Online Spatiotemporal Least-Squares Support Vector Machine

Article pubs.acs.org/IECR

Online Spatiotemporal Least-Squares Support Vector Machine Modeling Approach for Time-Varying Distributed Parameter Processes XinJiang Lu,* Feng Yin, and MingHui Huang State Key Laboratory of High Performance Complex Manufacturing, School of Mechanical & Electrical Engineering, Central South University, Hunan 410083, China ABSTRACT: Nonlinear and time-varying distributed parameter systems (DPSs) are challenging to accurately model due to potential spatiotemporal coupling, infinite-dimensional property, and time-varying dynamics. Although the least-squares support vector machine (LS-SVM) can effectively model lumped parameter systems, it is less effective to model the time-varying dynamics of a DPS, as it lacks the ability to incorporate time-varying spatiotemporal dynamics. Here, an online spatiotemporal LS-SVM approach is proposed to model a nonlinear and time-varying DPS. An adaptive spatial kernel function is first developed for online capture of the time-varying relationship between spatial locations. An online time coefficient model is then constructed to account for the time-varying temporal dynamics of the DPS. Combination of the adaptive spatial kernel function with the online time coefficient model allows for reconstruction of the complex DPS and ensures the model can reflect real-time spatiotemporal dynamics well. Experiments on a laboratory curing thermal process show the effectiveness of the proposed method.

1. INTRODUCTION Numerous practical processes, such as heating processes, fluid motion, and spray deposition physical properties of a system, can nonlinearly vary throughout space and time.1−3 These processes are considered as distributed parameter systems (DPSs).1,4−6 Within a DPS, the input, the output, and even the parameters are often dynamic with respect to space and time.4,7 Generally, it is difficult to model this type of process owing to the infinitedimensional nature of the problem and time-varying and nonlinear dynamics. In the recent years, much research has improved the ability to model a DPS. This research may be classified into two categories: one where the partial differential equation (PDE) of the DPS is known and one where the PDE is unknown. For the former, many approaches were developed to transform the PDE into an ordinary differential equation (ODE) in order to easily predict the dynamics of the DPS and design the controller of the DPS.8−17 The finite difference method (FDM),9,10 the finite element method (FEM),11,12 and the method of multiple scales13,14 are the most common methods used in this transformation. These methods discretize the PDE into the ODE. However, these methods require a known PDE and pay less attention to time-varying dynamics of DPSs, as well as produce a very high-order ODE that is difficult to use when designing a controller. Alternative methods are the spectral method2,15 and the approximate inertial manifold method,16,18 which produces a low-order approximate model. However, each of the methods mentioned above cannot model an unknown and time-varying DPS. © XXXX American Chemical Society

The second category of modeling uses data to derive a model for the DPS. A common modeling method uses an assumption of space basis functions from either experience or prior process knowledge, such as the orthogonal space basis function19 or Green’s function.20 However, using this method can lead to poor modeling accuracy when there is a lack of experience or prior process knowledge. The most common data-driven modeling method is the Karhunen−Loève (KL) decomposition.21−23 This method first determines the finite space basis functions from experimental data, which generates a low-order ODE model;18 then temporal dynamics are included using a mature method such as the neural network,24 the Hammerstein model,25 or the Wiener model.26 Although the KL method has been successfully used in many applications, it is a linear modeling method which neglects the influence of nonlinear space dynamics. Thus, the KL method can only be used to model linear or weakly nonlinear systems, and it is less effective for modeling a strongly nonlinear DPS. In addition to the recently developed KL-Fuzzy method,27 almost all data-driven modeling methods are not very effective for modeling a time-varying and nonlinear DPS. However, the KL-Fuzzy method is still only applicable to linear or weakly nonlinear DPSs, since the KL method is applied to gain the space basis function. Thus, an effective modeling method for nonlinear and time-varying DPSs is currently lacking. Received: Revised: Accepted: Published: A

