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Published by RMS Acoustics & Mechatronics 由 RMS 声学与机电一体化出版
1 Introduction ..... 3 1 简介 ...... 3
2 The Physical Meaning of Feedback ..... 3 2 反馈的物理意义 ...... 3
2.1 Passive Feedback ..... 3 2.1 被动反馈..... 3
2.2 Active Feedback ..... 4 2.2 主动反馈..... 4
3 Feedback Control Loop ..... 6
3.1 Interaction of Elements ..... 6 3.1 元素的相互作用...... 6
3.2 Properties of Feedback Control ..... 8 3.2 反馈控制的特性............. 8
3.3 Stability and Robustness in Feedback Control ..... 10 3.3 反馈控制的稳定性和鲁棒性............. 10
4 Model Based Feedforward Control ..... 15
4.1 Adaptive Learning Feedforward and Observer ..... 16 4.1 自适应学习前馈和观察者..... 16
1 Introduction 1 简介
This paper is meant to give some background knowledge that is used in the design of modern active controlled subwoofers. 本文旨在提供现代有源控制低音炮设计中使用的一些背景知识。
It is based on the theory of motion control as presented in the book: "The Design of High Performance Mechatronics"1. 它基于《高性能机电一体化设计》一书中介绍的运动控制理论。 For that reason the subject of motional feedback theory is treated in a limited way, concentrating on the typical stability requirements, error reduction and dynamics, that are related to the use of feedback of a loudspeaker. 因此，运动反馈理论的主题受到限制，集中于与扬声器反馈的使用相关的典型稳定性要求、误差减少和动态。 Although feedback is mainly addressed, also some words are spent to "Model Based Feedforward Control" as many people think that that will be the ultimate solution. 虽然主要是解决反馈问题，但也用了一些词来描述“基于模型的前馈控制”，因为许多人认为这将是最终的解决方案。
2 The Physical Meaning of Feedback 2 反馈的物理意义
For those people who are reluctant with the use of mathematics, related to motion control, first the concept of feedback in a mechanical system is explained by means of how it changes the properties of a dynamic system. 对于那些不愿意使用与运动控制相关的数学的人来说，首先通过它如何改变动态系统的属性来解释机械系统中的反馈概念。
2.1 Passive Feedback 2.1 被动反馈
In mechanical sense "feedback" relates to the application of a force, which counteracts a mechanical motion aspect, being a displacement, a velocity or an acceleration. 在机械意义上，“反馈”涉及力的施加，其抵消机械运动方面，即位移、速度或加速度。 In mechanics three dynamic elements are defined, which provide such feedback for each of the mentioned motion aspects. The first element, the spring, gives a counteraction force, which is proportional to the displacement between both ends of the spring according to Hooke's law. 在力学中定义了三个动态元素，它们为每个提到的运动方面提供这样的反馈。第一个元件，弹簧，提供反作用力，根据胡克定律，该反作用力与弹簧两端之间的位移成正比。 The second element, the damper, gives a counteracting force, which is proportional to the velocity between two sides of the damper. The third element, the mass, gives a counteracting force, which is proportional to the acceleration of an object. 第二个元件是阻尼器，它产生反作用力，该反作用力与阻尼器两侧之间的速度成正比。第三个元素，质量，给出反作用力，该反作用力与物体的加速度成正比。 These feedback phenomena are called passive because they do not imply any supply of energy from outside to the system. 这些反馈现象被称为被动，因为它们并不意味着从外部向系统提供任何能量。 It is useful to be aware of these intrinsic passive "feedback" properties of mechanical structures as these are directly comparable with the active feedback principles that will be described in the next section. 了解机械结构的这些固有的被动“反馈”特性是很有用的，因为它们可以直接与下一节中将描述的主动反馈原理进行比较。
As explained in the paper on "Low Frequency Sound Generation by Loudspeaker Drivers" the dynamic behaviour of a loudspeaker driver is determined by the mass of the moving part, the membrane and actuator coil, the spring stiffness of the surround suspension and the air within the enclosure and the damping, mainly caused by the motion EMF and the low impedance of the amplifier. 正如“扬声器驱动器产生低频声音”一文中所解释的那样，扬声器驱动器的动态行为由运动部件、振膜和致动器线圈的质量、环绕悬架的弹簧刚度以及扬声器驱动器内的空气决定。外壳和阻尼，主要是由运动电动势和放大器的低阻抗引起的。 The frequency response of such a dynamic system is characterised by two frequency ranges, divided by a resonance frequency. Below the resonance frequency the system will move with an amplitude, which is proportional to the force and the inverse of the stiffness. 这种动态系统的频率响应的特征在于除以共振频率的两个频率范围。在共振频率以下，系统将以一定幅度移动，该幅度与力和刚度的倒数成正比。
Above the resonance frequency the system will move with an amplitude, which is proportional to the force and the inverse of the mass. The resonance frequency is determined by the combined moving mass and stiffness in the following way: 高于共振频率时，系统将以一定幅度移动，该幅度与力和质量的倒数成正比。共振频率由组合移动质量决定 和刚度 通过以下方式：
2.2 Active Feedback 2.2 主动反馈
Active feedback in motion control aims to mimic dynamic elements by measuring a mechanical aspect and supplying a force by means of an electronic amplifier and actuator, which counteract the measured aspect. This means that proportional position feedback control creates a virtual stiffness, which drives the object to a wanted position. 运动控制中的主动反馈旨在通过测量机械方面并通过电子放大器和执行器提供力来模拟动态元素，从而抵消测量的方面。这意味着比例位置 反馈控制会产生虚拟刚度，将物体驱动到所需位置。 Proportional velocity feedback control creates a virtual damper, which reduces deviation of the velocity from a set value and proportional acceleration feedback control creates a virtual mass, which reduces deviations in the acceleration from a set value. 比例速度反馈控制创建虚拟阻尼器，减少速度与设定值的偏差，比例加速度反馈控制创建虚拟质量，减少加速度与设定值的偏差。
