Robust Observer-Based Control of an Aluminum Strip Processing Line

Transcription

1 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 36, NO. 3, MAY/UNE Robust Observer-Base Control of an Aluminum Strip Processing Line Prabhakar R. Pagilla, Member, IEEE, Eugene O. King, Member, IEEE, Louis H. Dreinhoefer, Senior Member, IEEE, an Srinivas S. Garimella Abstract Tension control of an aluminum strip in a strip processing line is the focus of this paper. A continuous strip processing line is truly a large-scale complex interconnecte ynamic system with numerous control zones to transport the strip while processing it. In this paper, two aspects affecting the tension behavior of the strip in the entire processing line have been stuie. First, a moel that accurately represents the ynamics of the strip in accumulator spans is erive from the first principles. Secon, an estimate ecouple state feeback controller is esigne for the linearize ynamics of controlle spans. The state estimates are obtaine using a Luenberger observer. Convergence of the state an estimation errors is shown. Some remarks on etection of actuator faults using a linear observer for interconnecte systems are also given. Inex Terms Accumulators, aluminum strip processing, moeling, observer, tension control. A L R K E B f tn Tn tn un un Un vn vn Vn M NOMENCLATURE Cross-sectional area of web. Polar moment of inertia of roller. Length of span. Raius of roller. Motor constants. Moulus of elasticity. Bearing friction. Operating value of strip tension. Change in strip tension force from operating value. Strip tension force. Input to riven motor. Input value at steay state. Change in input from steay-state value. Strip velocity. Steay-state operating web velocity. Change in velocity from steay state. Density of aluminum strip. Mass of the accumulator carriage. Paper PID 99 24, presente at the 1999 Inustry Applications Society Annual Meeting, Phoenix, AZ, October 3 7, an approve for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Metal Inustry Committee of the IEEE Inustry Applications Society. Manuscript submitte for review October 8, 1999 an release for publication December 27, P. R. Pagilla is with the School of Mechanical an Aerospace Engineering, Oklahoma State University, Stillwater, OK USA ( pagilla@ceat.okstate.eu). E. O. King an S. S. Garimella are with the Alcoa Technical Center, Pittsburgh, PA USA ( eugene.king@alcoa.com; srinivas.garimella@alcoa.com). L. H. Dreinhoefer is with Alcoa FRP Engineering, Knoxville, TN 3792 USA ( lou.reinhoefer@alcoa.com). Publisher Item Ientifier S () t h Time. Strain. Thickness of web. I. INTRODUCTION ACONTINUOUS aluminum strip processing line typically consists of an entry section, a process section, an an exit section. The entry section consists of an unwin stan, tension leveler, an an entry accumulator. Operations such as wash, coat, an quench on the strip are performe in the wash an coat section. The exit section consists of an exit accumulator an a rewin stan. The function of the entry an the exit accumulators is to store/release strip material. The accumulators facilitate continuous operation of the line when either a rewin roll or unwin roll change over takes place. Tension control of the aluminum strip in the entire processing line is crucial to maintaining tension of the strip at esire levels. This further assures the require quality of the finishe roll. The primary motivation for this work stems from observations mae on an Alcoa finishing process line. It has been observe that the ynamics of the accumulator plays an important role on the behavior of strip tension in the entire line. Tension isturbance propagation has been notice ue to motion of the accumulator carriage both upstream an ownstream of the accumulator. Our first preliminary work reporte in this paper was to look at the strip ynamics ue to carriage motion. Previous work has ignore the ynamics of the carriage motion on strip tension ynamics. In this work, we erive a mathematical moel of the strip tension ynamics from the first principles taking into account the time-varying nature of the length of the strip in the accumulator. The erive moel reflects not only the time-varying position of the accumulator carriage but also its spee changes. The secon aspect of this work eals with the esign of an observer-base