# Active Control of Broadband Noise Inside a Car Using a Causal Optimal Controller
## 1 Introduction  
The acoustic quality of a car determines its ride comfort and has been a significant marketing issue for car manufacturers [1]. Broadband road noise inside a car, caused by road-type interaction, is the main noise source when a car is driving at a relatively high speed [2]. Thus, measures have to be taken to isolate or reduce this noise to an acceptable level. Traditionally, sound-absorbing materials have been used to attenuate the noise. However, when they are used to reduce low-frequency noise, the mass and space they require can be out of scale. In order to overcome this problem, the active control system is applied [3]. Active control of the interior road noise of a car has been researched for over two decades, and a variety of active noise control (ANC) techniques have been proposed to improve ride comfort [4; 5; 6]. [PERSON] et al. proved that at least six vibrational reference signals are required to attenuate broadband road noise by using a filtered-x least mean square (FXLMS) algorithm [4]. [PERSON] et al. used a Hoe controller to cancel the noise around front seat, although only a reduction of 10 dB at 40 Hz was achieved [7]. [PERSON] et al. proposed a leaky constraint multiple FXLMS algorithm, and road-booming noise around 250 Hz was reduced about 6 dB at the error microphone positions [8]. [PERSON] et al. used an active structural acoustic control technique to reduce the vibration of suspensions, and a 4.6 dB attenuation was measured in the 50-250 Hz on a quarter-car test bench [9]. [PERSON] and [PERSON] show that a multi-input multi-output system was required to have a distinct reduction for the broadband noise [10; 11]. Further, [PERSON] et al. used a headrest system to reduce interior road noise around a listener's ears with remote microphone technique [12]. Although many algorithms were explored,the constraints limiting the improvement of control performance for broadband road noise still have not been solved [9; 13]. As we know, the performance of ANC systems relies heavily on the secondary path characteristics [14]. To improve control results, high modelling accuracy of the secondary path is required [15; 16]. However, if the secondary path is a nonminimum-phase system, it will generate a waterbed effect on the control system, which has a significant influence on the design of the controller and the deterioration of the control performance [13; 17], regardless of the fact that the controller is designed by FXLMS [18] or Hoo robust control algorithms [19]. Under this situation, even a perfect modelling cannot lead to a satisfactory result [20]. Although the above adaptive control algorithms have the ability to reduce noise with time-varying primary paths, the best control results cannot be achieved due to their large gradient estimation deviation of the cost function for time-varying noise. In this paper, a novel formulation for identifying primary paths with correlated excitation signals is presented, and the influence of a nonminimum-phase secondary path on the control system is investigated. Based on the theoretical analysis, a novel causal optimal controller is proposed to improve the control performance for the time-invariant system. Besides, the traditional method to predict the noise reduction is using the coherence between reference and noise signals [4; 11; 21]. The resulting prediction has a large deviation from the practical reduction, because it does not take causality constraint into account. Since the proposed controller considers the nonminimum-phase characteristics of secondary paths as well as the frequency responses of systems, it can also be used as a more accurate predictor to estimate the maximal achievable noise reduction and is more instructive in engineering. All the work is conducted under the assumption that the control systems are linear time-invariant. The organisation of this paper is as follows. In Section 2, the impact of a nonminimum-phase secondary path on the noise-reduction performance is analysed and its noncausal inversion is formulated.