JPH11265205A - Simultaneous optimization design system - Google Patents

Abstract

PROBLEM TO BE SOLVED: To simultaneously optimize the design parameters of a structure and control by using a computer by alternately executing the computation of structure optimization and the computation of control optimization while mutually transferring parameters. SOLUTION: A structure optimization design part 32 performs decomposition into finite elements for the structure variable of a step, obtains the intrinsic mode and intrinsic vibration number of the structure and obtains the state matrix of a state equation. The state matrix is delivered to a control optimization design part 35, the optimum design of a state is executed and a control parameter to be minimum is obtained. A judgement part 37 judges the minimum control parameter and delivers it to the structure optimization design part 32. The structure optimization design part 32 obtains the gradient of an object (evaluation function), decides the direction of a search and obtains the step length of a line search. The judgement part 33 judges whether or not a structure parameter is optimized. In such a manner, while performing transfer between the structure optimization design part 32 and the control optimization design part 35, a point for minimizing an evaluation value is searched.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a design system for optimally selecting design parameters for a structural system and a control system using a computer.

[0002]

2. Description of the Related Art In many structures today, active control is used for improving dynamic characteristics such as improving instantaneous response and reducing vibration. The conventional design method has attempted to optimize the control system after performing structural optimization as shown in FIG. 1A. Due to interference between the structural system and the control system, the parameters of the structural system and the control system obtained by the conventional method are not always optimal for the entire system.

For this reason, attention has been paid to simultaneous optimization of a structure system and a control system as shown in FIG.
l, Vol. 32, 1994, pp. 866-873, AIAA Journal, V
ol. 25, 1987, pages 1133 to 1138, "Measurement and Control", Vol. 36, No. 4, April 1997, and JP-A-6-324711.

[0004] However, the output performance of the conventional simultaneous optimization system is not sufficient, and a higher performance simultaneous optimization system is required.

[0005]

SUMMARY OF THE INVENTION Accordingly, an object of the present invention is to provide a design system for simultaneously optimizing structural and control design parameters using a computer.

[0006]

The present invention has the following structure to solve the above-mentioned problems. That is, the structure is optimized by a finite element method, and a structure optimization unit that calculates a state matrix of a state equation in which the evaluation function is minimized, and the evaluation function is minimized in a state vector provided by the structure optimization unit. Control optimization means for calculating a control parameter to become, based on the control parameters provided from the control optimization means, the structure optimization means step the structure parameters to calculate the state matrix, and Based on the state matrix obtained by the operation, the control optimization means alternately repeats the operation of calculating the control parameter, and the structural parameter and the control parameter when the evaluation function becomes the overall minimum value A simultaneous optimization design system that seeks.

According to the present invention, the calculation of the structure optimization and the calculation of the control optimization are executed alternately while exchanging parameters with each other, and the optimum control parameters in the state matrix calculated by the structure optimization means are obtained. Since the calculation of the optimal state matrix by stepping the structural parameters based on the control parameters is repeated, the local parameters do not fall to the local minimum value (minimum value).
An overall minimum can be reached.

According to a second aspect of the present invention, in the simultaneous optimization system of the first aspect, the control optimizing means uses a control parameter obtained in a previous operation as an initial value of a control parameter for an optimization operation. Take the configuration.

According to the second aspect of the present invention, the control parameters obtained in the previous calculation are used as initial values in the control optimization calculation, so that the calculation speed can be increased.

[0010] Further, the invention described in claim 3 is based on claim 2.
In the invention described in the above, the structure optimization operation uses a sequential quadratic form programming method, and the control optimization includes:
A configuration using a linear matrix inequality method is employed.

According to the third aspect of the present invention, the optimal structure parameters and control parameters can be efficiently calculated by a combination of the sequential quadratic programming method used for the structure optimization and the linear matrix method used for the control optimization. .

[0012]

Embodiments of the present invention will now be described with reference to the drawings. FIG. 2 shows a general block diagram for joint optimization. FIG. 3 is a block diagram obtained by subdividing the block diagram of FIG. 2, and the optimal design proceeds in parallel by the structural system optimal design unit 32 and the control system optimal design unit 35. FIG.
Is a flowchart showing a time series of these operations.

