Subspace Modeling Method for Integrated Design and Control in

Article pubs.acs.org/IECR

A KL/Subspace Modeling Method for Integrated Design and Control in Curing Processes Xin Jiang Lu* and Ming Hui Huang State Key Laboratory of High Performance Complex Manufacturing, and School of Mechanical & Electrical Engineering, Central South University, Hunan 410083, China S Supporting Information *

ABSTRACT: In this Article, a data-driven modeling approach is proposed for hybrid design/control variables in complex curing ovens. The coupling influence of the design/control variables is first separated into two subtasks: modeling respectively for the design-variable-dependent basis function and the control-variable-dependent temporal model. Each subtask only considers the nature of its corresponding variable and does not interact with the other variable, which will be less complex and more easily modeled. The original system is then reconstructed by synthesizing these two submodels. The advantage of this proposed method is that the well-developed design methods and control methods can respectively handle the design-variable-related model and the control-variable-related model, which will benefit to achieve the desirable overall curing performance by integration of design and control. Finally, the effectiveness of the proposed method is verified by the modeling of an actual curing oven. design/control performance,1,5 is fully dependent on its design variables, such as shape and size of the oven, and its control variable, such as power of heaters. On the one hand, the design variables are a discrete variable that is only able to take several discrete values due to the limitation of both cost and the manufacturing accuracy of the machines. Also, these several discrete values are often given beforehand, and cannot be taken arbitrarily according to the modeling requirements. On the other hand, the control variable is a continuous variable, which varies continuously with time and may take an arbitrary value within its bounds. In the design and control of the curing process, there are many first-principle modeling methods for both the design variables6,7 and the control variables.8,9 However, because all of these first-principle models are usually built under many simplifications and assumptions, they are inaccurate due to unknown dynamics and unknown boundary conditions and complexities of the curing process, because it includes conduction, convection, and radiation.10,11 However, data-driven modeling problems in complex curing ovens are usually depicted to obtain a model that contains either discrete design variables or continuous control variables,10,11,25 while modeling them together has never been considered. This leads to difficulty in obtaining the optimal curing performance by the integration of design and control. Therefore, there is still a need for an effective data-driven modeling method to be developed for the curing process. The Karhunen−Loéve (KL) method has become the most efficient technique to extract empirical eigenfunctions (EEFs) from the finite numerical or experimental data.12−15 Because the number of these EEFs is usually small, the model order is reduced.14,15 It has also strong factorization ability10,11 and has been widely applied in system modeling. For example, it was

1. INTRODUCTION The overall performance of the manufacturing actually takes into account both the control performance and the design performance,1−3 as shown in Figure 1. The system at the machine-level

Figure 1. Overall performance of the manufacturing.

is a nonlinear but continuous process to produce the required product by adjusting control variables. At the supervision level, the goal of the production is to meet the specific high-level criteria, for example, the economic criteria in manufacturing, by optimizing design variables (sometimes also including control variables). Thus, the overall performance of this manufacturing system depends on not only the low-level process control but also the high-level system design. This requires a model that must fully include both design variables and control variables. It is also desirable that this model includes two separate parts, design-variable-related part and controlvariable-related part, in the integration of design and control. This is because the well-developed design methods and control methods can effectively handle them, respectively, which will benefit to achieve the desirable overall performance of the manufacturing by integration of design and control. So far, there is little research on the development of this kind of model via data. Such a typical example is a curing oven in the semiconductor packaging industry. It is used to provide a required temperature distribution for setting compounds that are introduced onto electronic components.4,5,25 Its curing performance, integrated © 2014 American Chemical Society

Received: Revised: Accepted: Published: 14377

May 8, August August August

2014 21, 2014 21, 2014 21, 2014

dx.doi.org/10.1021/ie501895h | Ind. Eng. Chem. Res. 2014, 53, 14377−14384

Industrial & Engineering Chemistry Research

Article

Figure 2. Curing oven system.

