# \(L_{1}\) Adaptive Control for a Vertical Rotor Orientation System
## 1 Introduction  
Vertical rotation devices are widely applied in industrial, scientific and civilian fields, e.g., flywheels, centrifuges and washing machines. Many of these applications have a fixed bottom bearing with a free upper side. The one-side bearing structure will lead to vibration during operation and, consequently, cause collision, noise and failure. One of the traditional methods to solve this problem in washing machines is through the addition of a passive balancer [1; 2; 3]. However, passive control approaches, although having the advantages of simplicity and low cost, are unsatisfactory in vibration reduction because of their open-loop property. For the vertical rotation devices, an active non-contacted vibration reduction system, such as an active magnetic bearing (AMB), placed at the upper side to control the attitude of the rotor, offers a better solution to the vibration problem. The Active Magnetic Bearing (AMB) system is a nonlinear system, with many uncertainties during its operation, such as parameter uncertainty and external random disturbances. Therefore, the control of the AMB system has been an active research topic. For example, [4] presented a feedback linearization approach to control the position of a horizontal rotor; [5], aiming at a 1-kWh flywheel energy storage device, analyzed the advantages and disadvantages of different approaches, such as decentralized control, Linear Quadratic Regulator (LQR) control and cross-feedback control; [6] proposed an on-line parameter identification method to solve the situation in which the rotor mass is unbalanced; [7], based on an AMB test rig, used radial basis function networks to identify the uncertainties and to synthesize an \(H_{\infty}\) controller with the rotor setpoint; [8] presented an adaptiveback-stepped controller for a flywheel energy storage system; [9] designed a test rig consisting of a flexible rotor supported by AMBs with a \(\mu-\)synthesis controller. The research above mostly focused on AMBs for horizontal rotors. Research on a vertical rotor has been rather limited. For example, [10] presented an H\({}^{\infty}\) controller to regulate the rotor position for vertical AMB systems under 10K rpm. This research, for either horizontal or vertical rotors, has focused on high-speed rotors. However, as mentioned above, vertical low-speed rotating devices are widely used, and their dynamic behaviors will be rather complicated [11]. Therefore, it is necessary to propose a control approach for low-speed vertical rotor position control. Meanwhile, under a lower spinning rate, the influences from parameter uncertainties and external disturbances will be more severe. The \(L_{1}\) adaptive control theory [12] provides a powerful tool to overcome the problems of uncertainties and nonlinearities. This theory has been utilized in many applications, e.g., [13] presented a one-output drilling direction adaptive control system in the presence of time-delay, unexpected disturbance and other uncertainties; [14] proposed an underactuated robots control system with unmodeled dynamics. These studies have provided significant theoretical and simulation results for \(L_{1}\) adaptive controllers. Based on these studies, in this paper, the authors propose an \(L_{1}\) adaptive controller for a two-input-two-output vertical rotor orientation control system and design an experimental rig based on dSPACE to test the performance of the rotor orientation control system. The work in this paper may also be applied to twin-rotor mechanical system control [15; 16; 17; 18]. This paper is organized as follows. Section 2 establishes the mathematical model of the rotor and presents the control aim of the system. Section 3 proposes the architecture of the adaptive control system. Section 4 analyzes the stability and performance of the adaptive control system. Section 5 gives the simulation of the adaptive control system. Section 6 presents an experimental validation of the performance of the rotor orientation control system. Section 7 provides a summary of the work.