Inner-Phase Analysis Based Statistical Modeling and Online

Article pubs.acs.org/IECR

Inner-Phase Analysis Based Statistical Modeling and Online Monitoring for Uneven Multiphase Batch Processes Luping Zhao,†,∥,‡ Chunhui Zhao,*,†,∥ and Furong Gao*,†,‡,§ †

State Key Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, China ∥ Key Laboratory of System Control and Information Processing, Ministry of Education, Shanghai, 200240, China ‡ Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR § Fok Ying Tung Graduate School, Hong Kong University of Science and Technology, Hong Kong SAR ABSTRACT: The multiplicity of operation phases is inherent in the nature of many batch processes, and each phase exhibits significantly different underlying behaviors. In addition, within each phase, normal processes in general follow certain underlying operation rules, called inner-phase evolution here, which however have not been addressed before. In this paper, a new statistical modeling and online monitoring method is proposed for multiphase batch processes. A two-level phase division algorithm is proposed to capture the process trend and trace inner-phase evolutions. It reveals that the inner-phase process in general goes through three statuses sequentially, i.e., transition, steady phase, and transition. Principal component analysis (PCA) and qualitative trend analysis (QTA) are combined to distinguish different inner-phase process statuses. Their different characteristics are then modeled and monitored separately, revealing more accurate process operation information. Meanwhile, the problem of uneven-duration batches is effectively handled in different inner-phase process statuses. The application to a typical multiphase batch process, injection molding, illustrates the feasibility and performance of the proposed algorithm.

1. INTRODUCTION As one important type of production, batch processes are widely applied in industry. To meet the requirements of fast changing markets and to manufacture higher-value-added products in batch processes, operation safety has drawn people’s attention.1−4 Batch processes usually have complicated characteristics which make it difficult to build a first-principle model within a limited time period. On the other hand, with the development of computers and sensors, abundant process data are available covering much process information. Therefore, multivariate statistical process control (MSPC) methods,5−8 which can extract process characteristics only based on process data rather than prior knowledge of a process, have been successfully developed. Multiway principal component analysis (MPCA)9 was first proposed for batch process monitoring to handle the three-dimensional data structure of batch processes; however, it is difficult to reveal the changes of process correlations along the time direction since it takes the entirety of batch data as a single object. Also, it is difficult for online application since the whole of the new batch data is not available up to the concerned time so that the unknown future values have to be estimated. Multiphase is a significant feature of batch processes. In general, multiple operation steps are included in each batch cycle, resulting in different process segments, called stages or phases.10,11,17 In this work, phases are preferred. A series of phases comprise a whole batch cycle, and each phase has its own characteristic, which requires special attention for multiphase batch process monitoring. Some works10−19 have been done focusing on multiphase characteristics of batch processes since the 1990s. Different MPCA models were established to monitor different phases offline at the end of each batch or online with the estimation of process data.10,11 Lu and Zhao et al.12,13 came up with phase-based © 2013 American Chemical Society

statistical modeling methods where a batch process was divided into multiple phases according to the changes of process characteristics. It was based on such recognition that the underlying variable correlations are similar within the same phase and different across phases. The phase-representative model was then developed for each phase, and different models were used for different phases. Using their methods, the problem of data estimation was solved, and monitoring performance was improved. Considering the transition problem between neighboring phases, a soft-transition multiple PCA (STMPCA) method was proposed to detect and model transitions for online process monitoring.16,17 To handle the uneven-duration problem which widely exists in batch processes, Lu et al.18 proposed to develop two kinds of models, one for phase division and the other for process monitoring. Considering that there are both similarity and dissimilarity among different groups of uneven-length batches, Zhao et al. proposed to model group-common and specific information respectively to handle the serious uneven-length problem.19 Despite their wide application, the phase-based multivariate statistical methods ignore the process trend reflected by innerphase (within a phase) variations which may lose important information about process operation. In this work, statistical modeling and online monitoring of process evolution within each phase is addressed for batch processes. Batch processes usually have an ordered set of processing activities,20 where different phases are connected together in one batch cycle and Received: Revised: Accepted: Published: 4586

October 31, 2012 January 29, 2013 February 24, 2013 February 25, 2013 dx.doi.org/10.1021/ie302990n | Ind. Eng. Chem. Res. 2013, 52, 4586−4596

