CROSS- D I R E C T I 0 NA L CON TR 0 1

Closed-loop identification with MPC for an industrial scale CD-control problem

D.R. Saffer and F.J. Doyle

Abstract: An approach to implementing closed-loop identification in a model predictive-control framework for the cross-direction control of basis weight is proposed. The closed-loop- identification technique uses the concepts of Markov parameters to determine a step-response model that can then be used within a model predictive controller. The technique is applied to an industrial-scale paper-machine-simulation benchmark problem for cases of varying degrees of shrinkage. The performance of wet- and dry-end full-array sensors are compared. Also, the performance of the closed-loop identification is compared for a nominal case and two cases in which shrinkage occurs in the drying process.

1 Introduction

For a number of years, researchers have been exploring methods to determine control-relevant models for cross- directional (CD) control of sheet- and film-forming processes. Most identification techniques have considered open-loop bump tests of some or all of the actuators. One technique that has been studied by a number of researchers is to reduce the order of the model by transforming the spatial response of the actuators. These techniques have included the use of Gram polynomials [ 11, pseudosingular- value decomposition [ 2 ] , and principal-components analy- sis [ 3 , 41. On transformation, the parameters within the new model space have been identified using open-loop bump tests. A number of studies [5-81 have used a least- squares technique to fit step- or impulse-response data to a given shaping function.

While open-loop identification is theoretically possible, the techniques outlined above are not without their disad- vantages. First, the process under investigation would need to be operated in open loop for an extended period of time. Secondly, the required excitations may cause the paper that is produced to be of low quality. Since the fastest of today’s paper machines runs at speeds of over 1700 metres per minute [9], the loss of product could be costly for paper manufacturers.

An alternative approach involves closed-loop identifica- tion of a model, which calls for persistent excitation while collecting data and a method to separate the dynamics of the process from the known dynamics of the controller. A number of surveys have been published on the topic of closed-loop identification, e.g. [ lo , 111. There are three main approaches to identification in closed loop: direct, indirect, and joint-input-output. The direct approach refers

0 IEE, 2002 ZEE Plwceedirigs online no. 2002025 1 DO/: 10.1049/ip-cta:2002025 1

Paper received 19th July 200 1

The authors are with the Department of Chemical Engineering, University of Delaware, NEWARK, DE: 19716, USA

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to a technique that ignores control interaction and attempts to identify the open-loop system from input and output signals. Direct approaches yield consistent and optimally accurate models if a correct noise model is assumed [ 111. The indirect approach identifies a closed-loop inodel and uses knowledge of the controller to determine the open- loop model. While knowledge of the regulator is necessary for indirect methods, fixed-noise models can be used while guaranteeing consistency. Finally, the joint-input-output approach uses an extra input or set-point signal and noise to identify a system whose outputs are the outputs and the actual inputs of the process [ I 11. The estimates of the open-loop system are consistent regardless of the assumed noise model so long as the feedback has a certain linear structure [l I].

Some of the above mentioned approaches have been applied to paper-machine control. One researcher [ 121 has treated the closed-loop-identification problem for CD control with a method demonstrating the joint-input- output approach and proved that one could determine the open-loop sensitivity function from closed-loop data due to the inherent time delay between actuation and the sensor location in sheet and film processes. The author demon- strated the estimation of steady-state response gain and location in a framework of parallel SISO estimators. Earlier, a number of researchers applied various closed- loop-identification techniques to a machine-direction (MD) model for control of a paper machine [ 13-1 51. While [ 13, 141 use only direct techniques to identify a difference- equation model and a frequency-response model, respec- tively, [15] compares direct and indirect methods to identify a difference-equation model.

In this paper, an indirect closed-loop-identification tech- nique originally developed by Phan et al. [16] has been extended to develop an unconstrained model predictive controller for a paper machine. While this technique is an indirect method, it is unique in that an observer is explicitly used within the identification equations [16]. Markov parameters were identified and converted into a step- response model. A demonstration of closed-loop identifi- cation using the model predictive controller was carried out on an industrial-scale benchinark paper-machine simula-

IEE Proc.-Contlvl Theory Appl.. Vol. 149, No. 5, Sepreiiiher. 2002

tion that considers shrinkage of the sheet during drying. Finally, the newly identified model was compared with the original model that did not account for shrinkage within the MPC framework.