feeback controller. Again, the motivation comes from the fact that processing lines o not generally contain aequate number of sensors to measure all the state variables. In some cases, such as hot ovens, it may not be possible to get sensor information. In this work, a moel-base Luenberger observer is constructe for the interconnecte controlle spans, where only velocity measurements are available. A full-orer observer that estimates both tension an velocity has been constructe. It is shown that a ecouple feeback controller using estimate states for feeback results in a stable close-loop system. Early work escribing the longituinal ynamics of a web can be foun in the book by Campbell [1]. Campbell s mathematical /$1. 2 IEEE

2 866 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 36, NO. 3, MAY/UNE 2 Fig. 1. Typical process line layout an terminology. moel for longituinal ynamics oes not preict tension transfer, as he oes not consier tension in the entering span. An historical perspective of lateral an longituinal behavior of moving webs is given by Young an Rei [4]. Wolfermann [2] reviews several problems associate with tension control an highlights some focus areas for the future. Mathematical moel of multispan web transport systems with/without ancer subsystems was evelope by Shin [5]. A large boy of research in the area of large-scale interconnecte systems has been reporte by Siljak in his book entitle Decentralize Control of Complex Systems [6]. This paper is organize as follows. In Section II, a sketch of a typical aluminum strip processing line an its elements are shown. Dynamic moel of the unwin, rewin, controlle, an free spans are given in Section III. A ynamic moel for tension in accumulator spans is erive in Section III-A. Section III-B contains the linearize ynamics of the controlle fixe spans. In Section IV, a ecouple state feeback controller is esigne for a simple two span controlle system. Section IV-A gives some remarks on etection of faults. Conclusions an future work are given in Section V. Fig. 2. Simplifie sketch of a web line. L 1 _t 1 AE(v 1 v ) v 1 t 1 + K 1 R ( ) v u 1 (2) 1 _v 1 B f1 v 1 + R 2 1 (t 2 t 1 ) (3) L 2 _t 2 AE(v 2 v 1 ) v 2 t 2 + v 1 t 1 (4) 2 _v 2 B f2 v 2 + R 2 2 (t 3 t 2 )+R 2 K 2 u 2 (5) L 3 _t 3 AE(v 3 v 2 ) v 3 t 3 + v 2 t 2 (6) 3 _v 3 B f3 v 3 + R 2 3 (t 4 t 3 ) (7) L 4 _t 4 AE(v 4 v 3 ) v 4 t 4 + v 3 t 3 (8) 4 _v 4 B f4 v 4 R 2 4 t 4 + R 4 K 4 u 4 (9) II. ALUMINUM STRIP PROCESSING LINE A sketch of a typical continuous strip process line layout is given in Fig. 1. It is compose of an entry section that unwins unprocesse strip, an entry accumulator that releases web into the process section when the entry section is stoppe, a process section where strip processing is performe, an an exit accumulator that stores web when the exit is stoppe for a rewin changeover, an an exit section that wins the processe web into rolls. Briles shown in the figure are riven rolls an are either riven by ac or c rives. Brile rolls provie transport of the web in the line. Both accumulator carriages are controlle by hyraulic means that provie regulation of tension in the strip when the carriage is in motion. III. DYNAMICS OF TYPICAL ELEMENTS IN A PROCESSING LINE Consiering Fig. 2, the ynamics [3] of the unwin roller, web spans, an rewin roller are given by _v R K u 1 + R 2 t 1 (1) where 1 an 4 enote the time-varying inertia of the unwin an rewin, respectively. The time-varying raii of the unwin an rewin rolls are R 1 r R 2 1i v ht R 1 r R 2 2i v 4ht where R 1i an R 2i enote the initial raii or unwin an rewin rolls. Notice that the ynamics are nonlinear an time varying. For control esign purposes, it is typically assume that the inertia of the unwin an rewin rolls are changing slowly when compare to the ynamics of the strip. The nonlinearities in the ynamics appear only in the tension ynamics an as bilinear terms in states. Moreover, the interconnecting nonlinearities in a controlle span epen only on the neighboring spans. Hence, the strip processing line is a special class of a general large-scale system, wherein the interconnecting nonlinearities epen on neighboring subsystems only. Also, notice that the span length is assume to be constant. In accumulators, the span length varies with the motion of the