Industrial & Engineering Chemistry Research

Article

method.26 Also, new frameworks and applications have been reported covering a very wide area.21,27−35 Recently, a novel approach was proposed to automatically identify the qualitative shapes using a polynomial-fit based intervalhalving technique32−35 to capture process trends in both offline and online situations. It is used in the present work for inner-phase evolution analysis. The fundamental language of QTA is the primitives defined by the first and second derivatives of variables,23 shown in Figure 1. A trend is represented as a sequence (combination)

each of them goes through its particular evolution. In general, three sequential statuses, i.e., transition, steady part, and transition, are the basic structure to describe the process variation within a phase, called inner-phase evolution here. PCA and qualitative trend analysis (QTA)21−35 methods are effectively combined to trace the inner-phase evolutions of batch processes. PCA is used as a basic statistical analysis tool to obtain statistical information from process data. Then, QTA is used on statistics obtained by PCA to capture the inner-phase evolutions where each phase can be divided into three parts, transition, steady part, and transition. Different modeling strategies are proposed to analyze their different characteristics. For online monitoring, the inner-phase operation process trend is well traced where the affiliation of the current sample point is judged and its operation status is supervised. The underlying inner-phase evolutions are critical to unveiling how a process phase develops with time. Analyzing the inner-phase evolutions can help to establish more accurate monitoring models and provide improved monitoring results, which can also provide enhanced process understanding. The rest of this paper includes four parts: First, the basic algorithms of PCA and QTA are briefly revisited in section 2. Then, the proposed method is presented in section 3, including the description of two-level phase division, statistical modeling in different inner-phase parts, and online monitoring for different operation statuses. In section 4, the application of the proposed method to a typical multiphase batch process, injection molding, is presented, and discussions are conducted based on the illustration results. At last, the conclusion is drawn.

Figure 1. Seven primitives in qualitative trend analysis (QTA).23

of these seven primitives.33 The procedure identifies the qualitative trend as a sequence of piecewise unimodals or quadratic segments. To estimate the significance of fit-error, an estimate of the noises obtained from wavelet-based denoising is used. The least-order (among constant, first-order, and quadratic) polynomial with a fit error that is statistically insignificant compared to noises (as dictated by F-test) is used to represent the segment.33 If the fit error is large even for the quadratic polynomial, then the length is halved, and the process is repeated on the first half segment until the fit error is acceptable. A constrained polynomial fit is used to ensure the continuity of the fitted data, and an outlier detection methodology is used to detect any jump (step) changes in the signal. The whole procedure is recursively applied to the remaining data until the entire data record is covered. The detail can refer to previous work,32,33 which is not addressed here.

2. PRELIMINARY 2.1. PCA Algorithm. Principal component analysis (PCA) is a mathematical linear transformation performed on a two-dimensional data matrix X(N × J), where N is the number of samples and J is the number of variables. In detail, PCA decomposes X as follows: R

X = TPT + E =

∑ tipiT + E i=1

(1)

where ti(N × 1) is the ith principal component (PC), also named a score vector, revealing the systematic variation information; pi(J × 1) is the corresponding loading vector to calculate ti from X, which also reveals variable correlation information. T(N × R) and P(J × R) are the corresponding principal component (PC) matrix and loading matrix, respectively. E represents the residuals after PCA explanation. R is the number of retained PCs in PCA systematic subspace. By PCA, each data space is divided into two different subspaces, systematic subspace and residual subspace respectively, X̂ and E. For online process monitoring, two statistics are calculated in two subspaces, hotelling T2 and SPE, summarizing systematic variation information and residual information, respectively. The detailed properties and calculations can be found in refs 36−39. 2.2. QTA Algorithm. Process trend analysis is a useful approach to exploit the temporal information and reason about the process state, which is used in the present work to trace innerphase evolution. Basically, trend analysis means identifying and capturing the increases and decreases of time-series data.21 Since the 1980s, qualitative trend analysis (QTA), which is also widely known as dynamic trend analysis, has been developed and has played an important role in process monitoring and fault analysis.22−35 Two basic aspects of QTA theory are trend extraction and trend matching. A lot of extended techniques about these two aspects have been published, such as wavelet techniques,24 neural networks,25 and the dyadic B-spline-based

3. METHODOLOGY The frame of new statistical modeling strategy is shown in Figure 2. It includes three steps, phase division, inner-phase division, and development of monitoring models. 3.1. Two-Level Phase Division. To reveal time-varying underlying process characteristics, the concerned phase division includes two levels: conventional phase division and inner-phase division. In conventional phase division, the whole process is divided into multiple phases by indicator variables. The knowledge of a concrete process may be necessary to decide the indicator variables. It is noted that due to an uneven-length problem in batch processes, the phases may have different durations for different batches. So those uneven-length phases are clustered into different groups as indicated by the phase length, and the even-length phases are collected in the same group. After conventional phase division, process data of the cth phase in the gth group are arranged as a three-way data array Xc,g(Ig × J × Kc,g), where Ig, J, and Kc,g refer to the number of batches, process variables, and time intervals within the cth phase in the gth group. After variable-wise unfolding, the two-dimensional data matrix X (Kc,gIg × J) from different groups (where g = 1, ..., G) ∼ c ,g

4587

dx.doi.org/10.1021/ie302990n | Ind. Eng. Chem. Res. 2013, 52, 4586−4596


Article

Figure 3. The program for online monitoring.

