A large-scale spatial truss model reduction method considering convex set uncertainty

Abstract

This invention relates to a method for reducing the order of large-scale spatial truss models considering convex set uncertainties. This method provides a more efficient and accurate control approach for truss structures, applicable to structural control problems of large-scale spatial trusses, and belongs to the fields of truss structure modeling and analysis, and truss structure control system design. By considering the correlation between uncertain parameters, this invention derives the propagation law of uncertain parameters, providing a solution framework for the application of convex set uncertainty in the control of large-scale spatial truss structures. Traditional probabilistic methods are limited by their high dependence on experimental data, restricting their application in large-scale spatial trusses. Convex set uncertainty analysis, as an emerging non-probabilistic method, is more suitable for solving uncertain large-scale spatial truss structure control problems with limited data.

Description

Technical Field

[0001] This invention relates to a method for reducing the order of a large-scale spatial truss model that considers the uncertainty of convex sets. This method provides a more efficient and accurate control method for truss structures and is applicable to structural control problems of large-scale spatial trusses. It belongs to the fields of truss structure modeling and analysis, and truss structure control system design. Background Technology

[0002] Large-scale space trusses are a critical component of space engineering. These structures, composed of complex members and nodes, play a vital supporting and connecting role in spacecraft, space stations, and other space probes. Therefore, large-scale space trusses are not only fundamental to space exploration but also indispensable for ensuring the long-term operation of spacecraft and the success of missions. However, due to the complexity of large-scale truss structures, directly analyzing and controlling their dynamic models is an extremely difficult task. Model order reduction, as an effective method for handling complex models, is particularly applicable to the structural control problems of large-scale space trusses.

[0003] Model reduction is an important research area in control systems. With technological advancements and increasing system complexity, we face more and more dynamic systems with high dimensionality, complexity, and computational burden. In this context, reducing model complexity, improving computational efficiency, and preserving key characteristics have become crucial challenges. In the process of reducing the order of control models for large-scale truss structures, singular value decomposition (SVD) methods have been widely applied. This method first constructs equilibrium state equations using the original dynamic equations, and then performs singular value decomposition on these equilibrium state equations. Next, by extracting modal parameters that match the desired dimensionality of the reduced model, the final reduced-order model is obtained.

[0004] In practical engineering, large-scale truss structures inevitably exhibit uncertainties due to the presence of multiple uncertainties and disturbances. Traditional probabilistic methods, such as Monte Carlo simulations, obtain the range of uncertainty through multiple experiments; however, this method is computationally intensive, time-consuming, and requires stringent experimental data. Sampling for large-scale truss structures is costly and data is insufficient, making traditional probabilistic methods unsuitable for structural control problems of large-scale spatial trusses under uncertain conditions. Compared to probabilistic methods, convex set analysis, as an emerging non-probabilistic theory, offers higher computational efficiency, particularly suitable for evaluating uncertainties with limited data. Furthermore, convex set analysis considers the correlation between uncertain parameters, resulting in higher accuracy compared to traditional interval analysis methods. Therefore, model order reduction methods based on convex set analysis are of significant importance in solving the structural control of large-scale spatial trusses. Summary of the Invention

[0005] The technical problem solved by this invention is to overcome the shortcomings of the prior art and propose a method for reducing the order of a large-scale spatial truss model that considers the uncertainty of convex sets.

[0006] The technical solution of this invention is:

[0007] A method for order reduction of large-scale spatial truss models considering convex set uncertainties, the method comprising the following steps:

[0008] Step 1: Construct a deterministic dynamic model based on the large-scale spatial truss structure, and construct the deterministic state-space equations of the deterministic dynamic model;

[0009] Step 2: Represent the uncertainty parameters of the large-scale spatial truss using the convex set method;

[0010] Step 3: Based on the deterministic state-space equation constructed in Step 1 and the uncertainty parameters represented by the convex set method in Step 2, construct the state-space equation based on the convex set.

[0011] Step 4: Construct the transformation matrix of the deterministic equilibrium state equation based on the deterministic state space equation constructed in Step 1, and use the constructed transformation matrix to construct the convex set-based state space equation in Step 3 to obtain the convex set-based equilibrium state equation.

[0012] Step 5: Calculate the deterministic controllability matrix of the deterministic state-space equation in Step 1, and calculate the convex set-based controllability matrix of the convex set-based equilibrium state equation in Step 4.

[0013] Step 6: Perform singular value decomposition on the deterministic performance controllability matrix and the controllability matrix based on convex sets obtained in Step 5, and use the convex set method to determine the uncertainty range of each singular value and the correlation between each singular value.