Figure 3. Data-driven modeling framework for design/control variables.

parameters, such as the angle θ of the heating block and the distance H between the LF and the heating block as shown in Figure 2. Thus, modeling of this curing oven represents a typical type of modeling problem for design/control variables. They may be described as

often used in modeling of a distribution parameter system because it can obtain the spatial basis function from the spatialtemporal data.10,11 However, it has never been applied to separate the influence of the design/control variables on the system. In this study, the KL method is applied to separate these influences for making modeling easier. In this Article, a data-driven modeling approach is proposed for design/control variables in a complex curing oven. The influence of design/control variables on this system is first separated into two parts, modeling respectively for the designvariable-dependent basis function and the control-variabledependent temporal model, by the KL method. The response surface method and the subspace modeling method are then employed to obtain these two submodels. These two submodels are further synthesized for reconstruction of the system. Finally, the proposed method is verified by the comparison of simulation results.

y(d , t ) = f (d , u)

(1)

where y represents the performance, such as temperature T on the LF, f(d, u) is the unknown model, d is the design variable vector, such as the geometric parameters [θ, H]T, and u is the control variable vector, such as the power of the heaters. This system performance is a coupled effect of both design variables and control variables. These design variables can only take several discrete values, which cannot be arbitrarily chosen due to the limitations of both cost and manufacturing, while the control variable can vary continuously with time and may take an arbitrary value within its bounds. These factors will bring a greater challenge to build this model with hybrid design/ control variables in the whole design space and the continuous control space.

2. PROBLEM STATEMENT The curing oven as presented in refs 1,5,25 is shown in Figure 2. Its heaters, which are embedded in the heater block, first transfer heat to the heating block, and then the heating block heats the integrated circuit (IC) placed on the lead frame (LF). A controller is required to control the power of the heaters to adjust the temperature of the LF. This oven is also filled with nitrogen to avoid oxidation. A motion mechanism inside the chamber is employed to move a working plate up and down. The curing performance of the oven depends on its temperature distribution. Thus, to obtain the desirable curing performance, an accurate temperature model must be first built. Obviously, the temperature model incorporates not only the control variable, the power u of the heater, but also the design

3. MODELING METHOD A data-driven modeling method is proposed for the curing process with the hybrid design/control variables, as indicated in Figure 3. It includes the following key points. • Variable separation: Inspired from some separation methods,10,11,16,25 the original system is first separated into the design-variable-dependent basis function and the control-variable-dependent temporal model by the KL method. This decomposes the original modeling task into two simple subtasks: modeling respectively for the design variable and the control variable. Each subtask only considers the nature of its corresponding variable 14378



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and has no coupling with the other variable. Its modeling complexity will be greatly reduced as compared to the original modeling complexity. Also, the well-developed design methods and control methods can effectively handle the design-variable-related part and the controlvariable-related part, respectively. • The response surface method (RSM) for design variable: The RSM method is used to model the basis function related to the design variable. • The subspace method for control variable: The subspace method is used to model the temporal model related to control variables. This built model has a good linear structure that will make the design of the controller easier. • Model reconstruction: This system model is reconstructed by synthesizing the design-variable-dependent basis function and the control-variable-dependent temporal model. 3.1. The KL Method for Variable Separation. The design/control-variable-dependent output is first expanded onto an infinite number of orthonormal design-variable-dependent basis functions {φi(d)}i =∞ 1. These basis functions reflect the influence of the design variable on the system output:

Figure 4. Snapshots at different design values.

continuous temporal input signal at the discrete values d1, ..., dN of the design variable, and the system output {y(di,t)}i =N,L1,t = 1 is measured as the snapshots, as shown in Figure 4. The KL method as presented in ref 10 is then developed to obtain the design-variable-dependent basis functions {φi(d)}i =n 1 among these snapshots {y(di,t)}i =N,L1,t = 1. The dominant basis function can be obtained by solving the following optimal problem:

∞

y(d , t ) =

∑ φi(d)yi (t )

(2)

i=1

min⟨|| y(d , t ) − yn (d , t )||2 ⟩ φi(d)

where yi(t) (i = 1,...,∞) is the temporal coefficients that reflect the influence of the continuous control variable on the system output. They may be figured out by yi (t ) = (φi(d), y(d , t )),

i = 1, ..., ∞

subject to (φi , φi) = 1,

i = 1, ..., n (7)

