# Research on Frequency-Selective Output Constraint Algorithm for Active Vibration Control
## 1 Introduction  
Compared with passive vibration control and semi-active vibration control, the active vibration control (AVC) has advantages in suppressing low-frequency vibration and micro vibration that impact on extensive aspects in vehicle engineering, transportation, optical system and precision manufacturing [1, 2, 3], etc. Typically, the aim of AVC is to suppress the undesired vibration as much as possible. However, it is hard to achieve the theoretical control results in most applications because of limitations such as the maximum force that can be generated by actuators, the complexity of AVC algorithms, the performance of controllers, the cost, etc. For safety, stability and simplicity, the output constraints have to be adopted to protect actuators from exceeding the secure range or to keep the actuator working in the band with a good linearity in order to simplify the algorithms in practical applications. For example, if the actuator fails or deteriorates in some frequency bands due to accidents but still works properly in the rest of the frequency bands, applying output constraints to keep the AVC system going and to achieve the optimal performance is a better and practical choice. As the vibration signals generated by power systems such as engines, electrical motors, gear boxes and other rotating mechanical parts, etc. are mostly harmonic signals containing multiple frequency components [4, 5, 6, 7], a feasible strategy is to focus on the suppression of some key sub-frequency components like the frequency components close to the resonance frequency, and ignoring the frequency components that have less impact. In fact, ejecting some key frequency components could significantly improve the vibration and noise environment of the entire system in most narrowband active vibration control applications. The Filtered-x Least Mean Square (FXLMS) algorithm was derived by [PERSON] in 1981 and has been extensively used in active noise control (ANC) and active vibration control (AVC) researches and applications [8, 9, 10]. It adjusts the controller vector to adaptively minimize the error signal. In practical applications, the actuators have various limitations such as a stroke limit, force limit, effective bandwidth, nonlinearity, etc. However, thebasic FXLMS algorithm has no constraints applied when generating the output signal (or the drive signal). Thus, the adjustment process will not cease to achieve the control goals coresponding to the limitations of actuators, which may eventually lead to unexpected results. For example, if the control goals have not been achieved while the actuator has reached its stroke or force limit, the output signal will keep growing under the FXLMS control law, finally causing damages. Studies on output constraints without destroying the stability of the FXLMS algorithm have been launched accordingly. [PERSON] and [PERSON] studied an effort constraint based on the method of Lagrange multipliers and leaky LMS algorithms in the frequency domain for multi-channel feed forward control [11; 12; 13; 14]. It limits the value of either the sum of squared control signals (total effort) or the mean square value of each individual control signal (individual efforts). However, the method cannot guarantee that the control output remains within any specified constraint. The selection of the value of the leakage coefficient can only be done using a trial and error procedure in applications [15]. Compared with [PERSON], [PERSON] used an exponential penalty function to avoid computing the piecewise update equation [16]. However, the problem of specifying an accurate constraint remains, especially in applications with a large gain. [PERSON] introduced the rescaling algorithm to ANC [16; 17]. It could accurately keep the output signal under constraint without losing performance when compared with the leaky LMS algorithms, but a certain amount of high-order harmonics may be introduced to the system.