Pheromone Propagation Controller: The Linkage of
Swarm Intelligence and Advanced Process Control
Der-Shui Lee and An-Chen Lee, Member, IEEE
Abstract—Statistical process control (SPC) is traditionally used
in advanced process control (APC). However, SPC, which treats measurements as a series of isolated statistical data, employs dif-ferent methods to deal with difdif-ferent problems. In this paper, we present a new perspective on process control, which treats the in-tercepts of the process in different runs as a social insect colony. Our novel algorithm, called the pheromone propagation controller (PPC), is a meta-heuristic method based on the assumption that the intercepts of the linear regression model have their own be-havior and affect others nearby on different runs. The pheromone basket is an environment initially filled with intercepts, and then the “intercepts pheromones” in the basket propagate according to the modified digital pheromone infrastructure. After propagation, the intercept in the next run can be forecast by extrapolating the last two entities of the pheromone basket. Consequently, a revised process recipe can be obtained from the forecast intercepts and the linear regression model. We also propose a workable scheme for adaptively tuning the PPC propagation parameter. We discuss the PPC stability region and the strategy for tuning the propaga-tion parameter as well as the effects of size of pheromone basket, model mismatch on the performance. Our simulation results show that the standard deviation and the mean square error for PPC, whether fixed or self-tuning, are more consistent than that of the EWMA, the predictor corrector control (PCC), and the double EWMA for five types of anthropogenic disturbance. We also ex-amined a hybrid disturbance obtained from semiconductor fab-rication. When system drifts, the PPC was superior to the other candidate controllers for all values of the PPC propagation pa-rameters and weightings of the other controllers, whether fixed or self-tuning.
Index Terms—Digital pheromone infrastructure, pheromone
basket, pheromone propagation controller, process control, swarm intelligence.
I. INTRODUCTION
I
N PROCESS control, combining statistical process control (SPC) and feedback control [1] is a popular technique. Quin et al. [2] divided semiconductor manufacturing process control into four levels: equipment control, run-to-run con-trol, island concon-trol, and fab-wide control. The lowest level is equipment-level control, which holds the desired parameters of tools. Run-to-run control adjusts the recipe slightly based on in-line measurements to even out disturbances. Island control shares information among tools to achieve tool matching andManuscript received January 30, 2008; revised December 25, 2008; accepted May 04, 2009. First published July 07, 2009; current version published August 05, 2009.
The authors are with the Department of Mechanical Engineering, National Chiao-Tung University, Hsinchu City, Taiwan (e-mail: [email protected]. tw).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSM.2009.2024865
feed-forward/feedback control. The highest level is fab-wide control, which optimizes the desired electrical properties by adjusting the target of the lower level controller. This study ad-dresses advanced process control (APC), which is a run-to-run control.
The exponentially weighted moving average (EWMA) con-troller is widely used in semiconductor APC. EWMA weighs past data with an exponential discount factor [3]. The stability and sensitivity of EWMA have been analyzed [4]–[6] and some attempts have been made to solve the problem of tuning the discount factor of the EWMA controller in different ways [7]–[11]. Predictor corrector control (PCC) and double EWMA control have been proposed to improve the performance of EWMA when dealing with drifting processes [12], [13]. The stability of double EWMA and multiple-input multiple-output (MIMO) double EWMA has been demonstrated [14], [15], and the weightings for the double EWMA controller have been tuned using different methods [16], [17]. The initial intercept iteratively adjusted (IIIA) controller [18] is used to modify double EWMA with different update procedures to set initial intercepts under drifting disturbance conditions. However, the EWMA-based solvers reject only specific types of disturbance and cannot adapt to a complex environment.
In addition to EWMA and double EWMA, artificial neural networks (ANNs) [19] map the relationship between input and output directly. ANNs, ant colony optimization (ACO), and data mining have been integrated to illustrate the “black box” of ANNs in chemical mechanical polishing (CMP) processes [20]. However, ANNs require many parameters in a suitably con-structed network with appropriate training data. The recursive least square (RLS) technique [21] can model a constant mean and a linear trend or random walk for online estimation, but the RLS controller may be unstable if the system gain varies with time.
In this paper, we describe an algorithm that adapts easily to complicated disturbances and the uncertainty of the linear re-gression model. The concept comes from the observation that current errors are caused by previous errors, and the symptoms of future errors come from present errors in the real world. The new algorithm is a meta-heuristic method, which uses swarm intelligence by assuming that each intercept has its own be-havior that affects nearby intercepts at different runs. Specifi-cally, swarm algorithms or devices are inspired by the collective behaviors of social insect colonies and other animal societies [22]. ACO [23], for example, is motivated by the behavior of ants in finding paths from their colony to food. Particle swarm optimization (PSO) [24] is inspired by the social behavior of or-ganisms such as birds in a flock or fish in a school. Stochastic
Fig. 1. Concept of the pheromone propagation controller (PPC).
diffusion search (SDS) [25] employs the tandem calling mech-anism of ants to perform cheap, partial evaluations of a candi-date solution to a search problem. In the semiconductor industry, Jiang et al. employed pheromone rules of ACO to solve dynamic scheduling problems for a fabrication line [26].
This paper presents the pheromone propagation controller (PPC) based on the digital pheromone infrastructure [27], which was inspired by the chemical dynamics of pheromone transition and was recently used to describe the decentralized self-orga-nizing behavior of unmanned aerial vehicles (UAVs) and bat-tlefield tactics [27]–[31]. However, the digital pheromone infra-structure must be modified to avoid the end effect. Under the modified digital pheromone infrastructure, the intercepts of a linear regression model in different runs are modeled as a social insect colony. The interaction among intercepts is modeled by a propagation mechanism, which means that an intercept affects others nearby.