The necessary condition of the solution can be obtained as10

(3)

where (·,·) represents the inner product. Usually, an orthogonal projection operator on the dominant characteristic modes φm = [φ1,φ2,...,φn]T is used to obtain the temporal coefficients: n ⎧ ⎪ yn (d , t ) = ∑ φi(d)yi (t ) ⎪ i=1 ⎪ ∞ ⎪ ⎨ y (t ) = (φ , ∑ φ (d)y (t )) m i i ⎪m i=1 ∞ ⎪ ⎪ = ∑ (φm , φi(d))yi (t ) ⎪ ⎩ i=1

(φi , φj) = 0,

∫ ΩR(d , ζ)φi(ζ) dζ = λiφi(d)

(8)

where R(d,ζ) = ⟨y(d,t),y(ζ,t)⟩ is the correlation function of two design variables, and φi(d) and λi are the ith eigenfunction and its corresponding eigenvalue. The basis function can be expressed as a linear combination of the snapshots as follows: L

φi(d) =

∑ γity(d , t )

(9)

t=1

(4)

Substituting eq 9 into eq 8 gives the following eigenvalue problem:

At the result of this projection, we have ym(t) = [y1(t),...,yn(t)]T. The next task is to obtain the below dynamics f between u(t) and ym(t) from the data set {u(t),ym(t)}t L= 1.

∫ Ω L1 ∑ y(d , t )y(ζ , t ) ∑ γiky(ζ , t ) dζ = λi ∑ γity(d , t )

ym (t ) = f (u)

L

L

t=1

k=1

t=1

(10)

(5)

The time correlation function is defined as 1 Ctk = Ωy(ζ , t )y(ζ , k) dζ (11) L The eigenvalue problem can be changed to the following problem:10

Thus, the low-dimension model for the system with hybrid design/control variables may be determined as follows: n ⎧ ⎪ = y ( d , t ) ∑ φi(d)yi (t ) ⎪n ⎨ i=1 ⎪ ⎪ y (t ) = f (u) ⎩m

L

∫

Cγi = λiγi (6)

(12)

The solution of the above eigenvalue problem yields the eigenvectors γi = [γi1...γiL]T, which can be used in eq 9 to construct the eigenfunctions φi(d). Let us express the eigenvalues λ1 > λ2 > ... > λn and the corresponding eigenfunctions φ1,φ2,...,φn. The ratio of the sum of the n largest eigenvalues to the total sum is

The key problem is how to compute the dominant characteristic modes {φi(d)}in= 1 and find the unknown model f from the data set {u(t),ym(t)}t L= 1. Here, the dominant characteristic modes {φi(d)}i n= 1 will be found by the KL method. First, the system is excited by the 14379



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Table 1. All Values of Design Variables training sample of d θ H

160 5

160 7.5

160 10

170 5

170 7.5

testing sample of d 170 10

180 5

180 7.5

180 10

165 6.5

175 8.5

163 8.5

176 8

Figure 5. Input and output of the system. n

En =

i=1

3.4. Model Reconstruction. After the basis functions φi(d) and the temporal model f are obtained, the low-dimension model of the system with hybrid design/control variables may be determined as follows:

K

∑ λi / ∑ λi i=1

(13)

where the maximal number of nonzero eigenvalues is K = min(N, L). Usually, the sufficient number of eigenfunctions that capture 99% of the system “energy” is used to determine the value of n.10,16,25 3.2. RSM for the Design-Variable-Dependent Basis Function. The design-variable-dependent basis functions are only known at these design values used for training, or else are unknown. For system design, the basis functions {φi(di)}in= 1 under the arbitrary design variable vector di in the design space must be known. Here, the RSM,17−20,25 a common modeling method for the design variable, is employed to construct the relationship between the design variable vector di and its corresponding basis functions {φi(di)}in= 1 from the training data D = [d1,...,dN]T and the output data {φi(d)}i n= 1. This relationship can be represented as

n

yn (d , t ) =

i=1

⎧ x = Axk + Buk ⎪ k+1 ∼ φi(d) = G(d) and ⎨ ⎪ ⎩(ym )k = Cxk + Duk

(14)

where G is a polynomial function. The coefficients of this polynomial function can be identified from input/output data by the least-square method. 3.3. Subspace Method for Temporal Modeling. Generally, when a dynamic system with continuous control variable is unknown, the subspace modeling method is an advisible choice to obtain the model of this system by using the input/output data because it is computationally tractable and robust.21−23 For this reason, the following state space model built by the subspace identification algorithm is employed to describe the dynamics f between u(t) and ym(t) from the data set {u(t), ym(t)}t L= 1: ⎧ x = Axk + Buk ⎪ k+1 ⎨ ⎪ ⎩(ym )k = Cxk + Duk