A remark must be made here on the term "proportional". This indicates that there is no dynamic, frequency dependent relation between the measurement and the exerted force. 这里必须对“比例”一词作出评论。这表明测量值和施加的力之间不存在动态的、频率相关的关系。 Further on it will be explained that by differentiation and integration smooth transitions can be made between the different" "virtual" actively created mechanical elements by means of measuring only one of them. 进一步将解释，通过微分和集成，可以通过仅测量其中之一来在不同的“虚拟”主动创建的机械元件之间进行平滑过渡。
With these findings it is easier to imagine the impact of feedback on the dynamic behaviour of a loudspeaker driver. Figure 1 shows the characteristic frequency response of a loudspeaker driver, when mounted in a closed-box enclosure. 有了这些发现，就可以更容易地想象反馈对扬声器驱动器动态行为的影响。图 1 显示了安装在封闭箱体中时扬声器驱动器的特征频率响应。 Below the resonance frequency it shows a +2 slope with phase lead and above the resonance frequency it shows a flat response, because the radiated sound is proportional to the acceleration. 在共振频率以下，它显示出 +2 斜率 相位超前且高于共振频率时，它会表现出平坦的响应，因为辐射的声音与加速度成正比。
When applying proportional position feedback by measuring the position of the membrane and supplying a force that reduces the deviation from the wanted position, a virtual spring stiffness is added to the stiffness by the surround suspension and the enclosed air in the enclosure. 当通过测量膜的位置并提供减少与所需位置的偏差的力来应用比例位置反馈时，环绕悬架和外壳中封闭空气的刚度会添加虚拟弹簧刚度。 As a result the resonance frequency will increase and the low frequency response is decreased. This is not a favourable situation and one of the first traps people fall into when thinking about active feedback of loudspeakers. Using position sensors is useless! 结果，谐振频率将增加并且低频响应降低。这不是一个有利的情况，也是人们在考虑扬声器主动反馈时首先陷入的陷阱之一。使用位置传感器是没有用的！
When applying velocity feedback by measuring the velocity of the membrane and supplying a force that reduces the velocity a damper is created. As a result the peak of the resonance is decreased, which is useful. 当通过测量膜的速度并提供降低速度的力来应用速度反馈时，就会产生阻尼器。结果，共振峰值降低，这是有用的。 In fact the combination of a voltage source amplifier with a loudspeaker driver creates a velocity feedback 事实上，电压源放大器与扬声器驱动器的组合会产生速度反馈
Figure 1: Proportional position feedback increases the stiffness, resulting in an increasing resonance frequency (red arrow). Proportional velocity feedback increases the damping, resulting in an decreasing peak at the resonance frequency (yellow arrow). 图 1：比例位置反馈增加了刚度，从而导致共振频率增加（红色箭头）。比例速度反馈会增加阻尼，从而导致共振频率处的峰值降低（黄色箭头）。 Proportional acceleration feedback increases the mass, resulting in a decreasing resonance frequency (green arrow). 比例加速度反馈增加质量，导致共振频率降低（绿色箭头）。
system because the motion EMF generates a force that counteracts the velocity. This principle is described in another paper on a Sensorless Velocity Feedback Subwoofer. 系统，因为运动电动势会产生抵消速度的力。另一篇关于无传感器速度反馈低音炮的论文描述了这一原理。 Finally, when applying acceleration feedback by measuring the acceleration of the membrane and supplying a force that reduces the acceleration a mass is created. 最后，当通过测量膜的加速度并提供减小加速度的力来施加加速度反馈时，会产生质量。 As a result the resonance frequency will be lower, which is beneficial as that increases the frequency range with a flat response. It also decreases the output at higher frequencies and one might make the erroneous conclusion that it decreases the efficiency. 结果，谐振频率将会降低，这是有益的，因为这增加了平坦响应的频率范围。它还会降低较高频率下的输出，人们可能会得出这样的错误结论：它会降低效率。 This is not true because the added mass is virtual and the lower output is caused by a lower voltage signal at the loudspeaker terminals due to the feedback. This means that this effect can be compensated by increasing the input signal again (see the footnote at page 9 . 这是不正确的，因为增加的质量是虚拟的，并且较低的输出是由于反馈而在扬声器端子处的较低电压信号引起的。这意味着可以通过再次增加输入信号来补偿这种影响（参见第 9 页的脚注）。
In the following this more qualitative describing explanation of the principle of feedback is shortly presented in a more official "control-technology" way. 在下文中，反馈原理的这种更定性的描述性解释很快将以更官方的“控制技术”方式呈现。
3 Feedback Control Loop
Figure 2 from the above mentioned book shows a basic feedback control loop of a motion system. 上述书中的图 2 显示了运动系统的基本反馈控制回路。
The plant is a control engineering term for the physical motion system that needs to be controlled. 被控对象是一个控制工程术语，指的是需要控制的物理运动系统。 In a motional feedback system it consists of the power amplifier, actuator, and the mechanical structure of the moving parts of the driver, the actuator coil, cone, rubber surround and spider. 在运动反馈系统中，它由功率放大器、执行器以及驱动器运动部件的机械结构、执行器线圈、锥体、橡胶圈和星形轮组成。
In a motional feedback system the Input disturbance is the noise that is generated in the controller. It will be addressed in the design chapter. 在运动反馈系统中，输入干扰是控制器中生成的噪声。这将在设计章节中讨论。
The Process disturbance refers to nonlinearity and noise from the mechanics like rubbing and leaking holes. 过程扰动是指来自机械原理的非线性和噪声，例如摩擦和泄漏孔。 Nonlinearity causes distortion and should be kept minimal or reduced by feedback and the other disturbance sources should be avoided as hissing leaking holes are irritating, while rubbing implies wear. 非线性会导致失真，应保持最小或通过反馈减少，并且应避免其他干扰源，因为泄漏孔的嘶嘶声令人恼火，而摩擦则意味着磨损。
The Output disturbance represents the influence from the environment, like the sound pressure from another loudspeaker. When using two subwoofers for the same signal this mutual disturbance is in fact beneficial as it increases the total efficiency at low frequencies . 输出干扰代表环境的影响，例如来自另一个扬声器的声压。当使用两个低音炮处理同一信号时，这种相互干扰实际上是有益的，因为它提高了低频的总效率 。