Figure 2. The illustration of modeling strategy.

are put together to obtain a two-dimensional phase-representative data matrix X (∑Gg=1Kc,gIg × J), which are then normalized to have ∼c

zero mean and unit variance, Xc(ΣGg=1Kc,gIg × J). In this way, the normalized data can keep the process variation information of each group within each phase. They are prepared for inner-phase process trend extraction and analysis in the next step. In inner-phase division, phases are divided into different innerphase parts according to inner-phase evolution. On the basis of the combination of PCA and QTA, it can be found that within a phase, there are generally two kinds of evolutions, as indicated by slope curve and flat line, which represent transitions at the start or the end of each phase and steady operation statuses in the middle of the phase. In general, a typical phase can be further divided into three parts according the inner-phase evolution analysis: transition from the previous phase, a steady part, and transition to the next phase. Here, the transition before the steady part is called the initial transition, and the one after the steady part is called the terminal transition. The inner-phase division is conducted by applying the combination of PCA and QTA on phase-representative data. Applying PCA on Xc(ΣGg=1Kc,gIg × J) Xc =

T TP c c

Rc

+ Ec =

∑ j=1

tj , cp Tj , c

Figure 4. A simplified schematic diagram of an injection molding machine.

Tc , g , k = Xc , g , kPc

(3)

where Tc,g,k covers the systematic variation information at the kth time interval within the cth phases in the gth group. The average scores over all batches at the same time within the cth phase in the gth group are defined as below: tc , g , k

+ Ec (2)

Then, by projecting the time intervals Xc,g,k(Ig × J) (where k = 1, 2, ..., Kc,g) within the cth phases in the gth group onto Pc, the timeslice PCs are obtained:

1 = Ig

Ig

∑ tc ,g ,k ,i i=1

(4)

where tc,g,k,i is the ith row of Tc,g,k. The first value of tc,g,k, t1,c,g,k, shows the average variation information along the first PC of the kth time-slice. Then, t1,c,g,k at different time intervals within the 4588



Article

same phase (k = 1, 2, ..., Kc,g) comprises a vector t1,c,g = [t1,c,g,1, ..., t1,c,g,k, ..., t1,c,g,Kc,g], which shows the time-varying evolution of the first PC. The interval-halving algorithm for trend extraction is applied to t1,c,g. Thus, for each phase, multiple segments are separated using seven primitives of QTA. They are the basic analysis units for inner-phase evolution. To judge which part these segments belong to (transition or steady part), the deviations of these segments are calculated as

Dċ , g , s =

description

unit

1 2 3 4 5 6 7 8 9 10

nozzle temperature nozzle pressure screw stroke screw velocity injection pressure plastication pressure SV1 opening SV2 opening cavity pressure 1 cavity pressure 2

°C bar mm mm/s bar var % % bar bar

Lc , g , s

=

t1, c , g , kend,s − t1, c , g , kst ,s kend, c , g , s − kst, c , g , s + 1

(5)

where t1,c,g,Kst,s and t1,c,g,Kend,s are the score value at the beginning and the end of the sth segment and Dc,g,s denotes the difference between them; Ḋ c,g,s is deviation of the sth segment; Lc,g,s is the duration of this segment; and kst,c,g,s and kend,c,g,s are the time indices corresponding to the beginning and the end of this segment, respectively. A threshold Ḋ *c,g should be defined on the basis of training data so that Ḋ c,g * of all segments within the cth phase in the gth group can be divided into two clusters corresponding to the steady part and transitions, respectively. Here, the threshold is defined using the two indices, median (MED) and median absolute deviation (MAD). For each phase, since the steady part is the major operation mode, the MED value is * ) in the steady part, and dominated by the deviations (Ḋ c,g deviations of the segments in the steady part are near MED. The MAD index is also dominated by the segments in the steady part since it is the middle value of differences between all deviations and MED. The threshold Ḋ *c,g can be defined as MED ± αMAD, where α is a constant attached to MAD, termed the relaxing factor here, which determines how much the deviations of segments in the

Table 1. Process Variables for Injection Molding Process no.

Dc , g , s

Figure 5. The process trajectories and phase division results with velocities of (a) 24 mm/s, (b) 32 mm/s, and (c) 40 mm/s. 4589



Article

Figure 6. The QTA segment division results of the injection phase with a velocity of (a) 24 mm/s, (b) 32 mm/s, and (c) 40 mm/s (the dotted line refers to t1,c,g; the red solid line refers to polynomial fitting results of QTA; the vertical dashed line indicates the segments represented by seven primitives.).