The structure as a continuum can be converted into a discrete system by using the structural modeling finite element method (FEM). The equation of motion of a discrete system is expressed as follows.

[0014]

(Equation 1)

Here, M _{n * n} , K _{n * n} and Cd
_{n * n} is the n * n mass matrix, stiffness matrix and damping matrix, respectively. q _{n + 1} , w
_{s * 1, u t} * 1, the displacement, the external force (disturbance), control input,
y _{v * 1} and z _{v * 1} are outputs for calculating H ₂ and H _∽ norm. s is the number of disturbances, and t is the number of control inputs. θ _{wn * s} , θ _{un * t} , θ _{y v * n} , θ
_{z j * n} is a matrix of w, u, y, z relations. v is the number of outputs y and j is the number of measurement points for H _} .

Using the mode coordinate transformation, the natural frequency and natural mode of the structure are obtained from equation (1). If the higher order modes are ignored, the following mode matrix is obtained.

[0017]

(Equation 2)

Here, Φ _i is an eigenmode, and Φ _{n * m} satisfies the following equation.

[0019]

(Equation 3)

Here, λ _i and Φ _i are the i-th eigenvalue and eigenvector of the structure. The coordinates are converted using the following mode coordinate conversion formula.

[0021]

(Equation 4)

Here, p is a displacement in mode coordinates. By substituting the above equations into equations (1) to (3), the equation of motion in the mode coordinate system is obtained.

[0023]

(Equation 5)

[0024] Here, the attenuation is assumed to be represented by the mode damping ratio xi] _i. As described above, the continuum structure can be converted into a multi-degree-of-freedom discrete system by using the finite element method, and the equations (8) to (10) can be further converted by using the modal analysis method.
Is obtained. Expressions (8) to (1)
Rewriting 0) in the form of a state equation yields the following equation:

[0025]

(Equation 6) here,

[0026]

(Equation 7)

D _{11v * k} , D _{12v * t} , and D _{22v * t} represent w and u components reflected on outputs y and z. If the concept parameter is represented by a vector r, the above matrix is a function of r.

For convenience of co-optimization with the controller , a fixed gain state feedback controller is used. That is, the control input is represented by the following equation.

[0029]

U = K _cx (18) where Kc is a gain matrix of the controller.

In the system as shown in FIG. 2, the transfer function from the disturbance w to the outputs y and z is as follows.

[0031]

(Equation 9)

H ₂ norm of transfer function T _wy and H ₂ of T _{wz
The ∽} norm is defined as follows.

[0033]

(Equation 10)

Here, the superscript ^* represents the conjugate transpose of the matrix. The evaluation function (objective function) of the simultaneous optimization is defined as follows.

[0035]

[Equation 11]

Here, a and b are positive constants. H ₂
And the definition of H _ム norm, the evaluation function F (r, K
c) represents the energy relationship between the disturbance input w and the measured outputs y, z. This evaluation function is a unified evaluation function that can evaluate the performance of the structural subsystem, the control subsystem, and the entire system.

Here, the problem of simultaneous optimization of structural parameters and control parameters for minimizing output due to disturbance can be defined as follows.

[0038]

(Equation 12)

Here, e _i and g _j are the constraints of the structural parameter equations and inequalities, respectively. Equation (24)
(26) defines a mixed joint optimization problem of H ₂ and H _∽ norm. Here, two cases of (a, b) = (1, 0) and (a, b) = (0, 1) are considered. These two cases correspond to the optimization problem of H ₂ and H _{そ れ ぞ れ} respectively.

Solving the simultaneous optimization problem For the defined evaluation function (23), the parameter r of the structural system
Since it is difficult to simultaneously obtain the parameter Kc of the control system and the control system, an iterative method of structural optimization and control optimization is used.
In the iterative method, control parameters are fixed when performing structural optimization, and structural parameters are fixed when performing control optimization.