3 PAGILLA et al.: ROBUST OBSERVER-BASED CONTROL OF AN ALUMINUM STRIP PROCESSING LINE 867 Substituting (15) into (1), we obtain t u (x; t)a(x; t) x 1+" x (x; t) 1u(x; t); A 1u (x; t)v 1 1+" x1 (x; t) 2u(x; t); A 2u (x; t)v 2 : 1+" x2 (x; t) (16) Assuming the ensity () an the moulus of elasticity (E) of the web in the unstretche state are constant over the cross section, (16) can be written as Fig. 3. Sketch of an accumulator span. t 1 1+" x (x; t) x v 1 v 2 1+" x1 (x; t) 1+" x2 (x; t) : (17) carriage of the accumulator. It is conventional wisom to just take the ynamics of the fixe length span an make the length of the span time varying accoring to the carriage motion. In the following section, it is shown that the longituinal ynamics of a web span with variable span length is ifferent. A. Dynamics of a Web in Accumulator Spans Consier the sketch of a simplifie accumulator span shown in Fig. 3. The law of conservation of mass for a control volume in the first span of Fig. 3 gives "Z x2 (x; t)a(x; t) x t 1 A 1 v 1 2 A 2 v 2 (1) where x 1 an x 2 enote the coorinates or rollers 1 an 2, respectively, from a fixe reference frame. Notice that for the accumulator case roller 1 is fixe (x 1 ) an roller 2 moves along with the carriage (x 2 l), where l enotes the variable length of the span. If we consier an infinitesimal element of the strip in the machine irection, the geometric relations between unstretche an stretche element are given by x (1+" x ) x u (11) w (1+" w )w u (12) h (1+" h )h u (13) where subscript u inicates the unstretche state of the element, an w an h enote the with an height of the web, respectively. The elemental mass, m, in the unstretche an stretche state is equal, which gives m xwh u x u w u h u : (14) Combining (11) (14), we obtain (x; t)a(x; t) u (x; t)a u (x; t) 1 1+" x (x; t) : (15) Assuming that the strain is very small, " x < 1, we can neglect higher orer terms an write 1(1 + " x ) (1" x ). Then, (17) can be written as "Z x2 (1 " x (x; t)) x t v 1 [1 " x1 (x; t)] v 2 [1 " x2 (x; t)]: (18) Assuming that the strain oes not vary with x, i.e., " x (x; t) " x, the left-han sie of (18) can be written as (1 " x ) x t x t (1 " x) +(1 " x ) t x : (19) Notice that the secon term in the right-han sie of (19) is a ifferentiation of an integral with variable limits of integration. Hence, the integral can be ifferentiate using Leibnitz rule 1 of ifferentiating an integral. For simplicity, taking the accumulator case given by Fig. 3, i.e., x 1 an x 2 l, applying Leibnitz rule for (19) gives 1 Leibnitz rule is t Z Z "Z l (1 " x ) x t "Z l x t (1 " x) +(1 " x ) t"z l f (x; t) x x : t) x f(; t)+ f( ; t t

4 868 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 36, NO. 3, MAY/UNE 2 Substituting (2) into (18) an using Hooke s law, i.e., t2 AE x, gives out with a little more work. Consier the ynamics of the two spans. _t 2 AE l [v 2 v 1 ] AE l _l 1 l [t 1v 1 t 2 v 2 ] l t 2 _ l: (21) Span 1: _X 1 A 1 X 1 + B 1 U 1 + A 1 X + A 12 X 2 (26) Y 1 C 1 X 1 : (27) Notice that the last two terms in (21) appear in the tension ynamics of the strip ue to the variable length of the spans in accumulators. In fact, the ynamics of this variable length are given by the accumulator carriage ynamics, i.e., M ( 2 lt 2 ) P forces on the carriage. It shoul also be observe from (21) that the ynamics of the accumulator carriage is well reflecte in the tension ynamics of the spans. B. Linearize Dynamics of Controlle Spans The linearize ynamic moel aroun an operating point of a controlle span given by (4) an (5) is T_ n v n T AE n + V v n1 n + (22) L n L n L n _V n 2 R2 T n B f V n + R2 T n1 + R2 T n+1 + RK n U n : (23) In matrix form, linearize ynamics are given by where X n A n A n; n+1 _X n A n X n + B n U n + A n; n1x n1 + A n; n+1 X n+1 (24) Y n C n X n (25) Tn V n 2 64 v n L n 2 R2 " R 2 AE L n B f B n 3 75 A n; n1 1 2 v n1; 64 L n R 2 C n [ 1]: Notice that (A n ;B n ) is controllable an (A n ;C n ) is observable Span 2: _X 2 A 2 X 2 ++B 2 U 2 + A 21 X 1 + A 23 X 3 (28) Y 2 C 2 X 2 : (29) Notice that, if the span lengths an the raii of the rollers are the same, then the matrices A 1 an A 2 are the same. Consier the following observers: _^X 1 A 1 ^X 1 + B 1 U 1 + L 1 (Y 1 ^Y 1 ) (3) _^X 2 A 2 ^X 2 + B 2 U 2 + L 2 (Y 2 ^Y 2 ) (31) where L i ;i1; 2 enotes observer gain matrix. Defining e i X i ^X i, we obtain the observer error ynamics to be _e 