Nonlinear optimal control of rotating flexible shaft in active magnetic bearings

Abstract

The optimal control of nonlinear systems has been studied for years by many researchers. However, the application of optimal control problem to nonlinear non-affine systems needs more attention. In this paper we propose an optimal control design technique for a class of nonlinear and control non-affine equations. The dynamic equations of a flexible shaft supported by a pair of active magnetic bearings (AMBs) are used as the nonlinear control non-affine equations. Mathematical model for the flexible beam is chosen to be the well known Timoshenko beam model, which takes rotary inertia and shear deformations into account, and it is assumed that the shaft is supported by two frictionless bearings at the ends. The effective control of such systems is extremely important for very high angular velocity shafts which are a feature of many modern machines. The control must be able to cope with unbalanced masses and hence be very robust. We shall approach the problem by discretising the Timoshenko beam model and using standard difference formulae to develop a finite-dimensional model of the system. Then we use a recently developed technique for controlling nonlinear systems by reducing the problem to a sequence of linear time-varying (LTV) systems. An optimal control designed for each approximating linear, time-varying system and recent results show that this method will converge uniformly on compact time intervals to the optimal solution. approximation technique, nonlinear optimal control, timoshenko beam, active magnetic bearings Citation:Tombul G S, Banks S P. Nonlinear optimal control of rotating flexible shaft in active magnetic bearings.

Primary resonance analysis of a nonlinear flexible shaft supported by active magnetic bearings using analytical method

“…Equations (7) and (8) show that Θ ∝ and Θ ∝ 2 , i.e., the stiffness has linear relation with and quadratic relation with 2 . Therefore, an increase of number of winding in each electromagnet, N, and cross section area of the electromagnet, causes the system more harden.…”

“…Stability and bifurcations of a flexible rotor levitated by active magnetic bearings were investigated in [6]. Tombul et al [7] studied nonlinear optimal control of rotating flexible rotor in active magnetic bearings. They used Timoshenko beam model.…”

Chaos of flexible rotor system with critical speed in magnetic bearing based on the improved precise Runge–Kutta hybrid integration

“…It is important to conduct a research on the flexible rotor regarding how to make it work safely when crossing the critical speed. 13 The study of the chaotic characteristics of the flexible rotor system of magnetic bearing has been drawn considerable attention. The Lyapunov exponent as a particularly identified approach which is used in chaotic recognition to rotor has become one of the focuses of research.…”

Study on stability and rotating speed stable region of magnetically suspended rigid rotors using extended Nyquist criterion and gain-stable region theory

“…In recent decades, many control strategies have been applied to MSRs, such as decentralized PID [2][3][4], decentralized PID plus cross-feedback [5,6], centralized PID plus cross-feedback [1,[7][8][9][10], adaptive control [11], robust control [12,13], inverse control [14,15] and optimal control [16][17][18][19]. Among these control methods, centralized PID plus cross-feedback control is widely used in industrial applications because it is simple and convenient to implement.…”

“…The second class of methods is to examine the system by using Lyapunov's theory [14][15][16][17][18]23]. The Lyapunov's methods can give a perfect mathematical proof of the system stability with a lack of frequency domain information which is very useful in the MSR debugging process.…”