⎧ k=1 ⎪ 0, Δt1, c , g , k = ⎨ ⎪ ⎩ t1, c , g , k − t1, c , g , k − 1 , k = 2, ...Kc , g

steady part are allowed to change. In general, it can be set by trial and error so that each phase can be properly divided into a transition and steady part to ensure the sensitivity to the innerphase evolution. However, there is no definite criterion or uniform standard to strictly quantify it. Therefore, its determination is inevitably affected more or less by artificial subjectivity factors. The investigation on the determination of α is meaningful and deserves further devotion in the future. Here, it is set to be 0.5. If the deviation * is smaller than the threshold Ḋ c,g * , this segment is assigned to the Ḋ c,g,s steady part; otherwise, it belongs to transitions. By the above analysis, three inner-phase parts are separated from each phase. However, the above analysis cannot be applied online since QTA is performed on the entirety of phase data to find out the segments and study the deviations of each segment. To make it proper for online application, the segments within the steady part based on the above QTA analysis results are further analyzed here so that a new statistical index can be defined and applied online to indicate the affiliation (steady part or transition) for each new observation. The gradient of t1,c,g is denoted as Δt1,c,g = [Δt1,c,g,1, ..., Δt1,c,g,k, ..., Δt1,c,g,Kc,g],

(6)

where subscript k denotes the time within phase c. The vector t1,c,g within the steady part obtained by QTA is denoted as t1,c,g,sp, and the gradient of t1,c,g,sp is denoted as Δt1,c,g,sp, revealing the variations between neighboring time intervals. Then, Δt1,c,sp values across all G groups are deemed to be normally distributed, from which a 99% confidence region can be readily obtained. It defines a normal variation region for those segments within the steady part. For each time interval, compare Δt1,c,k with the predefined 99% confidence region. If the time interval shows Δt1,c,k beyond the region, it is assigned to the steady part; otherwise, it is assigned to transitions. Using the combination of QTA with PCA analysis for innerphase division, the process evolution can be tracked in detail within each phase. The steady part is separated from transitions which are then modeled differently because of their different underlying characteristics. The two-level phase division procedures are shown in Figure 2, where each process phase obtained in the first-level division procedure is further separated 4590



Article

Figure 7. The deviations of QTA segments within the injection phase with a velocity of (a) 24 mm/s, (b) 32 mm/s, and (c) 40 mm/s (the dotted line refers to the deviation of each QTA segment; the horizontal line refers to the threshold Ḋ *c,g).

where Pc,sp(J × Rc,sp) denote the variable correlations along the first Rc,sp PCs retained in the PCA model. The monitoring statistics are then calculated as below:

into three parts based on the effective combination of QTA and PCA so that the inner-phase process evolution can be tracked. Also, it is noted that for those batch processes where the operation process does not reach a steady status, only a single transition region may be identified from each phase by the proposed algorithm. For that case, how to better model and monitor the process status of each phase is another important issue which deserves further study. 3.2. Phase Division Based Process Modeling. After twolevel phase division, different statistical models should be developed for steady parts and transitions within each phase as shown in the third step in Figure 2. A common model for steady part is established based on variable-wise unfolding data matrix of steady part, while time-slice models are built for transitions. The details are introduced below. 1. Model Development for the Steady Part. The steady-part data of each uneven group are variable-wise unfolded and put together, comprising Xc,sp(ΣGg=1Kc,g,spIg × J). They are then normalized to have zero mean and unit variance. Then, a PCA model is built on Xc,sp: Xc ,sp = Tc ,spPcT,sp + Ec ,sp

tiT, c ,sp = x iT, c ,spPc ,sp

(8)

Ti2, c ,sp = ti , c ,sp TSc ,sp−1ti , c ,sp

(9)

eiT, c ,sp = x iT, c ,sp − x̂ iT, c ,sp = x iT, c ,sp − x iT, c ,spPc ,spPcT,sp

(10)

SPEi , c ,sp = ei , c ,sp Tei , c ,sp

(11)

where subscript i denotes the batch and Sc,sp is the covariance of Tc,sp. The calculation detail of confidence limits can be found in refs 36−39. 2. Model Development for Transitions. After the inner-phase division, those transitions from different uneven groups are synchronized by the curve fitting method.40 Therefore, the action of monitoring is retrospective in the transition region. Time-slice data are obtained by putting time-slice data from different groups together, Xc,tr,k(ΣGg=1Ig × J) (where subscript tr denotes transition), which is then normalized to have zero mean and unit variance.

(7) 4591



Article

Figure 8. The inner-phase division results of the injection phase with a velocity of (a) 24 mm/s, (b) 32 mm/s, and (c) 40 mm/s (the dotted line refers to the gradient of t1,c,g (Δt1,c,g); the horizontal dashed line refers to the 99% confidence region).