Fig. 1 shows the overview of PPC. In Fig. 1(a), the pheromone basket is a moving window that is initially filled with intercepts or pheromones; is the coordination within the pheromone basket and can map to run. The shape of the pheromone basket in Fig. 1(a) is a single-dimensional line, but could be different for other specific applications. Fig. 1(b) illus-trates an example of the input of pheromone basket. Then, the
intercepts in the pheromone basket propagate themselves into a steady state according to the modified digital pheromone infra-structure, which will be described in Section II. The final prop-agation result is shown in Fig. 1(c), which reflects the tendency of the external inputs (or intercepts) in the specific pheromone basket. Fig. 1(d) shows that the intercept for the next run can be forecast by extrapolating the last two entities of the final propagation result in the pheromone basket. Particularly, Figs. 1(a)–1(d) are all at the same time stamp. Finally, the process recipe for the next run can be obtained by the forecast intercept and the process model. We also propose a workable scheme for adaptively tuning the propagation parameter in Section III-F.
We conducted simulations to compare the performance of PPC with EWMA, PCC, and double EWMA. Five types of anthropogenic disturbance and semiconductor fabrication data were used as disturbances in the simulations with three perfor-mance indices (average, standard deviation and mean square error) to evaluate the performance. The controller parameters such as the PPC propagation parameter and the weights of the other controllers were fixed and were obtained from historical data with minimum sum square error. The proposed self-tuning PPC, the self-tuning EWMA [9], and the self-tuning PCC [17] were also included in the simulations.
The rest of this paper is organized as follows. Section II ex-plains our modifications to the digital pheromone infrastruc-ture. Section III illustrates the PPC structure including param-eter tuning to improve performance. Section IV analyzes the PPC stability region and compares the controller structure of PPC with other controllers. Section V shows the simulation re-sults for the proposed controller. The final section is the conclu-sion and describes related areas for future work.
II. THEMODIFIEDDIGITALPHEROMONEINFRASTRUCTURE FORPPC
In nature, pheromone, a chemical substance, is released by an insect or animal which causes another individual of the same species to react. When pheromone is released, it evaporates from one position and propagates itself by wind. So, pheromone has different states and transition dynamics. The digital pheromone infrastructure [27] imitated the mech-anism of pheromone by modeling the environment, states of pheromone and transition dynamics of a pheromone. While the natural pheromone is powered by wind and time, the digital pheromone infrastructure models the transition dynamics by transition parameters (evaporation and propagation param-eters), transition functions where iterations for pheromone propagation are practiced. Therefore, as compared the power of natural with digital pheromones, wind maps to transition parameters and transition functions; time maps to iterations.
This section introduces the modified digital pheromone infrastructure. We define pheromone basket, mapping to the environment, as a moving time window in the manufacturing process. The pheromone states and transition parameters are the same as defined in the digital pheromone infrastructure. Because of the end effect and the energy balance in our one-dimensional pheromone basket, we modified the transition functions in [27].
A. Pheromone Basket
The pheromone basket is the pheromone propagation envi-ronment. The environment of the modified digital pheromone infrastructure is a tuple , where is a finite set of
po-sitions M within the pheromone
basket, M is the size of the pheromone basket, and maps to the current run and maps to M runs beforehand. In addi-tion, is a finite set of neighbors of and
is the size of . In addition, the modified pheromone in-frastructure assumes that the propagation relationship between and is irreflective, which means that will accept propagation inputs from without preconditions.
From Fig. 1(a), the pheromone basket fills with “intercept pheromones” initially. Since the “intercept pheromones” propa-gate themselves into a steady state at a time stamp from Fig. 1(b) to Fig. 1(c), the measurement of the intercepts can be treated as the external impulse input to the pheromone basket. The ex-ternal input is a finite set
where is the run number of
the manufacturing process, is the number of iterations (or propagations) in the transition functions, and is the global limit of the external inputs in the environment . Note that iteration is executed to update transition functions but not means the process. We use the notation for the set of natural numbers and for the set of real numbers. Since the external input is the initial condition for launching transition functions of pheromone propagation, maps to the intercept at
the run on the process and is 0 when
is larger than 0. B. Pheromone States
The states in the pheromone basket are and , where
is a finite set of the propagated inputs at run and iteration ,
and is a
fi-nite set of the aggregated pheromones at run and iteration . Therefore, is regarded as the propagated input from to at iteration and run . Similarly, is re-garded as the aggregated pheromone of at iteration and run . In PPC, refers to the effectiveness of one intercept affecting its nearby intercepts; refers aggregation result which involves the influence of external input and prop-agation effect .
In addition, and are
as-sumed to be the default initial conditions. While launching transition functions as shown in the following sections, disseminates pheromones and aggregates pheromones simultaneously.
C. Pheromone Transition Parameters
Two transition parameters of the digital pheromone infra-structure [27] are the evaporation parameter and the propagation parameter . In PPC, the propagation pa-rameter describes the effect of a measurement on other nearby measurements, and the evaporation parameter indicates the weakening property of the monitored data. In other words, in-dicates that the importance of the measurement data will “evap-orate” with time. Because measurements in a short period can be treated as a nondissipative system, the modified pheromone infrastructure uses in a small pheromone basket (Proof is given in Appendix A).