March 8, 2017 June 5, 2017 June 8, 2017 June 8, 2017 DOI: 10.1021/acs.iecr.7b00984 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research One of the most popular data-driven modeling methods is the least-squares support vector machine (LS-SVM).28 It has gained many successful applications, due to its approximation ability and computational attractiveness.29−31 This method is generally used to model lumped parameter systems32 and is not inherently suited to spatiotemporal modeling, as it ignores spatial variations. Although a spatiotemporal LS-SVM was recently developed to model a nonlinear DPS,33 it was found to be less effective for modeling a time-varying DPS, as it is used to model the timeinvariant DPS and does not have an adaptive function for model updates, and the spatiotemporal calculations are also offline. In this paper, we develop an online spatiotemporal LS-SVM method to model a nonlinear and time-varying DPS. This method not only captures the temporal dynamics using an online time coefficient model but also considers the time-varying relationship between spatial locations through an adaptive spatial kernel function. Combination of the adaptive spatial kernel function with the online time coefficient model allows for reconstruction of unknown, time-varying, and nonlinear DPSs and ensures the model can reflect well real-time spatiotemporal dynamics. Experiments on a practical thermal curing process imply that the proposed method is successful in modeling a nonlinear and time-varying DPS.

∂y(x , t ) ∂x

y(x , t )|t = 0 = y(x , 0)

(4)

where y(x,t) is the output of position x at time t; C is a function of y(x,t); p(t) is an unknown time-varying parameter; the spatial coordinates (v1,v2,v3) for three-dimensional space are represented by x = (v1,v2,v3); F(·), Q(·), g1(·), and g2(·) are unknown time-varying nonlinear mappings; u(t) = [u1(t), u2(t), u3(t), u4(t)]T denotes the vector of manipulated inputs with the spatial distribution b(x); and xb represents the spatial boundaries. It is challenging to model this type of DPS because of the following factors: • time-varying property of the process; • strong nonlinearity of the spatiotemporal dynamics; • unknown and complex boundary conditions. Currently, although many discrete methods are used to transform the diffusion-reaction PDE of eq 1 into the ODE,9−18 they require a known PDE and pay less attention to the timevarying dynamics of the DPS, which causes them to be less effective for modeling of an unknown and time-varying DPS. Most existing data-driven modeling methods are linear; thus, nonlinear spatial variations are ignored. Linear models include the existing KL-based20−22 or SVD-based modeling methods21 and the PCA-based modeling method.34 These methods are also typically offline and rarely have any online adaptive function. Thus, these models are less effective for a nonlinear, time-varying DPS. An online data-driven method is needed for modeling this type of DPS.

3. ONLINE DPS MODELING APPROACH Here, we propose an online spatiotemporal LS-SVM approach to address the modeling problems of a nonlinear and time-varying DPS, as depicted in Figure 2 and with an update process described in Figure 3. The main ideas underlying this method are as follows: • The spatiotemporal LS-SVM model includes two parts: a spatial kernel function and a time coefficient (also called a time-related Lagrange multiplier). The spatial kernel function is used to represent the nonlinear relationship on space, while the time coefficient is used to represent the dynamics in time. • Using the modeling error, both the spatial kernel function and the time coefficient are updated for allowing the model to adapt to the time-varying dynamics of the DPS. In this way, this proposed method not only considers the time dynamics and spatial nonlinearities of a DPS but also has an online adaptive function. Thus, it can effectively model a nonlinear and time-varying DPS. The details of the proposed online spatiotemporal LS-SVM are presented in the following sections. 3.1. Spatiotemporal LS-SVM Modeling Method. A sample of data can be collected from the DPS as the data set {u(tj), y(xi,tj)}s,L i=1,j=1, where xi is the space location of the i-th sensor and tj is the j-th time, and s is the number of sensors and L is the length of the sampling time. The following spatiotemporal LS-SVM model33 may be used to describe this data set:

Figure 1. Curing process in the curing oven.

process. Heaters are applied to heat the integrated circuit (IC). A complex thermal transmission happens in this process, as it contains conduction, convection, and radiation. Thus, it is a typical DPS. In addition, there are complex time-varying boundary conditions and time-varying parameters, such as the variation effects of the epoxy on the IC. All these factors make modeling this system difficult. Generally, the time-varying nonlinear DPS may be described as follows: ∂y(x , t ) ∂F(y(x , t )) ⎫ ∂ ⎧ ⎨C(y(x , t )) ⎬ = ⎭ ∂t ∂x ⎩ ∂x (1)

and boundary conditions are the following: y(x , t )|xb = g1(x , y(x , t ))|xb

(3)

and the initial condition is below:

2. PROBLEM DESCRIPTION Lots of practical processes are unknown, time-varying, and nonlinear DPSs. Here, a curing process for chip manufacture,22,33 as shown in Figure 1, is used as an example to describe this type of

+ Q (y(x , t ), p(t )) + b(x)u(t )

= g2(x , y(x , t ))|xb xb

y(x , t ) = w(t )T ϕ(x) + b(t )

(2) B

(5) DOI: 10.1021/acs.iecr.7b00984 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Figure 2. Online spatiotemporal LS-SVM method.

relationship between spatial locations, w(t) is the weight matrix, and b(t) is the bias term, which is a function of u(t). The solution to the modeling problem (eq 5) can be found by the least-squares method with the temporal Lagrange multiplier αi(tj). Then the resulting LS-SVM model becomes s

y ̂( x , t j ) =

∑ αi(t j)K (x , xi) + b(t j)

(6)

i=1

where αi(tj) and b(tj) are the optimal solutions from the following equation, and K is the kernel function. ⎡ 0 1T ⎤⎡ b(t j) ⎤ ⎡ 0 ⎤ s ⎥=⎢ ⎢ ⎥⎢ ⎥ ⎢1 Q + γ −1I ⎥⎢ α(t )⎥ ⎣⎢ Y (t j)⎥⎦ ⎣s ⎦⎣ j ⎦

(7)

Here, 1s = [1, ..., 1] ∈ R , [α(tj), ..., αs(tj)] , Y(tj) = [y(x1, tj), ..., y(xs, tj)]T, I is an identity matrix, and the element of Q is Qil = ϕ(xi)T ϕ(xl) = K(xi,xl) (i, l = 1, ..., s). Data points collected from the system provide context to the model; however, at unsampled or untrained time points, the distributed LS-SVM model has an unknown temporal Lagrange T

Figure 3. Adaptive update process.

where y(x,t) is the temperature of the spatial location x at time t, ϕ(·) is a nonlinear function that represents the nonlinear

s

T

Figure 4. Two-step update method. C

DOI: 10.1021/acs.iecr.7b00984 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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We define the error performance index function as below:

multiplier. In order to forecast the temporal Lagrange multiplier at time intervals without experimental data, the time coefficient model that employs another LS-SVM model may be developed to represent the temporal dynamics:33 αi(t j) = ωiT φ(zi(t j − 1)) + ci ,

i = 1, ..., s

(8)

σ(tL + 2) = σ(tL + 1) + ηΔσ(tL + 1)

Δσ(tL + 1) =

(9)

(14)

∂Ey(tL + 1)

∂σ(tL + 1) s ⎛ = −2 ∑ ⎜⎜(y(xi , tL + 1) − y (̂ xi , tL + 1)) i=1 ⎝

where δ is the regularization factor,Ci(tL) = ci, Bi(tL) = [βi(t2), ..., βi(tL)]T, Ai(tL) = [αi(t2), ..., αi(tL)]T, and βi(tj) is the Lagrange multiplier. After obtaining the ci and βi by solving eq 9, the time coefficient can be modeled as

s

L

∑ αî (tL + 1)

i = 1, ..., s

|| xi − xl ||2

l=1

σ 3(tL + 1)

2

e− || xi − xl ||

(15)

(10a)

Then the time-varying relationship between space positions is represented by the optimal spatial kernel function:

Then the bias term b(tj) in model 6 can be described by33 s

s

i=1

s

∑ ∑ αî (t j)K (xi , xl))/s i=1 l=1

⎛ || x − x ||2 ⎞ i ⎟ K (x , xi , σ(tL + 2)) = exp⎜ − 2 ⎝ σ(tL + 2) ⎠

(10b)

Thus, the spatiotemporal LS-SVM can be represented as follows:

∑ αî (t )K (x , xi) + b(̂ t ) i=1

(11)

with eqs 10a and 10b

Although this model (eq 11) can accurately represent many complex DPSs, it cannot approximate the time-varying DPS well because it is used to model the time-invariant DPS and lacks any adaptive ability, and the spatiotemporal calculations are also offline. 3.2. Online Update Method. Here, a two-step update method is proposed to modify the spatiotemporal LS-SVM model for the time-varying DPS, as indicated in Figure 4. The core ideas are outlined below: • Based on modeling error, the spatial kernel function is updated; • Projecting the practical output on the updated spatial kernel function, the new Lagrange multiplier is obtained and used to update the model of the Lagrange multiplier; • The time-varying dynamics of the DPS is reconstructed by integrating the updated spatial kernel function with the updated Lagrange multiplier. A. Update of the Spatial Kernel Function. Here, the following radial basis function is employed to represent the spatial kernel function: ⎛ || x − x ||2 ⎞ i ⎟ K (x , xi , σ ) = exp⎜ − 2 σ ⎝ ⎠

(16)

B. Update of the Time Coefficient Model. Then, an online updating method is developed to adaptively adjust the time coefficient model, so that time-varying dynamics can be captured. It includes the following steps: Step 1: Collection of New Time Coefficient. Using the real system output y(xi,tL+1), the real time coefficient α(tL+1) can be computed according to eq 7:

s

y (̂ x , t ) =

⎞ ⎟ ⎠

/σ 2(tL + 1)⎟

k=2

b(̂ t j) = (∑ y(xi , t j) −

(13)

i=1

where σ(tL+1) is σ at tL+1 and η ∈ (0,1) is the learning rate, and

⎡0 ⎤⎡ C (t ) ⎤ ⎡ 0 ⎤ 1TL − 1 ⎢ ⎥⎢ i L ⎥ = ⎢ ⎥ −1 ⎥ ⎢1 ⎣ L − 1 Ω + δ I ⎦⎢⎣ Bi (tL)⎥⎦ ⎣ Ai (tL)⎦

∑ βi (tk)K̃ (z(t j), z(tk− 1)) + ci ,

s

∑ (y(xi , tL + 1) − y (̂ xi , tL + 1))2

where y(x,tL+1) is the real system output at time tL+1 and ŷ(x,tL+1) is the model output from eq 11. We can calculate the optimal value of σ at time tL+2 according to the following equations by using the gradient-based optimization method:

where ωi and ci are the weight vector and the bias term, and φ is the unknown mapping function. The model input is zi(tj−1) = [αi(tj−1)T, u(tj−1)T]T. The least-squares method is employed to solve the modeling problem (eq 8). Then we can gain the following linear equation:

αî (t j) =

1 2

Ey(tL + 1) =

⎡ 0 1T ⎤⎡b(t ) ⎤ ⎡ 0 ⎤ s ⎢ ⎥⎢ L + 1 ⎥ = ⎢ ⎥ ⎢1 Q (t ) + γ −1I ⎥⎢⎣ α(t )⎥⎦ ⎣ Y (tL + 1)⎦ L+1 ⎣s ⎦ L+2

(17)

Here, Q il(tL + 2) = K (xi , xl , σ(tL + 2)),

i , l = 1, ..., s

Solving eq 17 gives the real time coefficient α(tL+1). Step 2: Online Updating of Time Coefficient. The new time coefficient data α(tL+1) are added to the data set. Using the new L+1 data set {u (tj),αi(tj)}s,i=1,j=1 , eq 9 may be rewritten as follows: ⎡ 0 1T ⎤⎡Ci(tL + 1) ⎤ ⎡ 0 ⎤ L ⎥⎢ ⎥=⎢ ⎢ ⎥ ⎢⎣1L PL ⎥⎦⎣⎢ Bi (tL + 1)⎥⎦ ⎣ Ai (tL + 1)⎦

(18)

where Ai (tL + 1) = [Ai (tL), α(tL + 1)]T

Solving eq 18 gives (12)

Ci(tL + 1) =

Here, σ is the width of the Gaussian spatial kernel function. According to modeling error, the parameter σ is updated in order to reflect the time-varying relationship between space locations.