(16)

This model 16 integrates design/control variables together. Because this model includes two separate parts, design-variablerelated part and control-variable-related part, the welldeveloped design methods and control methods can effectively handle them, respectively. This will benefit the optimal design and control of the system to achieve the desirable overall performance of the curing oven by integration of design and control. Obviously, this proposed modeling is a data-driven modeling method. When the nonlinear system works in a large operating region, this modeling method will be less effective due to the limitation of the subspace modeling method.

∼

φi(d) = G(d)

∑ φi(d)yi (t ) with

4. CASE STUDIES Modeling of the practical curing oven as shown in Figure 2 is used to verify the effectiveness of the proposed method. The length, breadth, and height of the LF are 240, 90, and 0.2 mm, respectively. According to practical requirements, the design variables are set as θ ∈ [160,180] and H ∈ [5,10], and they, including the training samples and the testing samples, are only able to take some discrete values as shown in Table 1 that are collected by the design of experiment (DOE). The random continuous input u as shown in Figure 5a, including a group of training samples and two groups of testing samples, is used to excite the system. The output, as indicated in Figure 5b, is the temperature mean on the LF, one of the most important indices for the curing performance. These outputs are excited

(15)

Here, the state x and the matrixes A,B,C,D can be directly identified in the subspace modeling method.21−23 14380



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Figure 6. Basis functions.

Figure 7. Modeling performance comparison under training samples.

by the training samples of both the control input u and the design variables θ and H. These outputs offer the snapshots for modeling. 4.1. KL Modeling. First, the KL method is used to separate the influence of the design variables and the control variable

from the aforementioned snapshots. The eigenvalues obtained by the KL method are shown below: [ λ1 ... λ 9 ] = [34 315 832.7 0.014 1.2 × 10−7 1.1 × 10−10 8.2 × 10−12 4.7 × 10−12 1.2 × 10−13 7.4 × 10−14] 14381

(17)



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Figure 8. Modeling comparison under the first testing samples.

Figure 9. Modeling comparison under the second testing samples.

Because E4 = 99.9%, the first four design-variable-dependent basis functions are used for modeling. 4.2. RSM for Design-Related Basis Function Modeling. The RSM model 14 then is used to construct the design-related basis function. The training data of d and the corresponding output data, the first four design-variable-dependent basis functions obtained by the above KL method, are used to build the following model: ∼

∼

∼

Table 2. Error Comparison under Testing Samples error first group of testing samples of u

NN method proposed method

∼

φ1 = [a1a 2a3a4]X , φ2 = [b1b2b3b4]X , φ3 = [c1c 2c3c4]X , φ4 = [d1d 2d3d4]X

(18)

second group of testing samples of u

θ = 165, H = 6.5

θ = 175, H = 8.5

θ = 163, H = 8.5

θ = 176, H=8

111.3 35.3

73.3 3.09

36.2 0.57

37.9 13.8

These four design-variable-dependent basis functions in eq 18 are shown in Figure 6, respectively. 4.3. Subspace Modeling for Temporal Modeling. Furthermore, the subspace modeling method is used to construct the temporal model. The training data set {u(t), ym(t)}t L= 1 in the aforementioned snapshots is used to obtain the state space model 15, whose state matrixes A, B, C, and D are shown in Appendix A. 4.4. Modeling Verification. Finally, the effectiveness of the developed model 16 is verified by both training samples and testing samples, and is also compared to the common neural network method. A. Verification with Training Samples. First, the training samples are used to verify the developed model. Two experiments