First, the current phase is judged by indicator variables. For the current phase c, a new observation is denoted as xc,new,k, which is normalized by the mean and standard deviation used for innerphase division. To online judge which inner-phase part the current sample lies in, the new data is projected onto the subspace spanned by the loading matrix Pc obtained in eq 2

Time-slice monitoring models are then built by applying PCA to these time slices Xc ,tr , k = Tc ,tr , kPcT,tr , k + Ec ,tr , k

(12)

where k = 1, 2, ..., Kc,tr, Kc,tr is the total number of time intervals within the transition region in the cth phase. The monitoring statistics are then calculated at each time: tiT, c ,tr , k

Ti2, c ,tr , k

=

x iT, c ,tr , kPc ,tr , k T

tc ,new, k = xc ,new, kPcT

(13) −1

= ti , c ,tr , k Sc ,tr , k ti , c ,tr , k

The first value of tc,new,k, t1,c,new,k, is then compared with that from its previous sample, t1,c,new,k−1, and the gradient Δt1,c,new,k is obtained on the basis of the rule defined in eq 6. Then, compare Δt1,c,new,k with the predefined 99% confidence region. If Δt1,c,new,k is well within this region, the new sample is judged to belong to the steady part within the current phase; otherwise, it is assigned to the transition region. If the new sample xc,new,k is judged to belong to a steady part, it is renormalized using the data normalization information from training data used for development of the steady-part model before eq 7, denoted as xc,new,k,sp. The online monitoring model for steady part will be adopted. xc,new,k,sp is projected onto Pc,sp,

(14)

eiT, c ,tr , k = x iT, c ,tr , k − x̂ iT, c ,tr , k = x iT, c ,tr , k − x iT, c ,tr , kPc ,tr , k PcT,tr , k (15)

SPEi , c ,tr , k = ei , c ,tr , k Tei , c ,tr , k

(17)

(16)

where subscripts i, c, and k denote batch, phase, and time respectively; Sc,tr,k is the covariance of Tc,tr,k. 3.3. Online Monitoring. The strategy for online monitoring is shown in Figure 3, including online identification of innerphase parts and realtime monitoring of inner-phase parts. The details are introduced below.

tc ,new, k = xc ,new, k ,spPcT,sp 4592

(18)



Article

Figure 9. The online monitoring results of three uneven batches in the injection phase with velocity of (a) 24 mm/s, (b) 32 mm/s, (c) 40 mm/s (vertical dotted line refers to the boundary of three inner-phase parts in injection phase; dotted line and dashed line refer to the 99% and 95% control limits, respectively; solid line refers to the hotelling T2 or SPE monitoring values).

Hotelling T2 and SPE statistics are then calculated as below: Tc2,new, k = tc ,new, k TSc ,sp−1tc ,new, k T

SPEc ,new, k = ec ,new, k ec ,new, k

Hotelling T2 and SPE statistics are calculated at each time as below

(19) (20)

ecT,new, k = x cT,new, k ,st − x̂ cT,new, k ,st = x cT,new, k ,st − x cT,new, k ,trPc ,stPcT,st

(23)

SPEc ,new, k = ec ,new, k Tec ,new, k

(24)

ecT,new, k = x cT,new, k ,tr − x̂ cT,new, k ,tr = x cT,new, k ,tr − x cT,new, k ,trPc ,tr , k PcT,tr , k

(21)

(25)

Hotelling T2 and SPE statistics are compared with the control limits of the steady part or transitions at each time. If any one of the two monitoring statistics exceeds control limits, a fault is detected.

If the new sample xc,new,k is judged to belong to the transition, it has to wait until all the transition samples within the current transition region are available. At the end of each transition region, all new samples within the transition region are synchronized by a curve fitting method.40 Because transition length is shorter in comparison with that of the steady part, the delay for monitoring of the transition is acceptable. After data synchronization, the normalized sample at each time xc,new,k,tr is projected to time-slice PCA models obtained in eq 12 tc ,new, k = xc ,new, k ,trPcT,tr , k

Tc2, k = tc , k TSc ,tr , k −1tc , k

4. ILLUSTRATION AND DISCUSSION 4.1. Process Description. The research is illustrated by a typical multiphase batch process, injection molding, which is an important manufacturing process. Figure 4 shows a simplified schematic diagram of an injection molding machine. A typical injection molding process consists of three major operation phases, the injection of molten plastic into the mold, packing-holding of the

