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Identification, Estimation, and Control of Sheet and Film Processes
7c-Ol 1
Copyright © 1996 IFAC 13th Triennial World Congress, San Francisco, USA
IDENTIFICATION, ESTIMATION, AND CONTROL OF SHEET AND FILM PROCESSES
Richard D. Braatz * Babatunde A. Ogunnaike ** Andrew P. Featherstone *
* University of fllinois, 600 South Mathews Avenue, Box C-3, Urbana, 1L 61801 ** Advanced Controls Group, DuPont Experimental Station, Wilmington, DE 19880
Abstract. Sheet and film processes are of substantial industrial importance and include coating, papermaking, and polymer film extrusion processes. Recent efforts in developing state-of-the-art technologies for the identification, profile estimation, and cross directional control of sheet and film processes are described and evaluated in terms of their effectiveness at addressing practical data and operational constraints. Keywords. Large-scale systems, Paper industry, Constraints, Robust control, Predictive control, Model-based control, Identification, Estimation
1. INTRODUCTION
Sheet and film processes are of substantial industrial importance and include coating, papermaking, and polymer film extrusion processes. Coating processes are of great importance to manufacturing, especially in the photographic, magnetic and optical memory, electronic, and adhesive industries. The total capitalization of industries which rely on coating technology has been estimated to be over $500 billion worldwide (Cohen, 1990). Papermaking is the mainstay of the pulp and paper industries; and polymer film extrusion is used to make a variety of products from the manufacturing of plastic films for windshield safety glass to blow molding for making large plastic bags (Martino, 1991; Callari, 1990). Improved control of sheet and film properties can mean significant reductions in material consumption; greater production rates for existing equipment; improved product quality despite a high turnover rate in the work force resulting in inexperienced operators; elimination of product rejects; and reduced energy con-
sumption (Atkins, Rodencal and Vickery, 1982; Wallace, 1981; Wallace, 1986). Recent efforts in developing state-of-the-art technologies for the identification, profile estimation, and cross directional control of sheet and film processes are described and evaluated in terms of their effectiveness at addressing practical data and operational constraints. 2. CHARACTERIZATION The process control problem for sheet and film processes is typically separated into two main control objectives (see Fig. 1). One is the maintenance of the average sheet or film thickness, which is referred to as the machinedirection (MD) control problem. The MD problem and many related single loop problems have been studied extensively since the application of stochastic control to the problem in the late 1960s (Astrom, 1967). Surveys on MD control are available (Brewster and Bjerring, 1970; Dumont, 1986). The second main control objective
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When an actuator is manipulated, sheet or film properties invariably change for some distance on either side of the position directly downstream from the actuator. These interactions are caused by fluid flow within the sheet and film, processing between actuator and sensor banks, and/or physical connections between actuators within an actuator bank.
Fig. 1. Generic sheet and film process with scanning gauge (not drawn to scale). is the maintenance of flat profiles across the machinethis is referred to as the cross-directional (CD) control problem. The focus here will primarily be on identification, estimation, and control as related to the CD problem because this is much more difficult than the MD problem. First appropriate models for these processes are described.
2.1 Generic Models for Sheet and Film Processes The actuators are almost always located at evenly spaced points along the cross-direction. In coating applications the fluid flows through a slot, and the actuators vary the amount of fluid flow at a location by changing the width of this slot, often by thermal-expansion bolts or motor- or hydraulic-driven screws. A larger variety of actuators are found in paper machines, where actuation can be through slice lip variation, heat lamps, air hoses, or steam sprays. In some cases, multiple banks of actuators are used, with a substantial amount of space between actuator banks. No matter what mechanism is used for manipulation of sheet or film properties, the actuator dynamics are usually assumed, at least nominally, to be identical within an actuator bank.
The observed interactions are typically incorporated into the profile response model with n actuators and m interaction parameters through a constant matrix P!!:';;. Then the overall dynamic model P!!:';; (s) is given by the product of scalar dynamics (associated with the actuation, processing, and sensing) and the interaction matrix P!!:';;. The interactions in sheet and film processes usually take one of the following three forms: centrosymmetric, Toeplitz symmetric, and circulant symmetric. The assumptions regarding the nature of the interactions determines the appropriate model structure. Assumptions that accompany each model structure have been detailed (Laughlin, Morari and Braatz, 1993) and are summarized below. Whether or not the assumptions are accurate can mean success or failure of the control system design based on one of these models. The multi variable interaction matrices for sheet and film processes typically have one of the following three structures: Toeplitz symmetric, circulant symmetric, and centrosymmetric. Most models found in the literature are Toeplitz symmetric, whose structure follows from the assumption that changes observed downstream from one actuator caused by adjustments at the nearest neighboring actuators are independent of position across the machine. Circulant symmetric matrices represent interactions for circular sheet or film processes, as are used in some polymer extrusion applications (Callari, 1990; Martino, 1991). Centrosymmetric models include the other structures as special cases and can also take into account different effects near the edges.
