Improved Empirical Eigenfunctions Based Model Reduction for

Article pubs.acs.org/IECR

Improved Empirical Eigenfunctions Based Model Reduction for Nonlinear Distributed Parameter Systems Mian Jiang†,‡,§ and Hua Deng*,†,‡ †

The State Key Laboratory of High Performance and Complex Manufacturing, Changsha 410083, China School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China § Hunan Provincial Key Laboratory of Health Maintenance for Mechanical Equipment, Hunan University of Science and Technology, Xiangtan 411201, China ‡

ABSTRACT: Karhunen−Loève (KL) decomposition is a popular approach for determining the principal spatial structures from the measured data. Empirical eigenfunctions (EEFs) can generally generate a relatively low-dimensional model among all linear expansions. The current study proposes improved EEFs for model reduction of the nonlinear distributed parameter systems (DPSs) by the basis function transformation from initial EEFs. The basis function transformation matrix is obtained using the balanced truncation method. This performance is proved theoretically. The numerical simulations for the rescaled Kuramoto− Sivashinsky equations show that using the improved EEFs has an evidently better performance than using the same number of the initial EEFs.

1. INTRODUCTION Many technological needs such as semiconductor manufacturing, nanotechnology, and biotechnology have motivated control of material microstructure, spatial profiles, and product size distributions.1 These processes belong to distributed parameter systems (DPS), while their input, output, state, and even parameters may vary both temporally and spatially. The development of a low dimensional approximation model which describes the spatiotemporal nature of the systems is a key problem for control design. The first-principle modeling of DPS typically leads to various partial differential equations (PDE). Traditional methods such as finite difference method (FDM), finite element method (FEM), and spatial basis functions expansion combined with weighted residual methods 2 including the Galerkin method, the collocation method, and the approximate inertial manifold method,3 etc. can be applied to the model reduction of the process. However, conventional time/space discretization approaches, such as FDM and FEM for model reduction, often yield highorder models that are unsuitable for synthesizing implemental and real-time control. General basis functions in spatial basis functions expansion, including the Fourier series, orthogonal polynomials,4 and eigenfunctions of DPSs1,2,5 are also not optimal in the sense that the dimension of the reduced model is not lowest at a given accuracy. Kurhunen−Loeve (KL) decomposition6,7 is a representative approach to find the principal spatial structures from the data, which is also referred to as principle component analysis (PCA), empirical orthogonal function analysis (EOF), or proper orthogonal decomposition (POD). The amount of variance of a system represented by the leading KL basis functions is often taken as an indication of the quality of a reduced model using those first several KL basis functions. However, past studies8−10 have pointed out that EOF-based models can have difficulties reproducing behavior dominated by irregular transitions © 2012 American Chemical Society

between different dynamical states. Modes representing only a tiny amount of variance can be crucial in the generation of certain types of dynamics. So, it is necessary to keep the influence of the modes representing only a tiny amount of variance on dominated modes, which will give both qualitatively and quantitatively better results than the common KL based models. In our previous works,11 new spatial basis functions have been proposed for model reduction of nonlinear DPS with an openloop stable corresponding linear time-invariant (LTI) system. However, it can not be used for completely unknown processes or nonlinear DPS with an open-loop unstable corresponding LTI system. The present study derives improved eigenfunctions for the model reduction of the DPSs, which are completely unknown or have an open-loop unstable corresponding LTI system. After time/space separation based on improved EEFs, low dimensional models are obtained by traditional system identification methods, such as neural networks,12 spatiotemporal Volterra modeling,13−15 and least squares support vector machines (LSSVM).16 First, KL decomposition is used for time/space separation from measured spatiotemporal output. Thus, a linear ordinary differential equation (ODE) system can be used to approximate the dominant dynamics of the temporal input and the lowdimensional time coefficients at the equilibrium points. Hence, a basis function transformation matrix is obtained by balanced truncation model reduction11,17 for the obtained linear ODE system. Improved eigenfunctions are obtained from initial eigenfunctions of measured spatiotemporal output by basis function transformation. Each new spatial basis function is a linear combination of the initial EEFs. This performance is Received: Revised: Accepted: Published: 934

May 7, 2012 October 18, 2012 November 30, 2012 November 30, 2012 dx.doi.org/10.1021/ie301179e | Ind. Eng. Chem. Res. 2013, 52, 934−940

Industrial & Engineering Chemistry Research

Article N

proved theoretically, and the numerical simulations for model reduction of the rescaled Kuramoto-Sivashinsky equations show the effectiveness of the improved EEFs.