[0014] Step 7: Based on the range of uncertainty intervals of each singular value obtained in Step 6 and the correlation between each singular value, use the singular value truncation algorithm based on convex sets to determine the order of the reduced-order model.

[0015] Step 8: Based on the order of the reduced-order model obtained in Step 7, with the optimization objective of maximizing the approximation between the reduced-order model and the deterministic dynamic model, the state variables of the reduced-order model are used as independent variables to establish an optimization model and solve the constructed optimization model, finally obtaining the deterministic reduced-order model of the large-scale spatial truss.

[0016] Step 9: Use the convex set method to represent the reduced-order model obtained in Step 8 using convex sets, thus obtaining the reduced-order model based on convex sets.

[0017] In step 1, the deterministic dynamic model constructed is as follows:

[0018]

[0019] Where M is the mass matrix, P is the damping matrix, K is the stiffness matrix, and t represents the time variable. w(t) and u(t) are the acceleration, velocity, and position of the system state variables, respectively. u These are the input force vector and the corresponding position matrix, respectively. The state variables are set as follows: The deterministic state-space equations are obtained in the following form:

[0020]

[0021] in, It is a system matrix. C is the input matrix, and C is the output matrix. The C matrix is ​​determined by the definition of the output quantity.

[0022] In step 2, the uncertainty parameters of the large-scale spatial truss are represented as a convex set as follows:

[0023] b = [b1, ..., b k …,b m ] T ∈b CM ={b|(bb c ) T M(b r ,ρ b (bb) c )≤1}

[0024] Where m represents the number of uncertainty parameters of the large-scale spatial truss structure, and b is an m-dimensional vector. k The k-th element of b, where k = 1, 2, 3, ..., m, corresponds to the k-th uncertainty parameter of the large-scale spatial truss structure. CM Describing a convex set, and b These are the upper and lower bounds of vector b, respectively. and They are vectors b CM The mean vector, radius vector, and correlation coefficient matrix, and These are the k-th convex elements. The mean and radius elements. It is b CM The kth element, The i-th convex element and the j-th convex element Correlation coefficient, M(b) r ,ρb ) is a matrix that has a symmetric positive definite property for describing the dispersion and correlation among all convex variables, and is a matrix about b. r and ρ b The function, M(b) r ,ρ b It is expressed as:

[0025]

[0026] in, It is b CM The covariance matrix. yes and The covariance value. Hadamard operator. This represents the element-wise multiplication of corresponding positions in two vectors or matrices.

[0027] In step 3, the constructed state-space equation based on convex sets is decomposed into two parts: deterministic and uncertain partial derivatives.

[0028]

[0029]

[0030] Where, x c (t), and u c (b,t) represent the convex set x, respectively. CM (t), and u CM The deterministic part of (b,t). A c (b) and B c (b) represent convex set A respectively. CM (b) and B CM (b) The definite part. and Representing convex set x respectively CM (t), and u CM The partial derivative of (b,t) with respect to the k-th uncertainty parameter. and They represent convex set A respectively CM (b) and B CM (b) The partial derivative with respect to the k-th uncertainty parameter.

[0031] Among them, convex set A CM (b) Through the deterministic part A c (b) and the partial derivatives of the convex set with respect to each uncertainty parameter pass Calculated, where It is a convex set formed by the elements in the i-th row and j-th column of matrix A, through Obtained through calculation. Let represent the nominal value of the element in the i-th row and j-th column of matrix A. The orthogonal matrix H and the diagonal matrix Λ are obtained through the following Cholesky decomposition: H T M(b r ,ρ b H = Λ, H T H = I. ξ b =Λ 1 / 2 H T (bb c ) represents a hyperellipsoid composed of initial uncertainty parameters. A represents ij The convex set perturbation. Through the convex set uncertainty method, it can be seen that... The radius is calculated using the following formula:

[0032]

[0033] Each element is obtained based on the deterministic part and the uncertain partial derivative. After obtaining the convex set representation, we get The convex set representation.

[0034] Construct the following state-space equation based on convex sets:

[0035]

[0036] In step 4, the transformation matrix T between the deterministic state-space equation and the deterministic equilibrium state equation is obtained by solving T = Σ 1 / 2 Z T V -1 We obtain that matrix V can be obtained through P = VV T The solution yields Σ, which is a diagonal matrix composed of Hankel singular values ​​of the deterministic state-space equations, Z is the eigenvector of matrix G, and P and G are the controllability and observability Gram matrices of the deterministic state-space equations.

