Fig. 1: Schematic drawing of the systemDevelopment of the physical modelFig. 2: Schematic drawing and parameters of the dc motor drivesState space controlFig. 3: Electrical servo drives with DSFb and current loop, shown as torque modulators= M =Comparison of state space and fuzzy controlFelix BierbaumIntroduction This work will provide a comparison of state space and fuzzy control methods for a papermachine roll. The state space approach is most frequently used as long as it is possible to derivea mathematical model that describes the process precise enough. Nevertheless there are somecases where a mathematical model is either not easy to be developed because the physic of theproblem is not known well or the mathematical model would be to complex to be useful. In suchcases fuzzy control, based on rules can be of advantage. In general fuzzy controllers try toemulate human knowledge about the control task.The paper machine roll was chosen as an example because it is easy enough to be handledmathematically and still has some interesting problems. On of them is, that the control task ismultivariable i.e. not only a uniform thickness of the paper is desired but also maintaining aparallel alignment. Another problem is, that maintaining a uniform thickness is more importantthan maintaining the parallel alignment due to cost reasons.This work will first develop a mathematical model of the roll bite, describe how to control thesystem using a state space based approach and finally develop a fuzzy controller. A comparisonof the results will conclude the work.Development of the physical modelFigure 1 shows a schematic drawing of themachine roll to be controlled. In order tomanipulate the vertical position andorientation of the roll there are two standarddc permanent magnet servo drives available.Each of these drives is connected to a ballscrew that applies the force of the motors to the axis and also converts the rotational movementof the motors into the required axial movement. The mass and inertia of the roll are smallcompared to the effective reflected mass and inertia on the motors obtained through the ballFig. 1: Schematic drawing of the systemscrews and are therefore neglected. With p denoting the pitch of the ball screw, which isassumed to be 0.1cm/rev, we can derive the reflected mass mr and the reflected inertia Jr of theroll as follows: = = The conversion between the torques produced by the motor (TM) into a force applied at the roll (FM) is obtained using the power conservation law:Using these variables we can apply Newton’s Laws or the Lagrange method to come up with themotion laws for the given system: = = = = FM1 denotes the force applied by the right motor, FM2 the force of the left motor. The verticalposition of the roll, measured in the centre, is denoted by z, the orientation by .The modelling of the servo drives shown in Figure 2, which are identical for both sides, can bedone using Kirchhoff’s voltage and current laws as well as Newton’s Laws. Fig. 2: Schematic drawing and parameters of the dc motor drivesFor the given drives we end up with = = State space controlFrom the differential equations derived above we can easily develop a state block diagram of theelectrical servo drives. Figure 3 shows this block diagram and its inherent state feedbacksT K 1 M T 1 ˆ a K p L 1 s 1 P R a K 2 M T P J 1 s 1 e K 1 ˆ T K T K 1 ˆ a K p L 1 s 1 P R a K P J 1 s 1 e K a R 1 ˆ T K 1 2 2 i 1 i + - - - - - + - - + + - - - * 2 M T * 1 M T a R - + + - P R ˆ e K ˆ e K ˆ P R ˆ (black). These state feedbacks are removed by adding decoupling state feedback (DSFb) shownin blue. Fig. 3: Electrical servo drives with DSFb and current loop, shown as torque modulatorsRemoving the state feedbacks removes the unwanted cross coupling between the states of themotor. Herby denote K and Rour best estimates of the usually not precisely known physicalmodel parameters. Form the mathematical model we derived for the roll we know that our desired input signals areforces that can be translated into torques. Therefore in a second step we add a current loop toboth motors. This current loop gives us direct access to the desired input variables. In fact if wetune the current loops fast enough the assumption that the servo drives give exactly the outputtorque we require is nearly true and therefore simplifies the further design process. The currentloops are both tuned for a bandwidth of about 1000Hz, which is fast, compared to the desired rolldynamics. The tuning itself is done simply by evaluating the closed loop transfer function Giafter adding decoupling state feedback. = Evaluating this equation for the desired bandwidth leads to an active resistance of = 27. Figure3 shows the current loop in green.Now that we have achieved direct access to the physical system input variables the next thing wenee to take care about is the manipulated input coupling. In fact, both roll state variables areaffected by both input variables what will cause problems if not considered in the controllerdesign. Furthermore if we would not take the input coupling into account we would not be ableto achieve different dynamics for the vertical position and the orientation of the roll as requiredby the process. Therefore we decouple the manipulated inputs by applying the inverse of themanipulated input-coupling matrix M to the system.= M = So far the only things we have done are the necessary preparations for closing the control loopson z and . Figure 4 shows the state block diagrams for the roll with added state feedbackcontrollers for the roll angle and vertical position (green). The servo drives are shown as torquemodulators as explained above. The chosen controller structure with an integration of thedifference between the desired output and actual output instead of adding an additional integratorto the system has the advantage of not producing an unbounded number in the integration stateand is mathematical equivalent. For the calculation of the state feedbacks, we assume that weknow our models parameters exactly, i.e. we assume K= K. One should also recognize that although Figure 4 shows the motor as a torque modulator the mechanical assumption = is notvalid even though we tuned the current loops relatively fast. Therefore this assumption
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