2 Model identification and model predictive control

2.7 Background Model predictive control (MPC) has attracted considerable attention as an effective method for CD control [ 5 , 17-21 1. The key challenges for CD control include time delays between the headbox and the measurement area, produc- tion rate and grade-transition changes, plant-model mismatch due to paper shrinkage or shifting, and the need to compute control moves for a large number of

. actuators in a short period of time. The advantage to using MPC for CD control include the ability to handle constraints on the rate, magnitude and bending moment of the actuators, and the ability to compensate for the interactions between the different actuator positions in the CD as well as interactions of multiple headboxes in the machine direction (MD).

A number of methods have been developed to make MPC more efficient for use in large-scale CD control. One approach [5, 221 employs an unconstrained model predic- tive controller to solve an off-line quadratic-programming (QP) problem that results in a linear time-invariant (LTI) controller. With this method, constraint handling is applied by scaling the actuator moves to preserve the direction of the input. This is the same technique that has been used for other LTI CD control strategies such as H , or Dahlin controllers. For these controllers, however, the constraint- handling compensation often results in suboptimal perfor- mance as compared with algorithms that account explicitly for actuator limitations within the optimisation framework.

A number of on-line optimisation implementations of MPC have been suggested as well. Rao et al. [20] propose the use of deadbeat control when actuator-velocity (AU) constraints are not necessary, or a modified constrained QP MPC by exploiting the sparse model structure to reduce computational time. Those authors point out that dead- beat controllers are implementable in certain situations; however, systems that require actuator-velocity constraints would not benefit from this technique. The authors also note that constrained-QP techniques are still computation- ally infeasible for current computing platforms. VanAntwerp [23] showed that a reformulated constraint set, using ellipsoids to bound the constraint set, results in an efficient suboptimal MPC scheme for CD control. Model-reduction techniques, such as the adaptive princi- pal-component analysis [ 171, have been coupled with receding-horizon control to produce results of similar quality to those of the full MPC problem.

Some researchers have attempted linear-programming (LP)-based formulations of MPC. Doyle and coworkers [8, 191 have demonstrated a number of different linear- programming techniques to improve further the computa- tional efficiency within the on-line optimisation. Others have used an L , norm for both a maximum deviation from setpoint and as a method of minimising the range of deviations across the CD [24].

profile using a simple CD controller while determining a model to map the headbox-actuator movements to each of the possible full-array sensor banks: one at the end of the fourdrinier table to collect wet-end measurements, and one at the end of the machine to collect dry-end measurements (Fig. 1). The identification was performed for three condi- tions: no shrinkage in the cross direction, 2% shrinkage in the cross direction and 5% shrinkage in the cross direction.

Although other methods of model identification have been described, e.g. [ 5 , 171, the identification procedure used in this paper was adapted from Phan et al. [16] and is merely summarised below. The details of the study can be found in the original work which included using pure gain and dynamic-feedback controllers. As this paper will extend the technique to the MPC framework, a summary of the closed-loop-identification technique for a dynamic feedback controller will be shown. If a pure gain controller is used, such as an LQR design with full state feedback, the generalised procedure reduces back to the pure- gain-controller closed-loop identification case.

In this study, the open-loop process is assumed to be linear and can be expressed in discrete state-space form:

x(i + 1) = Ax(i) + Bu(i) (1)

y(i) = Cx(i) + Du(i)

where x is the state vector of size nx, u is the vector of inputs of size nu, and y is the vector of outputs from the system of size ny. The overall objective is to determine the discrete impulse H (or step S,) response coefficients for the model that can be represented as combinations of the state-space matrices, A , B, C and D:

H(O) = D ~ ( i ) = CA'-' B i = 1. . . . , p I

S,,(O) = D S,(i) = D + CA'-'B i = 1, . . . , p T== I

---.-. ,- sensor banks

L actuators

machine direction - headbox

2.2 Model identification Prior to the building of a model predictive controller that could manage sheet shrinkage, an identification study was performed. The goal was to maintain control of the CD