T̂ =

i=1

ai =

bi(x)ui(t )

denotes the vector of manipulated spatiotemporal inputs, where ui(t) is the ith temporal signal with certain spatial distribution bi(x). A and B are the two linear operators that involve linear spatial derivatives on the state variable and spatiotemporal input. F(·) is a nonlinear function containing spatial derivatives for T(x,t) and U(x,t). To collect the data that are assumed to be fully representative of the temporal progression of the system (eq 1) to obtain the EEFs, sufficient sensors should be used to measure the spatiotemporal variable depending on the required modeling precision. The data of spatiotemporal variable T d = {T(xh,tj)}N,L h = 1,j = 1 of the DPS are measured at N spatial locations x1,x2,...,xN and sampling times t1,t2,...,tL. Define the inner product and ensemble average as

∫Ω f (x)g(x) dx

(7)

λk (8) F where F denotes the sum of the eigenvalues of the covariance matrix. Assuming that the eigenvalues are sorted in descending order, the eigenfunctions are ordered from most to least energetic. Using only the first r most energetic eigenfunctions where r < N, an approximation of the data is constructed as

i=1

(f (x), g (x)) =

(ϕi ·ϕi)

Fk =

n

∑

(T̂ ·ϕi)

Each eigenfunction has an energy percentage that depends on the associated eigenvalues of the eigenfunction

(1)

subject to a number of boundary and initial conditions. In eq 1, T = T(x,t) denotes the vector of state variable, where t ∈ [0,∞] is the time variable, x ∈ Ω is the spatial coordinate, and only one spatial-dimension is considered here. U = U (x , t ) =

(6)

where ai(t)s are the data coefficients computed from the projection of a sample vector onto an eigenfunction

2. EEFS FOR MODEL REDUCTION Suppose that a nonlinear DPS is governed by a PDE with the following state description: ⎛ ∂T ∂T ∂U ⎞⎟ = AT + BU + F⎜T , , ..., U , , ... ⎝ ∂x ∂t ∂x ⎠

∑ ai(t )ϕi(x)

r

TE = T̅ +

∑ aiϕi i=1

(9)

For the above illustration, the common model reduction can be accomplished for the nonlinear DPS based on the initial EEFs.

3. IMPROVED EEFS BASED MODEL REDUCTION For the nonlinear DPSs, the improved EEFs for model reduction are obtained based on the initial EEFs from the measured spatiotemporal variable Td(x,t) and the corresponding basis function transformation matrix. Several steps for the necessary improvement are shown in Figure 1. First, KL decomposition is

(2)

L

T (x , t ) = (1/L) ∑ T (x , t j) j=1

(3)

For convenience, a variation of the spatiotemporal variable T with zero ensemble average is computed as T ̂ = T − T̅

(4)

where T̅ = T̅ (x,t) and T̂ = T̂ (x,t). The orthogonal eigenfunctions of the data are calculated as the following eigenvalue problem.

∫Ω D(x , ζ)ϕi(ζ) dζ = λiϕi(x)

(5)

Figure 1. Improved EEFs derived for DPSs.

subject to (ϕi(x),ϕi(x)) = 1, i = 1,2,...,N, where D = D(x,ζ) = ⟨T̂ (x,t)T̂ (ζ,t)⟩ is the spatial two-point correlation function. Given that the covariance matrix D is symmetric and positively definite, its eigenvalues λi,i = 1,2,...,N are real, and its eigenvectors ϕi,i = 1,2,...,N form an orthogonal set. Since the data are always discrete in space, one must numerically solve the integral eq 5. Discretizing the integral equation gives a N × N matrix eigenvalue problem. Thus, at the most N eigenfunctions at N sampled spatial locations can be obtained. One characteristic of the eigenfunctions is that they form an optimal basis for the expansion of a spatiotemporal data set. If a large enough N is given (i.e., N → ∞), T̂ can be expressed as6,7,11

used for time/space separation from measured spatiotemporal output to obtain the initial EEFs and the corresponding temporal coefficients. In general, only the first few eigenvalues and the corresponding EEFs are used to represent the dominant dynamics. In this step, N EEFs are selected to derive the r improved EEFs, where r < N. Second, a linear ordinary differential equation (ODE) system (eq 10) is introduced to approximate the dominant temporal dynamics of the input and the low-dimensional time coefficients at the equilibrium points. 935

dx.doi.org/10.1021/ie301179e | Ind. Eng. Chem. Res. 2013, 52, 934−940


Article r

v(̇ t ) = A 0v(t ) + B0 u(t )