Finally and most importantly the Sensor disturbance represents the measurement error by the sensor. In a feedback controlled loudspeaker it consists mainly of thermally induced noise and non-linear distortion. It is the most important source of disturbances as the controller will force the motion system to follow the erroneous measurement. 最后也是最重要的是，传感器干扰代表传感器的测量误差。在反馈控制扬声器中，它主要由热感应组成 噪声和非线性失真。它是最重要的干扰源，因为控制器将迫使运动系统遵循错误的测量结果。
3.1 Interaction of Elements 3.1 元素的相互作用
Each of the elements in the feedback control chain has its own inherent dynamic properties. They also interact in both directions in such a way that each element not only determines the input of the next element but also influences the previous element by its dynamic load. 反馈控制链中的每个元件都有其固有的动态特性。它们还在两个方向上相互作用，使得每个元素不仅决定下一个元素的输入，而且还通过其动态负载影响前一个元素。 The best example for this is the interaction between the actuator and the amplifier. 最好的例子是执行器和放大器之间的相互作用。
First of all the current from the amplifier will generate a force in the actuator following Lorentz' law, which can be written in two ways: 首先，来自放大器的电流将在执行器中产生遵循洛伦兹定律的力，该定律可以用两种方式表示：
where: 在哪里：
- the flux density ( ) of the magnetic field at the coil - 磁通密度（ ) 线圈处的磁场
- the current (A) - 电流（A）
Figure 2: Block diagram of a control system with feedback control. The plant consists of the moving parts of the loudspeaker driver. The diagram clearly shows the different places where interfering disturbances impact the system. 图 2：具有反馈控制的控制系统框图。该装置由扬声器驱动器的移动部件组成。该图清楚地显示了干扰扰动影响系统的不同位置。 The indicated variables are commonly used symbols in control engineering. 所指示的变量是控制工程中常用的符号。
= the total length of copper wire of the coil in the magnetic field = 线圈铜线在磁场中的总长度
the change in flux linkage, the amount of flux captured by all windings of the coil together. 磁链的变化，即线圈所有绕组共同捕获的磁通量。
Both notations are in principle correct and the first notation is best known, while the second is better as it prevents errors in designing actuators. For this report the first notation is sufficient, while also driver manufacturers use the term , also called the "Force Constant", as one of the relevant parameters for a driver. 两种表示法原则上都是正确的，第一种表示法最为人所知，而第二种表示法更好，因为它可以防止设计执行器时出现错误。对于本报告，第一个符号就足够了，驱动程序制造商也使用该术语 ，也称为“力常数”，作为驱动器的相关参数之一。
The above implies that the amplifier makes the moving part of the driver move by its current. 上面的内容意味着放大器通过其电流使驱动器的移动部分移动。
As a reaction a voice coil in a magnetic field will generate a voltage , which is proportional to the force factor and the motion velocity : 作为反应，磁场中的音圈会产生电压 ，与力因子成正比 和运动速度 :
A voltage source amplifier, as is used by all audio brands, will short circuit the coil for this motion induced voltage . Due to this short circuit will generate a current in the coil, which in its turn generates a force according to Equation (2) that is proportional to velocity and works in the opposite direction of the movement. This means that the use of a voltage source amplifier causes a damping effect, which is required for a non-controlled loudspeaker as otherwise the first resonance would have a very high value, where represents the level of the resonance above the non resonant response. The effect for different levels of damping is shown in Figure 3 and the shown peaking in output is very well audible. 所有音频品牌都使用的电压源放大器将使线圈因运动感应电压而短路 。由于这种短路 将在线圈中产生电流，电流又根据方程（2）产生与速度成正比的力 并沿运动的相反方向工作。这意味着使用电压源放大器会产生阻尼效应，这是非受控扬声器所必需的，否则第一谐振将具有非常高的频率。 值，其中 表示高于非共振响应的共振水平。图 3 显示了不同阻尼级别的效果，并且可以清楚地听到所示的输出峰值。
As a conclusion it is clear that the amplifier influences the dynamic behaviour of the driver. 结论很明显，放大器影响驱动器的动态行为。
Figure 3: Frequency response of the radiated sound of an electrodynamic loudspeaker, normalised to above the first resonance frequency with different damping settings. 图 3：电动扬声器辐射声的频率响应，归一化为 高于具有不同阻尼设置的第一共振频率。
3.2 Properties of Feedback Control 3.2 反馈控制的性质
In feedback control the actual status of the moving parts is monitored by a sensor and the controller is generating a control action based on the difference between the desired motion (reference signal) and the actual motion (sensor signal). 在反馈控制中，运动部件的实际状态由传感器监控，控制器根据期望运动（参考信号）和实际运动（传感器信号）之间的差异生成控制动作。
The output as shown in Figure 2 is measured and compared with (subtracted from) , which is the reference after input filtering. The result of this comparison, which is the error , is used as input for the feedback controller, which tries to keep this error as small as possible. 输出 如图2所示是测量并比较（减去） ，这是参考 输入过滤后。这个比较的结果，也就是误差 ，用作反馈控制器的输入，反馈控制器试图使该误差尽可能小。
Feedback control is also called closed-loop control, because the sensor signal is fed back in a closed-loop to the input of the system. 反馈控制也称为闭环控制，因为传感器信号以闭环方式反馈到系统的输入。
In control theory the mathematics of the Laplace transform are used to model the dynamic behaviour of a mechanical system in the frequency domain, based on the equations of motion in the time domain. Working with the resulting frequency 在控制理论中，拉普拉斯变换的数学用于基于时域中的运动方程在频域中对机械系统的动态行为进行建模。使用产生的频率
responses is more easy to work with as controllers can be made by simple filters. By using the Laplace variable , equations called "Transfer Functions" are derived from the equations of motion, which enable to generate Bode-plots, showing the frequency response in both amplitude and phase. 响应更容易使用，因为控制器可以通过简单的过滤器来制作。通过使用拉普拉斯变量 称为“传递函数”的方程源自运动方程，可以生成波特图，显示幅度和相位的频率响应。 Using the defined notions from Figure 2, the transfer function of a feedback loop is derived from the following equations starting with the error : 使用图 2 中定义的概念，反馈环路的传递函数可以从以下等式中导出，从误差开始 :