IEE Proc -Control Theory Appl , Vol 149 No 5. Septrmher 2002

c -30s d Fig. 1 The times represent approximate time delays at various points in the model

Papev machine with wet- and dry-endjiill-avray sensors

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The controller is also in state-space form:

s(i + 1) = Ps(i) + &(i) (2)

I , is an identity matrix the size of the number of outputs, ny. The state-space form of MPC results from combining (4) with a single shifted version of itself [25]:

u(i) = Rs(i) + Sy(i) where s is the controller state vector of size ns. An excitation signal v of size nu used in the identification procedure is added to the input u. The excitation signal is assumed to be persistently exciting and uncorrelated with measurement noise.

2.3 State-space MPC To implement the closed-loop identification routine, a state-space formulation of MPC is utilised in this work. Further, the development will be limited to an uncon- strained controller, which leads ultimately to an equivalent LTI formulation. State-space formulations have been considered by a number of researchers [25-271 and the particular step-response formulation by Li et al. [25] is employed in this study.

As with most predictive controllers, the algorithm can be decomposed into three discrete elements: prediction, correction and compensation. For unconstrained MPC, the compensation step takes the form of a fixed state- feedback law with a suitably defined state:

A W k ) = K,,,,E(k I k )

= Kinpc[Y(' I k) - '(')I (3)

where E(klk) is the predicted future error of the process, Y(klk) is the predicted hture outputs, and R(k) is the predicted desired trajectory. Since CD control is a regula- tory problem, the reference trajectory will be set to zero at all times in this formulation. The prediction and correction equations for a step response model are given by:

Y(k1k- 1 > = M Y ( k - 1 ~ k - 1 ) + S l , A l l ( k - 1 ) (4)

where

M =

0 I , 0 * . ' 0 0

0 . . I,. . . 0 0

. . . . . . . . . . . . . . . . . .

0 0 0 .. zny 0

01

where

The predicted future outputs Y(klk) become the states s(i) of (2). Similarly, the plant outputs y(k) and the previous control moves AU(k - 1) become the inputs y(i) to the controller and the current control moves AU(k) are the outputs u(i) from the controller of (2).

To complete the problem specification, the traditional quadratic objective function [26] is used:

This optimisation problem can be recast as the following least-squares problem [26]:

with the solution

AU = (s;r;rysll + r : r u i - ' s , T r , T r y ( - w ( k i k ) ) (9)

K~~~~ = rs/r,;r,s,, + r;rui-'sP~;rJ, (10)

The controller gain can thus be realised as

Equation (6) will be applied in the closed-loop identifica- tion routine of the following section as the controller of (2). The goal of the identification will be to determine the step response model SI, within the MPC framework.

2.4 Identification algorithm The identification scheme of from Phan et al. [16] uses the state-space form of the system (1) and the controller (2) in a combined form in a two-step process. First, the proposed state-space open-loop model and the state-space controller are combined to form a closed-loop state-space represen- tation (eqns. 30-32 of [ 161):

Y,G) = C,X,(i) + D,u,(i)

U,G> = F,y,(i) + V,G>

with a feedback controller

The addition and subtraction of an observer term M a , to the state equation in ( 1 1) yields the closed-loop observer form of the state-space system (eqn. 33 of [ 161):

x,(i + 1) = &X,(i) + B,J,(i) (12)

Y,( i ) = C,X,(i) + D,U,(i) M , is determined by a deadbeat condition (A,,+ M,C,)P = 0. This allows one to follow the dynamics related to the controller and the process independent of the observer dynamics. The deadbeat condition, however,

IEE P~.uc.-Control Tlieor?, Appl , Vol. 149, No. 5, September 2002

will impose certain limitations on minimum data sets and model orders that are allowable for closed-loop identifica- tion. The observer plays a key role in the identification of the system parameters. It is the observer gain that allows one to proceed from the closed-loop observer state-space model to the open-loop state-space model.

With the above state-space framework in mind, data of the closed-loop system in question are collected. From these data, particularly the v(i) , y( i ) , u(i) and U(i(i), one can determine the closed-looD observer Markov parameters Hc,c+ according to

Y , = H u c v u

i = 1,

) . . .