[0037] The deterministic equilibrium equation is expressed as: in

[0038] The partial derivative part of the equilibrium state equation based on convex sets is expressed as: in

[0039] The equilibrium state equation based on convex sets is:

[0040]

[0041] In step 5, the controllability matrix of the deterministic equilibrium state equation Solving the Lyapunov equations shown below yields:

[0042]

[0043] Considering uncertainty b, the above equation can be expressed as the deterministic part and the first-order partial derivative of uncertainty:

[0044]

[0045]

[0046] By solving the above two equations, the deterministic part of the controllability matrix of the equilibrium state equation is obtained. The first-order partial derivatives with respect to the uncertainty parameter are divided into the components. The controllability matrix of the equilibrium state equation based on convex sets is obtained using the convex set method.

[0047] In step 6, the singular value decomposition of the performance control matrix is ​​determined as follows: in Represents a deterministic singular value matrix, diag() n×n This indicates that an n-order diagonal matrix is ​​constructed using the input column vector as its elements. U c and V c These represent the left and right singular value vector matrices of deterministic nature, respectively.

[0048]

[0049] in, Represents matrix U c The i-th column vector Represents matrix V c The first-order partial derivatives of the i-th column vector are related as follows:

[0050]

[0051]

[0052] The first-order partial derivatives of the singular values ​​with respect to the uncertainty parameters are obtained by solving the above equation. Convex sets for obtaining singular values ​​using the convex set method. The range of uncertainty intervals for each singular value is obtained using the following convex set operation:

[0053]

[0054] in Based on convex set operations, the covariance relationship between the singular values ​​is obtained using the following operation:

[0055]

[0056] in

[0057] In step 7, the singular value truncation algorithm based on convex sets is described as follows:

[0058] Each singular value lies within its convex set obtained through the proposed method. For two adjacent singular values ​​of an uncertain convex set... and By setting two uncertain adjacent singular values ​​as two coordinate axes, three types of convex sets were plotted in two dimensions, with the angle bisector in the first quadrant representing σ. i+1 =σ i , σ i+1 =σ i The relationships between convex sets are distance, adjacency, and intersection. A cutoff criterion is designed for the intersection case.

[0059] To truncate the order of an uncertain control system with two densely distributed singular values ​​of the CMB (Central Block Model), i.e., how to select the truncation order position from i or i+1, a truncation criterion σ is applied. cr To determine the truncation order. Based on the probability of convex sets. and Defined as

[0060]

[0061]

[0062] Where Ω=Γ1∪Γ2∪Γ3 is the angle bisector σ of the CMB oblique ellipse in the first quadrant. i+1 =σ i The area cropped on the right. Ω i =Γ2∪Γ3 and Ω i+1 =Γ3 satisfies in Ω and The area. S Ω , and They are Ω and Ω respectively i and Ω i+1 The area.

[0063] To provide a singular value truncation criterion, the convex set probability ratio is defined.

[0064] The singular value truncation order w based on convex sets is set as follows:

[0065]

[0066] That is

[0067]

[0068] The optimization model established in step 8 is shown below:

[0069] find

[0070] min

[0071] st

[0072] 1≤w≤n-1

[0073] in This indicates the retained w-order state parameters, and the superscript "^" indicates a reduced-order model of the state-space equations. It is x T The sub-states of the corresponding state parameters, x T =[x1,…,x n ] T It represents the total nth order state of the complete original system. and These are the first w singular values ​​of the simplified and complete original systems, respectively.

[0074] In step 9, the convex set order reduction model is as follows: The obtained convex set reduced-order model is used for the structural control of large-scale spatial trusses and the parametric analysis of large-scale spatial trusses.

[0075] Beneficial effects

[0076] 1. This invention proposes a model order reduction algorithm based on convex set uncertainty analysis for large-scale spatial trusses. Considering the correlation between uncertainty parameters, the propagation law of uncertainty parameters is derived, providing a solution framework for the application of convex set uncertainty in the control of large-scale spatial truss structures.

[0077] 2. The high dependence of traditional probabilistic methods on experimental data limits their application in large-scale spatial trusses. In contrast, the convex set uncertainty analysis method, as an emerging non-probabilistic method, is more suitable for solving the control problem of large-scale spatial truss structures with limited data and uncertainties.

[0078] 3. Traditional probabilistic methods are computationally intensive and time-consuming when dealing with high-dimensional systems, making them difficult to apply. In contrast, convex set-based model order reduction methods only need to consider the propagation law of uncertainty parameters, which can complete the calculation more efficiently and is therefore more suitable for large-scale spatial truss structure control problems.