1 (A 1 L 1 C 1 )e 1 + A 12 X 2 (32) _e 2 (A 2 L 2 C 2 )e 2 + A 21 X 1 : (33) Since the pairs (A 1 ;C 1 ) an (A 2 ;C 2 ) are observable, the eigenvalues of matrices A 1 L 1 C 1 an A 2 L 2 C 2 can be arbitrarily place by choosing the observer gain matrices L 1 an L 2. Now, consier the following controllers base on estimate feeback U 1 K 1 ^X 1 (34) U 2 K 2 ^X 2 (35) where K 1 an K 2 are feeback gain matrices. With these control laws, the ynamics become Define the following: _X 1 A 1 X 1 B 1 K 1 ^X 1 + A 12 X 2 (36) _X 2 A 2 X 2 B 2 K 2 ^X 2 + A 21 X 1 : (37) X1 e 1 e 1 e 2 X2 Then, the close-loop ynamics become : e 2 _e 1 A 1 e 1 + A 12 e 2 (38) IV. OBSERVER-BASED FEEDBACK CONTROLLER The control objective can be state as follows. If there is a perturbation in the tension an/or velocity of a span ue to some isturbances, then fin the perturbation in control input that brings the states to their operating values. For controller esign, we assume that only velocity measurements are available. The output equation given by (25) reflects this choice. For simplicity, we show the esign of an observer-base controller consiering two controlle spans. Generalization can be carrie where _e 2 A 2 e 2 + A 21 e 1 (39) Ai B i K i A i A12 A 12 A 12 A21 A 21 A 21 B i K i A i L i C i :

5 PAGILLA et al.: ROBUST OBSERVER-BASED CONTROL OF AN ALUMINUM STRIP PROCESSING LINE 869 We now show convergence of the close-loop errors e 1 an e 2 to zero. Let T i be a similarity transformation for A i, i.e., 3 i : T 1 A i i T i is iagonal. The matrix 3 i is negative efinite. Define e 1 an e 2 such that e 1 T 1 e 1 an e 2 T 2 e 2. The error ynamics in e 1 an e 2 become _ e e 1 + T 1 1 A 12 T 2 e 2 (4) e _ e 2 + T 1 2 A 21 T 1 e 1 : (41) The error ynamics (4) an (41) can be written in matrix form as " _e1 3 1 T 1 1 A 12 T 2 e1 _ e 2 T 1 2 A 21 T e 2 {z } A : (42) The matrix A can be mae negative efinite by proper choice of the eigenvalues of 3 1 an 3 2. Hence, the errors converge to zero. A. Remarks on Detecting Faults It is well known that the Luenberger observers given by (3) an (31) can be use to etect faults. It can be shown that such an approach oes not work for interconnecte systems because the states of the neighboring subsystems appear in the observer error ynamics. Consier the moification of (26) an (28) to reflect actuator faults _X 1 A 1 X 1 + B 1 g 1 U 1 + A 1 X + A 12 X 2 (43) X_ 2 A 2 X 2 + B 2 g 2 U 2 + A 21 X 1 + A 23 X 3 : (44) In the above equations, g 1 1 an g 2 1 means the actuators are healthy. The observer error ynamics becomes _e 1 (A 1 L 1 C 1 )e 1 + A 12 X 2 + B 1 (g 1 1)U 1 (45) _e 2 (A 2 L 2 C 2 )e 2 + A 21 X 1 + B 2 (g 2 1)U 2 : (46) Fault etection can be carrie out as follows. If kc i e i k i, then no fault occurs in actuator i;ifkc i e i k > i, for any t t f, then fault has occurre at time t f, where i is a prespecifie threshol value. Notice that C i e i Y i ^Y i, an, hence, is known. This type of fault etection approach cannot be use to conclue an actuator fault in a particular span, because the error ky i ^Y i k might have exceee a prespecifie threshol value ue to the interconnection terms X i1 an X i+1. Moreover, in the linearize ynamics, (43) an (44), the control input U i is a perturbation to the actual control input u i. Hence, the linearize ynamics given above may not actually etect actuator faults. V. CONCLUSIONS AND FUTURE WORK In this paper, a ynamic moel for strip tension ynamics in accumulator spans has been evelope. This moel reflects the motion ynamics of the accumulator carriage. A Luenberger observer was propose for the linearize ynamics of interconnecte spans. An estimate state feeback controller was esigne for the linearize ynamics. Convergence of the states an estimation errors is shown. Our future work will focus on consiering the entire process line to investigate tension isturbance propagation from one span to others that are ownstream an upstream. In this paper, we mentione that the strip processing line is truly a large-scale interconnecte system. Although we have not worke with the ynamics of the entire line in this work, future work will focus on casting the entire process line ynamics as a large-scale interconnecte system. It appears that such a framework may not only help in preicting tension isturbance propagation in the entire line, but also in the supervision