Spectral-Approximation-Based Intelligent ... - Semantic Scholar
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 5, SEPTEMBER 2005
Spectral-Approximation-Based Intelligent Modeling for Distributed Thermal Processes Hua Deng, Han-Xiong Li, Senior Member, IEEE, and Guanrong Chen, Fellow, IEEE
Abstract—A spectral-approximation-based intelligent modeling approach is proposed for the distributed thermal processing of the snap curing oven that is used in semiconductor packaging industry. The snap curing oven can be described by a nonlinear parabolic distributed parameter system (DPS) in the time–space domain. After finding a proper approximation of the complex boundary conditions of the system, the spectral methods can be applied to time–space separation and model reduction, and neural networks (NNs) can be used for state estimation and system identification. With the help of model reduction techniques, the dynamics of the curing process derived from physical laws can be described by a model of low-order nonlinear ordinary differential equations with a few uncertain parameters and unknown nonlinearities. A neural observer can then be designed to estimate the states of the ordinary differential equation model from measurements taken at specified locations in the field. Using the estimated states, a hybrid general regression NN is trained to be a nonlinear model of the curing process in state–space formulation, which is suitable for the further application of traditional control techniques. Real-time experiments on the snap curing oven show that the proposed modeling method is effective. This modeling methodology can be applied to a class of nonlinear DPSs in industrial thermal processing. Index Terms—Curing process, distributed thermal process, neural networks (NNs), nonlinear distributed parameter systems (DPSs), spatial system identification, spectral methods, state estimation.
I. INTRODUCTION
T
O PRODUCE high-quality semiconductor devices, increasingly complex processes are used more and more in the semiconductor back-end packaging industry. After epoxy die attach, the bonded leadframe is cured in a snap curing oven, which provides the required curing temperature distribution. Traditional proportional-integral-derivative (PID) controllers often fail to maintain a consistent temperature distribution due to the complex time-spatial nature of the process and external disturbances. It is important for the industry to obtain a good model of the process at a reasonable cost and accuracy for funManuscript received June 19, 2003; revised April 30, 2004. Manuscript received in final form January 7, 2005. Recommended by Associate Editor G. Rosen. This work was supported in part by the RGC of Hong Kong SAR under Project CityU 1129/03E and in part by the China National Science Fund for Distinguished Young Scholars under Grant 50429501. H. Deng is with the School of Mechanical and Electrical Engineering, Central South University, Changsha, China, and also with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong. H.-X. Li is with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]). G. Chen is with the Department of Electronic Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong. Digital Object Identifier 10.1109/TCST.2005.847329
damental analysis and control-related issues including design, simulation, diagnosis, and supervision. The snap curing process is one kind of thermal process that follows the basic principles of heat transfer (conduction, radiation, and convection). Due to the complex boundary conditions and external disturbances, it is very difficult to obtain good models based on physical insights alone [1]. Even if this kind of structure can be derived from physical laws, the model will still contain many unknown parameters. The modeling issue in semiconductor thermal processing (i.e., rapid thermal processing (RTP) systems) has been seriously considered, and control-related problems have also been studied [2]–[8]. However, the basic requirement for the aforementioned control methods is that a known physical model of the RTP must be known. The radiative heat transfer is dominant for the RTP systems [5]. This is because the operating temperatures of RTP systems can be up to 1100 C. However, the curing temperatures of the epoxy considered are around 180 C and, thus, the radiative heat transfer is not dominant. Mathematically, the curing process belongs to nonlinear parabolic distributed parameter systems (DPSs) described by partial differential equations (PDEs) arising from heat, mass, and momentum balances [9]. A general mathematical description for DPSs usually consists of PDEs with mixed or homogeneous boundary conditions. The main characteristic of a DPS is that the outputs, inputs, process states and parameters can vary temporally and spatially. Controlling a DPS is usually accomplished by truncating the PDE with boundary conditions to a system of ordinary differential equations (ODEs) that permit the synthesis of a controller [10]. This transformation may result in high-order solutions of PDEs, which usually mean that a controller synthesis is not implementable due to such physical limitations as a finite number of sensors and actuators. If the mathematical model of a parabolic PDE is completely known, with mixed or homogeneous boundary conditions, then the resulting nonlinear DPS might be approximated by a lower dimensional ODE system through modal decomposition using the eigenfunctions of the associated linear operator [9], [11]. This method has been used to derive a lower dimensional ODE model for a nonlinear parabolic PDE with parameter uncertainties, for which the problem of designing robust controllers has been studied [11]–[13]. In addition, it has been used to determine low order approximations for spectral systems with guaranteed error bounds and to check the stability of feedback spectral systems [14], [15]. If the boundary conditions are too complicated to be used to obtain the eigenfunctions of the associated linear operator, then the techniques of proper orthogonal decomposition (POD) may be used to extract empirical eigenfunctions (EEFs) instead. Afterwards, lower dimensional ODE
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Fig. 2. Simplified configuration of the curing system in Cartesian space.
Fig. 1. Spectral approximation based modeling for the curing process.
systems can still be derived with modal decomposition techniques [16]–[18]. If the mathematical model of the DPS is totally unknown, then a lower order approximation can be derived for a linear DPS with the input–output data using the singular value decomposition (SVD) method [19], and for a nonlinear DPS using POD techniques [20] or in combination with SVD [10]. As only input–output data are used, the accuracy of the model is limited by the availability of sensors and actuators due to a limited number of EEFs obtained. Besides, the snapshots should be representative and the measurement time interval has to be carefully chosen for a proper modeling based on EEFs [16]. As there are only a finite number of sensors and actuators available in practice, it is unclear how these data-driven methods work with systems of higher dimensional spatial natures. Furthermore, all of the data-driven methods mentioned are only applicable offline. In practice, the process is nether completely known or totally unknown. Use of existing knowledge will make modeling easier and more accurate. Unlike the modeling approaches of DPSs mentioned previously, this paper proposes a simple but effective modeling method for snap curing processing by integrating spectral methods [11], [21] with neural networks (NNs). Based on the partial knowledge of the process, the spectral methods that are shown in Fig. 1 are used for time–space separation and model reduction, and NNs are used for state estimation and system identification. The dynamics of the snap curing process is established according to physical laws and knowledge of the process. Based on the dynamics of the curing process, a lower order nonlinear ODE model with uncertain parameters and unknown nonlinearities is then derived using model reduction techniques. Thereafter, a NN-based observer is designed to estimate the states of the ODE model using selected measurements. Finally, a hybrid general regression neural network (GRNN) is trained as a nonlinear model of the process in state–space formulation. The proposed method emphasises five
key points: an engineering-preferred model in reduced order is obtained with the help of the spectral approximation and partial knowledge of the process; intelligence-based methods are used to deal with the unknown nonlinearities and unmodeled dynamics; only a finite number of measurements are required; the developed model is of a time-spatial distributed nature that can satisfactorily describe the temperature distribution in the curing process; and the modeling method is applicable either offline or online. Using the established model, the temperature distribution related fundamental analyses, such as simulation, controller design, fault diagnosis and supervision, can be carried out for the curing process. II. CURING PROCESS AND ITS SPECTRAL MODELLING A. Fundamental Dynamics of the Curing Process After adhesive die attach, the adhesive is cured in a curing oven. Die attach is the process whereby the chip (die) is attached to the die pad or die cavity of the leadframe. Epoxy die attach uses epoxy to mount the die on the die pad or cavity. The epoxy is then cured at a specified temperature in a curing oven to form a tight bond between the die and the leadframe. A simplified configuration of the snap curing oven is shown in Fig. 2, with its data acquisition and control system shown in Fig. 3. The oven chamber has four heaters for heating and four thermocouples for temperature sensing. The parts to be cured are moved in and out from the inlet and outlet. For fundamental analysis and better control, a model for estimating the temperature distribution inside the chamber is required. As the temperature of the epoxy between the die and the leadframe is of interest, an energy balance of the epoxy may be constructed to analyze the system, which would make the whole analysis simpler. However, it is evident that the temperature distribution is not consistent inside the oven chamber. Thus, even though an energy balance of the epoxy can be constructed, it is still difficult to use because the boundary conditions of the epoxy (the temperatures around the epoxy) are unknown. The leadframe usually contains many chips (dies). In practice, it is difficult to place many sensors to measure the temperature around the epoxy. These are the main challenges and the main motivation for building a model of the oven and then using the model to estimate the temperature distribution of the curing process. For the kind of curing process considered, the leadframe is placed under the heaters as indicated in Figs. 2 and 3. The volume of the epoxy between
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Fig. 3. Data acquisition and control system of the curing process.