D. Transition Functions
While defining the pheromone basket , the parameters and , the external input , state , and , Brueckner introduced two transition functions to describe the propagated inputs and the aggregated pheromone [27]. The transition function of the propagated inputs
is
(1) where indicates that is the neighbor of ,
is the number of neighbors of , and
indicates that affects its neighbors equally. Because the shape of the proposed pheromone basket is a one-dimensional line, the basket has only one neighbor at the two ends of the basket and two neighbors in the other positions. Thus, (1) be-comes (2), found at the bottom of the page.
In addition, the transition of the aggregated pheromone is defined as [27]
(3)
Because propagates the ratio of
out to in (2), the
ratio of should be subtracted from
(3) to maintain its balance. Thus, we modify (3) to be
(4) With is 1, (4) becomes (5) if if if if if (2)
Fig. 2. Boundary effect of pheromone propagation without a modifier.
Moreover, the profile of the final transition result of
should be flat when the pattern of the external input is flat. Unfortunately, the profile of the final propagation result ob-tained from (2) and (5) is not flat due to the end effect as shown in Fig. 2. To overcome this, we modify the propagation-out ratio
in the two extremities from to (See the
proof in Appendix B) when the shape of the pheromone basket is a line and (2) and (5) become(6) and (7), found at the bottom of the page.
Fig. 3 illustrates the concept of (6) and (7). The final propa-gation results of (6) and (7) can be obtained analytically by (A9) in the Appendix C. For example, if M is 6, the final propagation results for and are (8) and (9), found at the bottom of the page.
III. PHEROMONEPROPAGATIONCONTROLLER
Fig. 4 illustrates the PPC block diagram for APC according to the modified pheromone infrastructure described in Section II. The PPC can be separated into four modules: plant, pheromone basket generator, intercept predictor, and recipe generator. This section describes each of these modules. In addition, a workable propagation parameter tuner in the intercept predictor is also proposed.
A. Plant
The plant is the process model in a simulation or in a real system, which can be obtained by the linear regression model. Because fabrication recipes usually have several parameters, the demonstrated MISO plant, which has recipes (inputs), is
(10) where
measurement at the end of run ;
recipes (inputs) of run and is a matrix; initial intercept of the process;
system gain of run and is a matrix; disturbance, which includes noise and uncontrolled terms of run .
In (10), the intercept of run , , becomes
(11) if if if if if (6) if if (7) (8) (9)
Fig. 3. Transitions with propagation parameterF and size M in a pheromone basket.
Fig. 4. Block diagram of a pheromone propagation controller.
Furthermore, this study considers two cases of system gain: i) The fixed system gain with bias:
(12) where is the fixed shift of the process gain and size of
is .
ii) The system gain shifts and drifts simultaneously, (13) where is the constant drifting rate of the process gain. B. Pheromone Basket Generator
The concept of the pheromone basket is the most important part of the PPC. Treating the intercepts of the linear regres-sion model for a sequence of runs as external inputs to the pheromone basket links process control with the modified digital pheromone infrastructure. The pheromone basket is initially filled with intercepts, which map the external input to the modified digital pheromone infrastructure. Then, states and are updated simultaneously by the transition functions of the modified infrastructure. Because the intercept of run can be obtained by the forecast intercept in
the previous run and the error in the current run, the external input of at run becomes
(14) where is the forecast intercept of run and is the error of run .
Equation (14) shows that the pheromone basket generator is a series of intercepts with a sequence of delays. The external input at of run maps the intercept of the previous runs.
In addition, the error of run in (14) is defined as
(15) where is a fixed process target.
C. Intercept Predictor
The intercept predictor propagates of the pheromone basket and the extrapolation of the last two entities of the aggregated pheromone to achieve one-step-ahead forecasting for the next run. In Fig. 4, the input to the intercept predictor func-tion block is a series of intercepts with a sequence of time delays
and propagation parameter , and the output of the intercept predictor is the forecast intercept at the next run.
The propagation of the pheromone basket obeys the modified transition functions in Section II. In addition, the final propa-gation result reflects the trend of the specific external input at run . Then, the forecast intercept at run can be inferred simply by extrapolating the last two entities of :
(16) Using (A9), the analytic solution of (16) yields
(17) where .. . .. .
is the external inputs of all of the positions at the pheromone basket. In addition, , which varies with the size of the pheromone basket M, is an algebraic expression of . For ex-ample, if M is 6, substituting (8) and (9) into (17) yields (18), found at the bottom of the page.
Figs. 5(a)–(c) shows an example of different forecast inter-cepts with an external ramp input and different propagation pa-rameters for a pheromone basket size . Fig. 5(a) shows the external ramp inputs, and Fig. 5(b) illustrates the final propagation results of Fig. 5(a) with different values of . Fig. 5(c) is the extrapolated result of Fig. 5(b). If is 0, the final propagation result is directly equal to one of the last two extrapolated entities of the external inputs. However, the extrap-olated result approaches the mean of the external inputs as approaches unity.
D. Recipe Generator
The recipe generator generates the recipe of the process for the next run. We use the linear regression model to produce
(19) where , which is a matrix, is the recipe (input) of run , is the estimator of the initial intercept , and
Fig. 5. Forecast intercept for different external inputs and pheromone parame-ters: (a) External inputs, (b) Final propagation results for different values ofF , and (c) Forecast intercepts for different values ofF .
, which is a matrix, is the estimator of the system (model gain).