) . . . . . .

0 z,(O) z,(l) . . .

In this case, p is the model memory and determines the order of the model to be identified. Also, N is the number of time samples that are used to identify the model. The value of p must be chosen such that p z (nx + ns)/(ny + as). The number of samples N necessary for the identification to proceed is bounded by N l p ( n u + P Z J ) ) / ~ Z L and may need to be significantly larger if a significant amount of noise is present. These limitations are necessarl for the open-loop impulse- response coefficients CA B, where k l p , to tend towards zero after a finite time as well as the need to have as many independent rows of Vas possible [28]. Also, each element of the excitation signal v = [ v l v 2 ] ~ where v, acts on the process input to excite the process directly and v2 on the setpoint to identify low-frequency behaviour, should be persistently exciting and uncorrelated. In the examples below, the excitation signals were selected to be pseudo- random binary sequences having a sample time greater than that of the sensor sample rate, each of which had different seeds for the random-number generator.

The least-squares solution to the equation above for the closed-loop observer Markov parameters is:

IEE Proc-Control Tlieory Appl.. Yol. 149, No. 5, Septenzher 2002

where @ is the pseudoinverse of the matrix Vu. The accuracy and efficiency of this algorithm is directly related to the size of the pseudoinverse problem, p(nu + ns) x N, and the number of closed-loop observer Markov para- meters that are to be identified. The closed-loop observer Markov parameters can be interpreted as the output and observer responses to an impulse input:

B,,(O) D, koc(i) = C,A:i'B,, i = 1, . . . , p

From the closed-loop observer Markov parameters, one can uniquely determine the so-called closed-loop system Markov parameters Hac by partitioning the closed-loop observer Markov parameters as

= [ H,,(O) f i g ( 1 ) figyl) . ' . fip(p> i@(p)]

and solving for the partitioned parameters

l f , , ( z ) = fiLB'(i) + 2 fi,!$(~)(I - fi,c(0)F,)-'H,c(i - z) T= I

(15) where Fa I S a matrix of the controller parameters:

The closed-loop Markov parameters parameters are equal to the closed-loop system's impulse response:

H,,(O) = D, HoC(i) = C,A:;'B,, Similarly, the open-loop system Markov parameters H , can be determined through

i = 1, . . . , p

I- 1

H O G ) = f f , L ( i ) V - F u H u c ( O ) l + c ff,,(T)F,H,(i - 7) I= I

(16)

The open-loop Markov parameters are used to determine the series of outputs of the system, given the excitation and the previous inputs to the system. From these Markov parameters, the open-loop impulse response

H(0) = D H ( i ) = CA'-'B i = 1,. . . , p

can be determined from the upper ny x nu portion of the H,(i) matrix. From the impulse-response coefficients H(i) , the step-response coefficients Sll(i) for use within the model predictive controller can be determined by

SAi) = f: H u ( 4 (17) T=l

3 Example and results

The benchmark problem studied in this paper is a proprie- tary Matlab and Simulink model of a paper machine [8]. Measurements of cross-direction variations can be achieved with either a full-array wet-end sensor at the end of the fourdrinier table or a full-array dry-end sensor located at the end of the machine (Fig. 1). Each of these sensor banks was modelled to contain roughly 360 independent sensor loca- tions distributed evenly across the machine while the head- box has 45 evenly spaced actuator locations. The objective in each case is to control the CD variation of basis weight in the presence of sustained disturbances.

Like many other sheet and film processes, the simu- lation model employs first-order-plus-time-delay dynamics

45 1

between the actuator and sensor with a Toeplitz symmetric Input-output interaction matrix. Prior to dry-end sensing, the outputs of the model can be nonlinearly remapped, based on the percentage of shrinkage. The time delays between the headbox and the sensor locations are included in Fig. 1. The model assumes that no significant shrinkage has occurred prior to the wet-end measurements.

While the theory of closed-loop identification allows one to identify each input-output relationship, the magnitude of inputs and outputs in an industrial paper machine prohibit one from identifying the model of every actuator position at every sensor location. As the number of inputs and outputs increases for a particular identification problem, it becomes increasingly difficult to excite all of the necessary directions, as well as save and perform the calculations on the large data matrices even on current computer hardware. For the present study, five actuator locations were selected across the span of the sheet and all were simultaneously identified using the Markov- parameter approach outlined in Section 2.4. The remaining actuator-to-sensor-location responses will be interpolated from the five identified responses.