a die and the leadframe is much smaller than that of the leadframe and of the oven chamber. The volume of the leadframe is also much smaller than the volume of the oven chamber. Thus, the effects of the epoxy and the leadframe on the temperature in the oven chamber are usually neglected in the modeling of the curing process. These effects will be considered as disturbances and will be compensated for during the control process of the temperature distribution. According to heat transfer laws [22], the fundamental heat transfer equation of the oven can be expressed as follows:
(2.1)
denotes the vector of manipulated inputs, with explanation
of
is a nonlinear function of which will be given later,
—the
with is the spatial distribution of the control action denotes measured outputs in spatial coordinates is the initial temperature distribution in the oven and with being associated with the efficiency and the power of the th heater, which is usually inaccurately given. Around a particular working point (i.e., 180 C), the specific heat of dry air under the atmospheric pressure can be considered to be constant. However, the thermal conductivity and the density are dependent on the temperature, which can be expressed as and
where density (kg/m ); specific heat (J/kg C); thermal conductivity (W/m C); temperature ( C) at time and loca; tion , spatial coordinates. and and are the effects of convection and In this model, represents the disturbances such radiation, as the effects of the moving parts inside the chamber and heat leaking from oven walls,
where and are nominal values around a working point, and and are functions of . As shown in Fig. 3, thyristors are used to modulate the input power, which means that there is a nonlinear static relationship and between the input pulsewidth modulation (PWM) ratio represents the power delivered to the chamber. Here, the nonlinear relationship, which can be divided into a linear and a nonlinear part , that is, part . As the curing temperature of the epoxy that we consider is around 180 C, which is different from the RTP systems,
DENG et al.: SPECTRAL-APPROXIMATION-BASED INTELLIGENT MODELING FOR DISTRIBUTED THERMAL PROCESSES
in which operating temperatures can be up to 1100 C, and the temperature difference is not very large during the curing process, the radiative heat transfer is not significant but cannot be neglected due to the consideration of modeling accuracy. The calculation of radiation is very difficult even within simple enclosures [18]. Because this term is a function of the temperature to the power of 4, it is treated as an unknown nonlinearity. It is well known that convection is conduction and/or radiation in a moving continuum [23]. In the oven chamber, the moving of the air is mainly due to its density difference, which is caused by temperature difference. However, as the leadframe is placed very close to the heaters and the chamber is quite small, around an operating temperature, the temperature difference in the chamber is not so large that the convection will dominate the heat transfer in the chamber. Because an analytical procedure to solve the problem of heat convection is quite complex and laborious, even for simple geometrical bodies and boundary conditions [22], the convection is treated as an unknown function of the temperature as radiation. Thus, one can approximately assume that the main mode of heat transfer is conduction in the curing process and take the other effects as unmodeled dynamics (unknown nonlinearities), leaving them for neural compensation. In terms of the previous analyses, (2.1) can be rewritten as
(2.2) where
and
are constants is the Laplacian operator
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where
with being unknown nonlinear functions. Thus, a complete model of the oven is described by a parabolic partial differential (2.2) with the boundary conditions (2.3a)–(2.3c). B. Model Reduction Analysis Due to the existing unknown nonlinearities, unknown parameters, and limited number of actuators and sensors, the oven model described by parabolic partial differential (2.2) and (2.3) is not applicable. A lower order ODE system is preferred not only for ease of controller design but also for accurate simulation and practical implementation. In this section, one of the model reduction techniques, the spectral methods, are used to reduce models (2.2) and (2.3) to a set of finitely many nonlinear ODEs. This lower order ODE system is then used as the framework for process identification and modeling. A parabolic PDE system typically involves spatial differential operators with eigenspectra that can be partitioned into a finite-dimensional (slow) and an infinite-dimensional (fast) complements [24]. This implies that the dynamical behavior of such a system can be approximately described by a finite-dimensional ODE system that captures the dynamics of the dominant (slow) modes of the PDE system. It is very convenient and effective to use the eigenfunctions of a spatial differential operator to derive a lower order ODE system for such a parabolic PDE system if the boundary conditions are homogeneous. However, the boundary conditions (2.3a)–(2.3c) include unknown parameters and nonlinear functions; therefore, a direct derivation of is impossible. In this eigenfunctions for the operator case, orthogonal polynomials can be used as basis functions instead of eigenfunctions. In doing so, however, the derived ODE system may have very high dimensions. Thus, models (2.2) and (2.3) are transformed into the following equivalent model, with a proof given in the Appendix
(2.4) The chamber walls of the oven are usually not well-insulated and the boundary conditions are not completely known. The free convection and radiation cause heat leakage from the exterior walls of the oven. As it is difficult to obtain reasonable boundary conditions using only physical insights, for simplicity one can set the boundary conditions to be unknown nonlinear functions of the boundary temperature, the space coordinates of the oven , and the ambient temperature , as follows:
(2.5a) (2.5b) (2.5c) where
(2.3a) (2.3b) (2.3c)
and
is the Dirac delta function.
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Now, the new boundary conditions (2.5a)–(2.5c) are homogeneous, although the system (2.4)–(2.5) is an infinite-dimensional nonlinear system. To further reduce it to a set of finitely many nonlinear ODE equations, the Galerkin method [11] can be applied as follows. 1) Separation of Time and Spatial Variables: According to the theorem of separation of variables [25], the solution of system (2.4)–(2.5) can be expressed in the following time–space decoupled form:
in which is rearranged in order of magnitude and then dewith being its correnoted as a new sequence of sponding eigenfunction, as follows: with
(2.6) Thus, the eigenvalue problem for operator be solved analytically [26], as
in (2.4) can
Then, (2.10) can be rewritten in a general nonlinear form as follows: (2.11a) (2.11b) where (2.7) where , and denote eigenvalues, and , denote the corresponding eigenfunctions. and Next, the solution of system (2.4)–(2.5) is expressed in an orthogonally decoupled series (2.8) where sion coefficients associated with
.. .
are expan. The residual and
is minimized with (2.9) . where is the domain , and , Due to the orthogonality of the eigenfunctions the integration of (2.9) after substituting (2.8) into (2.9) will give
(2.10a)
where
(2.10b)
represents nonlinearities and unmodeled dynamics. 2) Separation of Slow and Fast Modes: To reduce the infinite-dimensional nonlinear ODE system (2.11) to a finite set of nonlinear ODE equations, the following assumption is made in (2.4), where for the eigenspectrum of the operator is defined as the set of all eigenvalues of , i.e., . Assumption 1: [27] 1) , where denotes the real part of . can be partitioned as , 2) consists of the first (with fiwhere , and nite) eigenvalues, i.e., . and , where 3) is a small positive parameter. Based on Assumption 1, one can define and , which are the modal subspaces of . By choosing the and such that orthogonal projection operators and , and by decomposing the state of the system
DENG et al.: SPECTRAL-APPROXIMATION-BASED INTELLIGENT MODELING FOR DISTRIBUTED THERMAL PROCESSES
(2.11) as be equivalently expressed as
, the system (2.11) can (2.12a) (2.12b) (2.12c)
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, set consisting of slow eigenvalues, and a stable infinite set containing the remaining fast eigen. Thus, the dominant dynamics of values, the nonlinear parabolic PDE system can be described with only a few dominant modes [9]. The coefficients solved from (2.13a), , forms the temperature distribution (2.13c)
where diagonal matrix with dimension and in the form of ; unbounded differential operator that generates a strongly continuous exponentially stable semigroup [11]; ; and Lipschitz vector functions; ; denotes that the state is in an infinitedimensional space. After neglecting the fast modes in (2.12), the following finitedimensional nonlinear system is obtained: (2.13a) (2.13b) Remark: It is interesting to note that the eigenspectrum of nonlinear PDEs (2.2) with boundary conditions (2.3a)–(2.3c) can be separated into a finite-dimensional (slow) and an infinitedimensional stable (fast) complement based on the separation of in (2.4). This is because the spectrum of the linear operator the dynamical behavior of the system can be approximately described by a finite-dimensional ODE system that captures the dynamics of the dominant (slow) modes of the PDE system. For , and , the eigenfunctions derived from (2.7), , form a complete orthogonal basis. Thus, the solution of the nonlinear system can be expressed in terms of the basis as functions in the form of (2.8) with , and the expansion coefficients that are associated with . If the nonlinear system has the properties that are shown in Assumption 1, then its dynamics can be separated into a slow part and a stable fast part, and the dominant (slow) dynamics can be used to approximate the nonlinear system. Such nonlinear parabolic PDE systems arise naturally from the modeling of processes such as diffusion, heat conduction or viscosity, a class that encompasses reaction-diffusion systems, RTP units and many other systems of engineering interest [5], [9], [11], [28]. If a nonlinear parabolic PDE model is known, then this method is frequently used to derive a lower dimensional ODE model, and a controller for the nonlinear parabolic PDE model can be designed based only on the lower dimensional ODE model [9], [11], [13], [28]–[30]. The curing process is such a nonlinear parabolic PDE process that the method can be used to derive a finite-dimensional ODE system which captures the dynamics of the dominant slow modes of the curing process, as discussed previously. 3) Synthesis of Time and Spatial Variables: Assumption 1 states that the eigenspectrum of can be partitioned into a finite
4) Order Selection of the Nonlinear ODE System: So far, a lower order nonlinear ODE system (2.13a) and (2.13b) has been derived by using the Galerkin method for the approximation of the parabolic DPS (2.4) with boundary conditions (2.5a)–(2.5c). To determine the order of the ODE system, a tradeoff between the order and the modeling accuracy must be considered. As the exact model of the snap curing process is unknown, it is difficult to estimate the truncated error when deriving a lower order nonlinear ODE system for the parabolic DPS. The higher the accuracy required, the higher the order of the ODE system must be used. The following method is proposed to determine a suitable order for the derived nonlinear ODE system. First, Assumption 1 is used to check whether the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow and an infinite-dimensional stable fast complement. Second, a small ratio factor, ( 0 for the curing process) is used for the order determina0.1 (i.e., is 10 times ) is chosen, tion. For example, if (which is supposed which will lead to a particular value of to be 16) for the curing process (2.13), then an th order nonlinear ODE system for the process can be derived by following . the previously discussed Galerkin method, usually with 5) Discretization of the Nonlinear ODE System: For practical implementation, a discrete-time model is often used. The previous reduced model is now discretised by the Euler forward formula as follows:
(2.14a) (2.14b) with being the sampling time. with , choose For simplicity, one can replace and around an operating point, and express as with being the estimate of and being the unknown parameter vector. The approximation of (2.14) around a working point can then be developed as
(2.15a) (2.15b) with
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Fig. 4.
