PAPER NO. 8 SUMMARY
Estimating Web Strain, Modulus of Elasticity and Loss Torque Using an Extended Kalman Filter
1999 International Web Handling Conference
Brian Thomas Boulter
Applied Industrial Control Solutions LLC
4597 E Sprague Rd.
Independence, OH, USA 44131
© ApICS ® LLC 2000
Abstract: A set of sub-optimal Extended Kalman Filters (EKF's) are described. The filters are used to estimate web tension, modulus of elasticity, and loss-torque for all sections in a small web processing line located at Rockwell Automation's Euclid Research Facility. Conditions necessary for observability, and estimate convergence are discussed
B. Problem Definition
The EKF is often used in the aerospace industry to estimate states and/or important system parameters in noisy systems. Why do we need to concern ourselves with this kind of technology in the drive systems business?
There are many sources of uncertainty in a web processing system, for example, tension measurements can be corrupted by incorrect calibration of the load cell or strain gauge. The same can be said for the uncertainty associated with calculation of roll diameter as a function of an estimated surface speed and its ratio to the winder motor speed in ratio detector algorithms. Time varying frictional losses add uncertainty to friction compensation algorithms as do slight changes in material density to inertia compensation algorithms. The estimation of motor torque from a current reference in DC or torque reference in AC as compared to the actual motor shaft torque introduces another source of uncertainty. In the paper and film industries, time-varying modulus of elasticity adds another source of uncertainty. A Kalman filter that is designed to estimate states based on a known model of the uncertainty in the measured states is desirable.
A Luenberger observer, can be implemented [5,6] to estimate system states, but such an implementation is not robust in the presence of noise. This is especially true for metals applications. Because of the high gain in the relationship between stress and strain (Young’s modulus) in metals applications, noise in the speed feedback can result in large errors in the strain estimation. A Kalman filter that is designed to provide reliable estimates of important system parameters in the presence of well modeled system noise would be most useful. An additional benefit provided by such a filter, is that it could estimate unmeasured parameters, such as web strain, and loss-torque.
The Kalman filter can provide the user with a measure of how "good" the estimate is. This measure, the covariance of the estimate, is most accurate in systems where the uncertainty is Gaussian, and all process and sensor noise is band-limited and white. In systems where there are cyclical disturbances (e.g. machine resonances) and the resulting noise is pink at certain frequencies, a Kalman filter can be designed to filter the cyclic noise and calculate un-biased parameter and state estimates , however, the design of the Kalman filter in these instances is significantly more complex, for this reason pink noise sources will not be considered in this presentation.
The usual drawback to the implementation of any optimal EKF is the need to heuristically derive optimum noise covariance matrices Q and R. Another is the trade-off between the need for computing power, and the need for filter bandwidth (i.e. update time). Most implementations usually result in such long update times that the Filter estimates are of little practical use in real time systems.
Figure 7. Shows the optimal EKF implementation for the Euclid lab-line system. In this system there are 22 system states, and 12 system inputs. To execute the EKF algorithm with this many states on a 7010 Automax ® processor would take approximately 20 [sec] per update. Clearly, with such an update time, the tension estimates could not be used to close a sufficiently responsive tension loop.
However, if a set of sub-optimal filters are designed, and linked with estimated parameters, as shown in Figure 8, the resulting sub-optimal filter computational requirements are significantly reduced. In this case the most number of estimated states is 3 and the most number of inputs is 2. The resulting matrices are at most 3x3, and all matrix computations can be accomplished with simple routines. The EKF algorithm can now be executed in about 15 [msec] using a 7010 Automax processor. Given that the scan-time of a standard Rockwell Automation tension loop is 22 [msec], tension estimates, for example, can be used to close a tension loop.
Having overcome the implementation hurdle, the process of exercising the EKF on the Euclid lab-line revealed that the estimated web strain, loss-torque, and web modulus of elasticity states, in those sections where the tension feedback was not made available to the filter, would not converge. A dual state analysis performed by Angus Andrews at the Rockwell Science Center  provided a proof that tension feedback was required if these states were to be estimated. The conclusions drawn in the proof were extensible to Luenberger observers, implying that any observer or state estimator of web strain, loss-torque, or modulus of elasticity was not feasable if it is implemented without tension feedback. This was an important conclusion given that the impetus for the study was to investigate the possibility of developing tension sensor-less web process lines.
C. Paper Overview
The first section contains a brief description of the EKF algorithm. The second section describes the non-linear process model used in the final sub-optimal Kalman filter implemented on the Euclid lab-line. The third section describes the coding of the EKF for use with the Automax distributed control system, and results from the integration of the filter onto the Euclid lab-line. Conditions necessary for observability and estimate convergence, along with conclusions are presented in the fourth section.
The author corresponded with Angus Andrews at the Rockwell Science Center for assistance when it was observed that the state estimates of tension, strain factor, web modulus of elasticity, and loss-torque would not converge in those sections where a tension measurement was not fed back into the EKF. A dual state analysis of the system was performed  and a comparison of optimal and sub-optimal filters revealed that the above states were only observable if a measurement of at least one of the above mentioned states was made available to each the sub-optimal filters. Tension is the only state that can be practically measured. This conclusion implied that the implementation of a tension sensor-less system is not feasible. However, important states, such as web strain, or loss-torque, can be estimated if the tension feedback is made available, and an EKF is used to estimate these states, even in the presence of significant process and sensor noise.
The structure and results of the dual state analysis will be the subject of another paper on this topic. However, by inspection of the equations describing strain factor, tension, loss-torque (i.e. speed), it is clear that the above states exhibit dependence. For example, if the estimate of web tension is lower than the actual, the estimate of loss-torque must be higher than the actual, this is because the estimate of shaft torque is accompanied by a measurement. It can be considered known with a degree of certainty. At a steady state speed, the torque on the shaft minus loss-torque, will produce strain in the material. What is lost in one estimate must be made up in the other estimate.