The excitation signals chosen for the examples were a set of random binary sequences, each with a different seed value and each with a 5 s hold time. The prediction horizon within the MPC routine was chosen to be four owing to the low order of dynamic character and the need to minimise the number of states for the controller in (6). The move horizon was chosen to be one as it was found that more control moves made the controller too sluggish. The original model in the MPC routine was assumed to be a first-order-plus-time-delay model with a diagonal gain matrix with a gain that is only 80% of the actual process’s diagonal gain elements. Once the model is identified, the newly identified model will be tested against this original

controller to determine the improvement achieved with the identification procedure.

An example of the type of data that was collected for analysis is shown in Fig. 2. The Figure shows the data for analysis of a dry-end measured system that incurs only a minimal nonlinear-shrinkage effect, which in turn results in an uneven actuator-response distribution. Noise was simu- lated with a zero-mean random-number generator with variance of 1 (gm/cni2)2 and 20 : 1 signal-to-noise ratio, which is significantly more than values reported for indus- trial scanning sensors. While only the first 300 significant (when a change was noticed) sample times were used, 350 samples were collected to compensate for the significant deadtime between the headbox and the sensor locations. Deadtime is not included as part of the model as it is assumed to be known and is added to the full model after identification of the actuator response. The number of samples was chosen to accommodate the condition N>p(nu + ny)/nu, or N z 4(5 + 360)/5 = 292. Fig. 3 shows the identified actuator responses using only the first 300 significant time points from the data set. From the peaks of the waves from each response shape, the centres of the other actuator responses can be extrapolated. This centre identification is shown for both dry-end trials in Fig. 4. Note that the extrapolation routine is able to detect the 2% edge shrinkage between the two cases as well as the uneven distribution of centres in both cases.

Fig. 3J’ shows the steady-state response of the third identified actuator as compared with the expected profile. The mean-squared error for this actuator-response profile was found to be 1.01 x 1 0-6 (gm/cm2)2, roughly 1 % of the maximum response gain. The error is mainly concentrated in areas of the cross direction that the response actually has no real effect. This is potentially due to the attempt to identify many actuators at the same time and could

5 , i

i -5‘ ~

a o j

400 400 -.. - -

200 - - ” 40 6o time,s 0 o 2o

actuator position ,j sensor position

e sensor position

Fig. 2 The set was taken from a simulation of a machine that incurs only a small amount of nonlinear shrinkage during drying and is sensed using a dry-end sensor A number of extra samples are collected to conipensate for the deadtime between the actuator and sensor locations a Excitation signal V I

b Excitation signal v2 c Controlled input u d Dry-end basis-weight response y with noise

Representative data set used in the identification analysis

452 IEE Puoc.-Contuol Theorj Appl.. Vol. 149, No. 5, September 2002

0.02 1 0.02 * ? 001 -2 0011

5 0; - x 0 ,

- 6 -001 -0.01 "

sensor position 0 * time. s sensor position time, s

.__ -; 0-0

400 -- '6 400 & - " ~ - 206 *

0.02 > 0.02

7 j 001: > 001; 2 01 -

-i_ -0 01 - -- -; 6 400 ;c0-*. ~ ~ - ~ - -- ' " - 6 400

200"- ^ ^ ^ _ - * - 2

sensor position O O r time, s sensor position 0 O d * time, s

I 0027 ! 1

i i

-001' - - - - - 4 I

sensor position f 0 100 200 300 400 sensor position

Fig. 3 Stepresponse ident$caiion setfionz the data collected in Fjg. 2 Each response is labelled for the actuator nu that it represents, y(nu) u Actuator v i b Actuator vz c Actuator ZI

dActuator y with noisc e Actuator y without noise f Identified steady-state response of the third identified actuator compared with the expected response. The mean-squared error between the two is 1.01 x ~ o P (gm/mzI2

possibly be eliminated if each estimation was only performed on a window of the CD instead of the entire CD.