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 5, SEPTEMBER 2005
Spectral based model reduction procedure for the curing process.
The constant matrices , and are determined by the nominal values of the snap curing system, and and represent the uncertain parameters and unknown nonlinearities. Based on the previous analyses, the coefficients can be obtained by solving (2.15a), from which the temperature distribution at the th step in the oven can be estimated by
and being a normalized with Gaussian radial basis function [32]. This is actually a recurrent NN observer, as shown in Fig. 6, and output of the oven. Here, is with the input the state vector of the NN, is a constant matrix that is designed stable. When and are adjusted to make the matrix is the estimate of . by updating laws, 1) Learning Algorithm: The back propagation through time or the real time recurrent learning algorithms [31], [33] are widely used for the training of recurrent NNs. However, the gradient calculations used in the algorithm may become time-consuming and unstable [34]. The online NN learning requires both convergence speed and stability. For single-input single-output nonlinear systems, an adaptive learning algorithm [35] has been used to train RBF NNs for input–output system identification based on Lyapunov stability theory. This adaptive learning algorithm can be modified for multi-input–multi-output (MIMO) systems and used for the online training of the proposed recurrent NN observer, as follows. and are The updating laws for (3.2) (3.3) with (3.4)
(2.16) where The spectral method based model reduction procedure for the curing process is summarized in Fig. 4. III. HYBRID SPECTRAL/NEURAL MODELLING METHODOLOGY The nonlinear ODE model (2.15) is still not applicable because of the uncertain parameters and unknown nonlinearities . The states in (2.6) cannot be measured because only a finite number of sensors are available. To obtain an applicable model for the snap curing process, a hybrid spectral/neural modeling method is proposed, as shown in Fig. 5, where a neural-network observer is first used to estimate the coof (2.15) with the help of the spectral method, efficient and a hybrid GRNN is employed to establish the dynamic relaand the coefficient . This tionship between the input hybrid spectral/neural model can provide a good estimation of the temperature distribution.
and are two constants. To avoid singularities, the updating laws for modified as follows:
and
are (3.5) (3.6)
where
A. Spectral Method Based Neural Observer The structured recurrent radial basis function (RBF) NN [31], in combination with the available knowledge about the oven , and , is used to estimate of (2.15a) in the such as following form:
(3.1a) (3.1b)
with and being two small constants that are determined by Lyapunov stability theory. 2) Stability Analysis: It can be proved that for the NN will observer (3.1), the output error and with updating converge to zero when and , respectively, and the laws (3.2) and (3.3) for output error will converge to a bounded area of radius
DENG et al.: SPECTRAL-APPROXIMATION-BASED INTELLIGENT MODELING FOR DISTRIBUTED THERMAL PROCESSES
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Fig. 5. Hybrid spectral/NN modeling method for the curing process.
are bounded if the and are the errors bounded. The analysis of the stability is given in the following. Suppose that the system (2.15a) and (2.15b) can be described as the following recurrent NN: (3.7a) (3.7b) is an optimal parameter vector, is an optimal where weight matrix, is the modeling error and is a normalized activation vector. The neural-network observer for system (3.7) is in the form of (3.1a) and (3.1b). Let the estimation error be denoted by . Then Fig. 6.
Structure of the NN observer.
when and with and , respectively. A updating laws (3.5) and (3.6) for detailed proof is provided in the Appendix. Stability analysis is difficult for the observer because the structured recurrent NN is not linear in its parameters [31]. However, with the previous learning algorithm and the constant matrix that stabilises the matrix - , one can ensure that
(3.8) with
As
-
, and
and
are bounded, (normalized activation) and
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Fig. 8. Sensor locations for modeling and verification. (The shaded sensors will be used for modeling.) Fig. 7. Hybrid spectral/GRNN model for the curing process.
(normalized inputs), by letting and be bounded during the learning process of the NN observer [31], one must in (3.8), with being a bounded positive have number. Because is stable and is bounded, from (3.8) one can conclude that the estimation error is bounded. B. Spectral Based NN Model After the state estimation is completed, a hybrid spectral based GRNN can be used to model the snap curing process, as shown in Fig. 7. The hybrid GRNN is expressed as follows: (3.9) where
, and
are weights of the hybrid GRNN and , with being a normalized activation
function. This hybrid GRNN can be rewritten in the following compact form:
with
and
Evolved from the RBF network, the GRNN can improve function approximation with a minimal number of weights [32]. The recursive least square (RLS) algorithm [36] is used to guarantee a fast convergence as follows:
The hybrid GRNN can be trained either offline or online. After training, it establishes the dynamic relationship between the and the expansion coefficients , from which the input temperature distribution is developed using (2.16).
IV. REAL-TIME EXPERIMENT FOR MODEL IDENTIFICATION AND VERIFICATION A. Input–Output Considerations 1) Sensor Locations: Six extra sensors (s3–s6, s9, s10) are placed in the oven, as shown in Fig. 8, for model identification in the neural-network and verification. The matrix pair observer (3.1) must be observable. As the matrix depends on the sensor locations, the observability may not be guaranteed if the sensor locations are taken too arbitrarily. Thus, thermocouples (s1, s2, s3, s4) are chosen for modeling the curing process. These selected locations can meet the observable condition and be representative of the heating region of the oven. The rest sensors (s5–s10) are used to evaluate the modeling performance. 2) Persistent Excitation of Input Signal: It is the power delivered to the chamber that changes the temperature in the chamber. Hence, the quantity of the power inputted to the chamber should be carefully controlled. Heaters of the curing system are controlled with on/off relays, which generate PWM input signals at only two levels (0 or 1). However, the pseudorandom binary signal (PRBS) is inappropriate to identify the nonlinear process because its two levels cannot excite the system dynamics sufficiently [36]. To overcome this difficulty, multilevel signals should be designed. However, the power that is delivered to the chamber is controlled by the input PWM signals, and one cannot directly give multilevel power inputs by using the present equipment in the curing process. As the quantity of the power inputted to the chamber is proportional to the input PWM ratio and it is convenient to change that ratio, one can generate multilevel PWM ratio signals instead of multilevel power inputs. Fig. 9 shows the PWM input in one sampling with impulse period, where the period is assumed to be signals of period and the on-duration . The percentage of the on-period is defined as PWM ratio , with for . This ratio factor is randomised with the required mean, which is set at 0.13 in our case. Thus, the power signals to the heaters can be generated indirectly by letting be random within [0, 1]. Due to the equivalence between the PWM ratio and the power delivered to the chamber in each , the multilevel PWM ratio signals should sampling period excite the curing process sufficiently. The experimental results have confirmed this analysis. The multilevel PWM ratio signals (first 500 samples) that are used to identify the nonlinear curing process for Heater 1 are shown in Fig. 10.
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Fig. 9. PWM impulse signals.
Fig. 10.
Fig. 11.
Performance of the NN model at sensor location s1 with training data.
Fig. 12.
Performance of the NN model at sensor location s1 with test data.
Multilevel ratios of PWM for Heater 1.
B. Experiments and Analysis Based on the method proposed in Section III, a 32-order hybrid GRNN model is obtained and compared with a 32-order linear state–space model. A total of 1500 measurements are collected with a sampling time of 10 s around 180 C. The modeling and verification are carried out in the following three steps. Step 1) Modeling of the training data from the measured locations One thousand records of the measurements from sensors (s1, s2, s3, and s4) are used to train the hybrid GRNN. The result given in Table I, and for sensor location s1 in Fig. 11, shows that the developed hybrid spectral/neural model can provide a very good estimation after training. Step 2) Model verification of the untrained data from the measured locations The last 500 records of measurements from sensors (s1, s2, s3, and s4) are used to test the effectiveness of the model for untrained data. The good performance is maintained, as shown in Table I and in Fig. 12 for sensor location s1. Step 3) Model verification of the un-measured locations (untrained) All 1500 records of measurements at sensors (s5, s6, s7, s8, s9, and s10) are used to evaluate the
estimation performance of the developed model for the temperature distribution of the oven. A good performance is achieved, as shown in Table II and in Fig. 13 for sensor location s5. The temperature distribution in the oven chamber can be . Under working given based on the estimated coefficients conditions around 180 C, the temperature distribution at 0.025 m (where the leadframe is placed) inside the height 8000 s is provided by the NN model, oven chamber when as shown in Fig. 14. 1) Comparison: For comparison, a linear model of the curing process is also developed by neglecting the nonlinear 0 in (3.9), and then using the RLS part, i.e., letting
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Fig. 14.
Temperature distribution at t
= 8000 s.
V. CONCLUSION Fig. 13.
Performance of the NN model at sensor location s5 with test data. TABLE I PERFORMANCE COMPARISON BY RMSE
TABLE II RMSES AT SENSOR LOCATIONS S5–S10
algorithm for learning. The root of mean square error (RMSE) is defined as the performance criterion RMSE where is the data number, and and are temperature measurements and the model estimation. Tables I and II compare the performances of the two different models in terms of RMSE. It is apparent that the hybrid NN model has much better performance than the linear model with the same order, and is therefore more suitable for the nonlinear curing process.