2.2 Characteristics of Sheet and Film Processes Processing usually occurs between actuator banks and sensing banks, with typical processing including drying, pressing, steaming, heating, and stretching. During normal operation the processing is stable, and can usually be assumed to be linear. Typical sensor measurements include wet basis-weight, dry basis-weight, opacity, moisture, caliper, and organic content. Since these sensor measurements are taken after some form of processing, they are located some distance down the machine-direction from the actuation. This results in a significant time delay. Additional delay is often associated with the sensing, for example, due to the use of integrating-type sensors.
The characteristics which make the effective control of sheet and film processes especially interesting and difficult are discussed below. • The large process dimensionality makes control challenging because: (1) even with the processing speeds achievable by modern control hardware, the largescale and high speed nature of these machines put serious constraints on the amount of on-line computation available for the control algorithm (Chen and Wilhelm, Jr., 1986); and, (2) processes oflarge dimension tend to have plant matrices which are
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poorly conditioned (Braatz, 1995; Braatz and Morari, 1994). • Unknown disturbances and inaccurate values for the parameters of the physical system make it impossible to determine an exact model, either phenomenologically or via input-output identification. This model inaccuracy must be taken into account for high performance reliable control. • Actuator positions are constrained. "Clipping" control actions, which is common in practice, can cause disastrous performance or even instability (Braatz, 1995). Constraint-handling will be needed when the disturbances are sufficiently large and have sharp spatial variations across the sheet or film. • Due to the high cost of sensors (a single sensor with auxiliary equipment necessary to operate the sensor can cost $300,000), only a few sensors are used to measure the uniformity of the sheet or film. To provide measurements along the entire machine direction, these sensors are placed on tracks so as to continuously travel back and forth perpendicular to the movement of the sheet or film. Since the sheet or film is moving in the machine direction, each sensor measures only a zigzag portion of the sheet or film, as illustrated in Fig. 1. It is from these limited number of measurements that the entire sheet or film profile (that is, at all sensing locations) must be estimated at each sampling time for use by the control algorithm.
Toeplitz symmetric description (described in Section 2). The circulant symmetric description has more desirable mathematical properties for the analysis of model requirements and for design of experiments (Featherstone and Braatz, 1995). Other researchers have chosen to write process signals (e.g,. process output, disturbances) and the profile response to each actuator move in terms of a linear combination of basis functions, and to identify the coefficients corresponding to each basis function. Basis functions under study include splines (Halouskova, Karnyand Nagy, 1993) and discrete orthonormal Chebyshev polynomials, also referred to as Gram or discrete Legendre orthogonal polynomials (Kjaer, Heath and Wellstead, 1994; Kristinsson and Dumont, 1996). Probably the most critical issue associated with the identification of these processes is how to choose the input moves so as to provide data rich in process information relevant for closed loop control. Methods for addressing this problem of input design, and for calculating estimates on the accuracy of the model parameters, are being developed (Braatz and Featherstone, 1995), as this information is critical for providing reliable control of poorly conditioned processes.
4. PROFILE ESTIMATION FROM SCANNING MEASUREMENTS The cross directional profile must be determined from measurements taken from the traversing sensors. Common industrial practice is to either take the current estimate at each sensor position to be its previouslymeasured value, or a weighted sum of its previouslymeasured values (Wang, Dumont and Davies, 1993a). Recently more sophisticated estimation methods have been proposed which incorporate a process model (Dumont, Davies, Natarajan, Lindeborg, Ordubadi, Fu, Kristinsson and Jonsson, 1991; Wang et al., 1993a; Wang, Dumont and Davies, 1993b).
The large-scale, uncertain, constrained, limited-data nature of sheet and film processes makes their control especially interesting and challenging. Recent efforts in developing state-of-the-art technologies for the identification, profile estimation, and cross directional control of sheet and film processes are now described and rated in terms of their effectiveness at addressing one or some of the above aspects of sheet and film.