Each identified model was tested for its disturbance- rejection capability within the MPC framework. The disturbance introduced at the headbox to which each trial was subjected is shown in Fig. 5. A typical result from one o f the trials can be seen in Fig. 6. The results o f these siniulations are suminarised in Table 1 as a scaled value of the CD-variance calculation as outlined in [29]. The MPC controller formed with the identified model proved to reduce the cross-directional variance by roughly 76-78% if a wet-

300 -

200 - L :: a,

100 -

end full-array sensor was used. Also, for trials in which the sensor was assumed to be at the dry end, a reduction of 74- 82% of the CD variance was found after implementation of the identified model. It was also found that the identified model had a reduction in the amount of CD variance in the cases when a significant amount of edge shrinkage occurs.

There were some drawbacks to this identification tech- nique, however. While the amount of time necessary to collect the data needed for identification is small, 300 significant samples in this case, the amount of time necessary for identification would make this technique

x identified centres, no edge shrinkage - interpolated centres, no edge shrinkage 0 identified centres, 10% edge shrinkage ..-- interpolated centres, 10% edge shrinkage

40 0

0 10 20 30

actuator

Fig. 4

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Centtz identzjication,fiunz the data collected in Fig. 2 andfrom data representing a machine that has 2% edge shrinkage

453

3

2 1

1 1

: o i 2 1 3 -1 1

i -3 1

v) - D -2

-4 i -5

600 500

300 400 300

200 . " -- 200 100 100

time, s 0 0 sensor position

Fig. 5 Disturbance profile used in each of'the MPG case studies

400

time, s sensor position

1

-1 200

50

time, s actuator position

Fig. 6 corresponding to Figs. 2, 3 und 4 a Output profile b Input profile

Representative closed-loop trial showing the resulting outpt1t and input profiles for the identified model on the machine

Table 1: Performance of MPC controllers before and after closed-loop identification

Sensor Shrinkage Before After Improvement after location (%) ID uCD ID mCD identification (%)

Wet end 0 0.980 0.216 76

Wet end 2 0.994 0.214 78 Wet end 5 1.000 0.217 78 Dry end 0 0.767 0.027 74 Dry end 2 0.842 0.094 75 Dry end 5 0.849 0.032 82

The 2 - (icD test was used as the performance criterion and all values are scaled against the worst case within the set

454

implausible for on-line implementation. While the proce- dure consumed only 400MB of memory, the computa- tional demand limits the on-line capabilities of this numerical technique. For the problem above, the identification routine took nearly 6min on a 400MHz SunOS machine with 1 GB of RAM and 2.5 GB of swap space. Using a Linux based PC with an 800MHz processor and 256MB of memory and 500 MB of swap space, the same identifi- cation routine took 3min to produce the same results. Finally, using a WindowsXP-based PC with a processor running at 1.2 GHz with 5 16 MB of memory and dynamic swap space, the routine took only 1.5 min. While a faster processor will improve the efficiency of the algorithm, the times are still far too long to be implemented on-line. Implementation of this type of technique in its current form

IEE Proc.-Control Theory Appl. , Vol. 149, No. 5, September 2002

Table 2: Computation time and identification error in the steady-state profiles for varying levels of noise

Signal-to Minimum Computat ion Mean-squared error noise-ratio da t a t ime (sed in s teady-state

required (N) profile (gm/cm‘)’

20: 1 300 86.273 1.010 x 10-6

2: 1 475 124.059 1.679 IO-^ 4:3 600 178.777 4.437 IO-^

4: 1 400 96.739 2.314 x

The computation times are for a WindowsXP based PC with a processor running at 1.2 GHz with 516 M B of memory and dynamic swap space

could be performed on demand in response to changes in ,operation that may affect the parameters of the model. Potential improvements to the current numerical technique will be outlined in Section 4.

Finally, varying noise levels were tested to determine the amount of necessary data, N and the corresponding increase in computation time to identify the actuator responses. Table 2 summarises this study. While the mean-squared error is an order of magnitude higher for the levels of noise above 5%, the computational demand and the necessary amount of data increases linearly as a function of the noise while the computation time increases exponentially as a function of the data required.

