Metal Cutting Tool Position Control using Static Output Feedback and Full State Feedback

In this paper, a metal cutting machine position control have been designed and simulated using Matlab/Simulink Toolbox successfully. The open loop response of the system analysis shows that the system needs performance improvement. Static output feedback and full state feedback H 2 controllers have been used to increase the performance of the system. Comparison of the metal cutting machine position using static output feedback and full state feedback H 2 controllers have been done to track a set point position using step and sine wave input signals and a promising results have been analyzed.


Introduction
A metal cutting machine or cutter tool is any tool that is used to separate some metallic material from the work piece by means of cutting. Cutting may be accomplished by single-point or multipoint tools. Single point tools are used in turning, shaping, planing and similar operations, and remove material by means of one cutting edge. Cutting tool materials must be harder than the material which is to be cut, and the tool blade must be in accurate position. The Coordinate position of the blade might be 1D (dimension), 2D and 3D and it must be able to withstand the disturbances that arise for example from the force generated in the metal-cutting process. Also, the tool must have a specific geometry, with clearance angles designed so that the cutting edge can contact the work piece without the rest of the tool dragging on the work piece surface.

Mathematical Modeling
A solenoid system is fed with an electrical voltage. The force exerted by the solenoid system is proportional to the current. This force controls the hydraulic actuator input. The hydraulic actuator system is fed with fluid from a constant pressure source in which the compressibility of the fluid is negligible. An input displacement x moves the control valve; thus fluid passes in to the upper part of the cylinder and the piston is forced to move horizontally. A low power displacement of x(t) causes a large high power displacement y(t). The output displacement moves the cutter blade.
The system layout is shown in Figure 1 below.

Figure 1 Metal Cutter Machine
The solenoid coil circuit equation becomes The flow rate of the fluid Q is related to the input displacement x(t) and the differential pressure across the piston will be     ,7 Q g x P  Using Taylor series linearization technique we have ,, g g x P and x p  is the operating point, the piston developed a force which is the area of the piston A multiplied by the pressure P.
The applied force to the mass become   Therefore the transfer function between the output displacement and input displacement will be: Therefore, the overall transfer function between the input voltage and the output cutter blade position can be obtaining by multiplying Equation (2), (4), (6)  The state space representation of the system becomes

The Proposed Controllers Design 3.1 Static Output Feedback Controller Design
Consider a linear time invariant system: The block diagram of the cutter system with static output feedback controller gain matrix is shown in Figure 2 below. Figure 2 Block diagram of the cutter system with static output feedback controller gain matrix 6 It is well known that the fixed order dynamic output feedback control design problem is a special case of the static output feedback problem, since the closed-loop system for the fixed order case has exactly the same structure as the static case with appropriately augmented system matrices [5]. Therefore, study of static output feedback problem includes more general scope of control problems. To assess the performance quality a quadratic cost function known from LQ theory is often used. However, in practice the response rate or overshoot are often limited. Therefore, we include into the LQR cost function the additional derivative term for state variable to open the possibility to damp the oscillations and limit the response rate.   x Ax B d t B u t    8 Figure 4 Open loop response The open loop response of the machine shows that the system input is a high voltage and the system needs improvement.

Comparison of the Cutting Machine using Static Output Feedback and Full State
Feedback 2

H Controllers using Step Input Desired Position Signal
The Simulink model of the cutting machine using static output feedback and full state feedback H 2 controllers using step input desired position signal is shown in Figure 5 below. Figure 5 Simulink model of the cutting machine using static output feedback and full state feedback H 2 controllers using step input desired position signal The simulation result of the comparison with the input voltage to the cutting machine using static output feedback and full state feedback H 2 controllers are shown in Figure 6, Figure 7 and Figure  8 respectively. Figure 6 Step response of the comparison 9 Figure 7 Input voltage to the system with static output feedback controller Figure 8 Input voltage to the system with full state feedback controller The input voltages of the cutting machine system with the proposed controllers shows improvement in reducing the voltage amplitude but the system with full state feedback H 2 controller shows better improvement. The data of the rise time, percentage overshoot, settling time and peak value is shown in Table 2. As Table 2 shows that the cutting machine system with full state feedback H 2 controller improves the performance of the system by minimizing the percentage overshoot and settling time.

Comparison of the Cutting Machine using Static Output Feedback and Full State
Feedback 2

H Controllers using Sine Wave Input Desired Position Signal
The Simulink model of the cutting machine using static output feedback and full state feedback H 2 controllers using sine wave input desired position signal is shown in Figure 9 below. The input voltages of the cutting machine system with the proposed controllers shows improvement in reducing the voltage amplitude but the system with full state feedback H 2 controller shows better improvement.

Conclusion
In this paper, the design and position control of a metal cutting machine have been done using static output feedback and full state feedback H 2 controllers. The open loop response of the system analysis shows that for the system cutting position of 3.4m, 1400-volt input is needed which is a 12 high voltage so the system needs performance improvement. Comparison of the metal cutting machine position using static output feedback and full state feedback H 2 controllers have been done to track a set point position using step and sine wave input signals. Both responses show that the cutting machine system with full state feedback H 2 controller improves the performance of the system by minimizing the percentage overshoot and settling time.