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Simulation using the state-space representationof noisy dynamic systems to determine effectiveintegrated process control designsHARRIET BLACK NEMBHARD aa Department of Industrial and Systems Engineering , Auburn University , Auburn, AL,36849-5346, USAPublished online: 31 May 2007.

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IIE Transactions (1998) 30, 247-256

Simulation using the state-space representation of noisy dynamic systems to determine effective integrated process control designs

HARRIET BLACK NEMBHARD

Department of Industrial and Syslems Engineering, Auburn University, Auburn, AL 36849-5346, USA

Received September 1995 and accepted March 1997

There has been considerable interest in investigating control policies that can be applied to dynamic systems to take advantage of engineering process control (EPC) and statistical process control (SPC). We design two simulation models using SIMULINK to represent control of noisy dynamic systems given by a state-space representation. We use the models on systems with dynamic

. behavior that have been degraded by an additive ARMA noise process. Our findings show that the combination of EPC and SPC is a more effective policy on a complex dynamic system than on a simple dynamic system.

1. Introduction

The intent of this paper is to show that the combination of engineering process control (EPC) and statistical pro- cess control (SPC) is effective on dynamic systems. We develop two simulation models to facilitate insights into how dynamic systems respond and how they can be controlled.

The first simulation model represents a noisy dynamic system, having a dynamic process component and a noise process component that both act upon the system output. We consider dynamic processes that can be represented by a first-order or second-order model and noise pro- cesses that can be represented by an autoregressive moving average (ARMA) model. We solve the state- space representations of these processes. The state-space representation offers a convenient form for simulation modeling and analysis.

The second simulation model represents designs that integrate EPC and SPC. In industry applications, process control, in terms of manipulating the output, is often implemented via EPC using some form of feedback or feedforward regulation. SPC is primarily a set of moni- toring tools, e.g., Shewhart charts and exponentially weighted moving average (EWMA) charts, used to identify a change in the process and eliminate the source of the change where possible. (Since SPC is actually for monitoring, Montgomery et al. [I] and Box and Kramer [2] have indicated that SPC is perhaps a misnomer; sta- tistical process monitoring (SPM) may be a more apt appellation.) The difference between control and moni- toring relates to a key cost distinction: EPC is often

applied under the assumption that the cost of making adjustments to the process (i.e., regulation) is free while SPC is often applied under the assumption that the cost is nonzero.

The two simulation models just described allow us to represent a noisy dynamic system with different control policies. Presently, we will consider the major findings from these models to motivate the importance of investigating integrated control designs. Fig. 1 shows the result of simulating three policies on the simple noisy dynamic system. Fig. 2 shows the result of simulating these same three policies on a more complex noisy dynamic system. We will describe the specific character- istics of these systems and policies later. In both cases, Policy A and Policy B involve using integrated EPCISPC control designs while Policy C involves only EPC control. All of the policies can be judged based on the number of process adjustments required and the average squared error of the output from a specified target. For the simple dynamic system, Policies A and B require fewer adjust- ments but permit a higher average squared error as compared to Policy C; therefore, the decision to use an integrated control approach depends on whether the adjustment cost is prohibitive and whether the output is acceptable with the increased average squared error. For the complex dynamic system, Policies A and B require fewer adjustments and result in a lower average squared error as compared to Policy C; therefore, we would always choose an integrated control approach.

While these results are associated with the specific systems' model inputs and disturbances, they 'inay provide insight on designing integrated process control - q

0740-817X 0 1998 " I I E

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Fig. 1. Simulation results of using three different control pol- icies on a simple dynamic system.

policies for other systems. An open question is: How general are the 'simpje' and 'complex' noisy dynamic systems? As a practical matter, the noisy dynamic system we would like to address is an industrial system that ex- periences a shift in operating conditions (caused by fac- tors such as startup, shutdown, or changes in raw materials) that is followed by a transient period where the

a process is rendered out of statistical control. By simu- lating noisy dynamic systems and simulating control policies on such systems, we demonstrate how to use simulation as a decision-making tool to develop good integrated control policies for the transient period. The simulation models are developed using a program called SIMULINK [3] which offers a graphical user interface that

Fig. 2. Simulation results of using three different control poli- cies on a complex dynamic system.

allows the development of block diagrams and hierar- chical models. The parameters of each block are evalu- ated by MATLAB [4].