Including the input filter the total transfer function of the feedback loop from the reference signal to the output as shown in Figure 2 is given by: 包括输入滤波器、来自参考信号的反馈环路的总传递函数 到输出 如图 2 所示，由下式给出：
When considering that is the gain of the forward path from error to output and equals the forward gain times the sensor gain and leaving away the (s) terms after the equal sign for reasons of simplicity the transfer function can be written as : 当考虑到这一点时 是从错误到输出的前向路径的增益， 等于前向增益乘以传感器增益，为了简单起见，忽略等号后面的 (s) 项，传递函数可以写为 :
by dividing the numerator and denominator by the forward gain it becomes clear that a high gain in the forward path will cause the transfer function to be only dependent of the input filter and the sensor: 通过将分子和分母除以正向增益，可以清楚地看出，正向路径中的高增益将导致传递函数仅依赖于输入滤波器和传感器：
In control design one has the freedom to choose the prefilter , the sensor and particularly the controller such that the total transfer function fulfils the desired specifications. 在控制设计中，可以自由选择预过滤器 , 传感器 特别是控制器 使得总传递函数满足所需的规格。
With these properties feedback control has the following benefits for motional feedback of loudspeakers: 凭借这些特性，反馈控制对于扬声器的运动反馈具有以下优点：
- Reduction of the effect of disturbances: Disturbances of the controlled motion system like distortion by non-linearity and unwanted sounds by undamped resonances are observed in the sensor signal, and therefore the feedback controller can compensate for them. - 减少干扰的影响：在传感器信号中观察到受控运动系统的干扰，例如非线性失真和无阻尼共振产生的有害声音，因此反馈控制器可以对其进行补偿。
- Handling of uncertainties: Feedback controlled systems can also be designed to cope with changes and tolerances in the different properties of the elements. - 不确定性的处理：反馈控制系统也可以设计为应对元素不同属性的变化和公差。 This is called robustness, which means that the stability and performance requirements are guaranteed even for parameter variations of the controlled mechatronic system. 这称为鲁棒性，意味着即使受控机电系统的参数发生变化，也能保证稳定性和性能要求。
Although feedback control provides some very good features, it has of course also some pitfalls that have to be dealt with: 尽管反馈控制提供了一些非常好的功能，但它当然也有一些必须处理的缺陷：
A good sensor is required: The feedback loop is closed, based on information from a sensor. Therefore feedback control only can be as good as the quality of the sensor signal allows. 需要一个好的传感器：基于传感器的信息，反馈回路是闭合的。因此，反馈控制只能在传感器信号质量允许的情况下进行。 In precision positioning systems accurate sensors are required with high resolution and bandwidth, which are very costly. The measurement and sensing system often takes a substantial part of the total financial budget. 在精密定位系统中，需要具有高分辨率和带宽的精确传感器，但成本非常高。测量和传感系统通常占据总财务预算的很大一部分。
Limited reaction speed: A feedback controller only reacts on errors, differences between the reference signal and the measured system status, which means that the error has to occur first before the controller can correct for it. Without an error there is no output! 有限的反应速度：反馈控制器仅对错误、参考信号与测量的系统状态之间的差异做出反应，这意味着错误必须首先发生，控制器才能纠正它。没有错误就没有输出！
Feedback of noise: As mentioned earlier, by closing the loop, the sensor noise is also fed back, causing the sound to follow the noise instead of only the wanted reference input signal. 噪声反馈：如前所述，通过闭环，传感器噪声也会被反馈，导致声音跟随噪声，而不仅仅是想要的参考输入信号。
Can introduce instability: When the "negative" feedback becomes positive by phase delays in the loop the feedback system can (and will!) become unstable, thereby causing the system to resonate it its maximum power at a low or high frequency. 可能会引入不稳定：当环路中的相位延迟使“负”反馈变为正反馈时，反馈系统可能（并且将会！）变得不稳定，从而导致系统在低频或高频下谐振其最大功率。 Due to the continuous maximum power it can eventually destroy the driver! 由于持续的最大功率，它最终会毁掉驾驶员！
Feedback control is a very useful principle in loudspeakers for low frequencies as it reduces distortion and resonating effects as long as one takes precautions agains instability.. 反馈控制对于低频扬声器来说是一项非常有用的原理，因为只要采取预防措施防止不稳定，它就能减少失真和共振效应。
3.3 Stability and Robustness in Feedback Control 3.3 反馈控制的稳定性和鲁棒性
As mentioned in the last bullet of the previous section one should always consider phase relations when applying feedback. 正如上一节最后一个项目符号中提到的，在应用反馈时应始终考虑相位关系。 With loudspeakers two areas in the frequency range are giving problems. At high frequencies the natural inertia (slowness) of things will cause phase delays of which many are unavoidable. This poses a limit to the maximum frequency at which feedback can be applied with success. 对于扬声器来说，频率范围内的两个区域会产生问题。在高频下，事物的自然惯性（缓慢）将导致相位延迟，其中许多延迟是不可避免的。这对成功应用反馈的最大频率构成了限制。 With precision positioning systems as are presented in "the book" this is the only frequency range of phase problems because these systems should operate from DC. 对于“本书”中介绍的精密定位系统，这是相位问题的唯一频率范围，因为这些系统应该在直流下运行。 A loudspeaker however has also a limitation at DC due to the fact that the sound is proportional to the acceleration. This means that a loudspeaker is not able to generate DC sound. 然而，由于声音与加速度成正比，扬声器在直流时也有限制。这意味着扬声器无法产生直流声音。 Fortunately that is not even required but the fact that the frequency response shows a decline at lower frequencies automatically has impact on the phase. Where at higher frequencies the phase is lagging behind (delay) at lower frequencies the phase is advancing (leading). 幸运的是，这甚至不是必需的，但频率响应在较低频率下显示下降的事实会自动对相位产生影响。在较高频率下，相位滞后（延迟），在较低频率下，相位超前（超前）。 Both can cause the phase to change more than , changing negative feedback into positive and creating an unstable system. 两者都会导致相位变化超过 ，将负反馈变为正反馈，并创建一个不稳定的系统。