TIE(x , t ) = T̅ +

z(t ) = C0v(t )

For DPS eq 1, the following are derived:

Normally, linear ODE systems (eq 10) at the equilibrium points are well worked in case of plants subjected to mild nonlinear dynamics. For the characteristics of model structure, models (eq 10) can be treated as a time invariant model, where the parameters and order of linear models are constant at each operating point. The matrices B0 and C0 are determined by actuator and sensor locations, respectively. Let {x1,x2,...,xm} be the sensor locations. Then the matrices B0 and C0 can be calculated as follows:

r i=1

⎛ ∂⎜⎜T̅ + ⎝

i=1

⎞ ai̅ (t )ψi(x)⎟⎟ ⎠

r

∑ i=1

⎞ ⎠

r

∑ ai̅ (t )ψi(x), i=1

n

∑ ui(t )bi(x),

, ...

i=1

n ∂(∑i = 1 ui(t )bi(x))

∂x

⎞ , ...⎟⎟ ⎠

(14)

Using Galerkin method, the following are then obtained. a (̇ t ) = Aa ̅ ̅ (t ) + B̅ u(t ) + f (a ̅ , u)

(15)

T where a(t) ̅ = [a1̅ ,a2̅ ,···ar̅ ]

A̅ = (A̅ ij )r × r , B̅ = (Bij̅ )r × n f (a ̅ , u) = [f1 (a ̅ , u), f2 (a ̅ , u), ..., fr (a ̅ , u)]T A̅ ij = (ψi , A(ψj)); ⎛ ⎜ ⎜ fi (a ̅ , u) = ⎜ψi , ⎜ ⎜⎜ ⎝

Bij̅ = (ψi , B(bj(x)))

⎛ ⎜ ⎜ F ⎜T ̅ + ⎜ ⎜ ⎝

r

∑ ai̅ (t )ψi ,

n i=1

⎛ ∂⎜⎜T̅ + ⎝

i=1

∑ ui(t )bi(x),

⎛ n ⎞ ∂⎜⎜∑ ui(t )bi(x)⎟⎟ ⎝ i=1 ⎠ ∂x

r

∑ i=1

⎞ ai̅ (t )ψi ⎟⎟ ⎠

∂x

, ...

⎞⎞ ⎟⎟ ⎟⎟ , ...⎟⎟ ⎟⎟ ⎟⎟ ⎠⎠

4. MODEL REDUCTION PERFORMANCE USING IMPROVED EEFS Suppose that there are enough sensors for measurement, and TP(x,t) denote the predicted spatiotemporal variable for T(x, t). Let T j = [T(x 1 ,t j ),T(x 2 ,t j ),...,T(x m ,t j )] T and T Pj = [TP(x1,tj),TP(x2,tj),...,TP(xm,tj)]T be the data of spatiotemporal variable T(x, t) and TP(x,t) at m spatial locations x1,x2,...,xm and some sampling time tj, respectively. To evaluate the model reduction performance at any sampling time tj by using the improved EEFs and the initial EEFs, the root-square error (RSE) as a performance index is introduced for comparison.

(11)

N j=1

⎝

∂x

where {ψ1,ψ2,...,ψr} and {ϕ1,ϕ2,...,ϕN} denote the improved EEFs and initial orthogonal EEFs of the data Td, respectively. Each improved EEF is a linear combination of the initial EEFs, which can also be rewritten as follows:

∑ R jiϕj ,

r

⎛ ⎜ ⎛ n ⎞ ⎜ + B⎜⎜∑ ui(t )bi(x)⎟⎟ + F⎜T̅ + ⎜ ⎝ i=1 ⎠ ⎜ ⎝

where Bij = (bi(x),ϕi(x));Cij = (δ(x − xi),ϕj(x)). The unknown matrix A0 can be estimated easily by using the conventional approximation techniques such as least-squares algorithm and generalized inverse matrix. Thus, the corresponding basis function transformation matrix R of the improved EEFs can be obtained from the balanced truncation model reduction11 for the obtained linear ODE system in eq 10. For the linear time-invariant system (eq 10), there exists a balanced transformation and the transformed system is in balanced form. Meanwhile, the controllability gramian and observability gramian of the new system become the same as an N order Hankel singular matrix. The diagonal elements of the matrix are Hankel singular values, which provide a measure for the importance of the states because the state with the largest Hankel singular value is the one affected the most by control. Therefore, an r order models which can keep the dominant dynamics of the original system can be obtained by the truncation of the first r new states corresponding to the first r large singular values. The mathematical expression of the balance truncation methods can be found in ref 11, and the details for the computation of R are shown as follows. Square root decompositions of symmetrical controllability gramian P and observability gramian Q of the system of eq 10 can be easily obtained. Let P = GGT, Q = HHT be square root decompositions and defining GHT = WΣVT as a singular value decomposition, then using the MATLAB style colon notation, the transform matrix R = [GWΣ−1/2](:,1:r). Because of the balanced truncation model reduction,11 the obtained matrix R is of column-orthogonalty. Thus, RTR = Ir. The improved EEFs are derived as follows(r < N)

ψi =

⎛

∑ ai̇ (t )ψi(x) = A⎜⎜T̅ + ∑ ai̅ (t )ψi(x)⎟⎟

⎛ C11 ··· C1N ⎞ ⎜ ⎟ C0 = ⎜ ⋮ ⋱ ⋮ ⎟ ⎜ ⎟ ⎝Cm1 ··· BmN ⎠

{ψ1 , ψ2 , ..., ψr } = {ϕ1 , ϕ2 , ..., ϕN } ·R

(13)

i=1

(10)

⎛ B11 ··· B1n ⎞ ⎜ ⎟ B0 = ⎜ ⋮ ⋱ ⋮ ⎟ , ⎜ ⎟ ⎝ BN1 ··· BNn ⎠

∑ ai̅ (t )ψi(x)

i = 1, 2, ..., r (12)

Given the column-orthogonality of R and initial EEFs, the rimproved EEFs are shown to be orthogonal toward each other. As to the model reduction for DPS 1, the spatiotemporal variables T(x,t) can be expanded onto the proposed improved EEFs with the corresponding temporal coefficients ai̅ (t).

m

RSE =

∑ e(xh , t j)2 h=1

936

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Article

where e(xh,tj) = T(xh,tj) − TP(xh,tj). The model reduction performance with the improved EEFs is given in the following theorem. Theorem 1. Given an orthogonal matrix R obtained by balanced truncation model reduction for eq 10, then the RSE based on r improved EEFs is smaller than that based on r initial EEFs at any sampling time tj if RTÊ 1Ê 2R is negative semidefinite, where ⎡ Ir 0 ⎤ ⎥ − RRT ; E1̂ = ⎢ 0 0 ⎣ N−r ⎦

⎧ ⎛ ⎪

(17)

⎛ N ⎜⎜ ∑ ai(t j)ϕi + ⎝i=r+1

(18)

i=1

r

∑ ai̅ (t j)ψi

(19)

i=1

Combining 11 with 13, the corresponding time coefficients can be derived as (a1(t j), ..., aN (t j)) = (a1̅ (t j), ..., ar̅ (t j))RT

M1(t j) =

⎞⎫ ⎪

i=1

i=1

⎠⎭

⎪

N

r

⎛N ∑ ⎜⎜∑ ai(t j)ϕi − m ⎝ i=1

(20)

i=1

i=1

(M2(tj))

(

< M1(t j)

r

2

(M2(tj))

⎞2 ψ a ( t ) ∑ i̅ j i ⎟⎟ < ⎠ i=1 r

(23)

(

⎛ N ⎞2 ⎜ ϕ a ( t ) ∑ ⎜ ∑ i j i⎟⎟ ⎠ m ⎝i=r+1

⎛ a (t ) ⎞ ⎜ 1 j ⎟ ⎜ a (t ) ⎟ 2 j ⎟ E3̂ E2̂ ⎜ ⎜ ⋮ ⎟ ⎜ ⎟ ⎜ a (t )⎟ N j ⎝ ⎠

Then, we have ⎛⎛ N ∑ ⎜⎜⎜⎜∑ ai(t j)ϕi − m ⎝⎝ i = 1

2

)