Here, X = [1 θ H θH] , and these unknown parameters ai, bi, ci, di (I = 1, 2, 3, 4) are easy to obtain from the output data and their corresponding input data by the least-square method. They are shown below: T

[a1a 2a3a4] = [0.174 0.000014 −0.0033 0.0000044] [b1b2b3b4] = [−0.505 −0.00045 0.101 −0.00013] [c1c 2c3c4] = [1.85 −0.011 −0.181 0.00108] [d1d 2d3d4] = [6.36 −0.037 −0.399 0.0023] 14382



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from the aforementioned snapshots, which are obtained under the training samples of u and the design variables respectively as (θ = 160°, H = 7.5 mm) and (θ = 180°, H = 10 mm), are employed to check the effectiveness of the developed model. The outputs of these two experiments are respectively shown in Figure 7a and b in comparison with outputs of the model 16, from which this developed model fits the practical output well. Thus, the proposed modeling method is effective to obtain the model of an unknown complex system with hybrid design/ control variables. B. Verification with Testing Samples. Two groups of testing samples that have different mean and variance then are used to verify the effectiveness of the developed model under new designs. a. Testing under Two New Designs and the First Group of Testing Samples of u. The first group of testing samples of u from 1001 to 2000 in Figure 5a and two new designs (θ = 165°, H = 6.5 mm) and (θ = 175°, H = 8.5 mm), which are different from those in the aforementioned snapshots, were used to test the developed model. Under these conditions, from Figure 8, the output of the developed model fits the practical output well. Thus, the proposed modeling method is effective to obtain the model of an unknown system with hybrid design/control variables. b. Testing under Two New Designs and the Second Group of Testing Samples of u. The second group of testing samples of u from 2001 to 3000 in Figure 5a and two new designs (θ = 163°, H = 8.5 mm) and (θ = 176°, H = 8 mm), which are different from those in the aforementioned snapshots and the above testing, were used to test the developed model. Under these conditions, from Figure 9, the output of the developed model fits the practical output well. Thus, the proposed modeling method is effective to obtain the model of an unknown complex system with hybrid design/control variables. C. Performance Comparison with the Neural Network Method. The effectiveness of the proposed method is further verified by comparing it to the neural network (NN) method presented in ref 24. This NN model is trained alike by the training samples of both u in Figure 4a and d in Table 1. The modeling error is defined as follows: ⎡ 0.99333 − 0.00070002 ⎢ 0.99895 ⎢−0.021829 ⎢ 0.10941 0.027958 ⎢ ⎢ 0.092271 − 0.0021498 ⎢ 0.0067163 −0.0115 A=⎢ ⎢−0.047499 − 0.00029373 ⎢ 0.016053 ⎢−0.020708 ⎢ −0.23375 0.02896 ⎢ 0.028245 ⎢ −0.21061 ⎢ 0.008113 ⎣ −0.10363

0.0023387 0.0072345

0.012

error =

∑ (y(d , ti) − y ̂(d , ti))2 (19)

i=1

Here, y and ŷ are the practical value and the estimated value from the model, respectively. This error is used to show the modeling precision. The smaller is the error, the better is the model. The performance comparison under the testing samples was carried out, and the results are shown in Table 2. The proposed method has a better modeling performance than the NN method due to a smaller modeling error. The reason is that the proposed method considers the different features of the design variable and the control variable, while the NN modeling method does not. Generally, the NN modeling method requires that the input spectrum satisfies certain special properties to guarantee that the system is able to be identified.16 As a result, a persistent excitation signal for input is needed. However, because the design variables can only take several discrete values in the design space, the persistent excitation signal cannot be satisfied when modeling the design variables. Thus, the NN modeling method has a worse approximate accuracy for modeling of this curing oven.

5. CONCLUSION A data-driven modeling approach is proposed for design/ control variables in a curing oven. because the original modeling task is decomposed into two simple and easy-todetermine subtasks by the KL method, the proposed method can effectively reduce the modeling complexity. It can also effectively model the system with design/control variables. The model developed in this study will benefit the optimal design and control of the system because the well-developed design methods and control methods may respectively handle the design-variable-related part and the control-variable-related part. The effectiveness of the proposed method has been demonstrated and verified on the model of the curing oven.