Between these extreme frequency ranges it is necessary to create a maximum forward gain in order to reduce the errors. 在这些极端频率范围之间，有必要创建最大前向增益以减少误差。
The challenge in designing a controller for motional feedback is thus to optimise between a high loopgain at the frequencies that have to be controlled and a low loopgain at the other frequencies. The optimal tuning of a feedback loop is called loop-shaping design. 因此，设计运动反馈控制器的挑战是在必须控制的频率处的高环路增益和其他频率处的低环路增益之间进行优化。反馈环路的最佳调整称为环路整形设计。
The most important and characteristic frequency area for the analysis of a controlled mechatronic system is around the open-loop unity-gain cross-over frequency, as shown in Figure 4 for the upper frequency limit. 对于受控机电系统分析来说，最重要和最具特征的频率区域是在开环单位增益交叉频率附近，如图 4 所示的频率上限。 In a motional feedback system the closed-loop bandwidth is directly related to the unity-gain cross-over frequency as above this frequency the loop gain becomes smaller than one and consequently the feedback controller becomes no longer effective. Usually the term bandwidth 在动反馈系统中，闭环带宽与单位增益交叉频率直接相关，因为高于该频率，环路增益变得小于一，因此反馈控制器变得不再有效。通常术语“带宽”
Figure 4: Stability condition and robustness of a feedback controlled system for the high frequency limit. equals the loop gain (forward gain times , where ) is assumed to be equal to one in this figure), is the closed loop response and is the Sensitivity. 图 4：反馈控制系统在高频限制下的稳定性条件和鲁棒性。 等于环路增益（正向增益乘以 ， 在哪里 ) 在此图中假设等于 1)， 是闭环响应， 是灵敏度。 The desired shape of these curves guide the control design by optimising the levels and slopes of the amplitude Bode plot at low and high frequencies for suppression of the disturbances (sensitivity) and of the phase Bode plot in the cross-over frequency region. 这些曲线的所需形状通过优化低频和高频振幅伯德图的水平和斜率来指导控制设计，以抑制干扰（灵敏度）以及交叉频率区域中的相位伯德图。
is defined as the frequency band where the power of the output signal of a system becomes less than half the desired power level. In terms of signal amplitude the corresponding value is equal to . In decibels this value is equal to and this value is a well-known definition for the bandwidth of filters and loudspeakers. 定义为系统输出信号功率小于所需功率电平一半的频带。就信号幅度而言，相应的值等于 。以分贝为单位，该值等于 这个值是滤波器和扬声器带宽的众所周知的定义。 In the context of reduction of errors it is preferred to define the control bandwidth as the range between the low and high unity-gain cross-over frequencies, where the amplitude of the open-loop frequency response exceeds a value of one. 在减少误差的背景下，优选将控制带宽定义为低和高单位增益交叉频率之间的范围，其中开环频率响应的幅度超过值1。 It is this open-loop gain that determines the suppression of disturbances, also called "Sensitivity" which is equal to the following equation. 正是这个开环增益决定了干扰的抑制，也称为“灵敏度”，等于以下等式。
When used in full detail using mathematical modelling software like MATLAB it will show that with systems of a higher order than one (which is the case with motional feedback) the sensitivity is increased at the frequency area just above the unity gain cross over frequency. 当使用 MATLAB 等数学建模软件进行详细使用时，它将表明，对于阶数高于一阶的系统（运动反馈就是这种情况），在单位增益交叉频率上方的频率区域，灵敏度会增加。 This effect is called the Bode-integral theorem and cannot be avoided. It is a sacrifice for the error reduction at the frequencies where the loopgain is much higher than one. 这种效应称为波德积分定理，是无法避免的。这是为了在环路增益远高于 1 的频率下减少误差而做出的牺牲。 The only way to keep the effect small is to spread it over a larger range by optimising the phase and amplitude margins, which are explained in the following. 保持影响较小的唯一方法是通过优化相位和幅度裕度将其扩展到更大的范围，这将在下面进行解释。
The key condition for closed-loop stability is that the total phase-lag for high frequencies and phase lead for low frequencies of the open-loop system, consisting of the feedback controller in series with the mechanics, must be less than in the frequency region of the cross-over frequencies. A system that has exactly phaselag at the cross-over point is called marginally stable. In this situation the smallest additional time-delay or phase-lag would make the closed-loop system unstable. 闭环稳定性的关键条件是，由与机械装置串联的反馈控制器组成的开环系统的高频相位滞后和低频相位超前必须小于 在交叉频率的频率区域中。一个系统恰好具有 交叉点处的相位滞后称为边际稳定。在这种情况下，最小的附加时间延迟或相位滞后都会使闭环系统不稳定。 Even though most audio designers hardly use it, a Nyquist plot, like the example shown in Figure 5, is most suitable to analyse the robustness of a feedback system. 尽管大多数音频设计人员很少使用它，但奈奎斯特图（如图 5 所示的示例）最适合分析反馈系统的鲁棒性。 It is an analysis tool that examines the open-loop frequency response of the feedback system including phase to predict the stability and the closed-loop response after the loop is closed. 它是一种分析工具，可检查反馈系统的开环频率响应（包括相位）以预测稳定性以及闭环后的闭环响应。 Its use is based on the Nyquist stability theorem, stating that a closed loop system will be stable when the Nyquist plot of the open-loop transfer function does not show a net clockwise encircling of the -1 point on the real axis. 其使用基于奈奎斯特稳定性定理，该定理指出，当开环传递函数的奈奎斯特图未显示实轴上 -1 点的净顺时针环绕时，闭环系统将是稳定的。 In other words a stable system after closing the loop is recognised in the Nyquist plot when the -1 point on the real axis is kept at the left-hand side upon passing with increasing frequencies. 换句话说，当实轴上的 -1 点在随着频率增加而通过时保持在左侧时，在奈奎斯特图中可以识别闭环后的稳定系统。
