220 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 2, MARCH 1998 The Analysis and Design of Spatial Control Systems in Strip Metal Rolling Stephen R. Duncan, Member, IEEE, Julian M. Allwood, and Srinivas S. Garimella Abstract—Commercial pressures on metal strip manufacturers drive ever greater demands on the control of residual stress distributions within the finished strip. Rolling mills in current use have a range of actuators available to attempt this control and new designs are being offered with arrays of similar actuators distributed across the width of the mill. The interaction of these actuators motivates a thorough analysis of spatial control in strip metal rolling. This paper takes theory that has been successfully applied to the paper and plastics industry and applies it for the first time to the strip rolling process, giving a toolkit for the analysis and design of current and future cross-directional control systems. Data from a state of the art commercial mill is used to allow characterization of typical error signals in terms of orthogonal basis functions. Actuators are analyzed to show how much power they have within the “spectrum” of this basis function expansion. Sensors are analyzed to show their filtering effect within the spectrum. The consequent theory is used to give a rationale to future actuator design and a benchmark for the assessment of control performance with existing actuators. Two control strategies are investigated and compared—minimum variance and mini-max—and their achievements characterized. Control system sensitivity is assessed. Index Terms— Actuators, Chebyshev functions, metals indus- try, minimax control, orthogonal functions, shape control, shape measurement. I. INTRODUCTION M ETAL strip is manufactured by passing a cast ingot through a sequence of rolling mills. On each pass, the material is subject to high through-thickness forces which cause thickness reduction and strip elongation. Typically 30-40 passes occur in reducing a 0.5-m-thick ingot to a coil of 0.1- mm strip. Ideally each pass would give a perfectly uniform reduction in thickness (profile) across the width. However the high forces required to cause thickness reductions cause the rolling mills to deflect, resulting in nonuniform reduction across the strip. In hot-rolling where strip temperature is raised to reduce its yield strength, it is possible to make significant alterations to strip profile. However, in cold rolling, although deviations in profile are typically less than 1% of final strip thickness, the extra elongation required of thinner strip causes Manuscript received January 1997; revised October 1997. S. R. Duncan was supported by the U.K. Engineering and Physical Research Council under Grant GR/J65334. S. R. Duncan is with the Control Systems Centre, UMIST, Manchester, M60 1QD, U.K. J. M. Allwood is with the Department of Mechanical Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BX, U.K. S. S. Garimella is with the Alcoa Technical Center, ALCOA, PA 15069 USA. Publisher Item Identifier S 1063-6536(98)02066-1. significant longitudinal residual stresses. These stresses tend to cause the strip to buckle and are known as shape or flatness defects. Such defects may at worst cause process interruption and reduced factory output if the buckles occur on-line, but more frequently cause difficulty to subsequent forming operations and may thus reduce prices. Habberley [20] and Molotilov [29] give illustrations of such problems in the automotive industry where the use of adhesive bonding for car body panels requires tight tolerances on component edge geometry. Commercial pressures will continue to drive a need for faster strip rolling and each increase in speed makes the process more vulnerable to the adverse effects of poor flatness. The range of actuators available to counter flatness defects is growing, giving increased controllability within the mill. Therefore there is a need for a thorough analytical method with which systems of the future may be designed, analyzed and optimized. As flatness defects are inherent in strip rolling, various actuators have been developed to counter the natural tendency of the mill to deflect away from the strip. Those illustrated are: • screw settings which act as a flatness actuator when the stand is one of many, so that the proportion of reduction taken at each stand (the gauge change) may be varied; • hydraulic jacks applied between the bearings of roll pairs are used to force apart the ends of the rolls; • shifting continuously variable camber (CVC) rolls with cambers ground as antisymmetric cubic polynomials are oppositely shifted to give changes in effective crown described by a parabola with magnitude controlled by the degree of roll shifting [24]. • variable camber and spray actuators are also illustrated in Fig. 1 and will be described below. The system of Fig. 1 is completed by the addition of a sensor placed downstream of the mill to measure strip residual stresses. In cold rolling such a “shapemeter” will typically exist only after the last stand of a tandem mill and in hot rolling, although optical devices are available to measure flatness, they are not widely used. Thus the illustration of Fig. 1 is optimistic and in reality many mills rely simply on operator observation to detect early signs of wavy edges as an indication of a flatness defect. A. Existing Work on CD Control in Strip Rolling Early rolling mills relied solely on screw settings for flatness controls. Increasing quality and throughput requirements of the 1960’s led to the development of hydraulic roll bending 1063–6536/98$10.00  1998 IEEE