Performance Investigation of a Permanent Magnet DC Machines using Robust Control Technique

: Performance Investigation of a Permanent Magnet DC Machines using Robust


Introduction
The permanent magnet machines have the benefit of getting an excitation from the permanent magnet which is found at the stator part of the machines in order to get some efficiency benefit for that.Induction machines have a good and regulated fluxes that's why it is helpful to optimize the efficiency.Both systems are used for variable-speed drive.The specification of the performance and efficiency of the permanent magnet machine better has better cost minimization function with optimal for the range and performance target.In this paper, the modeling design and control of a permanent magnet Dc motor and generator for improving the angular speed and generating current have been done using robust control theory.

Mathematical Modelling 2.1 Permanent Magnet DC Motor Modelling
Consider the cross section of a motor shown in Figure 1 below that has several loops of wire.

Figure 1 PMDM cross section
The static magnetic field formed by a electromagnet with strength  going from left to right (North to South).The thickness of the magnet is defined by ℓ.The loops of the wire around a "rotor" which is an iron cylinder with radius "a" that is free to rotate about its center.There are "n" rounds of wire with current going into the page on the right side, and comes out of the page to the right).The rotational angle, θ, is positive in the clockwise direction, the resistance of the wire is given by R. The moment of inertia of the rotor, "J," with friction of rotation "Br."The mathematical model for this system is derived by separating the system as mechanical system and the electrical system as shown in Figure 2   Taking the Laplace transform with initial conditions set to zero and solve for the ratio of output to input yields: The parameters of the PMDM is shown in Table 1 below.

Permanent Magnet DC Generator Modelling
A permanent magnet DC generator is typically the same machine as a motor.The motor uses a mechanical input and generates an electrical output.

Proposed Controllers Design 3.1 Augmentations of the Model with Weighting Functions
The weighted control structure of the systems is shown in Figure 5, W1(s), W2(s), and W3(s) are weighting functions.The assumption that G(s), W1(s), and W3(s) G(s) are all proper systems.The weighting function W3(s) is not required to be proper system.In the state space structure of the systems, the output vector y1 = [y1a, y1b, y1c] T cannot be used directly to the control signal u.Clearly, Figure 5 represents a more general picture of optimal and robust control systems.The design of the H 2 optimal and H  synthesis controllers is done by using the idea of the augmented state space model.

Comparison of a PMDM with H 2 Optimal and H  Synthesis Controllers for Improving the Angular Position using a Step Reference Angular Position Input
The angular position output of a PMDM performance analysis is done by simulating the system with the proposed controllers for a step input reference input and the simulation result is shown in Figure 6 below.Figure 6 Step response of the PMDM The simulation result shows that the PMDM with H 2 optimal controller improves the angular position in minimizing the percentage overshoot and the settling time better than the system with H  synthesis controller.

Comparison of a PMDM with H 2 Optimal and H  Synthesis Controllers for Improving the Angular Position using a Random Reference Angular Position Input
The angular position output of a PMDM performance analysis is done by simulating the system with the proposed controllers for a random input reference input and the simulation result is shown in Figure 7 below.The simulation result shows that the PMDM with H 2 optimal controller improves the angular position in tracking the reference input signal with better amplitude than the system with H  synthesis controller.

Comparison of a PMDG with H 2 Optimal and H  Synthesis Controllers for
Improving the Generating Current using a Step Reference Current Input The angular position output of a PMDG performance analysis is done by simulating the system with the proposed controllers for a step input reference input and the simulation result is shown in Figure 8 below.Figure 8 Step response of the PMDG The simulation result shows that the PMDG with H 2 optimal controller improves the generating current in minimizing the percentage overshoot better than the system with H  synthesis controller.

Comparison of a PMDM with H 2 Optimal and H  Synthesis Controllers for
Improving the Generating Current using a Random Reference Current Input The angular position output of a PMDM performance analysis is done by simulating the system with the proposed controllers for a random input reference input and the simulation result is shown in Figure 9 below.The simulation result shows that the PMDG with H 2 optimal controller improves the angular position in tracking the reference input signal with better amplitude than the system with H  synthesis controller.

Conclusion
The modelling, design and control of a permanent magnet Dc motor for an improvement of angular position and a permanent magnet Dc generator for improving of the generating current using augmentation based H 2 optimal and H  synthesis controllers.A comparison of the proposed systems with the proposed controllers for the analysis of the performance improvement of the angular position and generating current using a step and random reference input signals.The simulation result of the PMDM for a step reference angular position input suggested that the PMDM with H 2 optimal controller improves the angular position in minimizing the percentage overshoot and the settling time while the simulation result of the PMDM for a random reference angular position suggested that the PMDM with H 2 optimal controller improves the angular position in tracking the reference input signal with better amplitude.The simulation results of the PMDG suggested that the PMDG with H 2 optimal controller improves the generating current in minimizing the percentage overshoot for a step input reference current and the PMDG with H 2 optimal controller improves the angular position in tracking the reference input signal with better amplitude.
Figure 1 PMDM cross sectionThe static magnetic field formed by a electromagnet with strength  going from left to right (North to South).The thickness of the magnet is defined by ℓ.The loops of the wire around a "rotor" which is an iron cylinder with radius "a" that is free to rotate about its center.There are "n" rounds of wire with current going into the page on the right side, and comes out of the page to the right).The rotational angle, θ, is positive in the clockwise direction, the resistance of the wire is given by R. The moment of inertia of the rotor, "J," with friction of rotation "Br."The mathematical model for this system is derived by separating the system as mechanical system and the electrical system as shown in Figure2below.

Figure 2
Figure 2 Mechanical and electrical systems The torque developed in the loop of wire is given by   21 2

Figure 3
Figure 3 PMDG cross sectionThe free body diagram of the mechanical and electrical system is shown in Figure4below.In this system consideration of the direction of the induced torque is essential.The positive current is coming out of the page on the left side of the rotor and the field is to the right.The back emf, is in the positive direction with counterclockwise direction of the wires on the left side have positive velocity downward.

Figure
Figure 4 Mechanical and electrical systems

Figure 5
Figure 5 weighted control structure with the proposed controllers

Figure 7
Figure 7 Random response of the PMDM

Figure 9
Figure 9 Random response of the PMDG

Table 1
r B 1.2 Nms Motor specification  1.8The transfer function numerically becomes