In (19), and are obtained from the linear regression model by the off-line DOE and are not equal to the process parame-ters and . The parameter comes from the intercept pre-dictor, and is the given target value. Then the recipe for run
is generated by
(20)
E. Propagation Parameter F Tuner
This section proposes two strategies for tuning the propaga-tion parameter . The first strategy uses historical data and ex-amines all possible values to obtain the best fixed propagation parameter, , with minimum mean square error:
(21) where is the index of the historical data. Then is used in the testing data.
The second strategy employs and available from the last run for adaptively tuning at run , . The adjusted propagation parameter is given by
(22) where all possible forecast intercepts are obtained after each run by (A9) and (17) to obtain , which can be interpreted as the best parameter for run .
Since noise will be included in (22), we conduct with a moving average filter to avoid overcorrection and the perturba-tion of noise:
(23) where is the size of the filter and its influence on performance is presented in Section V-A1.
F. Control Procedure
The PPC requires M historical intercepts, where M is the de-sired size of the pheromone basket. The whole control procedure is presented in the following sequence, including the case when the run number is less than M at the beginning of the process. Step 1) Set the following initial conditions: and are zero; input of run 1 is the initial recipe; if is self-tuning, let , , and approach 1.
Step 2) Get the process error , set to , and determine the recipe using (20).
Step 3) Get the process error and put , , , and into (14) to generate external inputs to the pheromone basket for run 2.
Step 4) Set to from (21) when is fixed and selected by historical data. When is self-tuning, calculate forecast propagation parameter using (23). Step 5) Use (A9) with propagation parameter to forecast
the steady-state value of the aggregated pheromone at run 3, and predict the process intercept using (17).
Step 6) Determine the recipe using (20) and obtain process error .
Step 7) Put and into (14) to generate the external input of the pheromone basket for run 3.
Step 8) Repeat steps 4–7 by replacing and with
and , respectively, for . This
produces the recipe and the error at run .
If the run is larger than the desired size of pheromone basket M, update the pheromone basket by (14).
Fig. 6. PPC stability region for different sizes M of the pheromone basket.
IV. STABILITYANALYSIS
In this section, we analyze the stability of the SISO PPC. We first derive its transfer function from the block diagram and then discuss the stability region under the conditions of different model mismatch and propagation parameters.
A. Transfer Function
Because the analytic solution of pheromone propagation has been determined, the transfer function of the PPC can be derived from Fig. 4. The figure shows that the pheromone basket gener-ator is modeled by a sequence of integer delays corresponding to a series of intercepts. The intercept predictor uses (17) to fore-cast the intercept of the next run, and the model gain is obtained using the linear regression model. Thus, the transfer function from target to output can be obtained by Mason’s gain formula: (24) where and are and with is one. In (24), the charac-teristic function has M poles, which can be used to check the stability region of the PPC as shown in the next section. In ad-dition, (24) and Fig. 4 indicates that PPC is an Mth order con-troller. The higher order controller has the capacity of elimi-nating higher order disturbances.
B. Stability Region
The stability region of the PPC with a constant model and process gain can be obtained by checking the pole locations of the characteristic function in (24), where the order of the char-acteristic function varies with the size of the pheromone basket M. We examined the PPC stability region in terms of model mis-match with different propagation parameters, , and the size of the pheromone basket, M. If all of the pole locations are within the unit circle, the point is a stable point. Fig. 6 shows the stability regions, which are located under each curve, for values of M in the range 5–10. Fig. 6 indicates that the sta-bility regions increase as increases, and the slight effect of M is observed when is less than 0.7. As approaches 1, the maximum stability value will converge to . For example,
TABLE I
CONTROLLERSTRUCTURECOMPARISON OFEWMA, PCC, D-EWMA,ANDPPC
if is 5, the maximum stability value is 6 as approaches 1. It seems counter-intuitive that a higher order PPC has a larger sta-bility region. Appendix D shows an example that a higher order moving average controller actually has a larger stability region than a lower order one.
C. Controller Comparison
This section compares PPC with EWMA, PCC and d-EWMA controller analytically. Table I lists the number of poles and zeros of the transfer function from disturbance to output, zero positions, controller type and allowable stable region for candi-date controllers. A controller can reject step disturbance when zeros of the transfer function include one and reject ramp dis-turbance when they include two ones.
In Table I, controller type of PPC varies with the propagation parameter . When is zero, PPC has only two zeros locating all in one, which is the same as PCC and d-EWMA controller
with and . When increases from zero, PPC
has M zeros, and still has a zero at one. As approaches to 1, the PPC is equal to a moving average filter of order M. Thus, PPC can deal with Mth order disturbance theoretically when is not zero. Furthermore, PPC always has a zero at one for every
, which means that PPC is able to reject step disturbance, same characteristics as EWMA. Note that PCC and d-EWMA can reject ramp disturbance with every and , while PPC can only deal with ramp disturbance when is zero. In summary, PPC is a scalable Mth order controller and its properties are similar to EWMA, PCC and d-EWMA under certain conditions. As for the stability region, it varies with the controller pa-rameters. The minimum acceptable stability region of PPC is the same as PCC and d-EWMA, while the maximum accept-able stability region of PPC is smallest among all controllers.
V. SIMULATIONRESULTS
This section compares the performance of PPC with EWMA, PCC, and double EWMA when subjected to different types of disturbance. This study used the training data to examine all pos-sible propagation parameters or weightings of other controllers to find a fixed optimal propagation parameter or the weight-ings of the other controllers using the minimum mean square error. Then the parameter or weightings were applied to testing data. Furthermore, the proposed self-tuning PPC was compared with self-tuning EWMA [9] and self-tuning PCC [17] directly through the testing data. Notably, there has been no literature to date regarding adaptive tuning of the double EWMA controller. Thus, seven candidate controllers were used in our simulation; we applied five types of anthropogenic disturbance and the data from semiconductor fabrication.