⎛ϕ T ⎞ ⎜ 1 ⎟ ⎜ T⎟ ϕ = ∑ (a1(t j), ...aN (t j))E1̂ ⎜ 2 ⎟ ⎜ ⋮ ⎟ m ⎜ ⎟ ⎜ϕT ⎟ ⎝ N⎠

⎛ a (t ) ⎞ ⎜ 1 j ⎟ ⎜ a (t ) ⎟ 2 j ⎟ = (a1(t j), a 2(t j), ...aN (t j) E1̂ (ϕ1 , ϕ2 , ...ϕN )E2̂ ⎜ ⎜ ⋮ ⎟ ⎜ ⎟ ⎜ a (t )⎟ ⎝ N j ⎠

Substituting eqs 21 and 22 into eq 23 yields ⎛N ∑ ⎜⎜∑ ai(t j)ϕi − m ⎝ i=1

(

− M1(t j)

(22)

2

)

(26)

Then eq 25 times eq 26 yields

To prove that 0 ≤ M2(tj) < M1(tj), only the following inequality has to be proved 2

⎠

⎛ ⎡ Ir ⎞ ⎛ ⎡ Ir 0 ⎤ ⎞ 0 ⎤ ⎥ − RRT ⎟⎟ ⎥ − RRT ⎟⎟ and E2̂ = ⎜⎜⎢ E1̂ = ⎜⎜⎢ ⎠ ⎝⎣ 0 0N − r ⎦ ⎝⎣ 0 2IN − r ⎦ ⎠

(21)

⎞2 a ( t ) ψ ∑ i̅ j i ⎟⎟ ⎠ i=1

⎞

∑ ai(t j)ϕi − ∑ ai̅ (t j)ψi ⎟⎟

The RSE with r improved EEFs is defined as M 2 (t j ) =

(24)

where

The RSE with r initial EEFs is defined as ⎛ N ⎞2 ⎜ ∑ ⎜ ∑ ai(t j)ϕi⎟⎟ ⎠ m ⎝i=r+1

r

⎛ a1(t j) ⎞ ⎜ ⎟ ⎜ a (t ) ⎟ 2 j ⎟ = (ϕ1 , ϕ2 , ..., ϕN )E2̂ ⎜ ⎜ ⋮ ⎟ ⎜ ⎟ ⎜ a (t )⎟ N j ⎝ ⎠

From eq 13, the predicted output based on r improved EEFs at sampling time tj can be expressed as follows TIEj = T̅ +

N

∑ ai(t j)ϕi − ∑ ai̅ (t j)ψi ⎟⎟⎬ < 0

(25)

r

∑ ai(t j)ϕi

⎠

⎛ a1(t j) ⎞ ⎟ ⎜ r ⎜ a (t ) ⎟ ⎛ r ⎞ ⎜⎜∑ ai(t j)ϕi − ∑ ai̅ (t j)ψi ⎟⎟ = (ϕ1 , ϕ2 , ···, ϕN ) ·E1̂ ⎜ 2 j ⎟ ⎜ ⋮ ⎟ ⎝ i=1 ⎠ i=1 ⎟ ⎜ ⎜ a (t )⎟ ⎝ N j ⎠

The predicted output based on r initial EEFs at the sampling time tj is as follows TEj = T̅ +

i=1

In inequality eq 24,

N i=1

⎩⎝ i = 1

⎛ N ⎜⎜ ∑ ai(t j)ϕi + ⎝i=r+1

⎡ Ir 0 ⎤ ⎥ − RRT E2̂ = ⎢ 0 2 I ⎣ N−r ⎦

∑ ai(t j)ϕi

⎞

⎪

m

Proof. From KL decomposition for measured spatiotemporal output at the sampling time tj, we have Tj = T̅ +

r

r

∑ ⎨⎜⎜∑ ai(t j)ϕi − ∑ ai̅ (t j)ψi ⎟⎟·

⎞2 ⎞ ⎞2 ⎛ N ⎜ ⎟ ∑ ai̅ (t j)ψi ⎟ − ⎜ ∑ ai(t j)ϕi⎟⎟ ⎟⎟ < 0 ⎠ ⎝i=r+1 ⎠⎠ i=1 r

And the following inequality can be derived:

)

(27)

where 937


Industrial & Engineering Chemistry Research ⎡ ϕ Tϕ ⎢∑ 1 1 ⎢m ⎢ ϕ Tϕ ⎢∑ 2 1 ̂ E3 = ⎢ m ⎢ ⋮ ⎢ ⎢ ∑ ϕNT ϕ1 ⎢⎣ m