A spectral approximation based intelligent modeling approach has been developed for the distributed thermal processing of the snap curing oven that is used in the semiconductor packaging industry. The snap curing oven is described by a nonlinear parabolic DPS in the time–space domain. After finding a proper approximation of complex boundary conditions of the system, the spectral method is applied to time–space separation and model reduction, and NNs are used for state estimation and system identification. With the help of model reduction techniques, the dynamics of the curing process derived from the physical laws are described by a model of low-order nonlinear ODEs with a few uncertain parameters and unknown nonlinearities. A neural-network observer is then designed to estimate the states of the ODE model from measurements taken at specified locations in the field. Using the estimated states, a hybrid general regression NN is trained to be a nonlinear model of the curing process in a state–space formulation. Real-time experiments on the snap curing oven have shown that the proposed modeling method is indeed effective. This modeling methodology can be applied to a class of nonlinear DPSs in industrial thermal processing. Further studies are underway for the optimal control design and fault diagnosis of the oven using the process model described here.
APPENDIX A EQUIVALENCE PROOF OF (2.4)–(2.5c) AND (2.2)–(2.3c) Using the solution (2.8) for (2.4) with boundary conditions (2.5a)–(2.5c), the integration (2.9) becomes
(A1.1)
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where
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and
Note that (A1.1) can be rewritten as
Substituting the previous equation into (A1.1), one has
The integration of the first term of the right-hand side in the previous equation is
(A1.2) On the other hand, using the solution (2.8) for (2.2) with boundary conditions (2.3a)–(2.3c), the integration (2.9) gives
Using the boundary conditions (2.5a)–(2.5c), the previous equation becomes
(A1.3) Note also that (A1.3) can be rewritten as
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The integration of the first term on the right-hand side of the previous equation is
APPENDIX B PROOF OF CONVERGENCE FOR THE UPDATING LAWS (3.2), (3.3) AND (3.4), (3.5) . Define 1) For the updating laws (3.2) and (3.3), one has
Using the boundary conditions (2.3a)–(2.3c), the previous equation becomes
where
Define a Lyapunov function
2)
Substituting the previous equation into (A1.3), one obtains the same result as (A1.2). Thus, one can conclude that the (2.4) with boundary conditions (2.5a)–(2.5c) is equivalent to the (2.2) with boundary conditions (2.3a)–(2.3c).
. Then
Thus, when . Therefore, the output error will converge to zero. For the updating laws (3.5) and (3.6), one has
where:
and .
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Define a Lyapunov function
In this case, since one has
Thus,
. Then
[35] and
,
when
and , where . Consequently, the error will converge to the ball centred at the origin of the error space with radius . This completes the proof. ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers and the associate editor for their valuable comments. REFERENCES [1] T. Soderstrom and P. Stoica, System Identification. Englewood Cliffs, NJ: Prentice-Hall, 1989. [2] H. Aling, R. L. Kosut, A. Emami-Naeini, and J. L. Ebert, “Nonlinear model reduction with application to rapid thermal processing,” in Proc. 35th IEEE Conf. Decision and Control, vol. 4, 1996, pp. 4305–4310. [3] S. Banerjee, J. V. Cole, and K. F. Jensen, “Nonlinear model reduction strategies for rapid thermal processing,” IEEE Trans. Semicond. Manuf., vol. 11, no. 2, pp. 266–275, 1998. [4] J. L. Ebert, G. W. van der Linden, R. L. Kosut, and A. Emami-Naeini, “Efficient CFD modeling of single wafer semiconductor fabrication systems for closed-loop evaluation,” in Proc. 36th IEEE Conf. Decision Control, vol. 1, 1997, pp. 830–831. [5] A. Emami-Naeini, J. L. Ebert, D. de Roover, R. L. Kosut, M. Dettori, L. M. L. Porter, and S. Ghosal, “Modeling and control of distributed thermal systems,” IEEE Trans. Contr. Syst. Technol., vol. 11, no. 5, pp. 668–683, Sep. 2003. [6] R. L. Kosut and M. G. Kabuli, “Robust control of thermal processes,” in Proc. 2nd Int. Rapid Thermal Processing Conf., 1994, pp. 296–297. [7] C. A. Lin and Y. L. Jan, “Control system design for a rapid thermal processing system,” IEEE Trans. Contr. Syst. Technol., vol. 9, no. 1, pp. 122–129, Jan. 2001. [8] K. M. Tao, R. L. Kosut, M. Ekblad, and G. Aral, “Feedforward learning applied to RTP of semiconductor wafers,” in Proc. 33rd IEEE Conf. Decision and Control, vol. 1, Lake Buena Vista, FL, 1994, pp. 67–72. [9] W. H. Ray, Advanced Process Control. New York: McGraw-Hill, 1981.
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[10] D. Zheng and K. A. Hoo, “Low-order model identification for implementable control solutions of distributed parameter system,” Comput. Chem. Eng., vol. 26, pp. 1049–1076, 2002. [11] P. D. Christofides, Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. Boston, MA: Birkhauser, 2001. , “Robust control of parabolic PDE systems,” Chem. Eng. Sci., vol. [12] 53, pp. 2449–2465, 1998. [13] P. D. Christofides and J. Baker, “Robust output feedback control of quasi-linear parabolic PDE systems,” Syst. Contr. Lett., vol. 36, pp. 307–316, 1999. [14] M. A. Erickson, R. S. Smith, and A. J. Laub, “Finite-dimensional approximation and error bounds for spectral systems with partially known eigenstructure,” IEEE Trans. Autom. Control, vol. 39, no. 9, pp. 1904–1909, Sep. 1994. [15] M. A. Erickson and A. J. Laub, “An algorithmic test for checking stability of feedback spectral systems,” Automatica, vol. 31, no. 1, pp. 125–135, 1995. [16] H. M. Park and D. H. Cho, “The use of the Karhunen-Loeve decomposition for the modeling of distributed parameter systems,” Chem. Eng. Sci., vol. 51, pp. 81–98, 1996. [17] R. A. Sahan, N. Koc-Sahan, D. C. Albin, and A. Liakopoulos, “Artificial neural network-based modeling and intelligent control of transitional flows,” in Proc. IEEE Int. Conf. Control Applications, Hartford, CT, 1997, pp. 359–364. [18] A. Theodoropoulou, R. A. Adomaitis, and E. Zafiriou, “Model reduction for optimization of rapid thermal chemical vapor deposition systems,” IEEE Trans. Semicond. Manuf., vol. 11, no. 1, pp. 85–98, Feb. 1998. [19] D. H. Gay and W. H. Ray, “Identification and control of distributed parameter systems by means of the singular value decomposition,” Chem. Eng. Sci., vol. 50, pp. 1519–1539, 1995. [20] K. A. Hoo and D. Zheng, “Low-order control-relevant models for a class of distributed parameter systems,” Chem. Eng. Sci., vol. 56, pp. 6683–6710, 2001. [21] D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications. Philadelphia, PA: SIAM, 1993. [22] A. J. Chapman, Fundamentals of Heat Transfer. New York: MacMillan, 1987. [23] V. S. Arpaci and P. S. Larsen, Convection Heat Transfer. Englewood Cliffs, NJ: Prentice-Hall, 1984. [24] B. Friedman, Partial Differential Equations. New York: Holt, Rinehart, 1976. [25] C. R. MacCluer, Boundary Value Problems and Orthogonal ExpansionsPhysical Problems from a Sobolev Viewpoint. Piscataway, NJ: IEEE Press, 1994. [26] D. L. Powers, Boundary Value Problems, 4th ed. Orlando, FL: Harcourt, 1999. [27] P. D. Christofides and P. Daoutidis, “Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds,” J. Math. Anal. Appl., vol. 216, pp. 398–420, 1997. [28] S. Y. Shvartsman and I. G. Kevrekidis, “Nonlinear model reduction for control of distributed systems: A computer-assisted study,” AICHE J., vol. 44, no. 7, pp. 1579–1595, 1998. [29] C. Antoniades and P. D. Christofides, “Integrated robust control and optimal actuator/sensor placement for uncertain transport-reaction processes,” Comp. Chem. Eng., vol. 26, pp. 187–203, 2002. [30] Y. Lou and P. D. Christofides, “Optimal actuator/sensor placement for nonlinear control of the Kuramoto-Sivashinsky equation,” IEEE Trans. Contr. Syst. Technol., vol. 11, no. 5, pp. 737–745, Sep. 2003. [31] D. Schroder, C. Hintz, and M. Rau, “Intelligence modeling, observation, and control for nonlinear systems,” IEEE/ASME Trans. Mechatronics, vol. 6, no. 2, pp. 122–131, Jun. 2001. [32] D. Schroder, Intelligence Observer and Control Design for Nonlinear Systems. Berlin, Germany: Springer-Verlag, 2000. [33] K. S. Narendra and K. Parthasarthy, “Gradient methods for the optimization of dynamical systems containing neural networks,” IEEE Trans. Neural Netw., vol. 2, no. 2, pp. 252–262, Mar. 1991. [34] L. Jin and M. M. Gupta, “Stable dynamic backpropagation learning in recurrent neural networks,” IEEE Trans. Neural Netw., vol. 10, no. 6, pp. 1321–1334, Nov. 1999. [35] K. P. Seng, Z. Man, and H. R. Wu, “Lyapunov-theory-based radial basis function networks for adaptive filtering,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 8, pp. 1215–1220, Aug. 2002. [36] O. Nelles, Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models. Berlin, Germany: SpringerVerlag, 2001.