(22) 4593



Article

material under pressure, and cooling of the plastic in the mold until the part becomes sufficiently rigid for ejection. In addition, plastication takes place in the barrel in the early cooling phase, where the polymer is melted and conveyed to the barrel front by screw rotation, preparing for next cycle. It can be readily implemented for experiments, in which all key process conditions such as the temperatures, pressures, displacement, and velocity can be online measured by their corresponding transducers, providing abundant process information. The material used in this work is high-density polyethylene (HDPE). Ten process variables as shown in Table 1 are selected for modeling, which can be collected online from measurements with a set of sensors. In this work, different operation recipes of injection are adopted by setting the injection velocity at 24, 32, and 40 mm/s, respectively, resulting in three different uneven groups regarding the injection phase. The packing-holding time is fixed to be 3 s; the cooling time is set to be 15 s. So for each batch process, all the other phases have the same duration except the injection phase. 4.2. Two-Level Phase Division. As introduced before, two levels of process divisions, conventional phase division and innerphase division, are performed to track the inner-phase evolution. First, conventional phase division is implemented. Using indicator variables, each batch process can be divided into four phases. The process variables, screw velocity, and SV1 opening are chosen to be indicator variables based on process knowledge. When the value of screw velocity is not zero, it means the screw moves forward, revealing the starting of the batch process. Also, screw velocity can be chosen to indicate the switch points of phases. Meanwhile, SV1 opening can be used to indicate the end of a batch cycle. In Figure 5, the phase division results are illustrated by three batches with different velocities, and the four phases (I−IV) are separated from each batch process, injection, packing-holding, plastication, and cooling. Second, inner-phase division is performed on the first-level phase division results where the injection phase is the focus since the uneven problem exits in this phase. t1,c,g values from the three uneven groups are analyzed by QTA, as shown in Figure 6. For the three uneven groups, denoted by a through c, t1,c,g obviously have a similar evolution, represented by two slopes before and after a flat line, revealing the fact that the process first evolves to the steady state and then departs from it within each phase. By analyzing the deviations of these segments obtained by QTA, as shown in Figure 7, ththe e steady parts of the three groups are identified, which are indicated by deviations above the threshold Ḋ c,g * . Further, to make it proper for online application, the segments within steady part are further analyzed. New statistics, Δt1,c,g,sp(g = 1,2,3), are calculated, and the 99% confidence region for online inner-phase division is defined, by which, the conventional phase is further divided into initial transition, the steady part, and terminal transition by comparing Δt1,c,g,k with the predefined 99% confidence region. The gradients Δt1,c,g,k are shown in Figure 8 together with inner-phase division results for the injection phase. The other even phases will also be analyzed for inner-phase division. The steps are similar to those for the injection phase but much simpler since they do not have the uneven-length problem. After inner-phase division, different PCA models are developed for each inner-phase part to capture their different characteristics and will be used for online application. Since the injection phase is the concerned phase, which has the uneven-length problem, the monitoring action will focus on this phase. 4.3. Online Monitoring. Three normal batches with injection velocities of 24, 32, and 40 mm/s are put into online

Figure 10. The online monitoring results of a fault batch with a velocity of 40 mm/s in the injection phase for (a) hotelling T2 and (b) SPE (vertical dotted line refers to the boundary of three inner-phase parts in injection phase; dotted line and dashed line refer to the 99% and 95% control limits, respectively; solid line refers to the hotelling T2 or SPE monitoring values).

testing. As shown in Figure 9, by the proposed algorithm, first, the injection phase of each batch can be divided into three innerphase parts, including initial transition, the steady part, and terminal transition, revealing the process evolution information in the injection phase. In detail, in Figure 9a, where the injection velocity is 24 mm/s, the separated inner-phase parts include initial transition (time intervals 1st−17th), the steady part (time intervals 18th−70th), and terminal transition (time intervals 71st−82nd). Then, these different inner-phase parts are monitored in different inner-phase parts using different monitoring models. The confidence limits of hotelling T2 and SPE are also defined corresponding to three parts. As shown in Figure 9, both hotelling T2 and SPE values stay well within the control limits defined in each inner-phase part, revealing that they are normal batch cycles following the normal inner-phase process evolution rules, which agrees well with the fact. One fault type is then used for online fault detection where the screw stroke is set to be abnormally larger than the normal case. The hotelling T2 and SPE online monitoring results in the 4594



Article

Figure 11. Contributions of variables to SPE for (a) all time intervals through the injection phase, (b) during time intervals 1−17, and (c) during time intervals 30−40.

injection phase are shown in Figure 10a and b, respectively. In Figure 10b, it is observed that the SPE values exceed control limits from the beginning of the initial transition, revealing the abnormality. Comparatively, this fault cannot be detected until the middle of the injection phase using the method proposed in previous work.19 The results are not shown here. The better fault detection performance may result from the use of inner-phase division where the transition patterns are separated from the steady part and they are then characterized more accurately. On the basis of the detection results, fault diagnosis can be conducted in time, which can result in more accurate and reasonable fault isolation results since the influences of fault may be propagated to other variables if fault detection is delayed. As shown in Figure 11, contributions of different variables to SPE are plotted in injection phase. To more clearly reveal the significance of timely fault detection for fault diagnosis, the changes of variable contributions along the time direction are plotted for each variable as shown in Figure 11a. Then, the variable contributions within the time intervals 1−17 (approximately initial transitions) are plotted in Figure 11b. In Figure 11b, it can be seen clearly that variable 3 contributes mostly to this fault.