3. IDENTIFICATION Standard identification techniques could be employed if the entire sheet or film profile was measured at each sampling instance. In general, the identification for sheet and film is complicated by the fact that measurements are obtained via the traversing sensor. Techniques are available for identifying the nominal model and disturbance characteristics for this problem of identification with missing data (Bergh and MacGregor, 1987; Jones, 1980). A more critical issue is that the quantity of online data available for sheet and film processes is low, so that a model with a limited number of parameters must be selected. The traditional model structure is the
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Several researchers (Bergh and MacGregor, 1987; Halouskova et al., 1993; Heath and Wellstead, 1995; Rigopoulos, Arkun, Kayihan and Hanczyc, 1996) have applied least squares optimal estimation theory (Astrom, 1970) to estimate the cross-directional profile. Standard multirate systems technique known as lifting (Tyler and Morari, 1995) and two-dimensional estimation theory (Heath and Wellstead, 1995) provide alternative methods for providing optimal estimated profiles. These estimated profiles have been used in accord with MPC (Campbell and Rawlings, 1994) and LQG controllers (Bergh and MacGregor, 1987; Heath and Wellstead, 1995; Tyler and Morari, 1995). It seems that no industrial ap-
plication of these optimal estimators have been reported in the literature. On the other hand, a sub-optimal modified Kalman filter derived assuming that the MD and CD variations have substantially different time scales has been applied to industrial processes (Chen, 1988). 5. CROSS DIRECTIONAL CONTROL Control techniques are characterized as being in one of the following categories: (1) linear control, (2) Model Predictive Control, (3) robust control, (4) robust control with dimensional reduction, and (5) linear control with directionality compensation. 5.1 Linear Control The plastic film industry began implementing multivariable control in the early 1970s. Such control began to be implemented on paper machines in the mid-1970's; however, widespread use of such systems did not occur until the 1980's (Wilhelm, Jr. and Fjeld, 1983). The two multivariable control techniques reported in the literature before 1988 were linear-quadratic-optimal (LQ) (Boyle, 1978; Wilhelm, Jr. and Fjeld, 1983) and model inverse-based control (Wilkinson and Hering, 1983). Although these linear control approaches have low on-line computational requirements, they do not explicitly address constraints and model inaccuracies. 5.2 Model Predictive Control
The application of Model Predictive Control (MPC) has been considered for the control of paper machines (Boyle, 1977), coating processes (Braatz, Tyler, Morari, Pranckh and Sartor, 1992), and polymer film extruders (Camp bell and Rawlings, 1994). In MPC, the control objective is optimized on-line subject to the constraints. A linear or quadratic optimization is solved at each sampling instance, and off-the-shelf software is available for performing these calculations. Because these optimization problems can be very large (over 300 variables and 300 constraints for a medium-size paper machine), for high speed machines it is not feasible to solve the optimization problem within the sampling interval (Braatz, 1995; Chen and Wilhelm, Jr., 1986). 5.3 Robust Control
The function, p" is a nonconservative measure for system robustness. No researcher has ever proposed the direct use of p,-theory to design controllers for sheet and film processes. This is due to the computational complexity associated with p,-theory. Proofs that any exact
algorithm for p,-theory must have exponential growth as a function of process dimension are available (Braatz, Young, Doyle and Morari, 1994). Although tight approximations allow computation which has polynomial growth as a function of process dimension, the order of this polynomial seems to be large ('" n 4 for controller synthesis) (Young, Newlin and Doyle, 1991). Published examples (Hovd, Braatz and Skogestad, 1993; Laughlin et al., 1993) illustrate that current p, software is inadequate for application to sheet and film processes. 5.4 Robust Control with Dimensional Reduction Researchers have studied how to exploit the structure of sheet and film processes to reduce the computational load associated with designing, analyzing, and implementing control systems. The control of circulant symmetric processes has been of interest to researchers since the early 1970s. Brockett and Willems (1974) used the discrete Fourier transform to decompose control problems for circulant symmetric processes into lower order problems. Others (Wall, Jr., Willsky and Sandell, Jr., 1979) used similar techniques to design decentralized controllers for these processes. Circulant matrix theory has also been used to develop methods for designing robust multivariable controllers based on the design of only one single loop controller (Laughlin et al., 1993), or a set of single loop controlls (Duncan, 1994). Circulant symmetric, Toeplitz symmetric, and centrosymmetric symmetric models were all covered by the theory. The controllers for circulant symmetric and Toeplitz symmetric models were decentralized; whereas centrosymmetric symmetric models were controlled by a decentralized controller in series with a constant decoupler matrix. A strong advantage of the approach was that no iterative design procedure (for example, like that required for p, synthesis (Doyle, 1985)) was necessary. Disadvantages of this approach were that: (1) only a limited uncertainty structure was allowed; (2) the controllers may be conservatively designed in some cases; and, (3) the method does not obviously generalize to nonsquare sheet and film processes. A related strategy was to use the properties of Unitaryinvariant norms to design nonconservative robust multivariable controllers via p, synthesis performed on an m x m diagonal plant (Hovd et al., 1993; Hovd, Braatz and Skogestad, 1994), where m is the number of interaction parameters which is typically much smaller than the process dimension n. This method was applicable to circulant symmetric processes (i.e., no end effects were allowed) with varying types of uncertainty descriptions.