[0079] 4. This invention takes into account the case where the uncertainty intervals of singular values ​​overlap, and designs a corresponding truncation criterion for this purpose.

[0080] 5. This invention designs an optimization model for selecting state variables under the uncertainty of convex sets, thereby obtaining a more accurate reduced-order model of large-scale spatial trusses. Attached Figure Description

[0081] Figure 1 A schematic diagram illustrating the relationship between singular values ​​of an uncertain convex set;

[0082] Figure 2 A schematic diagram illustrating the determination of the truncation order based on the truncation criterion;

[0083] Figure 3 It is a truss system comprising 5 spans;

[0084] Figure 4 The results are for the 17th and 18th order singular values;

[0085] Figure 5 This is a schematic diagram showing the upper and lower bounds of the original system response and the reduced-order system response. Detailed Implementation

[0086] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0087] A method for order reduction of large-scale spatial truss models considering convex set uncertainties, the method comprising the following steps:

[0088] Step 1: Construct a deterministic dynamic model based on the large-scale spatial truss structure, and construct the deterministic state-space equations of the deterministic dynamic model;

[0089] Step 2: Represent the uncertainty parameters of the large-scale spatial truss using the convex set method;

[0090] Step 3: Based on the deterministic state-space equation constructed in Step 1 and the uncertainty parameters represented by the convex set method in Step 2, construct the state-space equation based on the convex set.

[0091] Step 4: Construct the transformation matrix of the deterministic equilibrium state equation based on the deterministic state space equation constructed in Step 1, and use the constructed transformation matrix to construct the convex set-based state space equation in Step 3 to obtain the convex set-based equilibrium state equation.

[0092] Step 5: Calculate the deterministic controllability matrix of the deterministic state-space equation in Step 1, and calculate the convex set-based controllability matrix of the convex set-based equilibrium state equation in Step 4.

[0093] Step 6: Perform singular value decomposition on the deterministic performance controllability matrix and the controllability matrix based on convex sets obtained in Step 5, and use the convex set method to determine the uncertainty range of each singular value and the correlation between each singular value.

[0094] Step 7: Based on the range of uncertainty intervals of each singular value obtained in Step 6 and the correlation between each singular value, use the singular value truncation algorithm based on convex sets to determine the order of the reduced-order model.

[0095] Step 8: Based on the order of the reduced-order model obtained in Step 7, with the optimization objective of maximizing the approximation between the reduced-order model and the deterministic dynamic model, the state variables of the reduced-order model are used as independent variables to establish an optimization model and solve the constructed optimization model, finally obtaining the deterministic reduced-order model of the large-scale spatial truss.

[0096] Step 9: Use the convex set method to represent the reduced-order model obtained in Step 8 using convex sets, and obtain the reduced-order model based on convex sets.

[0097] In step 1, the deterministic dynamic model constructed is as follows:

[0098]

[0099] Where M is the mass matrix, P is the damping matrix, K is the stiffness matrix, and t represents the time variable. w(t) and u(t) are the acceleration, velocity, and position of the system state variables, respectively. u These are the input force vector and the corresponding position matrix, respectively. The state variables are set as follows: The deterministic state-space equations are obtained in the following form:

[0100]

[0101] in, It is a system matrix. C is the input matrix, and C is the output matrix. The C matrix is ​​determined by the definition of the output quantity.

[0102] In step 2, the uncertainty parameters of the large-scale spatial truss are represented as a convex set as follows:

[0103] b = [b1, ..., b k …,b m ] T ∈b CM ={b(bb c ) T M(b r ,ρ b (bb) c)≤1}

[0104] Where m represents the number of uncertainty parameters of the large-scale spatial truss structure, and b is an m-dimensional vector. k b is the k-th element of b, where k = 1, 2, 3, ..., m, corresponding to the k-th uncertainty parameter of the large-scale spatial truss structure. CM This represents a convex set. and b These are the upper and lower bounds of vector b, respectively. and They are vectors b CM The mean vector, radius vector, and correlation coefficient matrix. and These are the k-th convex elements. The mean and radius elements. It is b CM The kth element. The i-th convex element and the j-th convex element The correlation coefficient. M(b) r ,ρ b ) is a matrix that has a symmetric positive definite property for describing the dispersion and correlation among all convex variables, and is a matrix about b. r and ρ b The function. M(b) r ,ρ b ) can be expressed as:

[0105]

[0106] in It is b CM The covariance matrix. yes and The covariance value. Hadamard operator. This represents the element-wise multiplication of corresponding positions in two vectors or matrices.

