Motional Feedback Theory in a Nutshell
运动反馈理论简述

Copying of the complete document is allowed for personal use only. The author/publisher is not responsible for any problems that might arise by using the contents of this paper.
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Published by RMS Acoustics & Mechatronics
由 RMS Acoustics & Mechatronics 出版

The Netherlands 荷兰

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 无源反馈 .....3
2.2 Active Feedback ..... 4
2.2 主动反馈 .....4
3 Feedback Control Loop ..... 6
3 反馈控制回路 .....6
3.1 Interaction of Elements ..... 6
3.1 各要素之间的相互作用 .....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 反馈控制中的稳定性和鲁棒性 .....10
4 Model Based Feedforward Control ..... 15
4 基于模型的前馈控制 .....15
4.1 Adaptive Learning Feedforward and Observer ..... 16
4.1 自适应学习前馈和观测器 .....16

This paper is meant to give some background knowledge that is used in the design of modern active controlled subwoofers.
本文旨在介绍一些用于设计现代有源受控低音炮的背景知识。

It is based on the theory of motion control as presented in the book: "The Design of High Performance Mechatronics"1. 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.
它以 "高性能机电一体化设计 "1 一书中介绍的运动控制理论为基础："高性能机电一体化设计 "1 一书中介绍的运动控制理论为基础。因此，我们对运动反馈理论进行了有限的处理，重点关注与扬声器反馈使用相关的典型稳定性要求、误差减小和动态特性。
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.
虽然主要讨论了反馈问题，但也花了一些篇幅讨论 "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。

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有关的数学知识的人来说，首先要解释的是机械系统中的反馈概念，即反馈如何改变动态系统的特性。

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 "是指施加一个力，抵消机械运动的一个方面，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.
力学中定义了三个动态元素，它们为上述运动的每个方面提供反馈。第一个元素是弹簧，根据胡克定律，弹簧两端的位移与反作用力成正比。
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.
第二个要素是阻尼器，它产生的反作用力与阻尼器两侧的速度成正比。第三个要素是质量，它产生的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.
这些反馈现象之所以被称为被动反馈，是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被动 "反馈 "特性非常有用，因为这些特性可与下一节将介绍的主动反馈原理直接比较。

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 "扬声器驱动器产生的低频声音 "一文中所解释的，扬声器驱动器的动态特性由运动部件、膜片和致动器线圈的质量、环绕悬挂装置和箱体内空气的弹簧刚度以及阻尼决定，阻尼主要由运动电磁场和放大器的低阻抗引起。
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由两个频率范围和一个共振频率构成。在共振频率以下，系统的运动幅度与力成正比，与刚度成反比。

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:
在共振频率之上，系统将以与力和质量倒数成正比的振幅运动。共振频率由运动质量 和刚度 按以下方式确定：

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致动器提供与测量方面相抵消的力来模拟动态元素。这意味着，比例位置 反馈控制会产生一个虚拟刚度，从而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.
比例速度反馈控制产生一个虚拟阻尼器，可减少速度与设定值的偏差；比例加速度反馈控制产生一个虚拟质量，可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 "成比例 "一词加以说明。这表明测量值和作用力之间不存在动态的、与频率相关的关系。
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.
此外，我们还将解释，通过分化和整合，只需测量其中一个机械元件，就能在不同的 "虚拟 "机械元件之间实现平稳过渡。

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，我们就更容易想象反馈对扬声器驱动器动态特性的影响了。图 1 显示了安装在封闭箱体中的扬声器驱动器的频率响应特性。
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 的斜率， 相位引导，在共振频率以上，它显示出平坦的响应，因为辐射声与加速度成正比。

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.
当通过测量膜片的位置并提供一个力来减少与所需位置的偏差，从而应用比例位置反馈时，环绕悬挂装置和外壳中的封闭空气会在刚度之外增加一个虚拟弹簧刚度。
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!
因此，共振频率会增加，低频响应会降低。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最先陷入的陷阱之一。使用位置传感器毫无用处！

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.
通过测量膜的速度并提供一个降低速度的力来应用速度反馈时，就会产生阻尼。因此，共振峰值会降低，这一点非常有用。
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可产生速度反馈