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8. REFERENCES
5.5 Linear Control with Directionality Compensation The traditional method for dealing with constraints is to use simple static nonlinear elements (for example, selectors and overrides) which modify the linear control system only when necessary (Buckley, 1971). Two advantages of this method are that: (1) well-developed linear control methods can be applied to design the linear controller; and (2) such constraint-handling methods are almost as easy to implement as a purely-linear controller. The static nonlinear elements are simple operations requiring very little computational effort and are already standard in industrial control. One of these methods is referred to as directionality compensation (Braatz, 1995; Campo, 1990; Segall, MacGregor and Wright, 1991). When the output of the linear controller cannot satisfy the constraints on the actuator movements, the directionality compensator scales back the linear control output while keeping the same direction until the control action becomes feasible. It was shown for an industrial scale adhesive coater that directionality compensation can perform nearly as well as Model Predictive Control, but with much simpler computation (Braatz et al., 1992). The main weakness with directionality compensation as a general design tool is that the latest analysis tools (Braatz, 1993) are too conservative for processes with large dimensionality.
6. CONCLUSION A significant amount of research has been conducted on sheet and film in last few decades, and much work remains to be done. Efforts should be directed to developing control relevant identification algorithms which exploit model structure to extract as much process information as possible for use by the control algorithms. The practical usefulness of existing estimation techniques needs to be determined. Methods for designing computationally feasible control systems which effectively address both constraints and model inaccuracies need to be developed. Rapid advances can be made through collaborations between engineers from academia and industry-this allows the weaknesses of theoretical approaches to be quickly identified, directing the researchers to further developments for improving the state of the art in sheet and film process operations.
7. ACKNOWLEDGMENTS
Astrom, K. J. (1967). Computer control of a paper machine - an application of linear stochastic control theory, IBM J. of Res. and Dev. 11: 389405. Astrom, K. J. (1970). Introduction to Stochastic Control Theory, Academic Press, San Diego, CA. Atkins, J. W., Rodencal, T. E. and Vickery, D. E. (1982). Correcting a poor moisture profile improves productivity, Tappi 65: 49-52. Bergh, L. G. and MacGregor, J. F. (1987). Spatial control of sheet and film forming processes, Can. J. of Chem. Eng. 65: 148-155. Boyle, T. J. (1977). Control of cross-direction variations in web forming machines, Can. J. of Chem. Eng. 55: 457-461. Boyle, T. J. (1978). Practical algorithms for crossdirection control, Tappi 61: 77-80. Braatz, R. D. (1993). Robust Loopshaping for Process Control, PhD thesis, California Institute of Technology, Pasadena, California. Braatz, R. D. (1995). Control of sheet and film processes, Third SIAM Conf. on Control and Its Applications, St. Louis, Missouri. Braatz, R. D. and Featherstone, A. P. (1995). Identification and control of large scale paper machines, Weyerhaeuser Workshop on Modeling and Control of Kamyr Digesters and Paper Machines, Tacoma, Washington. Braatz, R. D. and Morari, M. (1994). Minimizing the Euclidean condition number, SIAM J. on Control and Optim. 32: 1763-1768. Braatz, R. D., Tyler, M. L., Morari, M., Pranckh, F. R. and Sartor, L. (1992). Identification and crossdirectional control of coating processes, AIChE J. 38: 1329-1339. Braatz, R. D., Young, P. M., Doyle, J. C. and Morari, M. (1994). Computational complexity of J.L calculation, IEEE Trans. on Auto. Control 39: 1000-1002. Brewster, D. B. and Bjerring, A. K. (1970). Computer control in pulp and paper 1961-1969, Proc. of the IEEE 58: 49-69. Brockett, R. W. and Willems, J. L. (1974). Discretized partial differential equations: examples of control systems defined by modules, Automatica 10: 507515. Buckley, P. S. (1971). Designing overide and feedforward controls, Control Eng. 18: 48-51. Callari, J. (1990). New extrusion lines run tighter, leaner, Plastics World pp. 28-31.
Support from W. David Smith and James Trainham of DuPont, and the UIUC Research Board are acknowledged.
