Modeling of nonlinear distributed parameter system for industrial thermal processes

Abstract

Distributed parameter system (DPS) becomes important because of advanced technological needs. Thermal processes in integrated circuit (IC) packaging and other industry are typical nonlinear DPS where the input and output vary both spatially and temporally. Modeling is required for prediction, control and optimization of temperature field. Their complex features: time/space coupling, nonlinear and unknown structure uncertainties make the modeling of this class of unknown nonlinear DPS very difficult and challenging. The physics-based DPS modeling often leads to nominal nonlinear partial differential equations (PDE) with unknown structure uncertainties. They are infinite-dimensional systems, thus the reduction to finite-dimensional systems is needed because of finite actuators/sensors and limited computing power. Though there are many model reduction methods, e.g., finite difference method (FDM), finite element method (FEM), spectral method and Karhunen-Loève (KL) method, they require an accurate nominal PDE model be available. Parameter estimation from input-output data is also not suitable because it requires the PDE structure be known. Though nonlinear DPS with unknown structure is very common in the industry, its identification receives a little attention. This problem is very difficult because both the parameter and structure need determine. Existing DPS identification approaches have some limitations. Green’s function based identification leads to a linear spatio-temporal kernel model, which is only a linear approximation for nonlinear DPS around a working point. FDM or FEM based identification will lead to a high-order model, which may result in an unpractical high-order controller. Spectral or KL based neural identification can lead to a low-order model. Compared with spectral method, KL method can obtain a lower-order model. However the complex nonlinear model structure may result in a complicated control design. On the other hand, the popularly used complexity restricted nonlinear models: Wiener, Hammerstein and Volterra, are temporal models and only studied for lumped parameter systems (LPS). Nonlinear principal component analysis (NL-PCA) is a nonlinear dimension reduction method in machine learning. It can achieve a lower-order and more accurate model than the KL method. Thus to overcome the limitations in the DPS identification, it is necessary to develop new nonlinear DPS modeling approaches by combining the advantages of different modeling and learning techniques. The objective of this thesis is to extend the traditional Wiener, Hammerstein, Volterra and neural modeling to DPS with the help of kernel idea, KL method or NL-PCA, develop some useful data-based spatio-temporal modeling approaches for unknown nonlinear DPS, and apply the proposed methods on typical thermal processes in IC packaging and other industry. Firstly, a KL based Wiener (KL-Wiener) modeling approach is proposed. Traditional Wiener model is popularly used because of its block-oriented nonlinear structure (a linear LPS followed by a static nonlinearity). For modeling nonlinear DPS, a spatiotemporal Wiener model is established with a linear DPS followed by a static nonlinearity. After the time/space separation, it can be represented by traditional Wiener system with a set of spatial basis functions. To achieve a low-order model, the KL method is used for the time/space separation and dimension reduction. Then the Wiener model is estimated using the least-squares estimation and the instrumental variables method, which can achieve consistent estimates under process noise. Secondly, a KL based Hammerstein (KL-Hammerstein) modeling approach is proposed. Traditional Hammerstein system is also a popular used block-oriented nonlinear model (a static nonlinearity followed by a linear LPS). To model nonlinear DPS, a spatio-temporal Hammerstein model (a static nonlinearity followed by a linear DPS) is constructed. After the time/space separation, it can be represented by traditional Hammerstein system with a set of spatial basis functions. To obtain a low order model, the KL method is used for the time/space separation and dimension reduction. Then the compact Hammerstein model structure is determined by orthogonal forward regression and the parameters are estimated with the leastsquares estimation and singular value decomposition. This approach can achieve a low-order parsimonious Hammerstein model. Thirdly, a kernel based multi-channel spatio-temporal Hammerstein modeling approach is proposed. A distributed Hammerstein system is constructed with a static nonlinearity followed by a spatio-temporal kernel. When the model structure is matched with the system, a basic identification approach using the least-squares estimation and singular value decomposition can work well. When there is some unmodeled dynamics, a multi-channel modeling framework is proposed to achieve a better performance, which can guarantee the convergence under noisy measurements. Fourthly, a Volterra kernel based spatio-temporal modeling approach is proposed. Traditional Volterra model consists of a series of temporal kernel. To reconstruct the spatio-temporal dynamics, a spatio-temporal Volterra model is constructed with a series of spatio-temporal kernels. It is also a nonlinear extension of Green’s function. To achieve a low-order model, the KL method is used for the time/space separation and dimension reduction. Then the model is estimated with a least-squares algorithm, which can guarantee the convergence under noisy measurements. Fifthly, a NL-PCA based neural modeling approach is proposed. The KL based dimension reduction is a linear approximation for a nonlinear system. To get a lowerorder or more accurate model, a NL-PCA based intelligent modeling framework is studied. One NL-PCA network is trained for nonlinear dimension reduction and nonlinear time/space reconstruction, and then the other linear/nonlinear separated neural model is to learn the system dynamics. With the powerful capability of dimension reduction and intelligent learning, this approach can model more complex nonlinear DPS. The simulations and experiments on typical thermal processes in IC packaging and other industry show that the proposed modeling approaches are effective. Different kinds of models may be suitable for different system and is usually selected by the performance. For the snap curing oven system in IC packaging, Volterra model achieves the best performance. The accuracy of KL-Wiener model is similar to Volterra model. Both Volterra and Wiener models are more suitable for this process than other models. This thesis provides several useful modeling frameworks for unknown nonlinear DPS. The proposed modeling approaches are data-driven and do not need a priori process knowledge, so they are very flexible and can be easily applied to other distributed processes. They can obtain low-order and simple nonlinear models. Though the modeling experiment may need more sensors, once the model has been established a few sensors will be enough for real-time applications.