∑ ϕ1T ϕ2

···

m

∑ ϕ1T ϕN ⎤⎥ ⎥

m

∑

ϕ2T ϕ2

···

⋮

⋮

Article

∑

ϕ2T ϕN ⎥⎥

⎥ ⎥ ⎥ ··· ∑ ϕNT ϕN ⎥ ⎥⎦ m

m

m

∑ ϕNT ϕ2 m

⋮

As (ϕ1,ϕ2,...,ϕN) are orthogonal each other, thus ⎡1 ⎢ 0 E3̂ = ⎢ ⎢⋮ ⎢⎣ 0

0 ··· 0 ⎤ ⎥ 1 ··· 0 ⎥ = IN ⋮ ⋱ ⋮⎥ ⎥ 0 ··· 1 ⎦

Figure 2. Improved EEFs based neural modeling. (28)

The advantage of the neural network is its ability to model complex nonlinear relationships without any assumptions on the nature of these relationships. The most often used neural networks include the radial basis function networks,18 backpropagation (BP) neural networks,19 among others. The present study employs a feedforward BP neural network to construct a low-dimensional approximate model for the dynamics of DPSs. The prediction output of nonlinear DPS is obtained by synthesis of the temporal predicted output and the proposed improved EEFs:

Substituting eqs 20 and 28 into eq 27 yields 2

(M2(tj))

(

− M1(t j)

2

)

⎛ a1(t j) ⎞ ⎜ ̅ ⎟ ⎜ a (t )⎟ j 2̅ ⎟ = a1̅ (t j), a 2̅ (t j), ···, ar̅ (t j) RTE1̂ E2̂ R ⎜ ⎜ ⋮ ⎟ ⎜ ⎟ ⎜ a (t ) ⎟ ⎝ r̅ j ⎠

(

)

r

(29)

TIE(x , k) = T̅ +

Since M1(tj) ≥ 0 and M2(tj) ≥ 0, M2(tj) < M1(tj) if RTÊ 1Ê 2R is negative semidefinite. This completes the proof.

6. NUMERICAL SIMULATIONS To evaluate the performance of the proposed improved EEFs for model reduction, the rescaled Kuramoto−Sivashinsky (K−S) equation20−22 in one space dimension is considered. The K−S equations is one of typical partial differential equations, which has first been derived in 1976 by Kuramoto and Tsuzuki20 as a model equation for interfacial instabilities in the context of angular phase turbulence for a system of a reaction-diffusion equation that models the Belouzov−Zabotinskii reaction in three dimensional space and independently, in 1977, by Sivashinsky21 to model thermal diffusion instabilities observed in laminar Mame fronts in two dimensional space. 2 ⎡ ∂ 2T 1 ⎛ ∂T ⎞ ⎤ ∂T ∂ 4T + 4 4 + α ⎢ 2 + ⎜ ⎟ ⎥ + U (x , t ) = 0 2 ⎝ ∂x ⎠ ⎦ ∂t ∂x ⎣ ∂x

··· T (x1 , tL) ⎞ ⎟ ··· T (x 2 , tL) ⎟ ⎟ ⋮ ⋮ ⎟ ⎟ ··· T (xm , tL)⎠

⎛ a1(t1) a1(t 2) ̅ ⎜ ̅ ⎜ a 2(t1) a 2̅ (t 2) = T̅ + (ψ1 , ψ2 , ···, ψr )⎜ ̅ ⋮ ⎜ ⋮ ⎜ ⎝ ar̅ (t1) ar̅ (t 2)

··· a1̅ (tL) ⎞ ⎟ ··· a 2̅ (tL)⎟ ⎟ ⋮ ⋮ ⎟ ⎟ ··· ar̅ (tL) ⎠

(33)

where α = 84.25; 4

U (x , t ) =

∑

bi(x)ui(t )

i=1

(30)

⎤ ⎡ 3π π bi(x) = δ ⎢x + − (i − 1)⎥ ; ⎦ ⎣ 4 2 ⎛ t i ⎞⎟ ui(t ) = 1.1 + 5 sin⎜ ; + ⎝ 10 10 ⎠