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Campbell, J. C. and Rawlings, J. 8. (1994). Gage control of film and sheet forming processes, AIChE Annual Meeting. Paper 224c. Studies in Robust Control Campo, P. J. (1990). of Systems Subject to Constraints, PhD thesis, California Institute of Technology, Pasadena. Chen, S.-C. (1988). Kalman filtering applied to sheet measurement, Proc. of the American Control Conj., pp. 643-647. Chen, S.-C. and Wilhelm, Jr., R. G. (1986). Optimal control of cross-machine direction web profile with constraints on the control effort, Proc. of the American Control Conj., pp. 1409-1415. Cohen, E. D. (1990). Coatings: Going below the surface, Chem. Eng. Prog. 86: 19-23. Doyle, J. C. (1985). Structured uncertainty in control system design, Proc. of the Conj. on Decision and Control, pp. 260-265. Dumont, G. A. (1986). Application of advanced control methods in the pulp paper industry-A survey, Automatica 22: 143-153. Dumont, G. A., Davies, M. S., Natarajan, K., Lindeborg, C., Ordubadi, F., Fu, Y., Kristinsson, K. and Jonsson, I. (1991). An improved algorithm for estimating paper machine moisture profiles using scanned data, Proc. of the Conj. on Decision and Control, pp. 1857-1861. Duncan, S. R. (1994). The design of robust cross directional control systems for paper making, Control Systems Centre Report 807, UMIST, Manchester, United Kingdom. Featherstone, A. P. and Braatz, R. D. (1995). Control relevant identification of sheet and film processes, Proc. of the American Control Conj. 2692-2696. Halouskova, A., Karny, M. and Nagy, I. (1993). Adaptive cross-direction control of paper basis weight, Automatica 29: 425-429. Heath, W. P. and Wellstead, P. E. (1995). Selftuning prediction and control for two-dimensional processes Part 1: fixed parameter algorithms, Int. J. of Control 62: 65-107. Hovd, M., Braatz, R. D. and Skogestad, S. (1993). On the structure of the robust optimal controller for a class of problems, Proc. of the IFAC World Congress, Vo!. IV, Sydney, Australia, pp. 27-30. Hovd, M., Braatz, R. D. and Skogestad, S. (1994). SVD controllers for H 2 -, H oo -, and J.t-optimal control, Proc. of the American Control Conj., Baltimore, Maryland, pp. 1233-1237. Jones, R. H. (1980). Maximum likelihood fitting of ARMA models to time series with missing observations, Technometrics 22: 389-395. Kjaer, A. P., Heath, W. P. and Wellstead, P. E. (1994). Identification of cross-directional behaviour in web
production: techniques and experience, Proc. of the IFAC Symp. on System Identification, Copenhagen, Denmark. Kristinsson, K. and Dumont, G. A. (1996). Paper machine cross directional basis weight using Gram polynomials, Automatica . in press. Laughlin, D., Morari, M. and Braatz, R. D. (1993). Robust performance of cross-directional basisweight control in paper machines, Automatica 29: 1395-1410. Martino, R. (1991). Motor-driven cams actuator flexible-lip automatic die, Modern Plastics 68: 2323. Rigopoulos, A., Arkun, Y., Kayihan, F. and Hanczyc, E. (1996). Estimation of paper machine full profile properties using Kalman filtering and recursive principal component analysis, Proc. of the Fifth Int. Conf. on Chemical Process Control (CPC- V), Tahoe City, California. Segall, N. L., MacGregor, J. F. and Wright, J. D. (1991). One-step optimal saturation correction, A utomatica 27: 135-139. Tyler, M. L. and Morari, M. (1995). Estimation of cross directional properties: scanning versus stationary sensors, AIChE J. 41: 846-854. Wall, Jr., J. E., WilIsky, A. S. and Sandell, Jr., N. R. (1979). Control of circulant systems, Proc. of the Joint Auto. Control Conj., pp. 854-860. Wallace, B. W. (1981). Economic benefits offered by computerized profile control of weight, moisture, and caliper, Tappi 64: 79-83. Cross-direction control: Wallace, M. D. (1986). profitability and pitfalls, Tappi J. 69: 72-75. Wang, X. G., Dumont, G. A. and Davies, M. S. (1993a). Estimation in paper machine control, IEEE Control Systems 13(8): 34-43. Wang, X. G., Dumont, G. A. and Davies, M. S. (1993b). Modelling and identification of basis weight variations in paper machines, IEEE 1'rans. on Control Systems Tech. 1: 230-237. Wilhelm, Jr., R. G. and Fjeld, M. (1983). Control algorithms for cross directional control, Proc. of the 5th IFAC PRP Conj., pp. 163-174. Wilkinson, A. J. and Hering, A. (1983). A new control technique for cross machine control of basis weight, Preprints of the 5th IFAC PRP Conj., Antwerp, Belgium, pp. 151-155. Young, P. M., Newlin, M. P. and Doyle, J. C. (1991). J.t analysis with real parametric uncertainty, Proc. of the Conj. on Decision and Control, pp. 1251-1256.