The remainder of this paper is outlined as follows. Section 2 has the definition of the system model, Section 3 gives its state-space representation, and Section 4 con- tains its simulation model. Section 5 contains the inte- grated process control policies for the system. Section 6 has the simulation model of the control mechanism and results obtained by using the control policies on the example cases, and Section 7 provides some broad guidelines on using the policies.

2. Noisy dynamic system model

In general, a process is a transformation between an input and an output. Our system model has two processes which are illustrated in Fig. 3. The first is a dynamic (industrial) plant process and the second is a noise pro- cess.

Dynamic behavior can occur when the output, Y,,, of a process does not respond instantaneously to a change in input, U,. The interested reader may refer to a process control text such as Seborg er al. [5] , for figures illus- trating the dynamic behavior of first-order and second- order processes. ((4) and (5) in the next section define these processes.) The characterization of a relationship that links the input to the output for a dynamic process is more difficult than for a steady-state process.

The modeling problem becomes more challenging with the presence of a noise process that affects the output. The noise may represent disturbances anywhere in the system; the complete effect of the noise is measured at the output and hence represented as affecting the output. The noise process, y,, is modeled as a series of random shocks, u,,, that pass through a linear filter.

We consider the combination of the dynamic plant process and the noise process as our system model. This type of model has been widely used to represent dynamic industrial processes that are sampled over time [5-71.

WlbNoLc

u. Filter

Fig. 3. Noisy dynamic system model.

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State-space representation of noisy dynamic systems

In the absence of process control, a disturbance would cause the output to drift away from the desired setpoint or. target value, T. We return to a discussion of process control after presenting the state-space representation and simulation of the noisy dynamic system model.

3. State-space representation of the system

A state-space representation is applicable to simulation analysis as well as to other modeling that involves nu- merical computation because it provides a convenient way to represent complex mathematical systems. A state- space representation will be used in the simulation model to represent the dynamic plant process and the noise process of the system shown in Fig. 3.

3.1. The dynamic plant process

Transfer functions for linear ordinary differential equa- tion (ODE) models provide a standard way to define the dynamic relationship between the input and output of a process. In the control literature, the commonly used notation for the Laplace transfer function form of the process is:

where Y(s) and U(s) are respectively the output and input of the system and the a's and b's are appropriate coeffi- cients (e.g., see (7J).

A continuous state-space representation

x = Ax+ Bu,

y = Cx+Du,

of the transfer function is given by the following two equations (a proof appears in [8]): -

-al -a2 - . - 0

-& . . .

The first equation is known as a state or transition equation and the second is known as an observation equation.

We consider dynamic plant processes that can be modeled by a first-order or second-order linear ODE. Examples of industrial systems that can be modeled by such ODES include continuous stirred-tank reactors, singlecomponent vaporizers, flash drums, mixing tanks, and flat-band ovens [5,9]. In the literature, the transfer function form of a first-order process is often expressed (using (1)) by:

where bo is taken to be zero, K is the steady-state process gain and 7 is the process time constant (or equivalently, the speed of the response) (see, e.g., [5,7,8]). Two first- order processes connected in series constitute a second- order process and can be expressed by:

3.2. The noise process

We investigate noise processes characterized by a station- ary stochastic autoregressive-moving average (ARMA) model. The ARMA process is a widely used mathematical model that is a special case of the linear filter of white noise [lo]. The stationary and invertible ARMA(p,q) process can be represented as a time-series [I I]:

where 4 and 9 are appropriate AR and MA coefficients. A discrete state-space representation:

for the ARMA(p, q ) process is given by the following two equations (a proof appears in [7]): D

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When using model identification methods for time series models it is difficult to discern second-order processes from third-order processes. Box er al. [6] discuss model identification and note that processes found in practice can frequently be represented by models in which p and q are not greater than 2. The next section concerns exam- ples involving first-order and second-order processes.

4. Simulation of the noisy dynamic system

Simulation is a powerful tool for solving the equations that describe noisy dynamic systems. Also, simulation can (approximately) solve system equations that are intrac- table by analytic methods.