4 Summary

A closed-loop identification routine has been developed in the model predictive-control framework. Simulation studies on an industrial benchmark simulator point to the efficiency of the approach. Starting with a simple model approximation, the closed-loop identification technique yielded a controller that outperformed the approximation- based controller in the cases of plant-model mismatch in the centre locations of the actuator responses.

While the identification technique only needed the minimum amount of data to identify the responses, a large amount of computation time was needed to manip- ulate the large data arrays that result from the low number of data points. This can be attributed to a number of issues regarding the implementation of this technique. First, the dimensionality of the states in the MPC routine, even for small move and prediction horizons, can become rather large. A reduction in the size of the state-space model used in MPC, such as that described in [27], could allow the identification routine to be performed much faster.

Some improvements could also be made to the algo- rithm implementation to increase the efficiency of the results, yielding a more tractable on-line technique. First, sparse-matrix mathematics could be employed to reduce the computational load of the algorithm. A majority of the computational demand is the recombination of the large Markov-parameter matrices from closed-loop observer, to closed-loop to open-loop. Similar sparsity techniques have been employed in recent constrained-MPC code for CD control [S, 301. Also, as mentioned above, in attempting to reduce inaccuracies in response identification at areas in the CD far away from the actuator location, windowing the CD to identify specific sections for each actuator would also decrease the number of samples needed, and thus the computational and memory demand.

While the technique described above has been formu- lated to allow only for unconstrained controller techniques,

IEE Proc.-Contivl Theor?; Appl., Vol 149, No. 5. September. 2002

it may be desirable, especially for CD control, to implement closed-loop identification with constrained controllers. A possible extension to the method described could be to use soft constraints, implementing system constraints as additional terms in the objective function. This technique would allow the process to react optimally while account- ing for the constraint set without the necessity for hard constraints that preclude the application of the described technique.

5 Acknowledgments

The authors acknowledge the financial support of the University of Delaware and the technical and financial support of Weyerhaeuser through the University of Delaware’s Process Control and Monitoring Consortium. Dr. Jay Lee of the Georgia Institute of Technology School of Chemical Engineering made valuable comments.

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References

KRISTINSSON, K., and DUMONT, G.A.: ‘Cross-directional control on paper machines using gram polynomials’, Automutica, 1996, 32, (4), pp. 533-548 FEATHERSTONE, A.P., and BRAATZ, R.D.: ‘Control relevant identi- fication of sheet and film processes’. Proceedings of American Control Conference, Seattle, WA, USA, 1995, pp. 2692-2696 RIGOPOULOS, A., ARKUN, Y., and KAYIHAN, F.: ‘Principal compo- nents analysis in estimation and control of paper machines’, Comput. Chem. Eng., 1996,20, pp. S1059-Sl064 RIGOPOULOS, A,, ARKUN, Y., and KAYIHAN, F.: ‘Model predictive control of CD profiles in sheet forming processes using full profile disturbance models identified by adaptive PCA’. Proceedings of American Control Conference, Albuquerque, NM, USA, 1997, pp. 1468-1472 BRAATZ, R., TYLER: M., MORARI, M., PRANCKH, E, and SARTOR, L.: ‘Identification and cross-directional control of coating processes’, AlClzE J., 1992, 38, pp. 1329-1339 GORINEVSKY, D., HEAVEN, M., HAGOUT-ALEXANDER, C., KEAN. M.. and MORGAN. S.: ‘New algorithms for intellieent identi- fication of ’paper alignment and nonlin&r shrinkage’. Proceedings of Control Systems’96, Halifax, Canada, 1996, pp. 335-348 GORINEVSKY, D., and HEAVEN, M.: ‘Automated identification of actuator mapping in cross-directional control of a paper machine’. Proceedings of American Control Conference, Albuquerque, NM, USA, 1997, pp. 3400-3404 SAFFER 11, D.R., DOYLE 111, EJ., RIGOPOULOS, A., and WISNEWSKI, P.: ‘MPC study for a dual headbox CD control problem’,