Therefore, variable 3 (screw stroke) is identified as the root cause of this fault, which agrees well with the real case. To reveal the influences of delayed fault detection on fault diagnosis, the variable contributions within the time intervals 35−50 are analyzed in Figure 11c. It is observed after the 36th time interval that the contributions of variable 3 are relatively small in comparison with those of variables 7 and 8 (SV1 opening and SV2 opening). Therefore, fault cause may not be correctly identified if fault detection is delayed. On the basis of the analysis of the inner-phase evolutions, more process operation information can be obtained within each phase. Their different characteristics can be better modeled, and thus faster fault detection is obtained. The uneven-length problem is also well handled. Both the online monitoring results and fault diagnosis results are satisfactory.

5. CONCLUSIONS In the present work, process trend based statistical modeling and an online monitoring strategy are proposed for multiphase batch processes which can track inner-phase evolution and handle the uneven-length problem. Two levels of a process division algorithm are developed which can separate transitions from the steady part 4595



Article

(16) Zhao, C. H.; Lu, N. Y.; Wang, F. L.; Jia, M. X. Stage-based softtransition multiple PCA modeling and on-line monitoring strategy for batch processes. J. Process Control 2007, 17, 728−741. (17) Yao, Y.; Gao, F. R. Phase and transition based batch process modeling and online monitoring. J. Process Control 2009, 19, 816−826. (18) Lu, N. Y.; Gao, F. R.; Yang, Y.; Wang, F. L. PCA-based modeling and on-line monitoring strategy for uneven-length batch processes. Ind. Eng. Chem. Res. 2004, 43, 3343−3352. (19) Zhao, C. H.; Mo, S. Y.; Gao, F. R.; Lu, N. Y.; Yao, Y. Statistical analysis and online monitoring for handling multiphase batch processes with varying durations. J. Process Control 2011, 21, 817−829. (20) Batch control. Part 1, Models and terminology; Instrument Society of America: Research Triangle Park, NC, 1995. (21) Melek, W. M.; Lu, Z.; Kapps, A.; Fraser, W D. Comparison of trend detection algorithms in the analysis of physiological time-series data. IEEE Trans. Biomed. Eng. 2005, 52 (4), 639−651. (22) Cheung, J. T.; Stephanopoulos, G. Representation of process trendspart I. A formal representation framework. Comput. Chem. Eng. 1990, 14 (4/5), 495−510. (23) Janusz, M.; Venkatasubramanian, V. Automatic generation of qualitative description of process trends for fault detection and diagnosis. Eng. Appl. Artif. Intell. 1991, 4 (5), 329−339. (24) Bakshi, B. R.; Stephanopoulos, G. Representation of process trendsIII. Multiscale extraction of trends from process data. Comput. Chem. Eng. 1994, 18 (4), 267−302. (25) Rengaswamy, R.; Hagglund, T.; Venkatasubramanian, V. A qualitative shape analysis formalism for monitoring control loop performance. Eng. Appl. Artif. Intell. 2001, 14 (1), 23−33. (26) Vedam, H.; Venkatasubramanian, V.; Bhalodia, R. A B-Spline based method for data compression, process monitoring and diagnosis. Comput. Chem. Eng. 1997, 21, S655−S660. (27) Haimowitz, I.; Phillip, P. L.; Kohane, I. Clinical monitoring using regression-based trend templates. Artif. Intell. Med. 1995, 7, 473−496. (28) Haimowitz, I.; Phillip, P. L.; Kohane, I. Managing temporal worlds for medical trend diagnosis. Artif. Intell. Med. 1996, 8, 299−321. (29) Hau, D.; Coiera, E.Learning qualitative models from physiological signals. In Proceedings of the AAAI Spring Symposium Series: Artificial Intelligence in Medicine; American Association for Artificial Intelligence: Menlo Park, CA, 1994; pp 67−71. (30) Hau, D.; Coiera, E. Learning qualitative models of dynamic systems. Mach. Learn. 1993, 26, 177−211. (31) Poirier, P. J.; Meech, J. A. Using fuzzy logic for on-line trend analysis. Second IEEE Conference on control applications; IEEE: New York, 1993; Vol. 9, pp 13−16. (32) Dash, S.; Maurya, M. R.; Rengaswamy, R.; Venkatasubramanian, V. A Novel Interval-Halving Framework for Automated Identification of Process Trends: Extended Version; Technical Report CIPAC-03-3, Purdue University: West Lafayette, IN, 2003. (33) Dash, S.; Maurya, M. R.; Rengaswamy, R.; Venkatasubramanian, V. A novel interval-halving framework for automated identification of process trends. AIChE J. 2004, 50 (1), 149−162. (34) Dash, S.; Rengaswamy, R.; Venkatasubramanian, V. Fuzzy-logic based trend classification for fault diagnosis of chemical processes. Comput. Chem. Eng. 2003, 27 (3), 347−362. (35) Maurya, M. R.; Rengaswamy, R.; Venkatasubramanian, V. Fault diagnosis by qualitative trend analysis of the principal components. Chem. Eng. Res. Des. 2005, 83 (A9), 1122−1132. (36) Jackson, J. A User’s Guide to Principal Components; Wiley: New York, 1991. (37) Jolliffe, I. Principal Component Analysis, 2nd ed.; Springer: New York, 2002. (38) Wold, S. Cross-validatory estimation of the number of components in factor and principal components models. Technometrics 1978, 397−405. (39) Nomikos, P.; MacGregor, J. Multivariate SPC charts for monitoring batch processes. Technometrics 1995, 41−59. (40) Coope, I. D. Circle fitting by linear and nonlinear least squares. J. Opt. Theory Appl. 1993, 76 (2), 381−388.

