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Determination of Domain of Attraction in Active Magnetic Bearing

Eshaghi, Jafar | 2012

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 43187 (05)
  4. University: Sharif University of Technology
  5. Department: Electrical Engineering
  6. Advisor(s): Mobed, Mohammad
  7. Abstract:
  8. Magnetic bearings have been introduced in recent decades to overcome the problem of energy loses due to friction in bearings of rotating machinery. In magnetic bearings, the rotor is suspended by magnetic forces without any friction. These bearings are of the two types of passive and active. In Passive Magnetic Bearings (PMB), magnetic forces are generated by permanent magnets and have the ability to tolerate a small distribution. In Active Magnetic Bearings (AMB) magnetic forces are generated by electromagnets and have the ability to tolerate larger distributions than the PMB can. Since AMB is inherently unstable, it is necessary to use controllers to stabilize it. Due to the nonlinear nature of the AMBs, controllers can generally stabilize the AMBs within a certain variance in their parameters and variables. To determine this permitted variance, a region (domain) of attraction (ROA (DOA)), sometimes called an attractive region (domain) for the equilibrium point of the closed loop consisting of the AMB and its controller should be estimated. Thus knowing an ROA estimate may in some sense help in assessing the reliability of a closed-loop system. The ROA can be estimated via numerical, analytical, or mixed methods. Numerical methods such as reverse trajectory are generally time-consuming for high-order systems and may encounter computational problems. Analytical methods can determine the ROA exactly; however, solving the associated equations may in some cases be impossible. Mixed i.e. numero-analytical methods often use the Lyapunov theorem, searching for multidimensional level sets V(x)=c along which the gradient V ̇ of V with respect to the nonlinear system x ̇=f(x) is negative. These methods often maximize the value of c so that the ROA estimate is tangent to the actual ROA. In one of these methods, the gradient function is considered as multiplying of the sum of squares (SOS) and positive definite polynomials, then the value of c is maximized using LMI optimization. In this thesis a voltage-controlled model of an active magnetic bearing system is used to first design stabilizing controllers for the closed loop. To that end, nonlinear equations of the AMB were Jacobian-linearized first. Then LQR and robust control techniques were used on the linearized plant model to arrive at two types of controllers for the sake of eventual comparisons. Subsequently, two methods of ROA estimation were employed. The reverse trajectory method was applied first where problems such as long computational times and errors were encountered. The ROA estimates were also found to be relatively large. Moreover, the ROA estimate corresponding to the robust controller was found to be larger than the estimate obtained for the LQR controller in our case. A Lyapunov-based method along with LMI techniques involving Sums-of-Squares (SOS) concepts was applied next. Here the computations were short and less prone to errors relative to the reverse trajectory method. Again it was found that the ROA estimate obtained for the robust controller was larger than the estimate corresponding to the LQR one in our case. Parallel coordinates were used as a means of displaying the ROA estimates in the 16-dimensional state space of the closed-loop system
  9. Keywords:
  10. Magnetic Bearing ; Linear Matrix Inequality (LMI) ; Linear Quadratic Requlator (LQR) ; Attractive Domain ; Reverse Trajectory ; Parallel Coordinates

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