New spatial basis functions for the model reduction of nonlinear distributed parameter systems

Deng, Hua; Jiang, Min; Huang, Changqing

doi:10.1016/j.jprocont.2011.12.008

Cited by 28 publications

(22 citation statements)

References 22 publications

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“…There are many calculation methods of transformation matrix S , such as balanced truncation method [4,11] and optimization method [5]. For the optimization method [12], the objective functions is given as follows…”

Section: New Basis Functions By Linear Transformationmentioning

confidence: 99%

“…In terms of the suitable choice of spatial basis functions, the nonlinear PDE (1) can be reduced to a finite-dimensional ordinary differential equation (ODE) system by Galerkin method, which is called model reduction for PDEs. A number of model reduction approaches for nonlinear PDEs have already been studied [4,5,6]. They result in the feasible implementation of control for practical applications in industrial processes.…”

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confidence: 99%

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confidence: 99%

“…To reduce the calculation difficulty of the PIPs, Deng [4] proposed a straightforward algorithm to derive new spatial basis functions. Like PIPs, the new spatial basis functions are linear combinations of general spatial basis functions, and the transformation matrix is derived from balanced truncation method [4] for a corresponding linear ODE system of nonlinear PDEs. Following the above methodology, Jiang and Deng [5] proposed a new algorithm to derive new spatial basis functions by linear transformation, and the transformation matrix is derived from optimization techniques of a spatial-temporal error functions.…”

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confidence: 99%

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New Basis Functions for Model Reduction of Nonlinear PDEs

Shuai

Han

2014

JNW

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The selection of the spatial basis functions is very important for model reduction of the nonlinear partial differential equations (PDEs) under time/space separation framework, which will significantly affect the accuracy and efficiency of the modeling. Using the spatial basis functions expansions and the Galerkin method, the finite-dimensional ordinary differential equation (ODE) systems can be obtained from the PDEs. However, the general basis functions are not optimal in the sense that the dimensions of the ODE system are not lowest at a given modeling accuracy. The current study proposes new basis functions for the model reduction of nonlinear PDEs, which are obtained by linear combinations of general spatial basis functions. The transformation matrix for the combination coefficients is derived from straightforward optimization techniques for an improved spatio-temporal error function between the approximation and the measured spatial-temporal output. The derivation of the improved error functions also considers the influence of the variance of the spatialtemporal error. Using the new basis functions expansions and Galerkin method, it can provide a lower dimensional and more precise ODE to approximate the dynamics of the nonlinear PDEs. The modeling performance are compared with the method proposed in reference, and the simulations shows the feasibility and effectiveness of the proposed new basis functions for model reduction of nonlinear PDEs.

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Section: New Basis Functions By Linear Transformationmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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New Basis Functions for Model Reduction of Nonlinear PDEs

Shuai

Han

2014

JNW

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“…However, this method will lead to a high dimension ODE model due to the complexity of distributed parameter system. And various orthogonal decomposition (Karhunen-Loève, KL) methods or singular value decomposition (SVD) methods are often used to obtain the principle features by a series of space basis functions and time factors [1,2,3]. However, no matter how to choose basis functions, the dimension is reduced linearly in terms of KL or SVD timespace separation method.…”

Section: Introductionmentioning

confidence: 99%

KPCA-ARX time-space modeling for distributed parameter system

Yang

Tao

2014

The 26th Chinese Control and Decision Conference (2014 CCDC)

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Modeling of distributed parameter systems (DPSs) is difficult because of their infinite dimensional time-space nature. For a class of nonlinear distributed parameter systems described by parabolic partial differential equations (PDEs), Kernel Principal Component Analysis (KPCA) method is utilized to extract the nonlinear basis functions in dominant space, and the time-space decomposition is carried out in terms of these basis functions to obtain the outputs in time domain. Since the dominant space extraction is influenced by the parameters of kernel functions, they are optimized by Genetic Algorithm (GA) to obtain more system information with less principal components. The input stimulation and time domain outputs are used to construct the ARX model, which is identified by the recursive least squares algorithm. The simulation results show that the proposed method can obtain more system information with less principal components and gain satisfying reconstruction accuracy.

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