IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 10, NO. 2, MARCH 1999 391 Improved Training of Neural Networks for the Nonlinear Active Control of Sound and Vibration Martin Bouchard, Member, IEEE, Bruno Paillard, and Chon Tan Le Dinh Abstract—Active control of sound and vibration has been the subject of a lot of research in recent years, and examples of applications are now numerous. However, few practical imple- mentations of nonlinear active controllers have been realized. Nonlinear active controllers may be required in cases where the actuators used in active control systems exhibit nonlinear characteristics, or in cases when the structure to be controlled exhibits a nonlinear behavior. A multilayer perceptron neural- network based control structure was previously introduced as a nonlinear active controller, with a training algorithm based on an extended backpropagation scheme. This paper introduces new heuristical training algorithms for the same neural-network control structure. The objective is to develop new algorithms with faster convergence speed (by using nonlinear recursive-least- squares algorithms) and/or lower computational loads (by using an alternative approach to compute the instantaneous gradient of the cost function). Experimental results of active sound control using a nonlinear actuator with linear and nonlinear controllers are presented. The results show that some of the new algorithms can greatly improve the learning rate of the neural-network control structure, and that for the considered experimental setup a neural-network controller can outperform linear controllers. Index Terms— Active control of sound and vibration, alter- native backpropagation schemes, nonlinear control, nonlinear actuators, nonlinear recursive-least-squares learning algorithms. I. INTRODUCTION T HE concept of active sound control has been known for more than 50 years. Active sound control works on the principle of destructive interference between an original “primary” disturbance sound field and a “secondary” sound field that is generated by some control actuators. For example, a classical application of active sound control is the control of sound waves in a small duct. Fig. 1 shows such a duct with an actuator and two sensors (a loudspeaker and two microphones in this case). In this example, sensor A (called a reference sensor) is used to measure an advanced information on the disturbance sound wave that propagates in the duct, and sensor B (called an error sensor) is used to monitor the performance of the active sound control system, thus providing feedback to a control algorithm. To keep this example simple, we assume that sensor A is not coupled with the actuator, so sensor A only measures the disturbance sound field. The control structure Manuscript received February 3, 1998; revised June 30, 1998 and December 1, 1998. This work was supported in part by the Institut de Recherche en Sant´ e et S´ ecurit´ e au Travail (I.R.S.S.T.), Quebec, Canada. M. Bouchard is with the School of Information Technology and Engineer- ing, University of Ottawa, Ottawa, Ontario K1N 6N5 Canada. B. Paillard and C. T. Le Dinh are with the Electrical Engineering and Computer Engineering Department, University of Sherbrooke, Sherbrooke, Quebec J1K 2R1 Canada. Publisher Item Identifier S 1045-9227(99)01909-8. Fig. 1. A classical application of active sound control: The control of sound waves in ducts. of Fig. 1 is called feedforward, because the controller feeds the actuator with a signal based on the advanced information obtained from sensor A. If the controller works properly, the signal sent in the actuator will generate a sound wave that will cancel the disturbance sound wave at the location of sensor B. Since usually only plane waves propagate in small ducts, the sound field will be uniform in any section of the duct, and the sound will be reduced from sensor B to the end of the duct. Detailed presentations of active sound control theory and applications can be found in [1]–[4]. Active sound control is certainly not limited to mono-channel feedforward one-dimensional sound field systems as in our simple example: there are also multichannel, feedback or three-dimensional active sound control systems. Also, the principle of destruc- tive interference is not limited to the control of acoustic waves, and it has been successfully applied to the control of vibration [5]. Even though the concept is simple, it is only with the development of fast low-cost digital signal processors during the last 15 years that the implementation of practical active sound and vibration control systems has become feasible. Digital technology is well suited for adaptive control systems or control configurations where a lot of precision is required in order to achieve a good performance. A good review of the different control techniques that have been used for the active control of sound and vibration can be found in [6]. Adaptive linear filtering techniques [7], [8] have been extensively used for the active control of sound and vibration, and many of today’s implementations of active control use those techniques. A popular adaptive filtering algorithm is the multichannel filtered-X LMS algorithm (or the multierror LMS algorithm) [9], because of its simplicity and its relatively low computational load. This algorithm is a steepest descent algorithm that uses an instantaneous estimate of the gradient of the cost function (i.e., the mean squared error). Fig. 2 shows a block-diagram of the mono-channel implementation of this algorithm: the filtered-X LMS [7]. 1045–9227/99$10.00  1999 IEEE