: constant matrix By solving the following eigenvalue problem, the eigenvalue ω _r ( _r
= 1 ... p) is obtained.

det (K−ω ² M) = 0 (1.2) Further, the eigenvector φ _{r with} respect to the eigenvalue ω _r can be obtained by solving the following equation.

Using the modal matrix φ having the obtained n (Np) eigenvectors φ _r as column vectors, the following coordinate transformation is performed.

Z = φξ (1.4) φ = [φ ₁ ,..., Φp] (1.5) where: ξ: Displacement vector after coordinate transformation (1.4) is substituted into equation (1.1), and both sides are substituted. and the equation of motion multiplied by the φ ^T from the left becomes the following equation.

^{φ T Mφ + φ T φ +} φ T Kφξ = φ T Du φ, M, the following relationship holds between the K.

It is also assumed that the following relationship holds.

φ ^T Cφ = diag [2ζ _r ω _r ] (1.9) where ζ _r : rth order damping ratio φ ^T Kφ =, φ ^T Cφ = Equation (1.6) can be rewritten as .

I ++ ξ = φ ^T Du (1.10) The equation of the multi-degree-of-freedom vibration system obtained as described above is as follows.

The equations of the control system and the feedback are as follows.

Where y: state variables of the control system where y: state variables of the control system r: input vector of the control system: output vector of the control system ,,,,,,,,
,: Constant matrix From equations (1.11) to (1.15), displacement vector z, velocity vector, control force vector u
Is eliminated, a state equation of the whole system is obtained. The obtained state equation is as follows.

FIG. 2 is a diagram showing a procedure for converting the state equation of the whole system into a diagonal shape by similarity conversion and then obtaining a frequency response.

 Next, the procedure will be described.

(F) Obtain the state equation of the whole system.

(G ₁₁₎ eigenvalues of the state equation, determine the eigenvectors.

(G ₁₂ ) Coordinate conversion is performed using the obtained eigenvalues and orthogonality of the eigenvectors to diagonalize the state equation, and the resulting equation is Laplace-transformed to obtain the transfer function of the system.

(H ₁ ) Transfer function s = 2πjf (j: imaginary unit, f: frequency)
To obtain the frequency response at each frequency fn.

Next, a procedure for deriving an expression that is the basis of the procedure shown in FIG. 2 will be described.

 Here, the state equation of the whole system is represented by the following equation.

Here, x: state variable vector (m-dimensional vector) y: output vector u: input vector A, B, C, D: constant matrix Solve the eigenvalue problem of equation (2.1) and obtain the system eigenvalue λ
_{Find i} . The eigenvector _{ｉ i} for the eigenvalue λ _i is obtained by solving the following equation.

(Λ _i IA) ψ _i = 0 (2.3) The eigenvectors are normalized so as to satisfy the following equation.

Here,: performs coordinate transformation of the following equation using the matrix Ψ with [psi _i the conjugate complex vector eigenvectors [psi _i as column vectors.

x = Ψη ...... (2.5) Ψ = [ψ _i, ..., ψ _m] ... (2.6) where, eta: variable vector (2.5) after the coordinate transformation by the coordinate transformation of equation (2.1) below, ( 2.2) Equation is as follows.

Here, the following relationship is established.

Ψ ⁻¹ AΨ = diag [λ _i ] (2.9) When Λ = Ψ ⁻¹ AΨ is replaced, the equations (2.7) and (2.8) become the following equations.

When the above equation is Laplace transformed, the following equation is obtained.

Here, H (s) = L (η (t)), U (s) = L (u (t)), Y (s) = L (y (t)), and the equation (2.12) is expressed as H (s) Solving for gives the following equation.

Substituting equation (2.12) into equation (2.13) gives the following equation.

Y (s) = {CΨ (sI + Λ) ⁻¹ Ψ ⁻¹ B + DU｝ U (s) (2.16) The frequency response is obtained by substituting s = jω = 2πjf into the above equation.

It can be seen from the above equation modification that the present method has the following features.

1) From the equation (2.15), Re (λ _i ) = 0 Im (λ _i ) = 2π
As long as it does not become jf, the frequency response can be obtained stably.

2) When the number of frequencies fn to be calculated from the equations (2.15) and (2.16) is large, the amount of calculation is smaller than when the frequency is not converted to a diagonal.

The above is the description of the method of converting the matrix A into a diagonal shape.