[0107] In step 3, combining the deterministic state-space equation from step 1 and the uncertainty parameter representation from step 2, the constructed state-space equation based on convex sets can be decomposed into two parts: deterministic and uncertainty partial derivatives.

[0108]

[0109]

[0110] Where x c (t), and u c (b,t) represent the convex set x, respectively.CM (t), and u CM The deterministic part of (b,t). A c (b) and B c (b) represent convex set A respectively. CM (b) and B CM (b) The definite part. and Representing convex set x respectively CM (t), and u CM The partial derivative of (b,t) with respect to the k-th uncertainty parameter. and They represent convex set A respectively CM (b) and B CM (b) The partial derivative with respect to the k-th uncertainty parameter.

[0111] Among them, convex set A CM (b) can be determined through the deterministic part A c (b) and the partial derivatives of the convex set with respect to each uncertainty parameter pass Calculated, where It is a convex set formed by the elements in the i-th row and j-th column of matrix A, which can be obtained by... Obtained through calculation. Let represent the nominal value of the element in the i-th row and j-th column of matrix A. Orthogonal matrix H and diagonal matrix Λ can be obtained through the following Cholesky decomposition: H T M(b r ,ρ b H = Λ, H T H = I. ξ b =Λ 1 / 2 H T (bb c ) represents a hyperellipsoid composed of initial uncertainty parameters. A represents ij The convex set perturbation. Through the convex set uncertainty method, it can be seen that... The radius can be calculated using the following formula:

[0112]

[0113] Therefore, each element is obtained based on the deterministic part and the uncertain partial derivatives. After representing the convex set, we can obtain The convex set representation.

[0114] Therefore, the state-space equations based on convex sets can be expressed in the form of partial derivatives of both deterministic and uncertain state-space equations. Considering that higher-order partial derivative terms have a relatively small impact on the system, they are ignored, and the following state-space equations based on convex sets are constructed:

[0115]

[0116] The above equation includes both the deterministic state-space equation and the partial derivatives of the state-space equation with respect to the uncertain parameters. The above equation can be simplified to the following expression, which is the state-space equation based on convex sets obtained in step 3, where... Representing a 2n-dimensional column vector space:

[0117]

[0118] The controllability proof of the state-space equation based on convex sets in step 3 is as follows:

[0119] Under the premise that the deterministic state-space equations are stable, i.e., Q c =[B c A cc B c … (A c ) n-1 B c The row is full rank, where Q is the controllability matrix of the deterministic state-space equation.

[0120] The controllability matrix based on the state-space equation of convex sets is Q = [B AB … A n-1 B], where

[0121]

[0122] therefore

[0123] in The rank of the Q matrix can be calculated using the following method:

[0124]

[0125] Where Δ = [Δ0 Δ1 … Δ n-1 ]. Due to Q c Having full-rank rows, the matrix [Q c 0] and [ΔQ c It also has full rank, which can be represented as r([Q]). c 0])=r([Δ Q c ]) = n. Because [Q c The row vector of [0] and [ΔQ] cThe row vectors of [Q] are linearly independent, therefore, r(Q) = r([Q]). c 0])+r([Δ Q c ])=2n. Therefore, matrix Q also has full rank, meaning the state-space equation based on convex sets is controllable. Q.E.D.

[0126] The stability proof in step 3 is as follows:

[0127] When the deterministic state-space equations are stable, all eigenvalues ​​of the state transition matrix have negative real parts, i.e., Re(λ). j (A c ))<0, where the Re() function represents extracting the real part of the input variable.

[0128] The eigenvalues ​​of the convex set state transition matrix A can be expressed as follows:

[0129] λ j (A)={λ j ||λ j I 2n -A|=0,j=1,2,3…2n}

[0130] Will Substituting into the above formula, we get:

[0131]

[0132] A c Each eigenvalue of is a double eigenvalue of matrix A. When the deterministic system is stable, matrix A c All eigenvalues ​​of matrix A have negative real parts, thus all eigenvalues ​​of matrix A have negative real parts, which shows that equation (10) is also stable.