For model identification by the neural network, there are n actuators with an implemental temporal signal u(t) and a certain spatial distribution. T(x,t) is measured at the m spatial locations x1,x2,...,xm and some sampling times t1,t2,...,tL. The temporal coefficients are calculated using eq 30 and a generalized inverse matrix. Using temporal signal u(t) and the temporal coefficients a(t), ̅ a feed forward neural network is employed to identify the dynamics: a(̂ k + 1) = NN (a(̂ k), u(k))

(32)

i=1

5. IMPROVED EEFS-BASED NEURAL MODELING In the Galerkin method, obtaining an exact analytical description of the low-dimensional ODE systems is difficult because of the nonlinearities in the system. Therefore, the neural network is used to identify long-term dynamical behaviors from temporal coefficients, as shown in Figure 2. ⎛ T (x1 , t1) T (x1 , t 2) ⎜ ⎜ T (x 2 , t1) T (x 2 , t 2) ⎜ ⋮ ⋮ ⎜ ⎜ ⎝T (xm , t1) T (xm , t 2)

∑ aî (k)ψi

i = 1, 2, 3, 4

Subject to the periodic boundary condition T (x , t ) = T (x + 2π , t )

In total, 41 sensors uniformly distributed in the space are used for measurement. A noise-free data set of 500 data points is collected from eq 1. The sampling interval Δt is 0.001 s and the

(31) 938



Article

simulation time is 0.5 s. The initial condition T0(x) is set to be cos(x). This size of data set used for training may be determined by the system complexity and the desired modeling accuracy. More complex system and higher modeling accuracy may need more data. For a real system, sufficient data are always required to obtain a satisfactory model. After the basis functions φ(x) are derived in terms of eq 5 by using the collected data, a linear system of the form of eq 10 can be identified to approximate the nonlinear DPS eq 33. According to the linear system, the basis function transformation matrix R that is of column-orthogonality can be derived by the balance truncation methods,11 and the improved basis functions ψ(x) can then be obtained. The calculation in the example shows that RTÊ 1Ê 2R is negative semidefinite and this means that the model reduction performance based on r improved EEFs is better than that based on r initial EEFs . For simplicity and convenience, the root-mean-square error (RMSE) at a time sampling interval over the testing data is used to evaluate the performance of the reduced models. As the energy percentage of the first 3 initial EEFs has reached 99%, the RMSE of the approximate models based on the first 3 EEFs and 3 improved EEFs are compared in Table 1.

Figure 4. First 3 improved EEFs.

Table 1. RMSE Comparisons for Initial and New Basis Functions RMSE

1 mode

2 modes

3 modes

EEFs improved EEFs

0.156 0.122

0.068 0.056

0.056 0.054

Figure 5. Measured output for testing.

As shown in Table 1, the value of RMSE using improved EEFs is much smaller than when using the same number of initial EEFs. The first 3 initial EEFs and first 3 improved EEFs are shown in Figures 3 and 4. A new set of 100 data is collected as shown in

Figure 6. Distribution error based on 3 initial EEFs on testing data.

Figure 3. First 3 initial EEFs.

Figure 5 for testing to compare the model reduction performance by using the two kinds of spatial basis functions. The spatio− temporal output of the K−S equation on testing data can be estimated from the synthesis of the approximate model and the discrete basis functions. In terms of the testing data, the predicted distribution errors based on two kinds of discrete basis functions are shown in Figures 6 and 7, respectively. The RMSEs of the approximate model based on the 3 improved EEFs and 3 initial EEFs are 0.054 and 0.056, respectively. Moreover, the RMSE of the approximate model with 2 improved EEFs is as small as that of the approximate model with 3 initial EEFs. The distributed error of the approximate model based on 2 improved EEFs is shown in Figure 8. It is evident that the control design for

Figure 7. Distribution error based on 3 improved EEFs on testing data.

approximate models with 2 orders is much simpler than that with 3 orders or more.

7. CONCLUSIONS The improved EEFs with lower computational cost were proposed for model reduction of nonlinear DPSs. The improved 939



Article

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Figure 8. Distribution error based on 2 improved EEFs on testing data.

EEFs were chosen in terms of the initial EEFs of measured output of the nonlinear DPSs, and the linear ODE is used to approximate the temporal dynamics. Each improved EEF is a linear combination of the initial EEFs. The balanced truncation method was used to derive the basis function transform matrix based on the linear ODE. The effectiveness of the improved EEFs was illustrated theoretically and numerically for the model reduction of the K−S equation.

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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 work was supported by the National Natural Science Foundation of China under Grants 51075404 and 51175170. REFERENCES

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