A. Anthropogenic Disturbance
The simulation settings with anthropogenic disturbance in-cluded the following: the process model was (1, 1.5), the 10% offset controller model was (1.1, 1.65), the initial input was 0, and the process target was 0, respectively.
Because PPC is a meta-heuristic method, the performance can only be assessed by different cases. This study examined five types of disturbance with as the backward shift operator and
as the Gaussian noise and uncontrolled terms of run : (a) IMA(1, 1)
(25) (b) ARMA (1, 1)
TABLE II
COEFFICIENTS OF THEANTHROPOGENICDISTURBANCE
(c) Random walks (27) (d) AR(3) (28) (e) ARI(3, 1) (29) The coefficients for the different types of anthropogenic dis-turbance are listed in Table II. IMA(1, 1) [5], [6], [8], [11], [13], [17], [21], ARMA(1, 1) [5], [8], [18], Random walk [1], [21] are typical disturbance models for the fabrication data in semi-conductor manufacturing and the high order disturbance ARI(3, 1) is also observed in sputter deposition process [32]. Addition-ally, AR(3) is employed to check the performance of PPC under high order disturbance.
We produced 100 sets of training data and 100 sets of testing data using (25)–(29) with , , and standard devia-tion . Each set of the training and testing data had 100 runs. In addition, this study employs three performance indices to evaluate the performance: average (Ave.), standard deviation (Std.) and mean square errors (MSE) of ’s (10) obtained from 100 sets of test data.
1) Influence of PPC Parameters on Output Performance: This section shows the influence of PPC parameters on output performance: 1) the size of pheromone basket M; 2) the stan-dard deviation of noise in (25)–(29); 3) the model mismatch; and 4) the filter size in (23).
The influence of the size of the pheromone basket is shown in Figs. 7(a)–7(b). In Fig. 7(a), when propagation parameter is fixed, the size of pheromone basket almost has no effect on MSE. In Fig. 7(b), when the propagation parameter is self-tuned, a small size of pheromone basket has benefits in dealing with the higher order disturbance. However, selecting M is a trade-off between stability and performance. One selects by the following considerations: (1) The PPC with has slightly smaller MSE than that of for ARMA(1,1) and IMA(1,1) as shown in Figs. 7(a)–7(b); (2) Stability region of the PPC with is larger than that of
as shown in Fig. 6.
For the minimum variance control [33], with a priori infor-mation about the disturbance structure, the Std. of the controlled
Fig. 7. Comparison of the size of pheromone basket with = 1:1: (a) PPC with fixed optimum propagation parameters and (b) Self-tuning PPC.
output is always equal to or larger than . The influence of the standard deviation of noise in (25)–(29) on performance is shown in Fig. 8(a)–(b), which shows output Std. is slightly larger than the setting of . It also indicates that the performance ap-proaches to that of minimum variance controller, which needs a priori information about the disturbance structure.
Fig. 9(a)–(b) examines the influence of different model mis-matches within the stability region, where no particular trend can be concluded, but it will increase with the increase of model mismatch for high order disturbances. From our observation, MSE with different model mismatches in Fig. 9(a)–(b) is in form of a parabolic curve when model mismatch is further extended; the minimum MSE is not always at for different types of disturbance.
The influence of the filter size in (23) on performance is shown in Fig. 10, where the size of pheromone basket of self-tuning PPC is 6. It shows that the perturbation of noise will appear when , i.e., no filtering. In addition, the result is slightly different when 3 and choosing larger needs more runs for initialization (it needs runs in simulation and real case). So, the self-tuning PPC in this study employs in the following simulations.
2) Performance Comparison of PPC With EWMA, PCC and Double EWMA: The fixed optimal propagation parameter or
Fig. 8. Comparison of the magnitude of with = 1:1 and M = 6: (a) PPC with fixed optimum propagation parameters and (b) Self-tuning PPC.
weights of the other controllers listed in Table III were ob-tained using the minimum MSE from 100 sets of training data. Next, the fixed optimal controller parameters were applied to the testing data. The simulation results from the 100 sets of testing data with fixed optimum parameters controllers are listed in Table IV. Table IV also compares consistency of the simula-tion results of five anthropogenic disturbances by mean, range and variance. One observed that the fixed PPC has better output Std. and MSE for ARMA(1, 1), AR(3) and ARI(3, 1) distur-bances and is consistent in Std. and MSE with smaller mean, range and variance. However, the Ave. of fixed PPC is worse than the other candidate controllers. Consequently, this study develops self-tuning PPC for improvement.
The three performance indices from the 100 sets of testing data with self-tuning controllers are listed at Table V. The Ave. index is improved over the one by fixed PPC. Table V also indicates that the self-tuning PPC, which is designed without a priori information about disturbance type, is superior to the other candidate self-tuning controllers in the sense of smallest variance in Std. and MSE. Summing up Table IV and 5, the self-tuning PPC is more consistent in output Std. and MSE than any other candidate controllers.
B. Fabrication Data
The test pattern, which was collected from semiconductor fabrication [32], is shown in Fig. 11, and is composed of ramp
Fig. 9. Comparison of the magnitude of the model mismatch: (a) PPC with fixed optimum propagation parameters and (b) Self-tuning PPC.