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Hua Deng received the B.Eng. degree from Nanjing Aeronautical Institute, Nanjing, China, in 1983, the M.Eng. degree from Northwestern Polytechnical University, Xi’an, China, in 1988, and the Ph.D. degree from the City University of Hong Kong, Hong Kong, in 2005. He was an Associate Professor at Changsha University of Science and Technology, China. He is currently a Professor in the School of Mechanical and Electrical Engineering, Central South University, Changsha, China. His research interests include modeling and control of complex dynamic systems, intelligent control, fault diagnoses, and computer-aided control system design.
Han-Xiong Li (S’94–M’97–SM’00) received the B.E. degree from the National University of Defence Technology, Changsha, China, in 1982, the M.E. degree in electrical engineering from Delft University of Technology, Delft, The Netherlands, in 1991, and the Ph.D. degree in electrical engineering from University of Auckland, Auckland, New Zealand, in 1997. He is currently an Associate Professor in the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong. He is holding “Lotus Scholar” in Central South University-an honorary professor endowed by Ministry of Hunan Province, China. He was awarded by the National Science Foundation for (overseas) distinguished young scholar in 2004. In the last twenty years, he has had opportunities to work in different fields including military service, industry, and academia. He has gained industrial experience in IC packaging as a senior process engineer for die-bonding and dispensing from ASM—a leading supplier for semiconductor process equipment. His research interests include modeling and intelligent control for complex industrial process with special interest to electronic packaging. Dr. Li serves as an Associate Editor for the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B.
Guanrong (Ron) Chen (M’89–SM’92–F’97) received the M.Sc. degree in computer science from Zhongshan University, China, in 1981 and the Ph.D. degree in applied mathematics from Texas A&M University, College Station, in 1987. After working at Rice University and the University of Houston for 15 years, he currently is a Chair Professor in the Department of Electronic Engineering, City University of Hong Kong. He was named IEEE Fellow in 1996, conferred more than 10 Honorary Guest-Chair Professorships from China and particularly an Honorary Professorship from the Central Queensland University of Australia. He is Chief, Advisory, Feature, and Associate Editors for eight IEEE Transactions and International Journals. Prof. Chen has (co)authored 15 research monographs and advanced textbooks, more than 300 journal papers, and about 180 conference papers, published since 1981 in the field of nonlinear systems dynamics and controls, with applications to Internet technology as well as intelligent control systems. He received the outstanding prize for the best journal paper award from the American Society of Engineering Education in 1998, the best transactions annual paper award from the IEEE Aerospace and Electronic Systems Society in 2001, and the best journal paper award from the Czech Academy of Sciences in 2002.
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 13, NO. 5, SEPTEMBER 2005
Spectral-Approximation-Based Intelligent Modeling for Distributed Thermal Processes Hua Deng, Han-Xiong Li, Senior Member, IEEE, and Guanrong Chen, Fellow, IEEE
Abstract—A spectral-approximation-based intelligent modeling approach is proposed for the distributed thermal processing of the snap curing oven that is used in semiconductor packaging industry. The snap curing oven can be described by a nonlinear parabolic distributed parameter system (DPS) in the time–space domain. After finding a proper approximation of the complex boundary conditions of the system, the spectral methods can be applied to time–space separation and model reduction, and neural networks (NNs) can be used for state estimation and system identification. With the help of model reduction techniques, the dynamics of the curing process derived from physical laws can be described by a model of low-order nonlinear ordinary differential equations with a few uncertain parameters and unknown nonlinearities. A neural observer can then be designed to estimate the states of the ordinary differential equation model from measurements taken at specified locations in the field. Using the estimated states, a hybrid general regression NN is trained to be a nonlinear model of the curing process in state–space formulation, which is suitable for the further application of traditional control techniques. Real-time experiments on the snap curing oven show that the proposed modeling method is effective. This modeling methodology can be applied to a class of nonlinear DPSs in industrial thermal processing. Index Terms—Curing process, distributed thermal process, neural networks (NNs), nonlinear distributed parameter systems (DPSs), spatial system identification, spectral methods, state estimation.
I. INTRODUCTION
T
O PRODUCE high-quality semiconductor devices, increasingly complex processes are used more and more in the semiconductor back-end packaging industry. After epoxy die attach, the bonded leadframe is cured in a snap curing oven, which provides the required curing temperature distribution. Traditional proportional-integral-derivative (PID) controllers often fail to maintain a consistent temperature distribution due to the complex time-spatial nature of the process and external disturbances. It is important for the industry to obtain a good model of the process at a reasonable cost and accuracy for funManuscript received June 19, 2003; revised April 30, 2004. Manuscript received in final form January 7, 2005. Recommended by Associate Editor G. Rosen. This work was supported in part by the RGC of Hong Kong SAR under Project CityU 1129/03E and in part by the China National Science Fund for Distinguished Young Scholars under Grant 50429501. H. Deng is with the School of Mechanical and Electrical Engineering, Central South University, Changsha, China, and also with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong. H.-X. Li is with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]). G. Chen is with the Department of Electronic Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong. Digital Object Identifier 10.1109/TCST.2005.847329
damental analysis and control-related issues including design, simulation, diagnosis, and supervision. The snap curing process is one kind of thermal process that follows the basic principles of heat transfer (conduction, radiation, and convection). Due to the complex boundary conditions and external disturbances, it is very difficult to obtain good models based on physical insights alone [1]. Even if this kind of structure can be derived from physical laws, the model will still contain many unknown parameters. The modeling issue in semiconductor thermal processing (i.e., rapid thermal processing (RTP) systems) has been seriously considered, and control-related problems have also been studied [2]–[8]. However, the basic requirement for the aforementioned control methods is that a known physical model of the RTP must be known. The radiative heat transfer is dominant for the RTP systems [5]. This is because the operating temperatures of RTP systems can be up to 1100 C. However, the curing temperatures of the epoxy considered are around 180 C and, thus, the radiative heat transfer is not dominant. Mathematically, the curing process belongs to nonlinear parabolic distributed parameter systems (DPSs) described by partial differential equations (PDEs) arising from heat, mass, and momentum balances [9]. A general mathematical description for DPSs usually consists of PDEs with mixed or homogeneous boundary conditions. The main characteristic of a DPS is that the outputs, inputs, process states and parameters can vary temporally and spatially. Controlling a DPS is usually accomplished by truncating the PDE with boundary conditions to a system of ordinary differential equations (ODEs) that permit the synthesis of a controller [10]. This transformation may result in high-order solutions of PDEs, which usually mean that a controller synthesis is not implementable due to such physical limitations as a finite number of sensors and actuators. If the mathematical model of a parabolic PDE is completely known, with mixed or homogeneous boundary conditions, then the resulting nonlinear DPS might be approximated by a lower dimensional ODE system through modal decomposition using the eigenfunctions of the associated linear operator [9], [11]. This method has been used to derive a lower dimensional ODE model for a nonlinear parabolic PDE with parameter uncertainties, for which the problem of designing robust controllers has been studied [11]–[13]. In addition, it has been used to determine low order approximations for spectral systems with guaranteed error bounds and to check the stability of feedback spectral systems [14], [15]. If the boundary conditions are too complicated to be used to obtain the eigenfunctions of the associated linear operator, then the techniques of proper orthogonal decomposition (POD) may be used to extract empirical eigenfunctions (EEFs) instead. Afterwards, lower dimensional ODE
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Fig. 2. Simplified configuration of the curing system in Cartesian space.
Fig. 1. Spectral approximation based modeling for the curing process.
systems can still be derived with modal decomposition techniques [16]–[18]. If the mathematical model of the DPS is totally unknown, then a lower order approximation can be derived for a linear DPS with the input–output data using the singular value decomposition (SVD) method [19], and for a nonlinear DPS using POD techniques [20] or in combination with SVD [10]. As only input–output data are used, the accuracy of the model is limited by the availability of sensors and actuators due to a limited number of EEFs obtained. Besides, the snapshots should be representative and the measurement time interval has to be carefully chosen for a proper modeling based on EEFs [16]. As there are only a finite number of sensors and actuators available in practice, it is unclear how these data-driven methods work with systems of higher dimensional spatial natures. Furthermore, all of the data-driven methods mentioned are only applicable offline. In practice, the process is nether completely known or totally unknown. Use of existing knowledge will make modeling easier and more accurate. Unlike the modeling approaches of DPSs mentioned previously, this paper proposes a simple but effective modeling method for snap curing processing by integrating spectral methods [11], [21] with neural networks (NNs). Based on the partial knowledge of the process, the spectral methods that are shown in Fig. 1 are used for time–space separation and model reduction, and NNs are used for state estimation and system identification. The dynamics of the snap curing process is established according to physical laws and knowledge of the process. Based on the dynamics of the curing process, a lower order nonlinear ODE model with uncertain parameters and unknown nonlinearities is then derived using model reduction techniques. Thereafter, a NN-based observer is designed to estimate the states of the ODE model using selected measurements. Finally, a hybrid general regression neural network (GRNN) is trained as a nonlinear model of the process in state–space formulation. The proposed method emphasises five
key points: an engineering-preferred model in reduced order is obtained with the help of the spectral approximation and partial knowledge of the process; intelligence-based methods are used to deal with the unknown nonlinearities and unmodeled dynamics; only a finite number of measurements are required; the developed model is of a time-spatial distributed nature that can satisfactorily describe the temperature distribution in the curing process; and the modeling method is applicable either offline or online. Using the established model, the temperature distribution related fundamental analyses, such as simulation, controller design, fault diagnosis and supervision, can be carried out for the curing process. II. CURING PROCESS AND ITS SPECTRAL MODELLING A. Fundamental Dynamics of the Curing Process After adhesive die attach, the adhesive is cured in a curing oven. Die attach is the process whereby the chip (die) is attached to the die pad or die cavity of the leadframe. Epoxy die attach uses epoxy to mount the die on the die pad or cavity. The epoxy is then cured at a specified temperature in a curing oven to form a tight bond between the die and the leadframe. A simplified configuration of the snap curing oven is shown in Fig. 2, with its data acquisition and control system shown in Fig. 3. The oven chamber has four heaters for heating and four thermocouples for temperature sensing. The parts to be cured are moved in and out from the inlet and outlet. For fundamental analysis and better control, a model for estimating the temperature distribution inside the chamber is required. As the temperature of the epoxy between the die and the leadframe is of interest, an energy balance of the epoxy may be constructed to analyze the system, which would make the whole analysis simpler. However, it is evident that the temperature distribution is not consistent inside the oven chamber. Thus, even though an energy balance of the epoxy can be constructed, it is still difficult to use because the boundary conditions of the epoxy (the temperatures around the epoxy) are unknown. The leadframe usually contains many chips (dies). In practice, it is difficult to place many sensors to measure the temperature around the epoxy. These are the main challenges and the main motivation for building a model of the oven and then using the model to estimate the temperature distribution of the curing process. For the kind of curing process considered, the leadframe is placed under the heaters as indicated in Figs. 2 and 3. The volume of the epoxy between
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Fig. 3. Data acquisition and control system of the curing process.