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，导致共振频率上升（红色箭头）。比例速度反馈会增加阻尼，导致共振频率峰值下降（黄色箭头）。
Proportional acceleration feedback increases the mass, resulting in a decreasing resonance frequency (green arrow).
比例加速度反馈会增加质量，从而降低共振频率（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.
因为运动电磁场会产生一个与速度相抵消的力。另一篇关于无传感器速度反馈式低音炮的论文对这一原理进行了描述。
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，通过测量膜的加速度并提供一个减小加速度的力来应用加速度反馈时，就会产生一个质量。
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.
因此，共振频率会降低，这是有好处的，因为这样可以增加频率响应范围，使频率响应更加平缓。但同时也会降低较高频率下的输出，因此可能会得出降低效率的错误结论。
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，而较低的输出是由于反馈导致扬声器终端的电压信号较低造成的。这意味着可以通过再次增加输入信号来补偿这种影响（见第 9 页脚注）。

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，我们将以更加正式的 "控制技术 "方式，对反馈原理进行更加定性的说明。

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.
工厂是一个控制工程术语，指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、激励器和驱动器运动部件的机械结构、激励器线圈、音盆、橡胶环绕和蛛网组成。

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产生的噪声。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.
过程干扰是指来自机械的非线性和噪音，如摩擦和漏孔。
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，因此应尽量减少或通过反馈来降低非线性，同时应避免其他干扰源，因为嘶嘶声漏孔会造成刺激，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 .
输出干扰代表来自环境的影响，例如来自另一个扬声器的声压。当使用两个超低音扬声器处理同一信号时，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，也是最重要的一点，传感器干扰代表传感器的测量误差。在反馈控制扬声器中，它主要包括热引起的 噪声和非线性失真。这是最重要的干扰源，因为控制器会迫使运动系统遵循错误的测量结果。

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。它们还以这样一种方式进行双向互动，即每个元件不仅决定下一个元件的输入，还通过其动态负载影响前一个元件。
The best example for this is the interaction between the actuator and the amplifier.
致动器和放大器之间的互动就是最好的例子。

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，根据洛伦兹定律，来自放大器的电流会在致动器中产生一个力，这个力有两种写法：

where: 在哪里？
- the flux density ( ) of the magnetic field at the coil
- 线圈磁场的磁通密度 ( )
- 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：带反馈控制的控制系统框图。设备由扬声器驱动器的运动部件组成。图中清楚地显示了干扰对系统产生影响的不同位置。
The indicated variables are commonly used symbols in control engineering.
所示变量是控制工程中常用的符号。

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.
这两种记法原则上都是正确的，第一种记法最为人所熟知，而第二种记法则更好，因为它可以避免在设计推杆时出现错误。在本报告中，使用第一种符号就足够了，而驱动器制造商也使用 （也称为 "力常量"）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.
这意味着放大器通过电流使驱动器的移动部件移动。

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 :
作为反作用力，音圈在磁场中会产生电压 ，该电压与力系数 和运动速度 成正比：

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.
电压源放大器（如所有音响品牌所使用的放大器）将使线圈短路，以获得运动感应电压 。由于这种短路 ，线圈中将产生电流，进而根据公式（2）产生一个与速度成正比的力 ，其作用方向与运动方向相反。这意味着电压源放大器的使用会产生阻尼效应，这对非受控扬声器来说是必需的，否则第一共振将具有非常高的 值，其中 代表高于非共振响应的共振水平。不同阻尼水平的效果如图 3 所示，输出峰值清晰可闻。

As a conclusion it is clear that the amplifier influences the dynamic behaviour of the driver.
总之，放大器显然会影响驱动器的动态性能。

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：不同阻尼设置下电动扬声器辐射声的频率响应，归一化为 ，高于第一共振频率。

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测，控制器根据预期运动（参考信号）和实际运动（传感器信号）之间的差值产生控制动作。

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.
如图 2 所示，对输出 进行测量，并与输入滤波后的参考 进行比较（相减）。比较的结果，也就是误差 ，被用作反馈控制器的输入，该控制器试图将误差保持在尽可能小的范围内。

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，因为传感器信号会以闭环方式反馈到系统的输入端。

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，拉普拉斯变换数学用于根据时域运动方程在频域中模拟机械系统的动态行为。利用所得到的频率
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.
由于控制器可以通过简单的滤波器制作，因此频率响应更易于操作。通过使用拉普拉斯变量 ，可以从运动方程推导出称为 "传递函数 "的方程，从而生成 Bode 图，显示振幅和相位的频率响应。
Using the defined notions from Figure 2, the transfer function of a feedback loop is derived from the following equations starting with the error :
利用图 2 中定义的概念，可以从误差 开始，通过以下方程推导出反馈回路的传递函数：