The complexity of the plot is in the fact that it is a complex plot with real and imaginary axis where phase and magnitude are combined and the frequency axis is not clearly shown. 该图的复杂性在于，它是一个具有实轴和虚轴的复杂图，其中相位和幅度组合在一起，并且没有清楚地显示频率轴。 It is a 2 dimensional vectorial plot where the distance from the origin indicates the gain and the angle of the line through the vector point and the origin with the right horizontal (positive real) axis determines the phase. 它是一个二维矢量图，其中距原点的距离表示增益，通过矢量点的直线的角度以及具有右侧水平（正实数）轴的原点确定相位。 An angle rotating clockwise is a negative phase relation and counterclockwise indicates a positive phase. For all frequencies such a vector point can be constructed and by connecting these points for all frequencies a curve is created (the blue line) along 顺时针旋转的角度为负相位关系，逆时针旋转的角度为正相位关系。对于所有频率，可以构建这样的向量点，并通过连接所有频率的这些点，创建一条曲线（蓝线）
Figure 5: The Nyquist plot of the open-loop response of a feedback system and its corresponding closed-loop frequency response of an example with HF limitation only. 图 5：仅具有 HF 限制的示例的反馈系统开环响应及其相应闭环频率响应的奈奎斯特图。 Stability is guaranteed when the -1 point on the real axis of the Nyquist plot is kept at the left-hand side of the open-loop response-line upon passing with increasing frequency. The dashed circles at the left graph determine the magnitude peak of the frequency response after closing the loop at the frequencies where the response-line crosses the circles. In this example and . 当奈奎斯特图实轴上的 -1 点在随着频率增加而通过时保持在开环响应线的左侧时，稳定性得到保证。左图的虚线圆圈确定幅度峰值 在响应线与圆相交的频率处闭合环路后的频率响应。在这个例子中 和 。
which the frequencies could be noted. An arrow alongside the blue line indicates increasing frequencies. Because both a phase of degrees and <-180 degrees is indicating potential trouble, most of the plot shows the left half from the origin. The fact that the frequency is not noted is overcome in practice because its first purpose is to show potential problems by means of computer simulation. 可以记录频率。蓝线旁边的箭头表示频率增加。因为两者都是一个阶段 度和 <-180 度表示潜在的麻烦，大部分图显示来自原点的左半部分。在实践中克服了未注明频率的事实，因为其首要目的是通过计算机模拟的方式显示潜在的问题。 A Nyquist plot is never made by hand while the modelling software immediately indicates the frequency, when pointing with a mouse to a place on the curve. 奈奎斯特图永远不会手工绘制，而当用鼠标指向曲线上的某个位置时，建模软件会立即指示频率。
The stability analysis with a Nyquist plot is done by examining the distance and direction of the plotted response graph of the open-loop system relative to the location of the -1 point on the real axis. 使用奈奎斯特图进行稳定性分析是通过检查开环系统绘制的响应图相对于实轴上 -1 点的位置的距离和方向来完成的。 The graph shows margin circles related to the capability of the closed-loop system to follow a reference input signal. 该图显示了与闭环系统跟随参考输入信号的能力相关的裕度圆。 Two values are shown in the Nyquist plot that are related to the robustness for stability of the closed-loop feedback system, the gain margin and the phase margin. 奈奎斯特图中显示了与闭环反馈系统稳定性鲁棒性、增益裕度和相位裕度相关的两个值。
The gain margin determines by which factor the open-loop gain additionally can increase before the closed-loop system goes unstable. It is defined by the distance between the loop-gain and unity-gain at the frequency where the phase-lag of becomes more negative than . The gain margin can have values between one and infinite. With first- and second-order transfer functions where the phase does never become more negative than the gain can be increased theoretically to infinite, corresponding to an infinite gain margin. 增益裕度决定了在闭环系统变得不稳定之前开环增益可以额外增加的因素。它由环路增益之间的距离定义 和频率单位增益，其中相位滞后 变得更加消极 。增益裕度的值可以介于 1 和无穷大之间。对于一阶和二阶传递函数，相位永远不会变得比 理论上增益可以增加到无限，对应于无限的增益裕度。
Figure 6: The Gain (GM) and Phase (PM) Margin in the Bode plot. At the LF bandwidth limit the phase-lead should be less than at gain, while the gain should be below at phase. At the bandwidth limit the phase-lag should be less than at gain and the gain should be below at phase. 图 6：波特图中的增益 (GM) 和相位 (PM) 裕度。在 LF 带宽限制下，相位超前应小于 在 增益，而增益应低于 在 阶段。在 带宽限制相位滞后应小于 在 增益且增益应低于 在 阶段。
The phase margin determines how much additional phase lag at the unity-gain cross-over frequency is acceptable before the closed-loop system becomes unstable. It is defined by the difference between the actual phase-lag of and at the unity-gain cross-over frequency. 相位裕度决定了在闭环系统变得不稳定之前，单位增益交叉频率处可以接受多少额外相位滞后。它由实际相位滞后之间的差异定义 和 在单位增益交叉频率。
When looking at the shown Nyquist plot the modelled example shows a phase that becomes more negative than -180 degrees at lower frequencies. This is counterintuitive but is as with increasing frequency the -1 point stays at the left hand side. If however the gain of one element in the feedback loop is reduced with more than a factor four the -1 point will be passed at the right side and the system will become unstable. 当查看所示的奈奎斯特图时，建模示例显示相位在较低频率下变得比 -180 度更负。这是违反直觉的，但 随着频率的增加，-1 点保持在左侧。然而，如果反馈环路中一个元件的增益减少超过四倍，则右侧将通过 -1 点，系统将变得不稳定。 This situation is called "conditionally stable" and is to be avoided when the gain can vary in the controlloop, as is the case with drivers with a large excursion range, like in subwoofers. 这种情况称为“条件稳定”，当控制环路中的增益可能变化时，应避免这种情况，就像具有大偏移范围的驱动器（如低音炮）的情况一样。