PIKULIK, I . , PORIER, N., and LEGER, E: ‘Papermaking in the third millennium’, Pulp Pup. Can., 1999, 100, (lo), pp. 23-27 GUSTAVSSON, I., LJUNG, L., and SODERSTROM, T.: ‘Survey paper: Identification of processes in closed loop-identificbility and accuracy aspects’, Automaticn, 1977, 13, pp. 59-75 FORSSELL, U., and LJUNG, L.: ‘Closed-loop identification revisitcd’, Automatica, 1999, 35, pp. 1215-1241 DUNCAN, S.R.: ‘Estimating the response of actuators in a cross- directional control system’. Proceedings of Control Systems 96, Halifax, Canada, 1996, pp. 19-22 BOHLIN, T.: ‘On the problem of ambiguities in maximum likclihood identification’, Automuticu, 1971, 7, pp. 199-210 TEE, L.H., and WU, S.M.: ‘An application of stochastic and dynamic models for the control of a papermaking process’, Technometrics, 1972, 14, pp. 481496 BOX.’ G., and MACGREGOR, J.F.: ‘The analysis of closed-loop dynamic-stochastic systems’, Technometrics, 1974, 16, pp. 391-398 PHAN, M., JUANG, J., HORTA. L., and LONGMAN, R.: ‘System identification from closed-loop data with known output feedback dynamics’, J: Guid. Control Dyn., 1994, 17, pp. 661-669 RIGOPOULOS, A,: ‘Application of principal component analysis in the identification and control of sheet-forming processes’. PhD thesis, Georgia Institute of Technology, Atlanta, GA, USA, 1999 CHEN, S., and WILHELM, R.: ‘Optimal control of cross-machine direction web profile with constraints on the control effort’. Proceedings of American Control Conference, Seattle, WA, USA, 1986, pp. 1409-1415 DAVE, P., WILLIG, D., KUDVA, G., PEKNY, J., and DOYLE 111, E: ‘LP methods in MPC of large scale systems - Application to paper machine CD control’, AIChEd, 1997, 43, pp. 1016-1031 RAO, C., CAMPBELL, J., RAWLINGS, J., and WRIGHT, S.: ‘Efficient implementation of model predictive control for sheet and film processes’. Proceedings of American Control Conference, Albuquerque, NM, USA, 1997, pp. 2940-2944 RAWLINGS, J., and CHIEN, I.: ‘Gage control of film and sheet forming processes’, A I C h E I , 1996, 42, pp. 753-766

Pulp PUP. Can., 2001, 102, pp. 97-101

455

22 DUMONT, G.: ‘Control techniques in the pulp and paper industry’, Control D,vn. Syst., 1990, 37, pp. 65-1 14

23 ANTWERP, J.G,V, and BRAATZ, R.: ‘Model predictive control of large scale processes, Dynamics and control of process systems’ (Elsevier Science, 1999), Vol. 1 , pp. 153-158

24 DUNCAN, S., and CORSCADDEN, K.: ‘Mini-max control of cross- directional variations on a paper machine’, IEE Proc., Control Theory

25 LI, S., LIM, K.Y., and FISHER, D.G.: ‘A state space formulation for model predictive control’, AIChE .L, 1,989, 35, pp. 241-249

26 LEE, J.H., MORARI, M., and GARCIA, C.: ‘State-space interpretation of model predictive control’, Aufomatica, 1994, 30, (4), pp. 707-717

Appl., 1998, 145, pp. 189-195

456

27 TVRZSKk DE GOUVEA, M., and ODLOAK, D.: ‘ROSSMPC: A new way of representing and analyzing predictive controllers’, Trans. IChemE, 1997,15, pp. 693-708

28 JUANG, J.: ‘Applied system identification’ (PTR Prentice Hall, Englewood Cliffs, NJ, 1994)

29 TAPPI, ‘Calculation and partitioning of variance using paper machinc scanning sensor measuremcnts’. TIP I 101 -01, Technical, 1997, Proccss Control Committee of the Process Control, Electrical and Information Division, TAPPI, Atlanta, GA

30 BACHSTROM, J., GHEORGHE, C., STEWART, G., and VYSE, R.: ‘Constrained model predictive control for cross directional multi-array processes’, Pidp Pap. Can., 2001, 102, (5), pp. 33-36

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