[0041] By fixing the optimization controller Kc structure, wherein the structure optimization problem (2
4), (25) and (26). To find the optimal structural parameter r, it is necessary to use the sequential
Next Form Programming (Sequential Quadratic Program
ming, SQP) method. In the SQP method, since a gradient of the evaluation function is required, a small perturbation is given to each component of the structural parameter r, and a numerical calculation is performed. The structure is changed according to the gradient, and the changed structure model is obtained.

The control optimization optimal controller, when securing the structural parameters r, it is the solution K _c of the formula (24). Equation (24) is a problem of mixed control of H ₂ and H _∽ , and a linear matrix inequality (LMI) for K _c is obtained. LM
Consider a controller design problem that satisfies the following equation in order to find the equation (1).

[0043]

(Equation 13)

Here, γ is a positive real number, and the structural parameter r is constant.

As mentioned above, two special cases are considered here. When (a, b) = (1, 0), the expression (2)
7) obtaining of H ₂ controller Kc that satisfies is equivalent to obtaining the Kc satisfying the following three inequalities.

[0046]

[Equation 14]

Here, Q and X ₂ are symmetric matrices. Therefore, the problem of designing an optimal controller under the condition of (a, b) = (1, 0) satisfies Expressions (28) to (30),
In addition, it is converted into a problem for obtaining Kc that minimizes γ.

When (a, b) = (0, 1), finding the _H∽ controller Kc that satisfies equation (27) is equivalent to finding Kc that satisfies the following two inequalities.

[0049]

(Equation 15)

Here, X _∞ is a symmetric matrix. So, is converted (a, b) = (0, 1) is the problem of designing an optimal controller conditions satisfying the formula (31) to (32), and the problem of finding the K _c where γ is minimized. From equations (28) to (3
0) and the equation (31) (32) are each H ₂ optimal control and H _∞
5 is an LMI expression of optimal control.

Equations (28) to (30) and Equations (31) to (32) are
Solved by MI method. Usually, controller design is easier than structural design. This is because the size of the control parameters K _c is smaller than the size of the structural parameters.

Combination of Two Optimization Algorithms In the embodiment of the present invention, a parallel method is used as a combination of subsystem optimization sequences.

In the serial method, structure optimization (SO) and control optimization (CO) are executed in the form shown in FIG. In the structure optimization process, the controller Kc is fixed,
The repetition is performed until the values of the structure parameters and the evaluation function sufficiently converge. Since the structural parameter r may be significantly modified in the structural optimization process where Kc is fixed, the system matrix may change significantly and the closed-loop system may become unstable.

To avoid this problem, a parallel method is proposed. In the parallel approach, structure optimization is used as the main process, and an optimal controller is designed for each modified structure in an iteration of the structure optimization. To design an efficient controller,
The optimal value _{Kc, i-1} of the controller in the previous step is
Used as initial parameters in the design of K _{c, i} .

Referring to FIG. 3 and FIG.
The terminating unit 31 sets the structural parameter r to r _o(10
1). The structural optimization design unit 32 determines the structural variable in step i.
And decompose it into finite elements (103).
Mode and eigenfrequency of (Φ, Λ)_i(10
4), state matrix of state equation (A, B, C, D)_iAsk for
(110).

This state matrix is the control optimization design unit 3
5 to execute the optimal control design. In the initial state, i.e., when i = 0, a feasible solution is obtained from equation (27), and this is set as the initial values of the control parameters _Kc and x (111, 11
2). When i is not 0, that is, after the initial state, the values of _Kc and x are set to the optimum values obtained in the previous cycle (113).

Under such conditions, the control optimization design unit 35
Solves the linear matrix inequality (LMI) described in “Optimization of control” (114) and finds the minimum K _ci (11).
Five). The determination unit 37 determines the minimum K _ci and passes it to the structure optimization design unit 32.

The structure optimization design unit 32 determines the gradient ∇ (H _i ) of the objective (evaluation function) (105), determines the search direction G _i (106), and determines the line search step length θ _i . (1
07). Step the structure parameter r by θ _i G _i (10
8), the determining unit 33 determines whether the optimization has been achieved (109). If the optimization has not been achieved, the changing unit 34
Changes the structural parameters to r _i = r _i-1 + θ _i G _i (10
2) Transfer to the structural optimization design unit 32. Thus, the above-described steps from step 103 onward are repeated.

