Cost function research for SPSA algorithm: An application of the cross-correlation function in periodic disturbance energy estimation

•A cost function vector of the simultaneous perturbation stochastic approximation algorithm is proposed.•The proposed method reaches a good convergence performance.•The vibration of the inlet pipeline is eliminated by using the proposed method. Active vibration control of stable rotating machinery with a time-variant structure is a research priority in the field. Simultaneous perturbation stochastic approximation (SPSA), a model-free active control strategy, has attracted a lot of attention from researchers. This paper studies the cost function of the SPSA algorithm to improve its performance in controlling periodic vibration. After the weight vector being perturbed by a perturbation vector, the existing cost functions of the SPSA algorithm can reveal the overall energy variation but hardly detail the variation of each harmonic wave. Therefore, the adaptive notch filter with the SPSA algorithm using these cost functions shows a low convergence speed and a bad control effect in active periodic disturbance controlling. In order to detail the energy variation of each harmonic, this paper proposes a cost function using the weighted cross correlation function. When the algorithm calculates the energy of each harmonic wave, the disturbances from other harmonics and background noise are reduced. The interference noises in the estimated gradient of the cost function are reduced indirectly. The algorithm convergence performance improves. To illustrate that, the simulations of controlling harmonic waves and the experiments of eliminating the vibration of the pump inlet pipeline are conducted. The results indicate that the adaptive notch filter with the proposed cost function has reached the fastest convergence speed and the best control effect so far among the use of other cost functions mentioned in this paper.