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APPENDIX A

0.0039094 − 0.0055318 −0.0076574 −0.0062824

0.0042287

0.9032

0.056177

0.15934

− 0.094137

0.97182

− 0.00028285

0.0036615

0.018

0.98878

0.013019

0.0070436 ⎤ ⎥ 0.0027481 0.0075958 ⎥ − 0.089252 − 0.0038154 ⎥ ⎥ − 0.015105 − 0.06324 ⎥ ⎥ − 0.00024945 0.013815 ⎥ , 0.046059 ⎥ − 0.009463 ⎥ 0.061702 ⎥ − 0.0034508 0.029401 0.19104 ⎥ ⎥ 0.87428 0.24527 ⎥ ⎥ 0.035906 1.0316 ⎦

−0.0030681 −0.001649 − 0.0049685 − 0.0056897 0.034978

0.030968

− 0.083632

0.042954

0.041854

− 0.020317

−0.012966 − 0.0040876 − 0.0038285 − 0.019254 − 0.0014488

0.049287

0.032709

0.019421

0.97659

0.0025533

0.038101

0.038651

−0.022839

0.98293

0.21263

0.1115

0.010497

−0.15263

−0.1103

1.0141

0.20941

0.19712

0.25844

−0.1183

−0.087

− 0.076697

0.070318

0.008124

− 0.056563

−0.050175

−0.040449

0.034627

− 0.038992

⎡ ⎡ −0.00020059 ⎤ ⎢ −81.17 11.818 ⎥ ⎢ 3.6846 1.5627 0.26382 −0.066386 −005 ⎢ ⎥ ⎢ 8.7314 × 10 ⎢ −56.72 −17.423 1.7219 1.7908 0.54667 −0.13601 ⎢ −0.0029804 ⎥ ⎢ −27.7 − 52.117 − 0.59305 2.1754 0.78131 −0.042801 ⎥ ⎢ ⎢ ⎢ −0.0005441 ⎥ 3.8314 1.552 0.23728 −0.051905 ⎢ −82.99 13.994 ⎢ −005 ⎥ −8.2862 × 10 ⎥ D = 0, B = ⎢ , C = ⎢−58.978 −14.722 1.9034 1.7717 0.5189 −0.12683 ⎢ ⎢ −0.00018654 ⎥ ⎢ −30.373 −48.922 − 0.37934 2.1444 0.75589 −0.044913 ⎥ ⎢ ⎢ ⎢ −0.00065606 ⎥ 3.9328 1.5451 0.21862 −0.041354 ⎢−84.246 15.497 ⎢ 0.00061437 ⎥ ⎢ −60.7 − 12.663 2.0419 1.7577 0.49714 −0.11885 ⎥ ⎢ ⎢ ⎢ −0.0034683 ⎥ ⎢ −32.543 −46.327 − 0.20569 2.1201 0.73407 −0.045101 ⎥ ⎢ ⎣ ⎦ ⎣ 0.0010696

14383

⎤ 1.4262 − 0.47158 ⎥ ⎥ 1.9225 3.5105 ⎥ − 1.0138 2.4849 8.2397 ⎥ 0.2063 ⎥ −0.34508 1.3465 1.3879 − 0.76774 ⎥ −0.097579 0.30268 1.8763 3.1428 ⎥ ⎥ −0.90008 2.4321 7.8043 ⎥ 0.1798 ⎥ −0.35874 1.4028 1.3613 − 0.97224 ⎥ −0.11561 0.3782 1.8408 2.8624 ⎥ ⎥ 0.15812 −0.80735 2.389 7.4508 ⎥ ⎦ −0.32538 −0.07407

1.2652 0.20397



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ASSOCIATED CONTENT

S Supporting Information *

SA, model of the curing oven; SB, training and testing samples of input u. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This project is partially supported by the National Basic Research Program (973) of China (2011CB706802), and the National Natural Science Foundation of China (51205420), the Program for New Century Excellent Talents in University (NCET-13-0593), the Hunan Provincial Natural Science Foundation of China (14JJ3011), and the Fund of the State Key Laboratory of Metal Extrusion and Forging Equipment Technology.

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REFERENCES

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