For an unconditionally stable system it is fortunately often sufficient to analyse the stability of the feedback loop by means of only the frequency and phase responses in the Bode plot, as shown in Figure 6. 幸运的是，对于无条件稳定的系统，通常仅通过伯德图中的频率和相位响应来分析反馈环路的稳定性就足够了，如图 6 所示。 As long as the phase margin at both ends of the open-loop bandwidth is in the order of or more and the gain margin is in the order of (factor 2) or more, a perfectly stable tuned feedback system is obtained. 只要开环带宽两端的相位裕度约为 或更多，增益裕度约为 （因子 2）或更多，获得完全稳定的调谐反馈系统。
4 Model Based Feedforward Control
Most of the actual research in industry and academia aims for "sensor-less" control of a loudspeaker by means of "Model-Based Feedforward Control" (MBFC). 工业界和学术界的大多数实际研究旨在通过“基于模型的前馈控制”（MBFC）对扬声器进行“无传感器”控制。 Indeed this is one of the most important research fields in the high-tech industry for one important reason: Feedback needs an error to act anyway, so it is a reaction with an unavoidable delay, the "settle time". 事实上，这是高科技行业最重要的研究领域之一，有一个重要原因：反馈无论如何都需要一个错误才能起作用，因此它是一种不可避免的延迟（“稳定时间”）的反应。 In high-tech mechatronic motion control it is a design rule to first compensate any error sources and only apply feedback for real unknown errors. 在高科技机电运动控制中，设计规则是首先补偿任何误差源，并且仅对真正未知的误差应用反馈。
MBFC starts with the idea that as long as one knows exactly how a system works and the system behaves reproducible, it is possible to control it by modifying the input signal in such a way that it compensates the deviations that are cause inside the (loudspeaker) system. MBFC 的出发点是，只要人们确切地知道系统是如何工作的并且系统的行为是可再现的，就可以通过修改输入信号来控制它，从而补偿（扬声器）内部引起的偏差。系统。 This compensation is easiest explained in mathematics. Assume is the output of a system in reaction to the input : 这种补偿最容易用数学来解释。认为 是系统对输入做出反应的输出 :
where is equal to the process of the system, which incorporates errors. If you desire output you only have to supply the system with times the inverse of as then the output will be: 在哪里 等于系统的过程，其中包含错误。如果你想要输出 您只需向系统提供 乘以 的倒数 那么输出将是：
Although this looks trivial, it requires the process to be invertible, which means that one has to be able to derive the input from the detected output. In mathematics there are methods to see if a process is invertible, its matrix should be square with a non-zero determinant. It goes, however, too far to do that here in this paper for a loudspeaker. 虽然这看起来微不足道，但它需要一个过程 是可逆的，这意味着必须能够从检测到的输出中导出输入。在数学中，有一些方法可以查看过程是否可逆，其矩阵 应该是具有非零行列式的平方。然而，在本文中对于扬声器来说这样做太过分了。 It is more easy (and I know the theoretical people will not be pleased) to look at the phenomena that play a role, the distortion sources as described in the paper "Distortion Sources in Subwoofers", the position dependent gain, the current dependent reluctance force, the temperature dependent resistance and the non-linear selfinducance. 更容易（我知道理论上的人不会高兴）查看起作用的现象，“低音炮中的失真源”论文中描述的失真源，位置相关增益，电流相关磁阻力、温度相关电阻和非线性自感。 For compensating (inverting) the position dependent gain one has to know the position of the membrane, which can only be derived by means of the dynamic model of the system with the input current. 为了补偿（反转）位置相关增益，必须知道膜的位置，而该位置只能通过具有输入电流的系统的动态模型来推导。 For low frequencies the dynamic model only consists of the mass of the moving diaphragm with coil and the total stiffness. It seems not too difficult to make the calculation, as long as the system does not change over time. 对于低频，动态模型仅包含带有线圈的移动膜片的质量和总刚度。只要系统不随时间变化，计算似乎并不太困难。 An error in the model, like a shift in the fundamental resonance frequency will easily cause the compensation to work out of phase with the problem, thereby increasing it rather than solving it. 模型中的误差（例如基本谐振频率的偏移）很容易导致补偿与问题异相，从而增加问题而不是解决问题。 Such a shift is well possible for instance due to the influence of temperature, which amongst others changes the stiffness of the surround. 例如，由于温度的影响，这种转变很可能发生，其中温度会改变周围的刚度。 The reluctance force is also position dependent so when necessary to compensate it one needs to know the position with the same risks for errors as with the gain. 磁阻力也与位置相关，因此当需要对其进行补偿时，人们需要知道位置，其误差风险与增益相同。 The temperature of the coil can be calculated from the current that passed over time and a previously determined thermal model of the system. And finally one has to derive a good model for the non-linear selfinductance, 线圈的温度可以根据随时间推移的电流和先前确定的系统热模型来计算。最后，我们必须导出一个好的非线性自感模型，
which can be sample dependent, because of the differences in magnetisation by the permanent magnets. 由于永磁体磁化强度的差异，这可能与样本相关。 Indeed it is possible to do this for one loudspeaker that has been measured and modelled but in the audio field the production of loudspeakers is not very strict with large tolerances on especially the magnetic part. 事实上，对于一个已经测量和建模的扬声器来说，这是可能的，但在音频领域，扬声器的生产并不是非常严格，尤其是磁性部件上的公差很大。 It would only work when the system could be regularly calibrated by means of a suitable.....sensor!!! In that case one might wonder why not use the sensor for active real-time feedback. 只有当系统可以通过合适的......传感器定期校准时，它才会起作用！在这种情况下，人们可能想知道为什么不使用传感器进行主动实时反馈。 The main drawback is then that one needs a real time sensor and these are expensive and critical with noise. 主要缺点是需要一个实时传感器，而这些传感器价格昂贵且噪声严重。
4.1 Adaptive Learning Feedforward and Observer 4.1 自适应学习前馈和观察者