In this section, we present the SIMULINK simulation models that represent the noisy dynamic system shown in Fig. 3. The models rely on the state-space representation of the system discussed in the previous section. The state- space form also allows us to make changes in the system easily as compared to the alternative of developing and solving a new mathematical model. Two examples illus- trate this point. (In the SIMULINK interface, Y, is shown as Y-n and x is shown as 2.)

order representation of the process. As part of the sim- ulation output, the effect of the input on the plant process is plotted via the GRAPH: DYNAMIC RESPONSE block and placed in a table via the TABLE: DYNAMIC RESPONSE block.

4.2. The noise process

Fig. 5 shows the block diagram of a SIMULINK model that simulates the ARMA noise process. (The dynamic plant process is also included.) The NORMAL RANDOM NUMBER

GENERATOR block provides pseudo-random shocks from a N(0,l) distribution. At the STANDARDDEVIATION block the output of the random shock is multiplied by the value of the standard deviation. The LINEAR FILTER block pro- vides the discrete state-space form of the linear filter process. The LEVEL block provides the y-axis center of the noise process. The s u ~ l block combines the linear filter with the process level; the output of the sum block is an ARMA(p, q) process. This ARMA process is plotted and placed in a table via the GRAPH: ARMA and TABLE: ARMA

blocks, respectively. The noisy dynamic system is modeled by adding the

output of the plant process to the output of the ARMA noise process using the SUM^ block. This combination is plotted and placed in a table using the GRAPH:OUTPWT and TABLE: OUTPUT blocks, respectively. The three GRAPH blocks are 'windows' that provide a real-time view of the process as the simulation progresses. The three TABLE blocks record the numerical values in a separate file for storage or further computation. The use of SIMULINK for building simulation models of this nature is discussed further by Nembhard and Nembhard [12].

4.1. The dynamic plant process 4.3. Examples

Fig. 4 shows the block diagram of a SIMULINK model that We will use the simulation model of a noisy dynamic simulates the dynamic plant process. The INPUT block system on two examples. These cases will be used to help provides a change in input shift a t a specified time. The determine and make generalizations about integrated PLANT PROCESS block provides the first-order or second- process control designs.

Fig. 4. Simulation model block diagram of a dynamic plant Fig. 5. Simulation model block diagram of a noisy dynamic process. system.

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State-space representation of noisy dynamic systems

4.3.1. Case 1: first-order dynamics with ARMA(1, I ) noise

A baking process uses a flat-band oven that continuously moves product along a conveyor. The relationship be- tween the output and input is described by the first-order transfer function

By (2) and (3), the matrices for the state-space represen- tation of the plant process are

A = [-1/41 B = [1] C = (51 D = (01.

This type of process can be influenced by a sudden input change that may characteristically occur when, for ex- ample, feedstock changes from one supply to another. A unit step input approximates such a change that occurs a t n = 10. The disturbance in the baking process is repre- sented by the ARMA (1, 1) process

yn = O.4yn-1 + un + 0.7~, -~ , (10)

with = 9. By (7) and (8), the matrices for the state- space representation of the disturbance are

The state-space representation of the noisy dynamic sys- tem was simulated for 50 time periods. Fig. 6(a-c) con- tains the graphical results from this simulation. Fig. 6a shows the first-order dynamic response for a unit-step input defined by the A, B, C, and D matrices for (9) with a step at n = 10; this response is represented as Y , in Fig. 3. It results from the GRAPH: DYNAMIC RESPONSE block win- dow of the simulation model and is comparable to those in [I51 (see Section 2). Fig. 6b is a realization of the ARMA(1, 1) process defined by the 2, B, c, and ma- trices for (10); this noise process is represented as y, in Fig. 3. It results from the GRAPH:ARMA block window and is comparable to those in Montgomery et al. [I 11. Fig. 6c is the system output - the combination of the dynamic response and the noise process. It results from the OUTPUT block in the simulation model. .

4.3.2. Case 2: second-order dynamics with ARMA (2. 2 ) noise

Two continuous stirred-tank reactors in series are used to transform a comonomer (which is a chemical building block used to make films for products like plastic bags and cups). The relationship between the output and input is described by the second-order transfer function

Fig. 6. First-order dynamics with ARMA(1, 1) noise (Case 1).

C = [0 16/31 D = [O].