FIG. 5 shows the distribution of the evaluation amounts in the contour system, where the vertical axis represents the structural parameters and the horizontal axis represents the control parameters. Valley 1 is a partial minimum (minimum value), and valley 2 is a global minimum. Now with respect to structural parameters r _o, the initial value of K _c is the point A in FIG. For r _o and K _c structure optimization design portion 32 is given obtaining the search direction G and step length of the search theta. Judgment unit 33
Determines whether optimization has been achieved at r _o and K _c . If no optimization is achieved, the changing unit 34 changes the structure parameters r = r _o + θG, passed to structural optimization design portion 32. This step of θG is indicated by 1P in FIG. The structure optimization design unit 32 obtains a state matrix by the SQP method and transfers the state matrix to the control optimization design unit 35. Based on this state matrix, the control optimization design unit 35
Find the point with the smallest evaluation value by the method. In FIG. 5, assuming that the point B is a point at which the evaluation value becomes the minimum under this condition, the control optimization design unit 35 passes the value of the control parameter K corresponding to the point B to the structure design unit 32.

The structure optimization design unit 32 for the passed K
Calculates the search direction G and the line search step length θ. The determining unit 33 determines whether the optimization has been achieved, and if the optimization has not been achieved, the changing unit 34 steps the structural parameter r by θG obtained this time. This step is indicated by 3P in FIG. The structure optimization design unit 32 calculates a state matrix using the stepped structure parameters, and transfers the state matrix to the control optimization design unit 35.

[0061] Control optimization design unit 35 obtains the control parameters K _c which has the smallest evaluation value under the given conditions of the previous optimum value as the initial value, pass to structure optimization design portion 32.

Thus, the calculation converges toward the valley 2 where the evaluation value is the overall minimum value. When the structure parameter is changed, the convergence from the initial value to the minimum value in the vicinity is always performed, so that convergence to the minimum value can be prevented, and the calculation can be well converged.
According to the algorithm, the minimum value for each calculation step is the closest extreme value as shown in FIG. 6, but the initial value is changed in successive calculation steps and the value shifts to an extreme value smaller than this extreme value. It is possible to avoid convergence to a local minimum and to converge to an overall minimum.

Model and Numerical Simulation As an example, a simultaneous optimization design is performed on the beam model shown in FIG. This beam is supported by two springs having a constant k, and has a concentrated mass ma at both ends. Disturbance w
At the center of the beam and apply a pair of moments u as control inputs symmetrically to the center. The beam is divided into six elements of equal length l, and the positions of the disturbance, control input, spring and mass are as shown in FIG. 8 (a). The displacement y1 at the left end of the beam is used as a measurement output. Referring to FIG. 8B, the dimensions t1, t2, and b of the cross section are constants, and the height h of the cross section is used as a design parameter, and the table shows its initial value hi. Table 1 shows the materials and geometric parameters of this structure.

[0064]

[Table 1] Young's modulus E 200GPa Width b 50mm Mounting mass ma 300Kg Thickness t1, t2 5mm Density ρ 7800kG / m3 Length l 200mm Spring rigidity k 3000N / mm Initial height hi 45mm

FIG. 9 shows two symmetric vibration modes in the initial state of the beam. Considering the symmetry of the beam, there are three independent variables of heights h ₁ , h ₂ , h ₃ of sections ₁ , ₂ , ₃ . The vector r of the structural design parameter is represented as follows.

[0066]

R = {r ₁ , r ₂ , r ₃ } = {h ₁ , h ₂ , h ₃ } (33)

In designing a controller, not only control performance but also control cost and actuator capability are important. To include the control cost in the cost function, the output vector is expressed in the following form.

[0068]

[Equation 17]

Where u is a control input and α is a positive real number. By substituting equations (34) and (35) into equation (23), an evaluation function for simultaneous optimization is obtained. When importance is placed on control performance, α is decreased, and when importance is placed on control cost, α is increased. α
The gain of the set controller K _c becomes very large in 0, in some cases occur errors on numerical calculation when H ₂ is used evaluation function.