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:
如图 2 所示，包括输入滤波器在内，从参考信号 到输出 的反馈回路总传递函数为

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 :
考虑到 是误差到输出的前向路径增益，而 等于前向增益乘以传感器增益，为简单起见，去掉等号后的 (s) 项，则传递函数可写成 ：

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:
将分子和分母除以前向增益，可以清楚地看出，前向路径中的高增益将导致传递函数只取决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，我们可以自由选择预滤波器 、传感器 ，尤其是控制器 ，从而使总传递函数满足所需的规格要求。

With these properties feedback control has the following benefits for motional feedback of loudspeakers:
凭借这些特性，反馈控制可为扬声器的运动反馈带来以下好处：
- 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.
- 减少干扰影响：在传感器信号中可以观察到受控运动系统的干扰，如非线性失真和无阻尼共振产生的杂音，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.
- 处理不确定性：反馈控制系统的设计还可应对元件不同特性的变化和公差。
This is called robustness, which means that the stability and performance requirements are guaranteed even for parameter variations of the controlled mechatronic system.
这就是所谓的鲁棒性，即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解决的缺陷：

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..
反馈控制是扬声器低频的一个非常有用的原理，因为它可以减少失真和共振效应，只要采取预防措施防止不稳定因素。

As mentioned in the last bullet of the previous section one should always consider phase relations when applying feedback. 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.
如上一节最后一项所述，在应用反馈时应始终考虑相位关系。扬声器在频率范围内有两个方面存在问题。在高频率下，事物的自然惯性（缓慢）会导致相位延迟，其中许多延迟是不可避免的。
This poses a limit to the maximum frequency at which feedback can be applied with success. 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.
这就限制了成功应用反馈的最高频率。对于 "本书 "中介绍的精密定位系统，这是出现相位问题的唯一频率范围，因为这些系统应从直流开始运行。
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，扬声器在直流时也会受到限制。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，但频率响应在低频时出现下降这一事实会自动对相位产生影响。在较高频率下，相位会落后（延迟），而在较低频率下，相位会前进（领先）。
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 ，从而将负反馈变为正反馈，造成系统不稳定。

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，有必要创建一个最大的前向增益，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难点在于如何在需要控制的频率上优化高环路增益，而在其他频率上优化低环路增益。反馈回路的优化调整称为回路整形设计。

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.
对于受控机电一体化系统的分析而言，最重要的特征频率区域是开环单增益交叉频率附近，如图 4 所示为频率上限。
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
在运动反馈系统中，闭环带宽与单增益交叉频率直接相关，因为超过这个频率，环路增益会变得小于 1，从而使反馈控制器不再有效。通常情况下，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.
等于环路增益（前向增益乘以 ，其中 ) 在本图中假定等于 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.
这些曲线的理想形状为控制设计提供指导，通过优化低频和高频的振幅 Bode 图的水平和斜率来抑制干扰（灵敏度），并优化交叉频率区域的相位 Bode 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水平一半的频段。就信号幅度而言，相应的值等于 。以分贝为单位，该值等于 ，该值是众所周知的滤波器和扬声器带宽的定义。
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.
在减少误差方面，最好将控制带宽定义为低频和高频单增益交叉频率之间的范围，在这个范围内，开环频率响应的振幅超过 1。
It is this open-loop gain that determines the suppression of disturbances, also called "Sensitivity" which is equal to the following equation.
正是这一开环增益决定了对干扰的抑制，也称为 "灵敏度"，等于下式

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.
如果使用 MATLAB 等数学建模软件进行详细分析，就会发现对于一阶以上的系统（运动反馈就是这种情况），灵敏度会在略高于统一增益交叉频率的频率区域增加。
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.
这种效应被称为波德积分定理，是无法避免的。在环路增益远大于 1 的频率上，它是减少误差的牺牲品。
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.
保持较小影响的唯一方法是通过优化相位和振幅余量，将其分散到更大的范围内，下文将对此进行解释。