[0133] In step 4, the transformation matrix T between the deterministic state-space equation and the deterministic equilibrium state equation can be obtained by solving T = Σ 1 / 2 Z T V -1 We obtain that matrix V can be obtained through P = VV T Solving for Σ, we find that Σ is a diagonal matrix composed of the Hankel singular values ​​of the deterministic state-space equations, Z is the eigenvector of matrix G, and P and G are the controllability and observability Gram matrices of the deterministic state-space equations. Therefore, the deterministic equilibrium state equations can be expressed as... in The partial derivatives of the equilibrium equations based on convex sets can be expressed as: in

[0134] Therefore, the equilibrium state equation based on convex sets is:

[0135]

[0136] The equilibrium state equation based on convex sets in step 4 can be obtained by simplifying the above equation:

[0137]

[0138] In step 5, the controllability matrix of the deterministic equilibrium state equation Solving the Lyapunov equations shown below yields:

[0139]

[0140] Considering the uncertainty b, the above equation can be expressed as a deterministic part and a first-order partial derivative part of the uncertainty:

[0141]

[0142]

[0143] By solving the two equations above, the deterministic part of the controllability matrix of the equilibrium state equation can be obtained. The first-order partial derivatives with respect to the uncertainty parameter are divided into the components. Furthermore, the controllability matrix of the equilibrium state equation based on convex sets in step 5 can be obtained through the convex set method.

[0144] In step 6, the singular value decomposition of the performance control matrix is ​​determined as follows: in Represents a deterministic singular value matrix, diag() n×n This indicates that an n-order diagonal matrix is ​​constructed using the input column vector as its elements. U c and V c These represent the left and right singular value vector matrices, respectively. According to classical singular value decomposition theory: in Represents matrix U c The i-th column vector Represents matrix V c The i-th column vector. Considering the propagation of uncertainty parameters and using the convex set uncertainty analysis method, the first-order partial derivative relationship can be obtained as follows:

[0145]

[0146]

[0147] The first-order partial derivatives of the singular values ​​with respect to the uncertainty parameters can be obtained by solving the above equation. Furthermore, the convex set of the singular values ​​in step 6 can be obtained through the convex set method. The uncertainty range of each singular value in step 6 can be obtained using the following convex set operation:

[0148]

[0149] in According to convex set operations, the covariance relationship between singular values ​​can be obtained using the following operation:

[0150]

[0151] in This allows us to obtain the correlation between the various singular values ​​in step 6.

[0152] In step 7, the singular value truncation algorithm based on convex sets is described as follows:

[0153] Each singular value lies within its convex set obtained through the proposed method. For two adjacent singular values ​​of an uncertain convex set... and like Figure 1 The following three relationships will appear. Three cases of convex sets are plotted in two dimensions by setting two uncertain adjacent singular values ​​as two coordinate axes. The angle bisector in the first quadrant represents σ. i+1 =σ i Therefore, in Figure 1 σ is shown in i+1 =σ i The relationship between three types of convex sets (far away, adjacent, and intersecting). Since the first two cases (far away and adjacent) are very clear, this invention designs a truncation criterion for the third case, namely...

[0154] To truncate the order of an uncertain control system with two densely distributed singular values ​​of the CMB (Constant Multiplication Table), i.e., how to select the truncation order position from i or i+1, a truncation criterion σ is applied. cr To determine the truncation order. Figure 2 The three straight lines in the diagram include the angle bisector σ in the first quadrant. i+1 =σ i Horizontal line σ i+1 =σ cr and vertical line σ i =σ cr Based on the possibility of convex sets and Defined as

[0155]

[0156]

[0157] Among them, such as Figure 2 As shown, Ω=Γ1∪Γ2∪Γ3 is the angle bisector σ of the CMB oblique ellipse in the first quadrant. i+1 =σ i The area cropped on the right. Ω i =Γ2∪Γ3 and Ω i+1 =Γ3 satisfies in Ω and The area. S Ω , and They are Ω and Ω respectively i and Ω i+1 The area.

[0158] To provide a singular value truncation criterion, the convex set probability ratio is defined.

[0159] Therefore, the singular value truncation order w based on convex sets is set as follows:

[0160]

[0161] That is

[0162]

[0163] The optimization model established in step 8 is shown below:

[0164] find

[0165] min

[0166] st

[0167] 1≤w≤n-1

[0168] in This indicates the retained w-order state parameters, and the superscript "^" indicates a reduced-order model of the state-space equations. It is x T The sub-states of the corresponding state parameters, x T =[x1,…,x n ] T It represents the total nth order state of the complete original system. and These are the first w singular values ​​of the simplified and complete original systems, respectively. Therefore, the optimal state choice can be determined, thus obtaining a deterministic reduced-order model;

[0169] In step 9, the convex set order reduction model is as follows:

[0170] The obtained convex set reduced-order model is used for the structural control of large-scale spatial trusses and the parametric analysis of large-scale spatial trusses.