The use of digital controllers with ample memory opened up the possibility to learn from previous errors, similar to the motion control of a human being. 使用具有充足内存的数字控制器开辟了从以前的错误中学习的可能性，类似于人类的运动控制。 Adaptive Feedforward Control (AFC) is well applicable in repeating actions and there is one thing for sure, A loudspeaker membrane is continuously repeating its motions. The repeating action allows for averaging the sensor signal, which reduces the impact of noise. 自适应前馈控制 (AFC) 非常适用于重复动作，并且可以肯定的是，扬声器振膜不断重复其运动。重复动作可以对传感器信号进行平均，从而减少噪声的影响。
Still, due to all dynamic effects and non-linearities the behaviour is different for many small frequency areas while not constant over time. 尽管如此，由于所有动态效应和非线性，许多小频率区域的行为是不同的，而且随着时间的推移并不恒定。 With modern control however an intermediate solution is possibly applicable using a real-time estimator, which is based on the model of the plant including the eigendynamics. 然而，通过现代控制，可以使用实时估计器来应用中间解决方案，该实时估计器基于包括特征动力学在内的对象模型。 Such an estimator is also called an observer while it observes the behaviour of a system by comparing it with the modelled behaviour and correcting its model parameters on this comparison in a process called innovation. 这样的估计器也称为观察者，它通过将系统的行为与建模的行为进行比较来观察系统的行为，并在称为创新的过程中根据这种比较纠正其模型参数。 Such an observer also allows a trade-off between the bandwidth (speed) of the estimation and the noise performance. 这样的观测器还允许在估计的带宽（速度）和噪声性能之间进行权衡。 An observer with an optimal trade-off between these two important properties is called a Kalman-filter, named after the Hungarian mathematician and electronic engineer Rudolph Emil Kálmán. 在这两个重要属性之间实现最佳权衡的观察者称为卡尔曼滤波器，以匈牙利数学家和电子工程师鲁道夫·埃米尔·卡尔曼 (Rudolph Emil Kálmán) 的名字命名。
Figure 7 shows the configuration of an observer in combination with state-feedback control of the observed system. The blocks in the dashed box represent the real mechatronic system. The blocks in the dotted box represent the mathematical model, which is implemented on a computer to simulate the behaviour of the mechatronic system in real-time. 图 7 显示了结合状态反馈的观察器的配置 对被观测系统的控制。虚线框中的块代表真实的机电系统。虚线框中的块代表数学模型，该模型在计算机上实现以实时模拟机电一体化系统的行为。 When both systems receive the same input signal , and both systems are identical, which means that a perfect model is available, both outputs and should be the same. However, in reality always modelling errors will occur while also the mechatronic system can be disturbed by external forces that are not taken into account and causing position and velocity errors. To compensate for these deviations the observer-gain matrix is introduced, which determines the innovation process by feedback of the prediction error to the observer, given by the difference between the output of the model and the output of the real system has to be designed such that the closed-loop system for the observer 当两个系统接收到相同的输入信号时 ，并且两个系统是相同的，这意味着可以使用完美的模型，两个输出 和 应该是一样的。然而，在现实中总是会出现建模误差，同时机电系统也可能受到未考虑的外力的干扰，从而导致位置和速度误差。为了补偿这些偏差，观察者增益矩阵 引入，它通过将预测误差反馈给观察者来确定创新过程，该误差由模型输出与真实系统输出之间的差异给出 必须设计为观察者的闭环系统
Figure 7: Model-based controller with an observer to estimate not measured values in a control system. The real-time feedback path is determined by the feedback matrix based on estimated values from within the model. The model is updated by the difference between the observer output and the real system output via the matrix . 图 7：基于模型的控制器，带有观测器，用于估计控制系统中的非测量值。实时反馈路径由反馈矩阵决定 基于模型内的估计值。模型通过观察者输出之间的差异进行更新 和实际系统输出 通过矩阵 。
part is stable. 部分稳定。
In spite of these new methods up till now no loudspeakers applying these technologies are developed, as far as the author knows. 尽管有这些新方法，但据作者所知，迄今为止还没有开发出应用这些技术的扬声器。 The main reason might be that it still requires a sensor, while real time feedback is proven to be a highly successful approach as applied by RMS Acoustics & Mechatronics. 主要原因可能是它仍然需要传感器，而实时反馈被证明是 RMS Acoustics & Mechatronics 应用的非常成功的方法。
The motion control section of the book is mainly written by Georg Schitter from TUVienna. 本书的运动控制部分主要由来自TUVienna的Georg Schitter撰写。
The use of the term "position feedback" or "position control" is more common instead of "displacement control". In fact displacement is the deviation from the wanted position, which is corrected by feedback. This wanted position can be stationary or changing. 术语“位置反馈”或“位置控制”的使用比“位移控制”更常见。事实上，位移是与所需位置的偏差，可以通过反馈进行校正。该所需位置可以是固定的或变化的。
Too few people are aware of the fact that an uncontrolled loudspeaker, so without a connected and working amplifier, will act as a resonator for sound coming from other loudspeakers! 很少有人意识到，不受控制的扬声器（因此没有连接且工作的放大器）将充当来自其他扬声器的声音的谐振器！ One should always short out the unused loudspeakers when judging and comparing different loudspeakers. 在判断和比较不同的扬声器时，应将未使用的扬声器短路。
These equations show that feedback reduces the response of the loudspeaker driver (which is included in the "forward gain") by the terms in the denominator. This effect on the forward gain can be compensated by increasing the gain of the pre-filter. 这些方程表明，反馈会按分母中的项降低扬声器驱动器的响应（包含在“前向增益”中）。这种对前向增益的影响可以通过增加增益来补偿 的预过滤器。
The following is also partly copied from "the book" and adapted to motional feedback. 以下内容也部分抄自“书本”，并根据动作反馈进行了改编。
State feedback is a mathematical method to design a discrete time digital controller. For more info see the book "The Design of High Performance Mechatronics" 状态反馈是设计离散时间数字控制器的数学方法。欲了解更多信息，请参阅《高性能机电一体化设计》一书