A unit step input occurs at n = 20. The disturbance in the process is represented by the ARMA (2, 2) process

with a: = 4. The discrete state-space representation of the The state-space representation of the plant process has disturbance has

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The state-space representation of the noisy dynamic sys- tem was simulated for 100 time periods. Similar to the previous example, we obtain the graphical results from this simulation shown in Fig. 7(a-c). Fig. 7a is the sec- ond-order dynamic response for a unit-step input. Fig. 7b

is a realization of the ARMA(2, 2) process. Fig. 7c is the system output response. By visual comparison with the first-order example, we observe that this second-order process has dynamic behavior for a longer time period. Hence, the simulation was longer for the second example to capture the full range of the system.

In the next section, we resume discussion of process control for the noisy dynamic system output represented in Figs. 6c and 7c.

5. Process control

The noisy dynamic system represented in Fig. 3 has no regulation; the examples in the previous section show what the output looks like without regulation. In general, such systems need regulation to operate profitably and satisfy quality or performance standards. Regulation is also used to reduce variation from the target. In Fig. 8, a feedback control mechanism compensates for distur- bances in the system model. Even with process control, there will be a deviation error, E , between the output and target.

The mechanism we apply to the system integrates EPC and SPC. Both approaches have an established base in the literature. and in practice. Texts on EPC policies in- clude [5,7,8]; and texts on SPC policies include [13,14]. MacGregor [15] shows that policies under each approach can minimize the variance of output deviations from the target depending on the type of process (e.g., steady-state, first-order dynamic, or higher-order dynamic), type of disturbance (e.g., stationary stochastic, nonstationary, or deterministic), and adjustment cost (i.e., free or nonzero). Recently, there has been an effort to combine EPC and SPC in order to minimize output variance beyond the

Feedback Control I Mechanism ~

Fig. 7. Second-order dynam~cs wlth ARMA(2.2) nose (Case 2). Fig. 8. Noisy dynamic system model with feedback control.

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State-space representation of noisy dynamic systems 253

individual capability of either approach [1,2,16]. How- ever, the simulation of noisy dynamic systems and the simulation of control policies on such systems have received little attention in the literature.

Specifically, we consider a proportional-integral (PI) controller as the EPC policy and a Shewhart chart as the SPC policy. These tools are widely used and familiar to engineers in industry. They also provide robust perfor- mance on complex processes. In particular, MacGregor [15] shows that although PI control is optimal for mini- mizing variance in first-order processes, it provides reasonable performance on higher-order processes. Both Box el al. [17] and MacGregor [15] have demonstrated that a Shewhart chart is optimal in steady-state processes but also has utility in monitoring other processes. These reasons make PI controllers and Shewhart charts good candidates for use in integrated EPC and SPC policies for controlling noisy dynamic systems. We will now consider the design of these policies and the performance measures to use in evaluating their effectiveness.

5.1. Control designs

A PI controller makes a compensatory adjustment for any disturbance in the process at every observation point because an underlying assumption is that all process adjustments are free. Hence, in the noisy dynamic system, the PI controller makes an adjustment before the transient phase due to the ARMA noise and after the transient phase due to the shift and the noise.

We have noted that a Shewhart chart is a monitoring tool and does not provide a control action. Rather, it identifies the point of a shift in the process mean to in- dicate'the effect of a so-called special cause because an underlying assumption is that each process adjustment in response to the special cause incurs a cost. Hence, in the noisy dynamic system, an SPC chart probably does not give a signal at each observation before the transient phase since the ARMA noise is not a special cause. It is likely to give a signal at the start of the transient phase since it is due to a shift.

In combining the PI controller and Shewhart chart, the assumption is that they are applied to a process with nonzero adjustment costs. Hence, the objectives in con- trolling the system are to minimize or reduce the adjust- ment costs and the variation from target. To reduce the number of adjustments to the system, the Shewhart policy is used to supervise the PI controller. We explore two such PIIShewhart policies.

5.1.1. Policy A: PIIShewhart-AE In the first integrated process control design, the Shew- hart chart allows the PI controller to make adjustments only after an indication of a special cause (i.e., operating shift). The Shewhart chart also monitors the process after the shift. This means that an upper control limit (UCL)

and a lower control limit (LCL) for the original target level are established and as long as the observation falls within the limits, no action is taken. When an observation exceeds the limits, the PI controller is activated to take over and make adjustments to the process. For an increase (upward shift) in the target, the UCL of the original target signals activation of the PI controller. For a de- crease (downward shift) in the target, the LCL of the original target signals activation of the PI controller. Shewhart control limits on the new target level (after the shift) evaluate whether the process stays within the limits once the new level is reached. This design combines a Shewhart policy to ac~iv.ate and evaluare (AE) the PI controller and is referred to as PIIShewhart-AE.