Fig. 10. The influence of the filter sizef in (23).
disturbance, step disturbance, and noise. Moreover, the test pat-tern has 197 runs, where the first 97 runs (runs 1–97) are des-ignated as training data and the last 100 runs (runs 98–197) are the testing data.
The other simulation settings were ,
, , initial ,
and ; the upper and lower specification limits of the controlled outputs were set at 25 and , respectively. The simulation assumed that the process had two variations; one was a fixed gain with bias , and the other
TABLE III
FIXEDOPTIMUMCONTROLLERPARAMETERS OF THEANTHROPOGENICDISTURBANCE INFIG. 10
TABLE IV
PERFORMANCECOMPARISON OF THEANTHROPOGENICDISTURBANCESWITHDIFFERENTFIXEDPARAMETERCONTROLLERS: (A) AVERAGE, (B) STANDARD
DEVIATION,AND(C) MEANSQUAREERROR
TABLE V
PERFORMANCECOMPARISON OF THEANTHROPOGENICDISTURBANCESWITHDIFFERENTSELF-TUNINGCONTROLLERS: (A) AVERAGE, (B) STANDARD
DEVIATION,AND(C) MEANSQUAREERROR
Fig. 11. Testing pattern composed of ramp disturbance, step disturbance, and noise.
drifted per run from the first run, which amounts to a model mismatch of approximately 1–2 for the training data and approximately 2–3 for the testing data. Thus, the simulation of the controllers with the fixed optimal controller parameters had eight scenarios with the permutation of hybrid disturbance, two types of system variation, and four control algorithms. The simulation of the self-tuning controllers included six scenarios, with permutations of the hybrid disturbance, two types of system variation, and three self-tuning controllers.
Fig. 12. Output of the controllers with fixed optimum controller parameters.
Fig. 12(a) and 1(b) shows the output of the controllers with the fixed optimal controller parameters. Table VI summarizes the simulation results of Figs. 12(a) and (b). From Table VI, PCC and double EWMA performs slightly better than PPC when system is fixed with a bias. When the system gain drifts, the PPC has a good output Std. and MSE. Fig. 13(a) and (b) shows the controlled output and the tuned parameters of the self-tuning controllers. The results reveal that the PPC offset observed with the fixed propagation parameters disappeared, while the self-tuning EWMA [9] still had an offset near the target and converged slowly. In addition, the transient time of the self-tuning PCC [17] was slower than that of the self-tuning PPC, as shown in Fig. 13(a), and oscillated [Fig. 13(b)]. Table VII summarizes the simulation results for the self-tuning controllers and shows that the self-tuning PPC was superior to the other controllers in output Ave., Std., and MSE. In addition, the performance of off-line searching for the best parameters was better than that of self-tuning controllers when the process had a fixed bias. When the system drifted, the self-tuning PPC had the best performance of all in our simulations. Furthermore,
the results are consistent whether the process had a fixed bias or drift.
VI. CONCLUSION
We used swarm intelligence to develop a new process con-troller, the PPC. PPC links swarm intelligence and APC by treating the intercepts of the linear regression model at different runs as a social insect colony and modeling the interactions among the intercepts in terms of propagation. We proposed a workable PPC scheme with the strategy of tuning the propaga-tion parameter adaptively. The method can be easily extended to the MIMO case. Dealing with high order disturbances is the advantage of PPC; in particular, no training time as for neural work and no special rules as for fuzzy logic are the advantages of the proposed self-tuning PPC. Smaller stability region is the dis-advantage. For the five anthropogenic disturbances, fixed PPC is more consistent than EWMA, PPC, and double EWMA, in sense of output Std. and MSE; self-tuning PPC improved Ave. index over fixed PPC and is the most consistent in Std. and MSE among candidate controllers. In short, PPC has advantage in higher order disturbance, such as AR(3) and ARI(3,1), and is as good as the competitors in lower order disturbance. For different disturbance model other than those in the paper, one can specu-late PPC will perform well since it is capable of deal with com-plex disturbance with a scalable Mth order controller. In addi-tion, for the semiconductor fabrication data, the double EWMA controller seems to perform slightly better than the PPC when system is fixed with a bias and the proposed self-tuning PPC was superior to the other candidate self-tuning controllers in our sim-ulations and had the best performance when the process drifted. Because PPC is a meta-heuristic method and not designed for any specific type of disturbance, the performance can only be as-sessed by different cases. Outstanding issues requiring further study include effect of the magnitude of the model mismatch on the output MSE, a two-dimensional pheromone basket, the role of the evaporation parameter in PPC, and extension of the dig-ital pheromone infrastructure to other fields such as parameter estimation or the disturbance observer.
APPENDIXA
We look at (2) and (4), and assume that the system is nondis-sipative over the short term. If the pattern of the external input to a pheromone basket is a series of ones, the final transition result of aggregated pheromone should also be a series of ones, regardless of the value of the propagation parameter . Therefore, the transition function should obey the energy balance law, which means that the summation of the aggregated pheromone before transition is equal to that of the final transition results. In (2), when and , . Thus, substituting (2) into (4) yields
(A1) and should be 1.
Fig. 13. Controlled output and the tuned parameters of the self-tuning controllers.
TABLE VI
SIMULATIONRESULTS OFCONTROLLERSWITHFIXEDPARAMETERSOBTAINED
FROMMINIMUMMSE
APPENDIXB
To overcome the end effect as shown in Fig. 2, we modify the propagation-out ratio in the two extremities from
to , which can be obtained by final value theorem in the
Ap-TABLE VII
SIMULATIONRESULTS OFSELF-TUNINGCONTROLLERS
pendix C. First, (2) and (5) are rewritten as (A2) and (A3), found at the bottom of the next page.