a die and the leadframe is much smaller than that of the leadframe and of the oven chamber. The volume of the leadframe is also much smaller than the volume of the oven chamber. Thus, the effects of the epoxy and the leadframe on the temperature in the oven chamber are usually neglected in the modeling of the curing process. These effects will be considered as disturbances and will be compensated for during the control process of the temperature distribution. According to heat transfer laws [22], the fundamental heat transfer equation of the oven can be expressed as follows:
(2.1)
denotes the vector of manipulated inputs, with explanation
of
is a nonlinear function of which will be given later,
—the
with is the spatial distribution of the control action denotes measured outputs in spatial coordinates is the initial temperature distribution in the oven and with being associated with the efficiency and the power of the th heater, which is usually inaccurately given. Around a particular working point (i.e., 180 C), the specific heat of dry air under the atmospheric pressure can be considered to be constant. However, the thermal conductivity and the density are dependent on the temperature, which can be expressed as and
where density (kg/m ); specific heat (J/kg C); thermal conductivity (W/m C); temperature ( C) at time and loca; tion , spatial coordinates. and and are the effects of convection and In this model, represents the disturbances such radiation, as the effects of the moving parts inside the chamber and heat leaking from oven walls,
where and are nominal values around a working point, and and are functions of . As shown in Fig. 3, thyristors are used to modulate the input power, which means that there is a nonlinear static relationship and between the input pulsewidth modulation (PWM) ratio represents the power delivered to the chamber. Here, the nonlinear relationship, which can be divided into a linear and a nonlinear part , that is, part . As the curing temperature of the epoxy that we consider is around 180 C, which is different from the RTP systems,
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in which operating temperatures can be up to 1100 C, and the temperature difference is not very large during the curing process, the radiative heat transfer is not significant but cannot be neglected due to the consideration of modeling accuracy. The calculation of radiation is very difficult even within simple enclosures [18]. Because this term is a function of the temperature to the power of 4, it is treated as an unknown nonlinearity. It is well known that convection is conduction and/or radiation in a moving continuum [23]. In the oven chamber, the moving of the air is mainly due to its density difference, which is caused by temperature difference. However, as the leadframe is placed very close to the heaters and the chamber is quite small, around an operating temperature, the temperature difference in the chamber is not so large that the convection will dominate the heat transfer in the chamber. Because an analytical procedure to solve the problem of heat convection is quite complex and laborious, even for simple geometrical bodies and boundary conditions [22], the convection is treated as an unknown function of the temperature as radiation. Thus, one can approximately assume that the main mode of heat transfer is conduction in the curing process and take the other effects as unmodeled dynamics (unknown nonlinearities), leaving them for neural compensation. In terms of the previous analyses, (2.1) can be rewritten as
(2.2) where
and
are constants is the Laplacian operator
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where
with being unknown nonlinear functions. Thus, a complete model of the oven is described by a parabolic partial differential (2.2) with the boundary conditions (2.3a)–(2.3c). B. Model Reduction Analysis Due to the existing unknown nonlinearities, unknown parameters, and limited number of actuators and sensors, the oven model described by parabolic partial differential (2.2) and (2.3) is not applicable. A lower order ODE system is preferred not only for ease of controller design but also for accurate simulation and practical implementation. In this section, one of the model reduction techniques, the spectral methods, are used to reduce models (2.2) and (2.3) to a set of finitely many nonlinear ODEs. This lower order ODE system is then used as the framework for process identification and modeling. A parabolic PDE system typically involves spatial differential operators with eigenspectra that can be partitioned into a finite-dimensional (slow) and an infinite-dimensional (fast) complements [24]. This implies that the dynamical behavior of such a system can be approximately described by a finite-dimensional ODE system that captures the dynamics of the dominant (slow) modes of the PDE system. It is very convenient and effective to use the eigenfunctions of a spatial differential operator to derive a lower order ODE system for such a parabolic PDE system if the boundary conditions are homogeneous. However, the boundary conditions (2.3a)–(2.3c) include unknown parameters and nonlinear functions; therefore, a direct derivation of is impossible. In this eigenfunctions for the operator case, orthogonal polynomials can be used as basis functions instead of eigenfunctions. In doing so, however, the derived ODE system may have very high dimensions. Thus, models (2.2) and (2.3) are transformed into the following equivalent model, with a proof given in the Appendix
(2.4) The chamber walls of the oven are usually not well-insulated and the boundary conditions are not completely known. The free convection and radiation cause heat leakage from the exterior walls of the oven. As it is difficult to obtain reasonable boundary conditions using only physical insights, for simplicity one can set the boundary conditions to be unknown nonlinear functions of the boundary temperature, the space coordinates of the oven , and the ambient temperature , as follows:
(2.5a) (2.5b) (2.5c) where
(2.3a) (2.3b) (2.3c)
and
is the Dirac delta function.
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Now, the new boundary conditions (2.5a)–(2.5c) are homogeneous, although the system (2.4)–(2.5) is an infinite-dimensional nonlinear system. To further reduce it to a set of finitely many nonlinear ODE equations, the Galerkin method [11] can be applied as follows. 1) Separation of Time and Spatial Variables: According to the theorem of separation of variables [25], the solution of system (2.4)–(2.5) can be expressed in the following time–space decoupled form:
in which is rearranged in order of magnitude and then dewith being its correnoted as a new sequence of sponding eigenfunction, as follows: with
(2.6) Thus, the eigenvalue problem for operator be solved analytically [26], as
in (2.4) can
Then, (2.10) can be rewritten in a general nonlinear form as follows: (2.11a) (2.11b) where (2.7) where , and denote eigenvalues, and , denote the corresponding eigenfunctions. and Next, the solution of system (2.4)–(2.5) is expressed in an orthogonally decoupled series (2.8) where sion coefficients associated with
.. .