5.1.2. Policy B: PIIShewhart-AR The second integrated process control design is an ex- tension of the first. In addition to allowing adjustments after an indication of a special cause, the Shewhart policy ceases to allow PI adjustments after s observations are within the control limits of the new target level. This means that control limits must be established for both the original target and the new target. One purpose for control limits in this design is to indicate the end of the transient period (and the beginning of a period of sta- tistical control). Shewhart [I81 defined a statistically in- control process as one for which a reasonable prediction can be made on future observations. A reasonable amount of data must be collected in order to make this prediction. For this particular example, we required 60 consecutive observations within the new target control limits to indicate the end of the transient period. This design combines a Shewhart policy to activate and regu- late (AR) the PI controller and is referred to as PI/ Shewhart-AR.

To provide a comparison for the two integrated process control designs, we also establish Policy C of using only a PI controller (i.e., only EPC) and Policy D of using no control.

5.2. Performance measures

The control policies are evaluated based on four perfor- mance measures: average squared error of the output from target; number of adjustments; average magnitude of adjustments; and number of alarms. (Others that could potentially be considered include mean absolute deviation and adjustment cost.)

The first two performance measures were mentioned in the introduction. Figs. 1 and 2 show that there is a tradeoff between average squared error and the number of adjustments. These two measures are likely to be tied to cost and quality in an industrial setting.

The remaining two performance measures have less of an impact on cost but provide additional dimensions in evaluating the integrated EPC/SPC designs. The average

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magnitude of adjustments indicates how aggressive the control action must be in order to compensate for the disturbance. In some cases, this range may be limited by the designer or by the physical equipment. The number of alarms indicates how many times the Shewhart control limits are exceeded during the transient period.

6. Simulation of integrated process control

Fig. 9 shows the block diagram of the SIMULINK model that simulates the control mechanism.

In the first level of detail, IN PORTS L-4 provide the noisy dynamic system output response to the control mechanism. Four duplicates of the output are needed to support the four actions of the controller. The EPC block implements the PI controller action and provides it to the system via the TO INPUT block. When there is no control action, the TO INPUT block simply supplies the original system output response. The CONTROL RESULTANT block determines whether control adjustments were used and supplies this information via the TO GRAPH: CONTROL AD- JUSTMENTS block.

In the second level of detail, logic operators activate or deactivate the controller at the PI ONIOFF block. The two SPC blocks provide the UCL and LCL of the Shewhart chart. The system output response is compared to the limits by the RELATION TO UCL and RELATION TO LCL blocks. The result of this comparison is made by the spc LIMITS TEST BLOCK. The count on the number of times the output response is within the control limits is accom- plished by the SUM and MEMORY blocks; the count is maintained via the s m and RESET blocks. This count is compared to the number of transient period samples (TPS) via the RELATION TO TPS block. The result of this comparison is made by the TPS TEST block.

F e E_dn Wbns -Ion Styb l- In PDm 1 4 Fmm NDR

6.1. Experipents and results

The simulation model for the control mechanism is used in conjunction with the simulation model for noisy dy- namic systems. This allows us to evaluate the perfor- mance of Policies A, B, C, and D on the Case 1 and Case 2 noisy dynamic systems in Section 4.3. The simulation of each policy on each case. has 15 independent replications using different pseudo-random number streams.

In short, we found that both of the PIIShewhart con- trol designs increased the average squared error over using only PI control for a first-order process and decreased the average squared error over using. only PI control for a second-order process. However, the number of adjustments required with the PIIShewhart control designs is significantly less in both cases. We now discuss these results and their significance in greater detail.

Table 1 shows the average performance results for the Case 1 system simulations. If no control is used, the av- erage squared error is 36.7. This is reduced to 2.8 by using PI control. To achieve this reduction, the process gets an average of 279 adjustments with an average magnitude of 19.8%. When the PI/Shewhart-AE design is used, the average squared error is 17.6 which is approximately a six-fold increase over only PI control but is better than the no control case. When the PIIShewhart-AR design is used, the average squared error is 16.6 which is also an increase over PI only control. In both of the integrated control designs, the average magnitude of adjustments is about 5% but far fewer adjustments are required when Shewhart limits are added to.the new target. In both of the integrated control designs, no Shewhart alarms occur during the transient period.