By (A2) and (A3), (A8) in Appendix C becomes
where .. . .. . .. . .. . .. . .. . .. . . .. ... .. . . .. ... .. . . .. ...
Without loss of generality, take “M equal to 5 and is the 5 1 matrix of ones” as an example. The final propagation result of obtained from (A9) must be the 5 1 matrix of ones.
(A5)
Solving (A5) yields
(A6) Relatively, when is the matrix of ones, (A9) in Appendix C becomes
(A7) Equation (A7) shows that the final propagation result is also an matrix of ones. Thus, the modified transition functions not only obey the energy balance law but also avoid the end effect.
if if if if if (A2) if if (A3)
(A11)
APPENDIXC
Because states and are updated simultane-ously, (6) and (7) can be rewritten in matrix form:
(A8) where .. . .. . .. . .. . .. . .. . .. . . .. ... .. . . .. ... .. . . .. ... and
In (A8), and can be obtained with the
z-transform and the final value theorem
(A9)
where , ,
and ... . Note
that if the above system is stable, in (A9) will converge to the matrix of 0 and will converge to . In addi-tion, the z-transform of the external input is equal to since is an impulse at by definition in Section II-A. Thus, is a function of and for a specific M. The final propagation results can be obtained ana-lytically using (A9).
APPENDIXD
When approaches 1, the PPC is equal to disturbance with a moving average (MA) filter; this result can also be observed from Fig. 5(c). In the following, we analyze the stability of MA(4) and MA(6) to show a higher order MA controller has a larger stability region than the lower order one.
1) The transfer function from disturbance to output of a MA(4) controller is
(A10)
where the stability region is 5.
2) The transfer function from disturbance to output of a MA(6) controller is (A11), found at the top of the page. where the stability region is 7.
REFERENCES
[1] E. Sachs, A. Hu, and A. Ingolfsson, “Run by run control: Combining SPC and feedback control,” IEEE Trans. Semicond. Manuf., vol. 8, no. 1, pp. 26–43, Feb. 1995.
[2] S. J. Qin, G. Cheery, R. Good, J. Wang, and C. A. Harrison, “Semi-conductor manufacturing process control and monitoring: A fab-wide framework,” J. Proc. Contr., vol. 16, pp. 179–191, 2006.
[3] J. S. Hunter, “The exponentially weighted moving average,” J. Qual.
Technol., vol. 18, pp. 203–209, 1986.
[4] A. Ingolfsson and E. Sachs, “Stability and sensitivity of an EWMA controller,” J. Qual. Technol., vol. 25, pp. 271–287, Oct. 1993. [5] S.-T. Tseng, R.-J. Chou, and S.-P. Lee, “A study on a multivariate
EWMA controller,” IIE Trans., vol. 34, pp. 541–549, Jun. 2002. [6] R. P. Good and S. J. Qin, “On the stability of MIMO EWMA run-to-run
controllers with metrology delay,” IEEE Trans. Semicond. Manuf., vol. 19, no. 1, pp. 78–86, Feb. 2006.
[7] T. Smith and D. Boning, “A self-tuning EWMA controller utilizing artificial neural network function approximation techniques,” IEEE
Trans. Compon., Packag., Manuf. Technol. C, vol. 20, no. 2, pp.
121–132, 1997.
[8] C. C. Hsu and C. T. Su, “A neural network-based adaptive algorithm on the single EWMA controller,” Int. J. Manufact. Technol., vol. 23, pp. 586–593, Apr. 2004.
[9] N. S. Patel and S. T. Jenkins, “Adaptive optimization of run-to-run con-trollers: The EWMA example,” IEEE Trans. Semicond. Manuf., vol. 13, no. 1, pp. 97–107, Feb. 2000.
[10] S. Adivikolanu and E. Zafiriou, “Extensions and performance/robust-ness tradeoffs of the EWMA run-to-run controller by using the internal model control structure,” IEEE Trans. Electron. Packag. Manufact., vol. 23, no. 1, pp. 56–68, Jan. 2000.
[11] S. T. Tseng, A. B. Yeh, F. Tsung, and Y. Y. Chan, “A study of the variable EWMA controller,” IEEE Trans. Semicond. Manuf., vol. 16, no. 4, pp. 633–642, Nov. 2003.
[12] S. Bulter and J. Stefani, “Supervisory run-to-run control of polysilicon gate etch using in situ ellipsometry,” IEEE Trans. Semicond. Manuf., vol. 7, no. 2, pp. 193–201, May 1994.
[13] A. Chen and R. S. Guo, “Age-based double EWMA controller and its application to CMP processes,” IEEE Trans. Semicond. Manuf., vol. 14, no. 1, pp. 11–19, Feb. 2001.
[14] S.-T. Tseng and N.-J. Hsu, “Sample-size determination for achieving asymptotic stability of a double EWMA control scheme,” IEEE Trans.
Semicond. Manuf., vol. 18, no. 1, pp. 104–111, Feb. 2005.
[15] E. D. Castillo and R. Rajagopal, “A multivariate double EWMA process adjustment scheme for drifting process,” IIE Trans., vol. 34, pp. 1055–1068, Dec. 2002.