are expan. The residual and
is minimized with (2.9) . where is the domain , and , Due to the orthogonality of the eigenfunctions the integration of (2.9) after substituting (2.8) into (2.9) will give
(2.10a)
where
(2.10b)
represents nonlinearities and unmodeled dynamics. 2) Separation of Slow and Fast Modes: To reduce the infinite-dimensional nonlinear ODE system (2.11) to a finite set of nonlinear ODE equations, the following assumption is made in (2.4), where for the eigenspectrum of the operator is defined as the set of all eigenvalues of , i.e., . Assumption 1: [27] 1) , where denotes the real part of . can be partitioned as , 2) consists of the first (with fiwhere , and nite) eigenvalues, i.e., . and , where 3) is a small positive parameter. Based on Assumption 1, one can define and , which are the modal subspaces of . By choosing the and such that orthogonal projection operators and , and by decomposing the state of the system
DENG et al.: SPECTRAL-APPROXIMATION-BASED INTELLIGENT MODELING FOR DISTRIBUTED THERMAL PROCESSES
(2.11) as be equivalently expressed as
, the system (2.11) can (2.12a) (2.12b) (2.12c)
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, set consisting of slow eigenvalues, and a stable infinite set containing the remaining fast eigen. Thus, the dominant dynamics of values, the nonlinear parabolic PDE system can be described with only a few dominant modes [9]. The coefficients solved from (2.13a), , forms the temperature distribution (2.13c)
where diagonal matrix with dimension and in the form of ; unbounded differential operator that generates a strongly continuous exponentially stable semigroup [11]; ; and Lipschitz vector functions; ; denotes that the state is in an infinitedimensional space. After neglecting the fast modes in (2.12), the following finitedimensional nonlinear system is obtained: (2.13a) (2.13b) Remark: It is interesting to note that the eigenspectrum of nonlinear PDEs (2.2) with boundary conditions (2.3a)–(2.3c) can be separated into a finite-dimensional (slow) and an infinitedimensional stable (fast) complement based on the separation of in (2.4). This is because the spectrum of the linear operator the dynamical behavior of the system can be approximately described by a finite-dimensional ODE system that captures the dynamics of the dominant (slow) modes of the PDE system. For , and , the eigenfunctions derived from (2.7), , form a complete orthogonal basis. Thus, the solution of the nonlinear system can be expressed in terms of the basis as functions in the form of (2.8) with , and the expansion coefficients that are associated with . If the nonlinear system has the properties that are shown in Assumption 1, then its dynamics can be separated into a slow part and a stable fast part, and the dominant (slow) dynamics can be used to approximate the nonlinear system. Such nonlinear parabolic PDE systems arise naturally from the modeling of processes such as diffusion, heat conduction or viscosity, a class that encompasses reaction-diffusion systems, RTP units and many other systems of engineering interest [5], [9], [11], [28]. If a nonlinear parabolic PDE model is known, then this method is frequently used to derive a lower dimensional ODE model, and a controller for the nonlinear parabolic PDE model can be designed based only on the lower dimensional ODE model [9], [11], [13], [28]–[30]. The curing process is such a nonlinear parabolic PDE process that the method can be used to derive a finite-dimensional ODE system which captures the dynamics of the dominant slow modes of the curing process, as discussed previously. 3) Synthesis of Time and Spatial Variables: Assumption 1 states that the eigenspectrum of can be partitioned into a finite
4) Order Selection of the Nonlinear ODE System: So far, a lower order nonlinear ODE system (2.13a) and (2.13b) has been derived by using the Galerkin method for the approximation of the parabolic DPS (2.4) with boundary conditions (2.5a)–(2.5c). To determine the order of the ODE system, a tradeoff between the order and the modeling accuracy must be considered. As the exact model of the snap curing process is unknown, it is difficult to estimate the truncated error when deriving a lower order nonlinear ODE system for the parabolic DPS. The higher the accuracy required, the higher the order of the ODE system must be used. The following method is proposed to determine a suitable order for the derived nonlinear ODE system. First, Assumption 1 is used to check whether the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow and an infinite-dimensional stable fast complement. Second, a small ratio factor, ( 0 for the curing process) is used for the order determina0.1 (i.e., is 10 times ) is chosen, tion. For example, if (which is supposed which will lead to a particular value of to be 16) for the curing process (2.13), then an th order nonlinear ODE system for the process can be derived by following . the previously discussed Galerkin method, usually with 5) Discretization of the Nonlinear ODE System: For practical implementation, a discrete-time model is often used. The previous reduced model is now discretised by the Euler forward formula as follows:
(2.14a) (2.14b) with being the sampling time. with , choose For simplicity, one can replace and around an operating point, and express as with being the estimate of and being the unknown parameter vector. The approximation of (2.14) around a working point can then be developed as
(2.15a) (2.15b) with
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Fig. 4.
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Spectral based model reduction procedure for the curing process.
The constant matrices , and are determined by the nominal values of the snap curing system, and and represent the uncertain parameters and unknown nonlinearities. Based on the previous analyses, the coefficients can be obtained by solving (2.15a), from which the temperature distribution at the th step in the oven can be estimated by
and being a normalized with Gaussian radial basis function [32]. This is actually a recurrent NN observer, as shown in Fig. 6, and output of the oven. Here, is with the input the state vector of the NN, is a constant matrix that is designed stable. When and are adjusted to make the matrix is the estimate of . by updating laws, 1) Learning Algorithm: The back propagation through time or the real time recurrent learning algorithms [31], [33] are widely used for the training of recurrent NNs. However, the gradient calculations used in the algorithm may become time-consuming and unstable [34]. The online NN learning requires both convergence speed and stability. For single-input single-output nonlinear systems, an adaptive learning algorithm [35] has been used to train RBF NNs for input–output system identification based on Lyapunov stability theory. This adaptive learning algorithm can be modified for multi-input–multi-output (MIMO) systems and used for the online training of the proposed recurrent NN observer, as follows. and are The updating laws for (3.2) (3.3) with (3.4)
(2.16) where The spectral method based model reduction procedure for the curing process is summarized in Fig. 4. III. HYBRID SPECTRAL/NEURAL MODELLING METHODOLOGY The nonlinear ODE model (2.15) is still not applicable because of the uncertain parameters and unknown nonlinearities . The states in (2.6) cannot be measured because only a finite number of sensors are available. To obtain an applicable model for the snap curing process, a hybrid spectral/neural modeling method is proposed, as shown in Fig. 5, where a neural-network observer is first used to estimate the coof (2.15) with the help of the spectral method, efficient and a hybrid GRNN is employed to establish the dynamic relaand the coefficient . This tionship between the input hybrid spectral/neural model can provide a good estimation of the temperature distribution.
and are two constants. To avoid singularities, the updating laws for modified as follows:
and
are (3.5) (3.6)
where
A. Spectral Method Based Neural Observer The structured recurrent radial basis function (RBF) NN [31], in combination with the available knowledge about the oven , and , is used to estimate of (2.15a) in the such as following form:
(3.1a) (3.1b)
with and being two small constants that are determined by Lyapunov stability theory. 2) Stability Analysis: It can be proved that for the NN will observer (3.1), the output error and with updating converge to zero when and , respectively, and the laws (3.2) and (3.3) for output error will converge to a bounded area of radius
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Fig. 5. Hybrid spectral/NN modeling method for the curing process.
are bounded if the and are the errors bounded. The analysis of the stability is given in the following. Suppose that the system (2.15a) and (2.15b) can be described as the following recurrent NN: (3.7a) (3.7b) is an optimal parameter vector, is an optimal where weight matrix, is the modeling error and is a normalized activation vector. The neural-network observer for system (3.7) is in the form of (3.1a) and (3.1b). Let the estimation error be denoted by . Then Fig. 6.
Structure of the NN observer.
when and with and , respectively. A updating laws (3.5) and (3.6) for detailed proof is provided in the Appendix. Stability analysis is difficult for the observer because the structured recurrent NN is not linear in its parameters [31]. However, with the previous learning algorithm and the constant matrix that stabilises the matrix - , one can ensure that
(3.8) with
As
-
, and
and
are bounded, (normalized activation) and
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Fig. 8. Sensor locations for modeling and verification. (The shaded sensors will be used for modeling.) Fig. 7. Hybrid spectral/GRNN model for the curing process.
(normalized inputs), by letting and be bounded during the learning process of the NN observer [31], one must in (3.8), with being a bounded positive have number. Because is stable and is bounded, from (3.8) one can conclude that the estimation error is bounded. B. Spectral Based NN Model After the state estimation is completed, a hybrid spectral based GRNN can be used to model the snap curing process, as shown in Fig. 7. The hybrid GRNN is expressed as follows: (3.9) where
, and
are weights of the hybrid GRNN and , with being a normalized activation
function. This hybrid GRNN can be rewritten in the following compact form:
with
and
Evolved from the RBF network, the GRNN can improve function approximation with a minimal number of weights [32]. The recursive least square (RLS) algorithm [36] is used to guarantee a fast convergence as follows:
The hybrid GRNN can be trained either offline or online. After training, it establishes the dynamic relationship between the and the expansion coefficients , from which the input temperature distribution is developed using (2.16).
IV. REAL-TIME EXPERIMENT FOR MODEL IDENTIFICATION AND VERIFICATION A. Input–Output Considerations 1) Sensor Locations: Six extra sensors (s3–s6, s9, s10) are placed in the oven, as shown in Fig. 8, for model identification in the neural-network and verification. The matrix pair observer (3.1) must be observable. As the matrix depends on the sensor locations, the observability may not be guaranteed if the sensor locations are taken too arbitrarily. Thus, thermocouples (s1, s2, s3, s4) are chosen for modeling the curing process. These selected locations can meet the observable condition and be representative of the heating region of the oven. The rest sensors (s5–s10) are used to evaluate the modeling performance. 2) Persistent Excitation of Input Signal: It is the power delivered to the chamber that changes the temperature in the chamber. Hence, the quantity of the power inputted to the chamber should be carefully controlled. Heaters of the curing system are controlled with on/off relays, which generate PWM input signals at only two levels (0 or 1). However, the pseudorandom binary signal (PRBS) is inappropriate to identify the nonlinear process because its two levels cannot excite the system dynamics sufficiently [36]. To overcome this difficulty, multilevel signals should be designed. However, the power that is delivered to the chamber is controlled by the input PWM signals, and one cannot directly give multilevel power inputs by using the present equipment in the curing process. As the quantity of the power inputted to the chamber is proportional to the input PWM ratio and it is convenient to change that ratio, one can generate multilevel PWM ratio signals instead of multilevel power inputs. Fig. 9 shows the PWM input in one sampling with impulse period, where the period is assumed to be signals of period and the on-duration . The percentage of the on-period is defined as PWM ratio , with for . This ratio factor is randomised with the required mean, which is set at 0.13 in our case. Thus, the power signals to the heaters can be generated indirectly by letting be random within [0, 1]. Due to the equivalence between the PWM ratio and the power delivered to the chamber in each , the multilevel PWM ratio signals should sampling period excite the curing process sufficiently. The experimental results have confirmed this analysis. The multilevel PWM ratio signals (first 500 samples) that are used to identify the nonlinear curing process for Heater 1 are shown in Fig. 10.
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Fig. 9. PWM impulse signals.
Fig. 10.
Fig. 11.
Performance of the NN model at sensor location s1 with training data.
Fig. 12.
Performance of the NN model at sensor location s1 with test data.