The structural design assumes that the total mass of the beam is constant. On the other hand, the following equation constraint is obtained.

[0071]

(Equation 18)

Assuming that the minimum value of the height is 5 mm, the inequality constraint ri> 5 mm (i = 1, 2, 3) is satisfied. Another one
As one of the inequality constraints, the static maximum elastic deformation of the beam (that is, the maximum displacement of the beam when the spring is replaced by simple support) is set to 2 mm.

The parameter α in the equation (34) was set to ^10-9 . In the SQP method of structure optimization, a gradient of an evaluation function is calculated by giving a small perturbation to each structure parameter. Limiting the relative values of the perturbation between 10 ^-6 and 10 ^-2. The relative error between the evaluation function and the parameter was used for convergence determination, and both convergence criteria were set to ^10-9 .

Table 2 shows the results of the H2 optimization. In this table
S₀C₀, S₁C₀, S₀C₁, S _TC_T, S_SC_SYou
And S_PC_PAre the open loop and the structured of the initial structure, respectively.
Open loop with initial structure, closed loop with initial structure, conventional optimal setting
Simultaneous optimal design by instrumentation method, serial method and optimal setting by parallel method
Represents the total. The conventional optimal design method is that the structural parameters are
Optimization and an optimal controller for the resulting structure.
A method of designing.

In the series method, repetition of structure optimization and control optimization was performed six times. Since the SQP method of the structure optimization also uses the iterative calculation, a total of 882 sets of structural parameters were calculated. When the system became unstable, the controller was redesigned during the optimization of the structure. A total of 142 controllers were designed. In the parallel method, the structural design was repeated 180 times until the convergence condition was satisfied, for which 180 controllers were designed.

[0076]

[Table 2]

Table 2 shows that the evaluation function can be sufficiently reduced only by optimizing the structural parameters. Once the structural parameters have been optimized, the heights r ₁ and r ₂
Becomes smaller, the height r ₃ becomes larger. That is, the center of the beam becomes thicker and both sides become thinner. In this shape of the beam,
The displacement at both ends due to the elastic deformation can be offset by the displacement due to the deformation of the spring, and the total displacement is reduced.

On the other hand, the result of S ₀ C ₁ shows that the H ₂ norm of the transfer function T _wy1 can be made smaller simply by adding an optimum controller to the initial structure. Further, target values obtained in S ₀ C ₁ from Table 2, it can be seen that smaller than the target value obtained in S _T C _T. That is,
When the control is added without performing the simultaneous optimization, the structure before the optimization is better than the optimized structure. As shown in the table, the target value can be further reduced by performing the simultaneous optimization. Also, within the range of significant figures used in the table, the results of the parallel method were the same as the results of the serial method. After the simultaneous optimization, the shape of the beam changed significantly. The cross section at the center of the beam has been reduced, and the control input has been made more efficient.

When the disturbance w is unit power white noise, || T _wy2 || ₂ is equal to the average power of the output component y ₂ = αu defined in equation (34). So || T
_The control cost can be evaluated by _wy2 || ₂ . Control cost of _S 0 _{C 1} from Table 2 it can be seen that less than control the cost of _S T _{C T.} The control costs after joint optimization _(S S _{C S} and _S P _{C P)} is smaller than _S 0 _{C 1.} By performing the simultaneous optimization in this manner, not only the response of the vibration but also the control cost can be reduced.

[0080]

[Table 3]

Table 3 shows that H_∞The result of optimization is shown. here
And S_SC_SResults are not optimized,
The value obtained by repeating manufacturing optimization and control optimization six times
You. H _∞Controller is H₂Donkey better than controller
Because of the stability of the
Does not cause system instability due to changes in structural parameters
won. Just H_∞The convergence of the optimization is H₂Slower than optimization
No. Even if structure optimization and control optimization are repeated 6 times, S_SC
_SIs the structural parameter of S_PC_PFar to the optimal value obtained with
Far away. Table 3 shows the values and transmissions of the evaluation function for each case.
Function T_xy1And T_xy2H_∞Table 2 shows the change in the norm.
It shows the same tendency.