[16] E. D. Castillo, “Long run and transient analysis of a double EWMA feedback controller,” IIE Trans., vol. 31, no. 12, pp. 1157–1169, 1999. [17] C. T. Su and C. C. Hsu, “A time-varying weights tuning method of the
double EWMA controller,” Omega, vol. 32, pp. 473–480, 2004. [18] S. T. Tseng, W. Song, and Y. C. Chang, “An initial intercept
itera-tively adjusted (IIIA) controller: An enhanced double EWMA feed-back control scheme,” IEEE Trans. Semicond. Manuf., vol. 18, no. 3, pp. 448–457, Aug. 2005.
[19] X. A. Wang and R. L. Mahajan, “Artificial neural network model-based run-to-run process controller,” IEEE Trans. Compon., Packag., Manuf.
Technol. C, vol. 19, no. 1, pp. 19–26, Jan. 1996.
[20] C. T. Su, J. T. Wong, and S. C. Tsou, “A process parameters deter-mination model by integrating artificial neural network and ant colony optimization,” J. Chinese Inst. Ind. Eng., vol. 22, no. 4, pp. 346–354, 2005.
[21] J. Wang, Q. P. He, S. J. Qin, C. A. Bode, and M. A. Purdy, “Recursive least square estimation for run-to-run control with metrology delay and its application to STI etch process,” IEEE Trans. Semicond. Manuf., vol. 18, no. 2, pp. 309–319, May 2005.
[22] E. Bonabeau, M. Dorigo, and G. Theraulaz, Swarm Intelligence: From
Natural to Artificial Systems, Santa Fe Institute Studies in the Sciences of Complexity. New York: Oxford Univ. Press, 1999.
[23] M. Dorigo, V. Maniezzo, and A. Colorni, “The ant system: Optimiza-tion by a colony of cooperating agents,” IEEE Trans. Syst. Man Cybern.
B, vol. 26, no. 1, pp. 29–41, Feb. 1996.
[24] J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proc.
IEEE Int. Conf. Neural Networks, Piscataway, NJ, 1995, vol. 4, pp.
1942–1948.
[25] J. M. Bishop, “Stochastic searching networks,” in Proc. 1st IEE Conf.
Artificial Neural Networks, London, U.K., Oct. 1989, pp. 329–331.
[26] H. Jiang, L. Li, F. Qiao, and Q. Wu, “The new method of dynamic scheduling in semiconductor fabrication line,” 8th ICARCV, vol. 3, pp. 1874–1878, Dec. 2004.
[27] S. Brueckner, “Return from the Ant: Synthetic Ecosystems for Man-ufacturing Control,” Ph.D. dissertation, Humboldt Univ. of Berlin, Berlin, Germany, 2000.
[28] H. V. D. Parunak, M. Purcell, and R. O’Connell, “Digital pheromones for autonomous coordination of swarming UAV’s,” in Proc. 1st AIAA
Unmanned Aerospace Vehicles, Systems, Technologies, and Operations Conf., May 2002.
[29] J. A. Sauter, R. S. Matthews, H. V. D. Parunak, and S. Brueckner, “Performance of digital pheromones for swarming vehicle control,”
AAMAS, pp. 903–910, Jul. 2005.
[30] B. Walter, A. Sannier, D. Reiners, and J. Oliver, “UAV swarm control: Calculating digital pheromone fields with the GPU,” in I/ITSEC, Dec. 2005.
[31] H. V. D. Parunak and S. Brueckner, “Swarming coordination of mul-tiple UAV’s for collaborative sensing,” in Proc. 2nd AIAA Unmanned
Unlimited Systems, Technologies, and Operations Conf., San Diego,
CA, Sep. 2003.
[32] J. H. Chen, T. W. Kuo, and A. C. Lee, “Run-by-run process control of metal sputter deposition: Combining time series and extended kalman filter,” IEEE Trans. Semicond. Manuf., vol. 20, no. 3, pp. 278–285, Aug. 2007.
[33] K. J. Astrom and B. Wittenmark, Computer-Controlled Systems:
Theory and Design, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1997, pp. 453–462.
Der-Shui Lee received the B.S. degree in
mechan-ical engineering from the Nation Chung-Hsing University, Tai-Chung, Taiwan, in 1997, and the M.S. degree in mechanical engineering from the Na-tional Chiao-Tung University, Hsin-Chu, Taiwan, in 1999. Currently, he is an on-the-job Ph.D. student in mechanical engineering at the National Chiao-Tung University.
After graduation, since 2000, he served as an As-sistant Researcher in Chung-shan institute of Science and Technology (CSIST). His research interests in-clude run-to-run (R2R) process control, swarm intelligence, and decision sup-port system.
An-Chen Lee (M’04) received the B.S. and M.S.
degrees in Power Mechanical Engineering from National Tsing-Hua University, Hsinchu, Taiwan, and the Ph.D. degree in 1986 from University of Wisconsin-Madison in mechanical engineering.
He is designated as Chair Professor of National Chiao Tung University and currently a Professor in the Department of Mechanical Engineering. His cur-rent research interests are CNC machine tool con-trol technology, Magnetic bearing technology, Rotor dynamic and control, and Semiconductor manufac-turing process control.
Prof. Lee served as an Editorial Board member of International Journal of Precision Engineering and Manufacturing, Chinese society of Mechanical En-gineers, and International Journal of Applied Mechanics and Engineering. He is the recipient of National Science Committee (NSC) Excellent Research Award (1991–1992), NSC Distinguished Research Award (1993–1994, 1995–1996, 1997–1998), NSC research fellow (1999–2001, 2002–2004), NSC research fellow Award (2005), Chinese Society of Mechanical Engineers Distinguished Engineering Professor Award (1995), and Chinese Society of Mechanical Engineers Distinguished Engineering Professor Award (2001).