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Copyright © 1996 IFAC 13th Triennial World Congress, San Francisco, USA
IDENTIFICATION, ESTIMATION, AND CONTROL OF SHEET AND FILM PROCESSES
Richard D. Braatz * Babatunde A. Ogunnaike ** Andrew P. Featherstone *
* University of fllinois, 600 South Mathews Avenue, Box C-3, Urbana, 1L 61801 ** Advanced Controls Group, DuPont Experimental Station, Wilmington, DE 19880
Abstract. Sheet and film processes are of substantial industrial importance and include coating, papermaking, and polymer film extrusion processes. Recent efforts in developing state-of-the-art technologies for the identification, profile estimation, and cross directional control of sheet and film processes are described and evaluated in terms of their effectiveness at addressing practical data and operational constraints. Keywords. Large-scale systems, Paper industry, Constraints, Robust control, Predictive control, Model-based control, Identification, Estimation
1. INTRODUCTION
Sheet and film processes are of substantial industrial importance and include coating, papermaking, and polymer film extrusion processes. Coating processes are of great importance to manufacturing, especially in the photographic, magnetic and optical memory, electronic, and adhesive industries. The total capitalization of industries which rely on coating technology has been estimated to be over $500 billion worldwide (Cohen, 1990). Papermaking is the mainstay of the pulp and paper industries; and polymer film extrusion is used to make a variety of products from the manufacturing of plastic films for windshield safety glass to blow molding for making large plastic bags (Martino, 1991; Callari, 1990). Improved control of sheet and film properties can mean significant reductions in material consumption; greater production rates for existing equipment; improved product quality despite a high turnover rate in the work force resulting in inexperienced operators; elimination of product rejects; and reduced energy con-
sumption (Atkins, Rodencal and Vickery, 1982; Wallace, 1981; Wallace, 1986). Recent efforts in developing state-of-the-art technologies for the identification, profile estimation, and cross directional control of sheet and film processes are described and evaluated in terms of their effectiveness at addressing practical data and operational constraints. 2. CHARACTERIZATION The process control problem for sheet and film processes is typically separated into two main control objectives (see Fig. 1). One is the maintenance of the average sheet or film thickness, which is referred to as the machinedirection (MD) control problem. The MD problem and many related single loop problems have been studied extensively since the application of stochastic control to the problem in the late 1960s (Astrom, 1967). Surveys on MD control are available (Brewster and Bjerring, 1970; Dumont, 1986). The second main control objective
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When an actuator is manipulated, sheet or film properties invariably change for some distance on either side of the position directly downstream from the actuator. These interactions are caused by fluid flow within the sheet and film, processing between actuator and sensor banks, and/or physical connections between actuators within an actuator bank.
Fig. 1. Generic sheet and film process with scanning gauge (not drawn to scale). is the maintenance of flat profiles across the machinethis is referred to as the cross-directional (CD) control problem. The focus here will primarily be on identification, estimation, and control as related to the CD problem because this is much more difficult than the MD problem. First appropriate models for these processes are described.
2.1 Generic Models for Sheet and Film Processes The actuators are almost always located at evenly spaced points along the cross-direction. In coating applications the fluid flows through a slot, and the actuators vary the amount of fluid flow at a location by changing the width of this slot, often by thermal-expansion bolts or motor- or hydraulic-driven screws. A larger variety of actuators are found in paper machines, where actuation can be through slice lip variation, heat lamps, air hoses, or steam sprays. In some cases, multiple banks of actuators are used, with a substantial amount of space between actuator banks. No matter what mechanism is used for manipulation of sheet or film properties, the actuator dynamics are usually assumed, at least nominally, to be identical within an actuator bank.
The observed interactions are typically incorporated into the profile response model with n actuators and m interaction parameters through a constant matrix P!!:';;. Then the overall dynamic model P!!:';; (s) is given by the product of scalar dynamics (associated with the actuation, processing, and sensing) and the interaction matrix P!!:';;. The interactions in sheet and film processes usually take one of the following three forms: centrosymmetric, Toeplitz symmetric, and circulant symmetric. The assumptions regarding the nature of the interactions determines the appropriate model structure. Assumptions that accompany each model structure have been detailed (Laughlin, Morari and Braatz, 1993) and are summarized below. Whether or not the assumptions are accurate can mean success or failure of the control system design based on one of these models. The multi variable interaction matrices for sheet and film processes typically have one of the following three structures: Toeplitz symmetric, circulant symmetric, and centrosymmetric. Most models found in the literature are Toeplitz symmetric, whose structure follows from the assumption that changes observed downstream from one actuator caused by adjustments at the nearest neighboring actuators are independent of position across the machine. Circulant symmetric matrices represent interactions for circular sheet or film processes, as are used in some polymer extrusion applications (Callari, 1990; Martino, 1991). Centrosymmetric models include the other structures as special cases and can also take into account different effects near the edges.
2.2 Characteristics of Sheet and Film Processes Processing usually occurs between actuator banks and sensing banks, with typical processing including drying, pressing, steaming, heating, and stretching. During normal operation the processing is stable, and can usually be assumed to be linear. Typical sensor measurements include wet basis-weight, dry basis-weight, opacity, moisture, caliper, and organic content. Since these sensor measurements are taken after some form of processing, they are located some distance down the machine-direction from the actuation. This results in a significant time delay. Additional delay is often associated with the sensing, for example, due to the use of integrating-type sensors.