[0171] Example

[0172] For example Figure 3 The diagram shows a truss system with 5 spans, where each horizontal and vertical member is 1m long, and the inclined members are [length missing]. The material density is 7.67 × 10⁻⁶. 3 kg / m 3 Young's modulus is 2×10 11 Pa, the cross-sectional area of ​​the rod is 1×10 - 4 m 2 Young's modulus has a 1% uncertainty.

[0173] The original truss system is a 34th-order system, which is relatively high. This invention reduces the order of the original truss system to 18th-order. To demonstrate the effectiveness of the method, a comparison is made with 1e4 Monte Carlo simulations, where the results for the 17th and 18th order singular values ​​are as follows: Figure 4 As shown;

[0174] After entering u 12 When 100 × δ(0)(N) is applied to the 12th degree of freedom in the truss diagram, where δ(0) represents the unit impulse signal at time 0. The upper and lower bounds of the original system response and the lower and lower bounds of the reduced-order system response are as follows: Figure 5 As shown.

[0175] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A large-scale spatial truss model reduction method considering convex set uncertainty, characterized in that The steps of this method include: Step 1: Construct a deterministic dynamic model based on the large-scale spatial truss structure, and construct the deterministic state-space equations of the deterministic dynamic model; Step 2: Represent the uncertainty parameters of the large-scale spatial truss using the convex set method; Step 3: Based on the deterministic state-space equation constructed in Step 1 and the uncertainty parameters represented by the convex set method in Step 2, construct the state-space equation based on the convex set. Step 4: Construct the transformation matrix of the deterministic equilibrium state equation based on the deterministic state space equation constructed in Step 1, and use the constructed transformation matrix to construct the convex set-based state space equation in Step 3 to obtain the convex set-based equilibrium state equation. Step 5: Calculate the deterministic controllability matrix of the deterministic state-space equation in Step 1, and calculate the convex set-based controllability matrix of the convex set-based equilibrium state equation in Step 4. Step 6: Perform singular value decomposition on the deterministic performance controllability matrix and the controllability matrix based on convex sets obtained in Step 5, and use the convex set method to determine the uncertainty range of each singular value and the correlation between each singular value. Step 7: Based on the range of uncertainty intervals of each singular value obtained in Step 6 and the correlation between each singular value, use the singular value truncation algorithm based on convex sets to determine the order of the reduced-order model. Step 8: Based on the order of the reduced-order model obtained in Step 7, with the optimization objective of maximizing the approximation between the reduced-order model and the deterministic dynamic model, the state variables of the reduced-order model are used as independent variables to establish an optimization model and solve the constructed optimization model, finally obtaining the deterministic reduced-order model of the large-scale spatial truss. Step 9: Use the convex set method to represent the reduced-order model obtained in Step 8 using convex sets, thus obtaining the reduced-order model based on convex sets.

2. The method for order reduction of a large-scale spatial truss model considering the uncertainty of convex sets according to claim 1, characterized in that: In step 1, the deterministic dynamic model constructed is as follows: where, is the mass matrix, is the damping matrix, is the stiffness matrix, denotes the time variable, , and are the acceleration, velocity and position of the system state variable, respectively, and are the input force vector and the corresponding position matrix, respectively, setting the state variable to the deterministic state space equation is in the following form: in, It is a system matrix. It is the input matrix. It is the output matrix. The matrix is ​​determined by the definition of the output quantity.

3. The method for order reduction of a large-scale spatial truss model considering the uncertainty of convex sets according to claim 2, characterized in that: In step 2, the uncertainty parameters of the large-scale spatial truss are represented as a convex set as follows: in, The number of uncertainty parameters representing a large-scale spatial truss structure. It is dimensional vector, yes No. The element, k=1, 2, 3, ..., m, corresponds to the nth element of the large-scale spatial truss structure. One uncertain parameter, Describing a convex set, and They are vectors The upper and lower bounds, , and They are vectors The mean vector, radius vector, and correlation coefficient matrix, and They are the first Convex elements The mean and radius elements; yes The One element, No. Convex elements and the Convex elements The correlation coefficient, It is a matrix that has a symmetric positive definite property for describing the dispersion and correlation among all convex variables, and is a matrix about and The function, It is expressed as: in, yes The covariance matrix; yes and The covariance value, the Hadamard operator "" indicates the multiplication of corresponding elements in two vectors or matrices.