within each phase. Consequently, different statistical modeling and online monitoring strategies are developed for different inner-phase parts. In this way, the process trend within each phase is clearly captured. In the application to the injection molding process, the proposed strategy works well for the online monitoring of the uneven-length phase and meanwhile offers satisfactory results of fault detection.

■

AUTHOR INFORMATION

Corresponding Author

*Tel: 86-571-87951879 (Z.C.), 852-23587139 (F.G.). Fax: 86571-87951879 (Z.C.), 852-2358 0054 (F.G.). E-mail: chhzhao@ zju.edu.cn (Z.C.), [email protected] (F.G.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is supported by Hong Kong Research Grant Council, General Research Fund (No. 612512), the National Natural Science Foundation of China (No. 61273166), the Fundamental Research Funds for the Central Universities (2012QNA5012), Project of Education Department of Zhejiang Province (Y201223159), Technology Foundation for Selected Overseas Chinese Scholar of Zhejiang Province and the Foundation of Key Laboratory of System Control and Information Processing, Ministry of Education, P.R. China.

■

REFERENCES

(1) Wold, S.; Kettaneh, N.; Fridén, H.; Holmberg, A. Modelling and diagnostics of batch processes and analogous kinetic experiments. Chemom. Intell. Lab. Syst. 1998, 44, 331−340. (2) Martin, E.; Morris, A.; Papazoglou, M.; Kiparissides, C. Batch process monitoring for consistent production. Comput. Chem. Eng. 1996, 20, 599−604. (3) Lee, J.; Dorsey, A. Monitoring of batch processes through statespace models. AIChE J. 2004, 50, 1198−1210. (4) Lane, S.; Martin, E.; Kooijmans, R.; Morris, A. Performance monitoring of a multi-product semi-batch process. J. Process Control 2001, 11, 1−11. (5) Jackson, J. A User’s Guide to Principal Components; Wiley: New York, 1991. (6) Wold, S.; Ruhe, A.; Wold, H.; Dunn, W., III The collinearity problem in linear regression. The partial least squares (PLS) approach to generalized inverses. SIAM J. Sci. Stat. Comput. 1984, 5, 735−743. (7) Geladi, P.; Kowalski, B. Partial least-squares regression: a tutorial. Anal. Chim. Acta 1986, 185, 1−17. (8) Nomikos, P.; MacGregor, J. F. Monitoring batch processes using multiway principal component analysis. AIChE J. 1994, 40, 1361−1375. (9) Kosanovich, K. A.; Piovoso, M. J.; Dahl, K. S. Multi-Way PCA applied to an industrial batch process. Proc. Am. Control Conf. 1994, 2, 1294−1298. (10) Dong, D.; McAvoy, T. J. Multi-stage batch process monitoring. Proc. Am. Control Conf. 1995, 3, 1857−1861. (11) Ü ndey, C.; Cinar, A. Statistical monitoring of multistage, multiphase batch processes. IEEE Control Syst. Mag. 2002, 22, 40−52. (12) Lu, N. Y.; Wang, F. L.; Gao, F. R. Sub-PCA modeling and on-line monitoring strategy for batch processes. AIChE J. 2004, 50, 255−259. (13) Zhao, C. H.; Wang, F. L.; Gao, F. R.; Lu, N. Y.; Jia, M. X. Adaptive Monitoring Method for Batch Processes Based on Phase Dissimilarity Updating with Limited Modeling Data. Ind. Eng. Chem. Res. 2007, 46, 4943−4953. (14) Zhao, C. H.; Wang, F. L.; Gao, F. R.; Lu, N. Y.; Jia, M. X. Improved knowledge extraction and phase-based quality prediction for batch processes. Ind. Eng. Chem. Res. 2008, 47, 825−834. (15) Zhao, C. H.; Wang, F. L.; Mao, Z. Z.; Lu, N. Y.; Jia, M. X. Quality prediction based on phase-specific average trajectory for batch processes. AIChE J. 2008, 54, 693−705. 4596


Inner-Phase Analysis Based Statistical Modeling and Online

Recommend Documents