The characteristics which make the effective control of sheet and film processes especially interesting and difficult are discussed below. • The large process dimensionality makes control challenging because: (1) even with the processing speeds achievable by modern control hardware, the largescale and high speed nature of these machines put serious constraints on the amount of on-line computation available for the control algorithm (Chen and Wilhelm, Jr., 1986); and, (2) processes oflarge dimension tend to have plant matrices which are
6639
poorly conditioned (Braatz, 1995; Braatz and Morari, 1994). • Unknown disturbances and inaccurate values for the parameters of the physical system make it impossible to determine an exact model, either phenomenologically or via input-output identification. This model inaccuracy must be taken into account for high performance reliable control. • Actuator positions are constrained. "Clipping" control actions, which is common in practice, can cause disastrous performance or even instability (Braatz, 1995). Constraint-handling will be needed when the disturbances are sufficiently large and have sharp spatial variations across the sheet or film. • Due to the high cost of sensors (a single sensor with auxiliary equipment necessary to operate the sensor can cost $300,000), only a few sensors are used to measure the uniformity of the sheet or film. To provide measurements along the entire machine direction, these sensors are placed on tracks so as to continuously travel back and forth perpendicular to the movement of the sheet or film. Since the sheet or film is moving in the machine direction, each sensor measures only a zigzag portion of the sheet or film, as illustrated in Fig. 1. It is from these limited number of measurements that the entire sheet or film profile (that is, at all sensing locations) must be estimated at each sampling time for use by the control algorithm.
Toeplitz symmetric description (described in Section 2). The circulant symmetric description has more desirable mathematical properties for the analysis of model requirements and for design of experiments (Featherstone and Braatz, 1995). Other researchers have chosen to write process signals (e.g,. process output, disturbances) and the profile response to each actuator move in terms of a linear combination of basis functions, and to identify the coefficients corresponding to each basis function. Basis functions under study include splines (Halouskova, Karnyand Nagy, 1993) and discrete orthonormal Chebyshev polynomials, also referred to as Gram or discrete Legendre orthogonal polynomials (Kjaer, Heath and Wellstead, 1994; Kristinsson and Dumont, 1996). Probably the most critical issue associated with the identification of these processes is how to choose the input moves so as to provide data rich in process information relevant for closed loop control. Methods for addressing this problem of input design, and for calculating estimates on the accuracy of the model parameters, are being developed (Braatz and Featherstone, 1995), as this information is critical for providing reliable control of poorly conditioned processes.
4. PROFILE ESTIMATION FROM SCANNING MEASUREMENTS The cross directional profile must be determined from measurements taken from the traversing sensors. Common industrial practice is to either take the current estimate at each sensor position to be its previouslymeasured value, or a weighted sum of its previouslymeasured values (Wang, Dumont and Davies, 1993a). Recently more sophisticated estimation methods have been proposed which incorporate a process model (Dumont, Davies, Natarajan, Lindeborg, Ordubadi, Fu, Kristinsson and Jonsson, 1991; Wang et al., 1993a; Wang, Dumont and Davies, 1993b).
The large-scale, uncertain, constrained, limited-data nature of sheet and film processes makes their control especially interesting and challenging. Recent efforts in developing state-of-the-art technologies for the identification, profile estimation, and cross directional control of sheet and film processes are now described and rated in terms of their effectiveness at addressing one or some of the above aspects of sheet and film.
3. IDENTIFICATION Standard identification techniques could be employed if the entire sheet or film profile was measured at each sampling instance. In general, the identification for sheet and film is complicated by the fact that measurements are obtained via the traversing sensor. Techniques are available for identifying the nominal model and disturbance characteristics for this problem of identification with missing data (Bergh and MacGregor, 1987; Jones, 1980). A more critical issue is that the quantity of online data available for sheet and film processes is low, so that a model with a limited number of parameters must be selected. The traditional model structure is the
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Several researchers (Bergh and MacGregor, 1987; Halouskova et al., 1993; Heath and Wellstead, 1995; Rigopoulos, Arkun, Kayihan and Hanczyc, 1996) have applied least squares optimal estimation theory (Astrom, 1970) to estimate the cross-directional profile. Standard multirate systems technique known as lifting (Tyler and Morari, 1995) and two-dimensional estimation theory (Heath and Wellstead, 1995) provide alternative methods for providing optimal estimated profiles. These estimated profiles have been used in accord with MPC (Campbell and Rawlings, 1994) and LQG controllers (Bergh and MacGregor, 1987; Heath and Wellstead, 1995; Tyler and Morari, 1995). It seems that no industrial ap-
plication of these optimal estimators have been reported in the literature. On the other hand, a sub-optimal modified Kalman filter derived assuming that the MD and CD variations have substantially different time scales has been applied to industrial processes (Chen, 1988). 5. CROSS DIRECTIONAL CONTROL Control techniques are characterized as being in one of the following categories: (1) linear control, (2) Model Predictive Control, (3) robust control, (4) robust control with dimensional reduction, and (5) linear control with directionality compensation. 5.1 Linear Control The plastic film industry began implementing multivariable control in the early 1970s. Such control began to be implemented on paper machines in the mid-1970's; however, widespread use of such systems did not occur until the 1980's (Wilhelm, Jr. and Fjeld, 1983). The two multivariable control techniques reported in the literature before 1988 were linear-quadratic-optimal (LQ) (Boyle, 1978; Wilhelm, Jr. and Fjeld, 1983) and model inverse-based control (Wilkinson and Hering, 1983). Although these linear control approaches have low on-line computational requirements, they do not explicitly address constraints and model inaccuracies. 5.2 Model Predictive Control