Table 2 shows the average performance results for the Case 2 system simulations. If no control is used, the average squared error is 342.4. This is reduced to 34.4 by

Fig. 9. Simulation model block diagram of EPC/SPC control mechanism.

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State-space representation of noisy dynamic systems 255

Table 1. Average performance results for the Case 1 system (first-order dynamics with ARMA(1, 1) noise)

Control Policy A Policy B Policy C Policy D mechanism PIIShewhart-AE PIIShewhart-AR PI-only none

Squared error 17.6 16.6 2.8 36.7 Number of adjustments 212 120 279 0 Average adjustment 0.052 0.058 0.198 0 Number of alarms 0 0 N/ A N/ A

Table 2. Average performance results for the Case 2 system (second-order dynamics with ARMA(2, 2) noise)

Control mechanism

Policy A Policy B Policy C Policy D PIIShewhart-AE PIIShewhart-AR PI-only none

Squared error 19.0 25.7 34.4 342.4 Number of adjustments 548 302 724 0 Average adjustment 0.081 0.086 0.267 0 Number of alarms 5 9 N/A N/A

using PI control. To achieve this reduction, the process gets an average of 724 adjustments of an average mag- nitude of 26.7%. When the PIIShewhart-AE design is used, the average squared error is 19.0 which is approx- imately half the error over only PI control. When the PI/Shewhart-AR design is used, the average squared error is 25.7 which is also a decrease over PI only control. In both .of the integrated control designs, the average magnitude of adjustments is about 8% but far fewer adjustments are required when Shewhart limits are added to the new target. In using Shewhart limits to activate and evaluate the PI controller, five alarms were signaled during the transient period exclusive of the activation signal. In using Shewhart limits to activate and regulate the PI controller, nine alarms were signaled during the transient period.

7. Summary and conclusions

A transient period occurs after an operating shift in the process (e.g:, startup, shutdown, grade changes, etc.). There is a significant need for better control of the tran- sient period in dynamic systems. For instance, food and beverage and pharmaceutical manufacturers attribute a large portion of their out-of-specification product to this period. The noisy dynamic system model is typical of many practical processes with a transient period.

In pursuing this investigation, we had an interest in addressing the general issue of controlling simple and complex noisy dynamic systems. For. this paper, we consider a simple system to be represented by a first-order process with ARMA(1, 1) noise, and a comparatively complex system to be represented by a second-order process with ARMA(2, 2) noise. Given this framework, we may use the cases presented here to help establish

some expectations to provide guidelines for using inte- grated control polices in such environments. .

7.1. Recommendations for simple noisy dynamic systems

We elected to use a PI controller and a Shewhart chart in the integrated process control designs partially because of their broad applicability. However, a PI co,ntrol policy is optimal for minimizing variance in first-order processes; the Shewhart policy is not.

Given the PI controller design, it is consistent that the PI policy outperforms integrated PI/Shewhart policies on average squared error in this system. That is, in general for first-order processes, the action of the PI controller should be sufficient to maintain the process at the target level. Therefore, we do not recommend using an inte- grated process control policy for first-order processes if minimum variance is the primary concern.

It seems that the only practical reason to use an inte- grated process control policy for a first-order process is in environments with significant adjustment costs. In this event, Shewhart limits used to activate and regulate the PI controller (PIlShewhart-AR) may give the best perfor- mance. The design of the PI controller complements the Shewhart chart so no alarms would be expected with this policy.

7.2. Recommendations for complex noisy dynamic systems

Neither a PI controller nor Shewhart chart is optimal for minimizing variance in second-order systems. However, using them together in an integrated process control policy is recommend for a complex system where mini- mum variance and/or number of adjustments is a concern.

Since PI control is not optimal for second-order sys- tems, the Shewhart limits will more likely be exceeded in

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256 Nembhard

this system. To reduce this possibility, using the Shewhart limits only to activate the controller (PIIShewhart-AE) is preferable to using it to also regulate the PI controller (PI/Shewhart-AR).