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，由反馈控制器与机械装置串联组成的开环系统在交叉频率区域内的高频总相位滞后和低频总相位领先必须小于 。在交叉点处，相位差恰好为 的系统称为边际稳定系统。在这种情况下，最小的额外时间延迟或相位滞后都会导致闭环系统不稳定。
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奈奎斯特曲线图，但如图 5 所示，奈奎斯特曲线图最适合用来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，用于检查反馈系统的开环频率响应（包括相位），以预测稳定性和闭环后的闭环响应。
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.
奈奎斯特稳定性定理指出，当开环传递函数的奈奎斯特图不出现实轴上-1 点的顺时针净环绕时，闭环系统将保持稳定。
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，当实际轴上的 -1 点随着频率的增加而保持在左侧时，在奈奎斯特曲线图中就能识别出闭环后的稳定系统。

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.
这幅图的复杂性在于它是一幅复数图，有实轴和虚轴，相位和幅值结合在一起，频率轴没有清楚地显示出来。
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，其中离原点的距离表示增益，通过矢量点和原点的直线与右侧水平轴（正实数）的夹角决定相位。
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
顺时针旋转的角度表示负相位关系，逆时针旋转的角度表示正相位关系。对于所有频率，都可以构建这样一个矢量点，将所有频率的这些点连接起来，就形成了一条曲线（蓝线），沿着

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：反馈系统开环响应的奈奎斯特图，以及仅有高频限制的示例的相应闭环频率响应。
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 .
随着频率的增加，奈奎斯特曲线图实轴上的 -1 点保持在开环响应线的左侧，这就保证了稳定性。左图中的虚线圆圈确定了在响应线与圆圈交叉的频率上闭环后频率响应的幅度峰值 。本例中 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。蓝线旁的箭头表示频率在增加。由于 度和 <-180 度的相位都表示潜在的问题，因此图中大部分显示的是从原点开始的左半部。在实践中，由于其首要目的是通过计算机模拟来显示潜在的问题，因此没有注意到频率这一事实是可以克服的。
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.
奈奎斯特曲线图从来不需要手工绘制，而当用鼠标指向曲线上的某个位置时，建模软件会立即显示出频率。

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.
使用奈奎斯特曲线图进行稳定性分析时，需要检查所绘制的开环系统响应图相对于实际轴上 -1 点位置的距离和方向。
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踪参考输入信号的能力有关的余量圈。
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相关的两个值，即增益裕度和相位裕度。

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出现不稳定之前，开环增益可以额外增加多少系数。它的定义是，在 的相位滞后变得比 更负的频率上，环路增益 与单增益之间的距离。增益裕度的取值范围从 1 到无限大。在一阶和二阶传递函数中，相位永远不会负于 ，理论上增益可以增加到无限大，即增益裕量为无限大。

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：博德图中的增益（GM）和相位（PM）裕度。在低频带宽限制下， 增益时的相位滞后应小于 ，而 相位时的增益应低于 。在 带宽限制下， 增益时的相位滞后应小于 ，而 相位时的增益应低于 。

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，在单位增益交叉频率上可接受的额外相位滞后程度。它由 和 在单位增益交叉频率上的实际相位滞后之差定义。

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.
从图中的奈奎斯特图来看，建模示例显示，在频率较低时，相位会变得比 -180 度更负。这与直觉相反，但 ，因为随着频率的增加，-1 点保持在左手边。但是，如果反馈环路中一个元件的增益降低超过 4 倍，-1 点就会通过右侧，系统就会变得不稳定。
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.
这种情况被称为 "条件稳定"，当增益在控制环中变化时应避免使用，例如超低音扬声器中具有较大偏移范围的驱动器。

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. 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.
幸运的是，对于无条件稳定的系统，通常只需通过 Bode 图中的频率和相位响应来分析反馈回路的稳定性即可，如图 6 所示。只要开环带宽两端的相位裕度在 或更大的量级，增益裕度在 （系数 2）或更大的量级，就能得到一个完全稳定的调谐反馈系统。

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部分实际研究旨在通过 "基于模型的前馈控制"（MBFC）实现扬声器的 "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，原因很重要：反馈无论如何都需要有误差才能起作用，因此它是一种具有不可避免的延迟的反应，即 "安顿时间"。
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所有误差源，然后才对真正的未知误差进行反馈。

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，只要我们确切地知道系统是如何工作的，并且系统的行为具有可重复性，就有可能通过修改输入信号来控制它，从而补偿（扬声器）系统内部产生的偏差。
This compensation is easiest explained in mathematics. Assume is the output of a system in reaction to the input :
数学最容易解释这种补偿。假设 是一个系统对输入 的反应输出：