Multilevel ratios of PWM for Heater 1.
B. Experiments and Analysis Based on the method proposed in Section III, a 32-order hybrid GRNN model is obtained and compared with a 32-order linear state–space model. A total of 1500 measurements are collected with a sampling time of 10 s around 180 C. The modeling and verification are carried out in the following three steps. Step 1) Modeling of the training data from the measured locations One thousand records of the measurements from sensors (s1, s2, s3, and s4) are used to train the hybrid GRNN. The result given in Table I, and for sensor location s1 in Fig. 11, shows that the developed hybrid spectral/neural model can provide a very good estimation after training. Step 2) Model verification of the untrained data from the measured locations The last 500 records of measurements from sensors (s1, s2, s3, and s4) are used to test the effectiveness of the model for untrained data. The good performance is maintained, as shown in Table I and in Fig. 12 for sensor location s1. Step 3) Model verification of the un-measured locations (untrained) All 1500 records of measurements at sensors (s5, s6, s7, s8, s9, and s10) are used to evaluate the
estimation performance of the developed model for the temperature distribution of the oven. A good performance is achieved, as shown in Table II and in Fig. 13 for sensor location s5. The temperature distribution in the oven chamber can be . Under working given based on the estimated coefficients conditions around 180 C, the temperature distribution at 0.025 m (where the leadframe is placed) inside the height 8000 s is provided by the NN model, oven chamber when as shown in Fig. 14. 1) Comparison: For comparison, a linear model of the curing process is also developed by neglecting the nonlinear 0 in (3.9), and then using the RLS part, i.e., letting
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Fig. 14.
Temperature distribution at t
= 8000 s.
V. CONCLUSION Fig. 13.
Performance of the NN model at sensor location s5 with test data. TABLE I PERFORMANCE COMPARISON BY RMSE
TABLE II RMSES AT SENSOR LOCATIONS S5–S10
algorithm for learning. The root of mean square error (RMSE) is defined as the performance criterion RMSE where is the data number, and and are temperature measurements and the model estimation. Tables I and II compare the performances of the two different models in terms of RMSE. It is apparent that the hybrid NN model has much better performance than the linear model with the same order, and is therefore more suitable for the nonlinear curing process.
A spectral approximation based intelligent modeling approach has been developed for the distributed thermal processing of the snap curing oven that is used in the semiconductor packaging industry. The snap curing oven is described by a nonlinear parabolic DPS in the time–space domain. After finding a proper approximation of complex boundary conditions of the system, the spectral method is applied to time–space separation and model reduction, and NNs are used for state estimation and system identification. With the help of model reduction techniques, the dynamics of the curing process derived from the physical laws are described by a model of low-order nonlinear ODEs with a few uncertain parameters and unknown nonlinearities. A neural-network observer is then designed to estimate the states of the ODE model from measurements taken at specified locations in the field. Using the estimated states, a hybrid general regression NN is trained to be a nonlinear model of the curing process in a state–space formulation. Real-time experiments on the snap curing oven have shown that the proposed modeling method is indeed effective. This modeling methodology can be applied to a class of nonlinear DPSs in industrial thermal processing. Further studies are underway for the optimal control design and fault diagnosis of the oven using the process model described here.
APPENDIX A EQUIVALENCE PROOF OF (2.4)–(2.5c) AND (2.2)–(2.3c) Using the solution (2.8) for (2.4) with boundary conditions (2.5a)–(2.5c), the integration (2.9) becomes
(A1.1)
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where
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and
Note that (A1.1) can be rewritten as
Substituting the previous equation into (A1.1), one has
The integration of the first term of the right-hand side in the previous equation is
(A1.2) On the other hand, using the solution (2.8) for (2.2) with boundary conditions (2.3a)–(2.3c), the integration (2.9) gives
Using the boundary conditions (2.5a)–(2.5c), the previous equation becomes
(A1.3) Note also that (A1.3) can be rewritten as
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The integration of the first term on the right-hand side of the previous equation is
APPENDIX B PROOF OF CONVERGENCE FOR THE UPDATING LAWS (3.2), (3.3) AND (3.4), (3.5) . Define 1) For the updating laws (3.2) and (3.3), one has
Using the boundary conditions (2.3a)–(2.3c), the previous equation becomes
where
Define a Lyapunov function
2)
Substituting the previous equation into (A1.3), one obtains the same result as (A1.2). Thus, one can conclude that the (2.4) with boundary conditions (2.5a)–(2.5c) is equivalent to the (2.2) with boundary conditions (2.3a)–(2.3c).
. Then
Thus, when . Therefore, the output error will converge to zero. For the updating laws (3.5) and (3.6), one has
where:
and .
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Define a Lyapunov function
In this case, since one has
Thus,
. Then
[35] and
,
when
and , where . Consequently, the error will converge to the ball centred at the origin of the error space with radius . This completes the proof. ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers and the associate editor for their valuable comments. REFERENCES [1] T. Soderstrom and P. Stoica, System Identification. Englewood Cliffs, NJ: Prentice-Hall, 1989. [2] H. Aling, R. L. Kosut, A. Emami-Naeini, and J. L. Ebert, “Nonlinear model reduction with application to rapid thermal processing,” in Proc. 35th IEEE Conf. Decision and Control, vol. 4, 1996, pp. 4305–4310. [3] S. Banerjee, J. V. Cole, and K. F. Jensen, “Nonlinear model reduction strategies for rapid thermal processing,” IEEE Trans. Semicond. Manuf., vol. 11, no. 2, pp. 266–275, 1998. [4] J. L. Ebert, G. W. van der Linden, R. L. Kosut, and A. Emami-Naeini, “Efficient CFD modeling of single wafer semiconductor fabrication systems for closed-loop evaluation,” in Proc. 36th IEEE Conf. Decision Control, vol. 1, 1997, pp. 830–831. [5] A. Emami-Naeini, J. L. Ebert, D. de Roover, R. L. Kosut, M. Dettori, L. M. L. Porter, and S. Ghosal, “Modeling and control of distributed thermal systems,” IEEE Trans. Contr. Syst. Technol., vol. 11, no. 5, pp. 668–683, Sep. 2003. [6] R. L. Kosut and M. G. Kabuli, “Robust control of thermal processes,” in Proc. 2nd Int. Rapid Thermal Processing Conf., 1994, pp. 296–297. [7] C. A. Lin and Y. L. Jan, “Control system design for a rapid thermal processing system,” IEEE Trans. Contr. Syst. Technol., vol. 9, no. 1, pp. 122–129, Jan. 2001. [8] K. M. Tao, R. L. Kosut, M. Ekblad, and G. Aral, “Feedforward learning applied to RTP of semiconductor wafers,” in Proc. 33rd IEEE Conf. Decision and Control, vol. 1, Lake Buena Vista, FL, 1994, pp. 67–72. [9] W. H. Ray, Advanced Process Control. New York: McGraw-Hill, 1981.
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Hua Deng received the B.Eng. degree from Nanjing Aeronautical Institute, Nanjing, China, in 1983, the M.Eng. degree from Northwestern Polytechnical University, Xi’an, China, in 1988, and the Ph.D. degree from the City University of Hong Kong, Hong Kong, in 2005. He was an Associate Professor at Changsha University of Science and Technology, China. He is currently a Professor in the School of Mechanical and Electrical Engineering, Central South University, Changsha, China. His research interests include modeling and control of complex dynamic systems, intelligent control, fault diagnoses, and computer-aided control system design.
Han-Xiong Li (S’94–M’97–SM’00) received the B.E. degree from the National University of Defence Technology, Changsha, China, in 1982, the M.E. degree in electrical engineering from Delft University of Technology, Delft, The Netherlands, in 1991, and the Ph.D. degree in electrical engineering from University of Auckland, Auckland, New Zealand, in 1997. He is currently an Associate Professor in the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong. He is holding “Lotus Scholar” in Central South University-an honorary professor endowed by Ministry of Hunan Province, China. He was awarded by the National Science Foundation for (overseas) distinguished young scholar in 2004. In the last twenty years, he has had opportunities to work in different fields including military service, industry, and academia. He has gained industrial experience in IC packaging as a senior process engineer for die-bonding and dispensing from ASM—a leading supplier for semiconductor process equipment. His research interests include modeling and intelligent control for complex industrial process with special interest to electronic packaging. Dr. Li serves as an Associate Editor for the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B.
Guanrong (Ron) Chen (M’89–SM’92–F’97) received the M.Sc. degree in computer science from Zhongshan University, China, in 1981 and the Ph.D. degree in applied mathematics from Texas A&M University, College Station, in 1987. After working at Rice University and the University of Houston for 15 years, he currently is a Chair Professor in the Department of Electronic Engineering, City University of Hong Kong. He was named IEEE Fellow in 1996, conferred more than 10 Honorary Guest-Chair Professorships from China and particularly an Honorary Professorship from the Central Queensland University of Australia. He is Chief, Advisory, Feature, and Associate Editors for eight IEEE Transactions and International Journals. Prof. Chen has (co)authored 15 research monographs and advanced textbooks, more than 300 journal papers, and about 180 conference papers, published since 1981 in the field of nonlinear systems dynamics and controls, with applications to Internet technology as well as intelligent control systems. He received the outstanding prize for the best journal paper award from the American Society of Engineering Education in 1998, the best transactions annual paper award from the IEEE Aerospace and Electronic Systems Society in 2001, and the best journal paper award from the Czech Academy of Sciences in 2002.