an fault iagnosis of the entire processing line. Further, using linear observer-base strategies for etection an iagnosis of faults is not conclusive for interconnecte systems. Focusing on the nonlinear ynamics to construct nonlinear observers may open up new avenues. Also, notice that this ynamic moel for strip ynamics assumes only one-imensional motion of the carriage. It has been observe that accumulator carriage may sway uring its motion. This may cause a moment on the strip in contact with the rollers on the accumulator carriage. We plan to investigate the effects of this on the strip ynamics in the future. Also, this moel oes not inclue the slip effects on the roller an its role in strip ynamics in accumulator spans. We also plan to explore this in our future work. REFERENCES [1] D. P. Campbell, Process Dynamics. New York: Wiley, [2] W. Wolfermann, Tension control of webs A review of the problems an solutions in the present an future, in Proc. 3r Int. Conf. Web Hanling, 1995, pp [3] K. N. Rei an K. C. Lin, Control of longituinal tension in multi-span web transport systems uring start up, in Proc. 2n Int. Conf. Web Hanling, 1993, pp [4] G. E. Young an K. N. Rei, Lateral an longituinal ynamic behavior an control of moving webs, ASME. Dynam. Syst., Meas., Contr., vol. 115, no. 2, pp , une [5] K. H. Shin, Distribute control of tension in multi-span web transport systems, Ph.D. issertation, Oklahoma State University, Stillwater, OK, May [6] D. D. Siljak, Decentralize Control of Complex Systems. New York: Acaemic, [7] B. Sohlberg, Monitoring an failure iagnosis of a steel strip process, IEEE Trans. Contr. Syst. Technol., vol. 6, pp , Mar Prabhakar R. Pagilla (S 92 M 96) receive the B.Engg. egree from Osmania University, Hyeraba, Inia, an the M.S. an Ph.D. egrees from the University of California, Berkeley, in 199, 1994, an 1996, respectively, all in mechanical engineering. He is currently an Assistant Professor in the School of Mechanical an Aerospace Engineering, Oklahoma State University, Stillwater. His current research interests lie mainly in web hanling processes, aaptive control, time-varying systems, control of robotic surface finishing processes, large-scale systems, an mechatronics.

6 87 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 36, NO. 3, MAY/UNE 2 Eugene O. King (M 69) receive the B.S.E.E. egree from Carnegie-Mellon University, Pittsburgh, PA, the M.S.E.E. egree from Rensselaer Polytechnic Institute, Troy, NY, an the M.B.A. egree from the University of Pittsburgh, Pittsburgh, PA, in 1964, 1967, an 1973, respectively. He has been with Alcoa for more than 3 years an is presently a member of the Manufacturing Systems Technology Platform at the Alcoa Technical Center, Pittsburgh, PA. Over the years, he has mae contributions to the company s manufacturing processes in the areas of process control, instrumentation, performance monitoring, an iagnostics. Srinivas S. Garimella receive the B.Tech. egree from the Inian Institute of Technology, Maras, Inia, an the M.S. an Ph.D. egrees from The Ohio State University, Columbus, in 1985, 1987, an 1994, respectively, all in mechanical engineering. Since 1989, he has been with the Alcoa Technical Center, Pittsburgh, PA, where he is currently a Technical Specialist. His research interests inclue manufacturing, system ynamics, an control. He is currently leaing research an evelopment projects at Alcoa in the areas of moeling an control of metal rolling processes an of processing equipment. Dr. Garimella receive a University Fellowship an a Presiential Fellowship uring his grauate stuies at The Ohio State University. Louis H. Dreinhoefer (M 75 SM 91) receive the B.S.E.E. egree from the University of Missouri, Rolla. In his career with Alcoa, he has hel positions in plant engineering, construction, equipment evelopment, research, an as a Corporate Staff Engineer. Currently, he is an Alcoa Corporate Resource Specialist for process furnaces an control systems in Knoxville, TN. Mr. Dreinhoefer is an active member of the IEEE Inustry Applications Society (IAS) an its Metal Inustry Committee. He wrote an presente a Metal Inustry Committee Awar Paper in He is the Metal Inustry Committee Session Organizer for the 2 IAS Annual Meeting to be hel in Rome, Italy. He is a Registere Professional Engineer in the State of Pennsylvania.