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:
其中 等于包含误差的系统过程。如果您希望输出 ，您只需向系统提供 乘以 的倒数即可，因为此时的输出将是：

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，但它要求过程 是可逆的，这意味着我们必须能够从检测到的输出推导出输入。在数学中，有一些方法可以看出一个过程是否可逆，即它的矩阵 应该是方形的，行列式不为零。然而，在本文中，对于扬声器而言，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.
更简单的方法是（我知道理论界人士会不高兴）研究起作用的现象，即 "超低音中的失真源 "一文中描述的失真源、与位置相关的增益、与电流相关的磁阻力、与温度相关的电阻和非线性自感应。
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.
要补偿（反转）与位置相关的增益，就必须知道膜的位置，而这只能通过输入电流的系统动态模型得出。
For low frequencies the dynamic model only consists of the mass of the moving diaphragm with coil and the total stiffness.
对于低频，动态模型只包括带线圈的运动膜片的质量和总刚度。
It seems not too difficult to make the calculation, as long as the system does not change over time.
只要系统不随时间发生变化，计算起来似乎并不困难。
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，如基频共振频率的偏移，很容易导致补偿工作与问题不同步，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，这种变化是完全可能的，因为温度除其他外还会改变周围的刚度。
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.
磁阻力也与位置有关，因此在需要对其进行补偿时，需要了解位置，并承担与增益相同的误差风险。
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,
线圈的温度可根据随时间流逝的电流和先前确定的系统热模型计算得出。最后，我们还必须推导出非线性自感的良好模型、
which can be sample dependent, because of the differences in magnetisation by the permanent magnets.
这可能与样品有关，因为永久磁铁的磁化率不同。
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，对一个扬声器进行测量和建模是可以做到这一点的，但在音频领域，扬声器的生产并不十分严格，尤其是磁性部分的公差很大。
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一个合适的..... 传感器进行定期校准时，它才会起作用！在这种情况下，人们可能会问，为什么不使用传感器进行主动实时反馈呢？
The main drawback is then that one needs a real time sensor and these are expensive and critical with noise.
其主要缺点是需要一个实时传感器，而这些传感器价格昂贵，且容易产生噪音。

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.
使用具有大量内存的数字控制器，可以从以前的错误中吸取教训，这与人类的运动控制类似。
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，有一点可以肯定，扬声器膜片正在不断重复其动作。重复动作可以使传感器信号平均化，从而减少噪音的影响。

Still, due to all dynamic effects and non-linearities the behaviour is different for many small frequency areas while not constant over time.
不过，由于所有的动态效应和非线性因素，许多小频率区域的表现是不同的，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.
不过，在现代控制技术中，有可能采用一种中间解决方案，即使用实时估算器，该估算器以工厂模型为基础，包括外生动力学。
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.
这种估计器也被称为观测器，它通过将系统行为与模型行为进行比较来观测系统行为，并根据比较结果修正模型参数，这一过程被称为创新。
Such an observer also allows a trade-off between the bandwidth (speed) of the estimation and the noise performance.
这种观测器还可以在估计带宽（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.
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 显示了观测器与状态反馈 控制相结合的观测系统配置。虚线框内的图块代表真实的机电系统。虚线框内的图块代表数学模型，该模型在计算机上实现，用于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
当两个系统接收相同的输入信号 时，并且两个系统完全相同，这意味着有一个完美的模型，两个系统的输出 和 应该是相同的。然而，现实中总会出现建模误差，同时机电一体化系统也会受到外力的干扰，而这些干扰并没有被考虑在内，从而导致位置和速度误差。为了补偿这些偏差，我们引入了观测器增益矩阵 ，该矩阵通过向观测器反馈预测误差来决定创新过程，而预测误差由模型输出与实际系统输出之间的差值给出 ，因此必须设计观测器的闭环系统。

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：带观测器的基于模型的控制器，用于估算控制系统中的非测量值。实时反馈路径由反馈矩阵 根据模型内的估计值确定。观察器输出 与实际系统输出 之间的差值通过矩阵 更新模型。

part is stable. 部分是稳定的。

In spite of these new methods up till now no loudspeakers applying these technologies are developed, as far as the author knows.
尽管采用了这些新方法，但据作者所知，迄今为止尚未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一个传感器，而 RMS Acoustics & Mechatronics 公司采用的实时反馈被证明是一种非常成功的方法。