These recommendations are based on using the simu- lation model for a state-space representation of noisy dynamic systems and a simulation model for the inte- grated process control policies developed in this paper. The models were programmed in SIMULINK and can be downloaded until December 1998 from http:// www.eng.auburn.edu/~nembhard. They can support investigation of other types of systems and controllers. In future work, they will be used to model higher-order dynamic systems with ramp disturbances and to deter- mine the effectiveness of integrated control designs that include moving averages. Additionally, these designs may be compared to other controllers that incorporate ad- justment cost concerns directly into the mathematical model of the system.

Acknowledgment

Many thanks to the editor and the three referees whose comments and helpful suggestions greatly improved this paper.

[7] Ogata, K. (1987) Discrete-Time Control Systems, Prentice-Hall, Englewood Cliffs, NJ.

[&I Ogata, K. (1990) Modern Control Engineering, 2nd edn., Prentice- Hall. Englewood Cliffs, NJ.

(91 Bartholomal, A. (ed.) (1 987) Food Factories: Processes. Equip- ment, Costs, VCH, Weinheim, Germany.

[ I@] Chui, C.K. and Chen, G. (1991) Kalman Filtering with Real-T~me Appitcarions, 2nd edn., Spnnger-Verlag, New York, NY.

[I l l Montgomery, D.C., Johnson, L.A. and Gardiner, J.S. (1988) Forecasting and Time Series Analysis. 2nd edn., McGraw-Hill, New York, NY.

[I21 Nembhard, H.B. and Nembhard, D.A. (1996) Dynamic simula- tion for time series modeling, in Proceedings of the 1996 Winter Simulation Conference, Charnes, J.M ., Morrice, DJ.. Brunner, D.T. and Swain, J.J.. (eds). 1407-1412.

[13] Grant, E L and Levenworth, R S (1996) Statutrcal Quailty Control, 7th edn., McGraw-HiH, New York, NY.

[I41 Montgomery, D.C. (1991) htroductlon to Statrctlcal Quality Control, 2nd edn., John Wiley, New York, NY.

[IS] MacGregor, J.F. (1988) On-line statistical process control. Chemlcal Engmeerlng Progress, 21-3 1 .

1161 Ogunnaike, B.A. (1995) A stat~strcai appreclatron of englneenng process control, in ASQC Fall Technical Conference, St. Louis, MO.

[I71 Box, G.E.P., Jenkins, G.M. and MacGregor, J.F. (1974) Some recent advances in forecasting and control, Part 11. Journal oJthe Royal Statrrt~cal Soclety. Serles C . Appkd Statistics, 23(2), I 58- 179.

[I81 Shewhart, W.A. (1980) Economic Control o j Quality of Manu- factured Product, American Society for Quality Control, Mil- waukee, WI.

References Biography

[I] Montgomery. D C , Keats, J.R., Runger, G C and Messina, W.S. (1994) Integrating statrstlcal process control and engrneerrng process control. Journal of Quality Technology, t6(2), 79-87.

[2] Box, G.E.P. and Kramer, T. (1992) Statist~cal process monltorlng and feedback adjustment - a discuss~on. Technometrrcs, 34(3), 251-257.

[3] S I M U L I N K ~ (1994) A program for simulating dynamic systems, computer software. The MathWorks, Inc.

[4] MATLAB" (1994) Cornpurer sofiware. The MathWorks, Inc. [5j Seborg, D.E., Edgar, T.F. and Mell~champ, D.A. (1989) Process

Dynamics and Control, John Wiley, New York, NY. [6] Box, G.E P., Jenkins, G M. and Reinsel, G C (1994) Time Series

Analysts: Forecasrmg and Control. 3rd edn., Yrentrce Hall, En- glewood Cliffs. NJ.

Hamet Black Nembhard IS an Assrstant Professor of lndustnal and Systems Engineering at Auburn University. She has previously held positrons wrth Pepsl-Cola, General Mllls, and Dow Chemrcal. Her B.A. is in Management from Ciaremont McKenna College and her B.S.E. is in Industr~ai and Management Systems Engineering from Arizona State University. Her Ph.D. and M.S.E. degrees are in Industrial and Operations Engineering from the Unrversity of M~chigan. Professor Nembhard's current research interests involve formulating and solving models for manufacturing systems to address control, production, quahty, and economic concerns that artse while a process IS In a transient phase. She is a member of IIE as well as ASQC and INFORMS.

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