An embodiment in which the matrix A is converted into the Hessenberg form will be described with reference to FIG. Matrix A
The purpose of converting to the Hessenberg form is to reduce the frequency response calculation time. In this example, the frequency response is obtained by the following procedure.

(F) Obtain the state equation of the whole system.

(G ₂₁ ) Perform a Householder transformation on the equation of state to convert it into a Hessenberg form.

(G ₂₂ ) Laplace transform the equation obtained by the above transformation.

(H ₂ ) Substituting S = 2πjf into the equation obtained by the Laplace transform, and solving the simultaneous equations to obtain the frequency response.

The procedure for deriving the formula that is the basis of the procedure shown in FIG. 3 will be described.

It is assumed that the state equation of the whole system is the equation (2.1).
It is assumed that the matrix A is transformed into the Hessenberg form by the transformation matrix T.

 Equation (2.1) is converted to equation (3.1).

Here, X = Tη.

T- ¹ AT is Hessenberg type Becomes

 When the equation (3.1) is Laplace transformed, the following equation is obtained.

Substituting ω = 2πjf into the equation (3.3.1) gives the following equation.

(2πjfI−T ⁻¹ AT) L (η) = T ⁻¹ BL (u)
(3.4.1) At this time, the following equation holds from equation (3.2).

The frequency response is obtained by solving the equation (3.4.1) for L (η) and substituting it into the equation (3.3.2).

When the matrix A is converted to the Hessenberg form according to the equation (3.5), the calculation for obtaining (2πjfI−T ⁻¹ AT) ⁻¹ is simplified. Therefore, compared to the case where no conversion is performed, it is possible to reduce the calculation amount of the frequency response for a large number of frequencies fn.

Although the embodiments of the present invention have been described above, the calculations excluding the steps of FIGS. 1 b and c can be performed in any of the embodiments.

〔The invention's effect〕

According to the present invention, a state equation in which a multi-degree-of-freedom vibration system given as a control target is coordinate-transformed to reduce the degrees of freedom,
By deriving the state equation of the entire system in which the state equation of the control system and the state equation of the feedback system are combined, the order of the state equation of the entire system can be reduced, and the effect of shortening the subsequent calculation time can be obtained.

Further, by performing a similarity transformation of the state equation of the entire system into a diagonal state equation, the determinant becomes singular and a stable calculation can be performed with less problems that the inverse matrix cannot be obtained.

Then, the state equation of the entire system is further converted into a Hessenberg state equation by analogy, so that a simple simultaneous linear determinant is directly solved, which is numerically stable and reduces the time required for the operation.

[Brief description of the drawings]

FIG. 1 is a flowchart for explaining an overall operation sequence according to an embodiment of the present invention, FIG. 2 is a flowchart for explaining an embodiment of similarity conversion shown in FIG. 1, and FIG. 3 is a flowchart shown in FIG. 11 is a flowchart illustrating another embodiment of the similarity conversion.

──────────────────────────────────────────────────続 き Continued on the front page (58) Fields investigated (Int. Cl. ⁶ , DB name) G01M 19/00 G01M 7/00 G01H 1/00-17/00 G06F 15/60 G05D 3/00 G05H 19 / 00

Claims (3)

(57) [Claims] 1. An N-degree-of-freedom vibration system having N degrees of freedom,
A control system for providing a control input to a control system for controlling the N-degree-of-freedom vibration system and a feedback system for feeding back the characteristics of the N-degree-of-freedom vibration system and the control system to perform frequency response analysis In the frequency response analysis method of (1), a step of deriving a vibration system given as a control target into an equation of an N-degree-of-freedom vibration system; a step of calculating P eigenvectors smaller than N existing in a control band; Deriving a state equation of a vibration system having P degrees of freedom in which the degrees of freedom are reduced from N to P using coordinate transformation; deriving a state equation of the control system and a state equation of the feedback system; Combining the state equation of the vibration system, the control system state equation, and the feedback system state equation to derive the state equation of the entire system; A step of performing a similarity conversion of the equation into a standard form; a step of performing a Laplace conversion of the standard form state equation; frequency response analysis method of a control system, characterized in that to obtain the frequency response given frequency f _n as the control input. 2. The method according to claim 1, wherein the similarity transformation calculates an eigenvector of a state equation of the entire system, and performs coordinate transformation using orthogonality of the eigenvector to derive a diagonal state equation. 3. A frequency response analysis method for a control system according to item 1. 3. The frequency response analysis method according to claim 1, wherein said similarity transformation derives a Hessenberg-type state equation by performing a Householder transformation of the state equation of the entire system. .