This means that attaching the controller to the initial structure is better than attaching it after performing a structural optimization. Output displacement y from disturbance w
Each frequency response of the transfer function T _xy1 to ₁ for the case of the H _∞ Simultaneous Optimization For H ₂ joint optimization Figure 1
0 and FIG. Response _S 0 _{C 1} in the low frequency region of the _S T _{C T} smaller than the simultaneous optimization _S P _{C P} responses smallest compared to other cases. Also, the response of the H _{∽ joint} optimization is smaller than that of H ₂ .

FIGS. 11 and 13 show the frequency response of the transfer function from the disturbance w to the control input u in the case of the H ₂ simultaneous optimization and the case of the _H∞ simultaneous optimization, respectively. The joint optimization flattens the frequency curve of the control input and reduces its maximum value.

[0084]

According to the present invention, the calculation of the structure optimization and the calculation of the control optimization are executed alternately while exchanging parameters with each other, and the optimum control parameters in the state matrix calculated by the structure optimization means are calculated. Obtaining and repeating the calculation of the optimal state matrix by stepping the structural parameters based on this control parameter,
The overall minimum can be reached without falling into a local minimum (minimum).

According to the second aspect of the present invention, the control parameters obtained by the previous calculation are used as initial values in the control optimization calculation, so that the calculation speed can be increased.

Further, according to the third aspect of the present invention, it is possible to efficiently calculate optimal structure parameters and control parameters efficiently by a combination of a sequential quadratic programming method used for structure optimization and a linear matrix method used for control optimization. it can.

[Brief description of the drawings]

FIG. 1 is a block diagram of a conventional design system.

FIG. 2 is a conceptual block diagram of a joint optimization system.

FIG. 3 is a block diagram of a simultaneous optimization system.

FIG. 4 is a flowchart of simultaneous optimization.

FIG. 5 is a diagram showing a distribution of local minimum values.

FIG. 6 is a diagram showing a state in which convergence to a minimum value is avoided.

FIG. 7 is a block diagram showing the concept of a simultaneous optimization system using a serial method.

FIG. 8 is a diagram showing a beam model.

FIG. 9 shows the shape of the first two symmetric modes of the original structure.

FIG. 10 is a diagram showing a frequency response of a transfer function from a disturbance to an output displacement in H ₂ optimization.

FIG. 11 is a diagram showing a frequency response of a transfer function from a disturbance to a control input in H ₂ optimization.

FIG. 12 is a diagram showing a frequency response of a transfer function from a disturbance to an output displacement in _H∽ optimization.

FIG. 13 is a diagram showing a frequency response of a transfer function from a disturbance to a control input in _H∽ optimization.

[Explanation of symbols]

 31 Structural initial value setting unit 32 Structural optimum design unit 33 Judgment unit 34 Change unit 35 Control optimum design unit 36 Change unit 37 Judgment unit

 ────────────────────────────────────────────────── ─── Continued on the front page (72) Koji Motoya Inventor Honda Technical Research Institute, 1-4-1 Chuo, Wako-shi, Saitama

Claims (3)

[Claims] 1. The structure is optimized by a finite element method,
A structure optimization unit that calculates a state matrix of a state equation that minimizes an evaluation function; and a control optimization unit that calculates a control parameter that minimizes the evaluation function in a state vector provided by the structure optimization unit. Based on the control parameters provided by the control optimization means, based on the operation of the structure optimization means to step the structure parameters to calculate the state matrix, based on the state matrix obtained by the operation, A simultaneous optimization design system wherein the control optimizing means alternately repeats the calculation for calculating the control parameter, and obtains a structural parameter and a control parameter when the evaluation function reaches an overall minimum value. 2. The simultaneous optimization design system according to claim 1, wherein said control optimization means uses a control parameter obtained in a previous operation as an initial value of a control parameter for an optimization operation. 3. The simultaneous optimization design system according to claim 2, wherein the operation of the structure optimization uses a sequential quadratic form programming method, and the control optimization uses a linear matrix inequality method.