A Multi-objective Adaptive Controller for Magnetic Bearing Systems

The paper considers three issues in flexible rotor and magnetic bearing systems, namely control of rotor vibration, control of transmitted forces, and prevention of rotor contact with auxiliary bearings. An adaptive multi-objective optimization method is developed to tackle these issues simultaneously using a modified recursive open loop adaptive controller. The proposed method involves automatic tuning of the weighting parameters in accordance with performance specifications. A two-stage weighting strategy is implemented involving base weightings, calculated from a singular value decomposition of the system’s receptance matrices, and two adjustable weighting parameters to shift the balance between the three objective functions. The receptance matrices are functions of rotational speed and they are estimated in situ. The whole process does not require prior knowledge of the system parameters. Real-time implementation of the proposed controller is explained and tested by using an experimental flexible rotor magnetic bearing system. The rotor displacements were measured relative to the base frame using four pairs of eddy current displacement transducers. System stability is ensured through local PID controllers. The proposed adaptive controller is implemented in parallel and the effectiveness of the weighting parameters in changing the balance between the transmitted forces and rotor vibrations is demonstrated experimentally. l correspondence to this author. 1 Nomenclature CQ Base scaling for displacement measurements CT Base scaling for transmitted forces J Cost function JA Augmented cost function JM Maximum displacement at magnetic bearing locations JQ Overall vibration cost function JT Transmitted force cost function t Time Fo Vector of unbalance forces in the frequency domain FT Vector of transmitted forces in the frequency domain G Transfer functions matrix of local PID controllers H Complex control gain matrix HA Augmented complex control gain matrix I Unity matrix Km Matrix of magnetic bearing negative stiffness coefficients Q Vector of measured displacements in the frequency domain q Vector of measured displacements QA Vector of augmented measurements in the frequency domain Qm Vector of measured displacements at magnetic bearing locations in the frequency domain Qo Vector of measured displacements at other than magnetic bearing locations in the frequency domain Copyright c © 2009 by ASME R Receptance matrix related to control forces RA Augmented receptance matrix Rm Sub-matrix of R corresponding to Qm Ro Sub-matrix of R corresponding to Qo U Vector of adaptive control forces in the frequency domain W Weighting matrix WA Augmented weighting matrix W f Weighting matrix related to FT Wm Weighting matrix related to Qm Wo Weighting matrix related to Qo Z Receptance matrix related to unbalance forces α1 Integral constant for ROLAC β Adjustable weighting parameter ∆(·) Change in (·) γ Adjustable weighting parameter σ[·] Maximum singular value of [·] τ Time variable Ω Rotational speed ω Frequency ω0 Fundamental frequency ( ˆ ) Optimum ( ̃ ) Predicted ( )T Transpose (Hermitian for a complex matrix) INTRODUCTION Magnetic bearings, when used to levitate a rotating shaft, permit relative motion without friction or wear. They are used in many industrial applications such as compressors, turbines, pumps, motors and generators [1, 2]. The future of magnetic bearings in critical applications depends on successfully addressing safety and reliability issues [3]. In addition to vibration control and levitation functions, magnetic bearings can be used to fulfil other functions, such as monitoring, auto-tuning, parameter identification, fault detection and tolerance, e.g. [4–6]. The versatility of magnetic bearings is important in the development of smart rotating machinery [7]. Various closed loop controllers have been used to control the rotor vibrations [8, 9]. Burrows et al. [10] reported pole placement techniques for synchronous vibration control of a rotor/bearing system. An optimisation approach was presented by Keogh et al. [11] to minimise the influence of forcing disturbances, modelling error, and measurement error. The application of multivariable design methodologies, such as H∞ [11] and μsynthesis [12], emphasise robustness issues of feedback control of active magnetic bearing systems. An open-loop strategy for controlling synchronous vibration under varying operating conditions was introduced by Burrows and Sahinkaya [13, 14]. The key features of this open-loop adaptive control (OLAC) strategy, also referred to automatic balancing, are its simplicity and the ability to apply self-tuning in 2 situ with no prior knowledge of the system model or parameter values. This approach has also been adopted in a broader context [15–19]. However, OLAC is not fast enough to respond to sudden changes in operating conditions e.g. due to mass unbalance or rapid transient excitation, because it relies on performing a Fourier transform of the measured steady state response. A recursive open-loop adaptive control (ROLAC) scheme was therefore developed [20]. This utilises a recursive version of the Fourier transform to update optimum control force components at every sampling interval. In some applications, e.g. turbomolecular pumps, gas turbines and compressors, it is important to minimise the transmitted forces from the rotor to the support structure. The control of transmitted forces and rotor vibrations imposes conflicting requirements and necessitates the use of multi-objective optimization. Active multi-objective control strategies usually involves design of separate controllers for each objective function, and switching between them in accordance with an algorithm based on speed [21] or base acceleration [22]. This paper discusses a unified adaptive approach, and extends the application of the ROLAC to minimise a multi-objective cost function. The method does not require any additional measurements to those required by ROLAC. Experimental results are presented to show that the proposed controller can effectively shift the balance between different objective functions according to the performance specifications.