1TL PL−1Ai (tL + 1) 1TL PL−11L

,

Bi (tL + 1) = PL−1(Ai (tL + 1) − Ci(tL + 1) ·1L ) D

(19)


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Figure 5. Snap curing oven system.

⎤ ⎡P ΩT where PL = ⎢ L − 1 L −1 ⎥ , ⎣ ΩL ΩL , L + δ ⎦

This model is updated according to a generated temporal Lagrange multiplier at each time point, allowing the model to be able to adapt to time-varying dynamics. • By integrating the spatial kernel function and the temporal Lagrange multiplier, the online spatiotemporal LS-SVM model can accurately represent the time-varying spatiotemporal dynamics of a DPS. Discussion. The proposed method is successful at modeling of a nonlinear, time-varying DPS. It not only captures the temporal dynamics using an online time coefficient model but also considers the time-varying relationship between spatial locations through an adaptive spatial kernel function. Combination of the adaptive spatial kernel function with the online time coefficient model allows for reconstruction of unknown, time-varying, and nonlinear DPSs and ensures the model can reflect well real-time spatiotemporal dynamics. Of course, this proposed method can require large computational time when being applied for a complex and large-scale modeling problem due to using the gradient-based optimization method. In the future, we will further improve its modeling performance.

ΩL = [ ΩL ,1 ... ΩL , L − 1]

Here, ΩL and ΩL,L are the kernel functions corresponding to z(tL). According to the theories of Partitioned Matrix Inverse and Matrix Inversion Lemma,35 the inverse may be written as ⎡ P−10 ⎤ ⎡ P−1ΩT ⎤ PL−1 = ⎢ L ⎥ + Δ⎢ L L ⎥[ ΩLPL−1 −1] ⎣ 00 ⎦ ⎣− 1 ⎦

(20)

−1

where Δ = (ΩL , L + δ −1 − ΩLPL−−1 1ΩTL )

From eqs 19 and 20, the model of the time coefficient at time tL+2 is updated as follows: α̂i(tL + 2) = Ci(tL + 1) + ΩL + 1Bi (tL + 1), i = 1, ..., s

(21)

Equation 21 may be rewritten: L+1

αî (tL + 2) =

∑ βi (t j)K̃ (z(tL + 1), z(t j − 1)) + ci , i = 1, ..., s

4. EXPERIMENTAL VALIDATION Here, we use the curing thermal process happened in the snap oven,19,31 as shown in Figure 5, to testify the proposed modeling method. This curing system consists of a computer with a dSPACE 1102 controller, a PCLD-789 signal condition board, a PCL-855 relay board, and a snap oven. The software for the snap oven control includes Matlab and dSPACE. Chips placed on the lead-frame cure best at a set temperature profile. In the thermal process, we adjust heat in the oven by using four heaters (h1− h4). The temperature distribution inside the chamber is required for the fundamental analysis and better curing quality. For modeling this system, 16 J-type thermocouples (range: −40 °C ∼ 375 °C, maximal measurement error: 1.5 °C) placed on the lead-frame are used to measure temperature, as shown in Figure 6.

j=2

(22)

Finally, by integrating the updated kernel function and the updated model of the time coefficient, the online model is given by s

y (̂ x , tL + 2) =

∑ αî (tL + 2)K (x , xi , σ(tL + 2)) + b(̂ tL + 2) i=1

(23) s

with b(̂ tL + 2) = (∑ y(xi , tL + 1) i=1 s

−

s

∑ ∑ αî (tL + 2)K (xi , xl , σ(tL + 2))) i=1 l=1

/sand eq 22, eq 16

This model (eq 23) can effectively describe the time-varying spatiotemporal dynamics of a DPS. 3.3. Summary. The proposed method may be briefly summarized as the following steps: • The spatial kernel function represents the nonlinear relationship between spatial positions. The online update of parameter σ allows the model to adapt to variation of spatial relationships. • The Lagrange multiplier incorporates the temporal dynamics of the system and is modeled by the LS-SVM.