The application of Model Predictive Control (MPC) has been considered for the control of paper machines (Boyle, 1977), coating processes (Braatz, Tyler, Morari, Pranckh and Sartor, 1992), and polymer film extruders (Camp bell and Rawlings, 1994). In MPC, the control objective is optimized on-line subject to the constraints. A linear or quadratic optimization is solved at each sampling instance, and off-the-shelf software is available for performing these calculations. Because these optimization problems can be very large (over 300 variables and 300 constraints for a medium-size paper machine), for high speed machines it is not feasible to solve the optimization problem within the sampling interval (Braatz, 1995; Chen and Wilhelm, Jr., 1986). 5.3 Robust Control
The function, p" is a nonconservative measure for system robustness. No researcher has ever proposed the direct use of p,-theory to design controllers for sheet and film processes. This is due to the computational complexity associated with p,-theory. Proofs that any exact
algorithm for p,-theory must have exponential growth as a function of process dimension are available (Braatz, Young, Doyle and Morari, 1994). Although tight approximations allow computation which has polynomial growth as a function of process dimension, the order of this polynomial seems to be large ('" n 4 for controller synthesis) (Young, Newlin and Doyle, 1991). Published examples (Hovd, Braatz and Skogestad, 1993; Laughlin et al., 1993) illustrate that current p, software is inadequate for application to sheet and film processes. 5.4 Robust Control with Dimensional Reduction Researchers have studied how to exploit the structure of sheet and film processes to reduce the computational load associated with designing, analyzing, and implementing control systems. The control of circulant symmetric processes has been of interest to researchers since the early 1970s. Brockett and Willems (1974) used the discrete Fourier transform to decompose control problems for circulant symmetric processes into lower order problems. Others (Wall, Jr., Willsky and Sandell, Jr., 1979) used similar techniques to design decentralized controllers for these processes. Circulant matrix theory has also been used to develop methods for designing robust multivariable controllers based on the design of only one single loop controller (Laughlin et al., 1993), or a set of single loop controlls (Duncan, 1994). Circulant symmetric, Toeplitz symmetric, and centrosymmetric symmetric models were all covered by the theory. The controllers for circulant symmetric and Toeplitz symmetric models were decentralized; whereas centrosymmetric symmetric models were controlled by a decentralized controller in series with a constant decoupler matrix. A strong advantage of the approach was that no iterative design procedure (for example, like that required for p, synthesis (Doyle, 1985)) was necessary. Disadvantages of this approach were that: (1) only a limited uncertainty structure was allowed; (2) the controllers may be conservatively designed in some cases; and, (3) the method does not obviously generalize to nonsquare sheet and film processes. A related strategy was to use the properties of Unitaryinvariant norms to design nonconservative robust multivariable controllers via p, synthesis performed on an m x m diagonal plant (Hovd et al., 1993; Hovd, Braatz and Skogestad, 1994), where m is the number of interaction parameters which is typically much smaller than the process dimension n. This method was applicable to circulant symmetric processes (i.e., no end effects were allowed) with varying types of uncertainty descriptions.
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8. REFERENCES
5.5 Linear Control with Directionality Compensation The traditional method for dealing with constraints is to use simple static nonlinear elements (for example, selectors and overrides) which modify the linear control system only when necessary (Buckley, 1971). Two advantages of this method are that: (1) well-developed linear control methods can be applied to design the linear controller; and (2) such constraint-handling methods are almost as easy to implement as a purely-linear controller. The static nonlinear elements are simple operations requiring very little computational effort and are already standard in industrial control. One of these methods is referred to as directionality compensation (Braatz, 1995; Campo, 1990; Segall, MacGregor and Wright, 1991). When the output of the linear controller cannot satisfy the constraints on the actuator movements, the directionality compensator scales back the linear control output while keeping the same direction until the control action becomes feasible. It was shown for an industrial scale adhesive coater that directionality compensation can perform nearly as well as Model Predictive Control, but with much simpler computation (Braatz et al., 1992). The main weakness with directionality compensation as a general design tool is that the latest analysis tools (Braatz, 1993) are too conservative for processes with large dimensionality.
6. CONCLUSION A significant amount of research has been conducted on sheet and film in last few decades, and much work remains to be done. Efforts should be directed to developing control relevant identification algorithms which exploit model structure to extract as much process information as possible for use by the control algorithms. The practical usefulness of existing estimation techniques needs to be determined. Methods for designing computationally feasible control systems which effectively address both constraints and model inaccuracies need to be developed. Rapid advances can be made through collaborations between engineers from academia and industry-this allows the weaknesses of theoretical approaches to be quickly identified, directing the researchers to further developments for improving the state of the art in sheet and film process operations.
7. ACKNOWLEDGMENTS
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Support from W. David Smith and James Trainham of DuPont, and the UIUC Research Board are acknowledged.
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