4. The method for order reduction of a large-scale spatial truss model considering the uncertainty of convex sets according to claim 3, characterized in that: In step 3, the constructed state-space equation based on convex sets is decomposed into two parts: deterministic and uncertain partial derivatives. in, , and Representing convex sets respectively , and The definite part; and Representing convex sets respectively and The definite part; , and Representing convex sets respectively , and For the first Partial derivatives of the uncertain parameters; and They represent convex sets respectively. and For the first Partial derivatives of the uncertain parameters; Among them, convex set Through the deterministic part The partial derivatives of the convex set with respect to each uncertainty parameter pass Calculated, where It is a matrix The Middle Line 1 A convex set composed of column elements, through Calculated; Representation matrix The Middle Line 1 Nominal values ​​of column elements; orthogonal matrix and diagonal matrix Obtained through the following Cholesky decomposition: ; This represents a hyperellipsoid composed of initial uncertainty parameters; ; express The convex set perturbation; through the convex set uncertainty method, it can be seen that... The radius is calculated using the following formula: Each element is obtained based on the deterministic part and the uncertain partial derivative. After obtaining the convex set representation, we get convex set representation; Construct the following state-space equation based on convex sets: 。 5. The method for order reduction of a large-scale spatial truss model considering convex set uncertainties according to claim 4, characterized in that: In step 4, the transformation matrix between the deterministic state-space equation and the deterministic equilibrium state equation... By solving We obtain, where the matrix It can be done Solving for the result, It is a diagonal matrix composed of Hankel singular values ​​of the deterministic state-space equations. It is a matrix eigenvectors, and It is the Gram matrix representing the controllability and observability of the deterministic state-space equations; The deterministic equilibrium equation is expressed as: ,in , ; The partial derivative part of the equilibrium state equation based on convex sets is expressed as: ,in , ; The equilibrium state equation based on convex sets is: 。 6. The method for order reduction of a large-scale spatial truss model considering the uncertainty of convex sets according to claim 5, characterized in that: In step 5, the controllability matrix of the deterministic equilibrium state equation Solving the Lyapunov equations shown below yields: Considering uncertainty The above equation can be expressed as the deterministic part and the first-order partial derivative part of the uncertainty: By solving the above two equations, the deterministic part of the controllability matrix of the equilibrium state equation is obtained. The first-order partial derivatives with respect to the uncertainty parameter are divided into the components. The controllability matrix of the equilibrium state equation based on convex sets is obtained through the convex set method. .

7. The method for order reduction of a large-scale spatial truss model considering convex set uncertainties according to claim 6, characterized in that: In step 6, the singular value decomposition of the performance control matrix is ​​determined as follows: ,in Represents a deterministic singular value matrix. This indicates that the elements are constructed using the input column vector. A diagonal matrix of order 1; and These represent the left and right singular value vector matrices of deterministic nature, respectively. in, Representation matrix The column vectors Representation matrix The Given column vectors, the first-order partial derivatives are related as follows: The first-order partial derivatives of the singular values ​​with respect to the uncertainty parameters are obtained by solving the above equation. Convex sets for obtaining singular values ​​using the convex set method The uncertainty intervals of each singular value are obtained using the following convex set operations: in According to convex set operations, the covariance relationship between the singular values ​​is obtained using the following operation: in .

8. The method for order reduction of a large-scale spatial truss model considering convex set uncertainties according to claim 7, characterized in that: In step 7, the singular value truncation algorithm based on convex sets is described as follows: Each singular value lies within its convex set obtained by the proposed method; for two adjacent singular values ​​of an uncertain convex set... and By setting two uncertain adjacent singular values ​​as two coordinate axes, three types of convex sets were plotted in two dimensions, with the angle bisector of the first quadrant representing... , The relationships between convex sets are distance, adjacency, and intersection. A cutoff criterion is designed for the intersection case. ; To truncate the order of an uncertain control system with two densely distributed CMB singular values, i.e., how to... or Select the truncation order position and apply a truncation criterion. To determine the truncation order; based on the probability of convex sets. and Defined as in, It is the angle bisector of the CMB oblique ellipse in the first quadrant. The area cropped on the right; and Is China satisfies and The area; , and They are , and The area; To provide a singular value truncation criterion, the convex set probability ratio is defined. Singular value truncation order based on convex sets Set as That is 。 9. The method for order reduction of a large-scale spatial truss model considering the uncertainty of convex sets according to claim 7, characterized in that: The optimization model established in step 8 is shown below: in Indicates reservation The state parameters corresponding to the order, indicated by the superscript " " represents a reduced-order model of the state-space equations, yes The sub-states corresponding to the state parameters in the middle. It is the total of the complete original system first state, and These are the simplified and complete versions of the original system, respectively. A singular value.

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