Figure 6. Sensor placements for modeling of snap curing oven. E


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Industrial & Engineering Chemistry Research In the experiment, we use random input signals to power the heaters for exciting the thermal process. Since the temperature in the oven changes slowly, a sampling interval Δt = 10 s is set and a total of 2800 measurements are collected. Since the LS-SVM itself has a good robustness to noise and noise in the experimental process is extremely small compared to signal as the high-accuracy data collection system is used, the experimental data is directly used to construct the model. Otherwise, common filtering methods will be employed to deal with the experimental data. A sample of the input signal in heater h2 is shown in Figure 7. The model was constructed by using Figure 9. Update process of α6.

Figure 10 and Figure 11 show the real output and the estimated model output at s1 and s10, respectively. Figure 12 and

Figure 7. Input signals of heater h2 in the experiment.

experimental data from sensors s1−s6, s8−s9, and s11−s16, and its performance was evaluated by data from the remaining sensors (s7, s10). In the modeling process, the parameter σ is updated via eqs 13−15 with the modeling error, and the temporal coefficient α6 is updated via eqs 17−21 with new time coefficient data. The update process of σ and the change process of the temporal coefficient α6 are shown in Figure 8 and Figure 9, respectively.

Figure 10. Model and real output at s1.

Figure 11. Model and real output at s10.

Figure 13 show the model output and the relative error at the 200th time point, respectively. From these figures, it is clear that the proposed method can model the time-varying curing process well. Then, we compare several commonly used DPS modeling methods, including the KL-Hammerstein method,25 the KLWiener method,26 the traditional LS-SVM method,28 the NLPCA-RBF method,36 and Spatiotemporal LS-SVM,33 with the proposed method on modeling of the curing processes. The TNAE(x) obtained from each of the six modeling methods are compared in Figure 14, and the minimum, maximum, and average values are listed in Table 1. Based on this comparison, the proposed method has the smallest minimum, maximum, and average values of TNAE(x), indicating superior model performance. This is because the proposed model can capture nonlinear spatiotemporal dynamics and has an online adaptive function.

Figure 8. Update process of σ.

Define the relative error and the temporal normalized absolute error (TNAE(x)) as follows: relative error =

y(x , t ) − y (̂ x , t ) × 100% y(x , t )

(24)

L

TNAE(x) =

∑t = 1 (y(x , t ) − y ̂(x , t ))

(25) L Here, y(x,t) and ŷ(x,t) are the real output and the model output, respectively. The smaller the relative error and TNAE(x) are, the better the modeling performance is. F


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represented by using a spatial kernel function. In addition, it adaptively adjusts the spatial kernel function and the time Lagrange multiplier in order to accurately model spatiotemporal dynamics in the system. Our results indicate that the online spatiotemporal LS-SVM can effectively model a nonlinear timevarying DPS. A practical curing experiment demonstrates the effectiveness of this method, and its superior modeling performance is also highlighted by comparing to other frequently used DPS modeling methods.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Figure 12. Model output at the 200th time.

ORCID

XinJiang Lu: 0000-0002-5100-1092 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is partially supported by the Project of Innovationdriven Plan in Central South University (2016CX009), and National Natural Science Foundation of China (51675539), Hunan province science and technology plan (2016RS2015).

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Figure 13. Relative error at the 200th time.

Figure 14. Performance comparison of TNAE.

Table 1. Modeling Performance Comparison method

MIN

MAX

AVG

LS-SVM28 KL-Wiener26 KL-Hammerstein25 NL-PCA-RBF36 Spatiotemporal LS-SVM33 Online Spatiotemporal LS-SVM

2.1 0.74 1.45 1.41 0.20 0.16

4.02 1.81 2.4 1.71 1.65 1.56

2.44 0.99 1.76 1.59 0.82 0.73

REFERENCES

5. CONCLUSION In this paper, we propose an online spatiotemporal LS-SVM modeling method for nonlinear and time-varying distributed parameter systems. While most traditional DPS modeling methods are linear and offline, the proposed model incorporates nonlinear effects and online adaptations. In the proposed method, the nonlinear relationship between spatial positions is G


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Online Spatiotemporal Least-Squares Support Vector Machine

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