Lecture 4 - Sensorless Control and Observer

Motivation / Big Picture

Sensorless does not mean information-lessSensorless 不等于没有信息

In Chapter 3, the controller wants quantities such as $i_{\alpha\beta}$, rotor speed $\Omega$, active-flux angle $\theta_d$, and sometimes disturbance or load torque. But real hardware gives us only a subset directly. Chapter 4 asks: how can we recover the missing states from the signals we do measure? 在 Chapter 3 里，控制器想用到 $i_{\alpha\beta}$、转速 $\Omega$、有功磁链角 $\theta_d$，有时还想知道扰动或负载转矩。但真实硬件只能直接给出其中一部分。Chapter 4 问的是：如何从可测信号中恢复那些不可直接测的状态？

Usually measured通常可测

Phase current, dc-bus voltage, encoder angle, command signals.相电流、母线电压、编码器角度、控制指令。

Usually needed控制想要

Rotor speed, active-flux angle, stator flux, disturbance torque, disturbance EMF.转速、有功磁链角、定子磁链、负载扰动、扰动反电势。

Observer job观察器任务

Combine physics, measured output, and error correction to reconstruct the missing states online.把系统模型、测量输出和误差校正组合起来，在线重建缺失状态。

One-sentence takeaway一句话结论

Observer = predictor + corrector. The predictor uses the model; the corrector injects the mismatch between measured and estimated outputs. Observer = 预测器 + 校正器。 预测器来自模型，校正器来自“测量输出”和“估计输出”的不一致。

speed flux disturbance sensorless

Why this matters for sensorless control这和 sensorless control 的关系 If speed is not directly measured, speed control does not disappear. It simply becomes a control + observer problem instead of a pure feedback problem. 如果转速不能直接测量，速度闭环并不会消失，它只是从“纯反馈控制问题”变成了“控制器 + 观察器”的耦合问题。

Sensor Imperfection / Measurement Limits

Why we cannot just differentiate or integrate blindly为什么不能简单微分或积分就完事

Angle to speed: differentiation amplifies defects角度求速度：微分会放大缺陷

Encoder angle is quantized. If you directly differentiate angle or pulse count, any quantization, jitter, or windowing choice enters the speed estimate. High bandwidth gives faster speed reconstruction but worse ripple and noise. 编码器角度是量化量。若直接对角度或脉冲计数做微分，量化误差、抖动和窗口长度都会直接进入速度估计。带宽高一些，重建更快；但波动和噪声也更重。

Voltage/current to flux: integration drifts电压电流求磁链：积分会漂移

Open-loop flux estimation uses $\psi_{\alpha\beta s}=\frac{1}{s}(u_{\alpha\beta s}-Ri_{\alpha\beta s})$. Any current offset or voltage bias is integrated forever, so even a tiny sensor drift can destroy the estimate. 开环磁链估计依赖 $\psi_{\alpha\beta s}=\frac{1}{s}(u_{\alpha\beta s}-Ri_{\alpha\beta s})$。只要电流零漂或电压偏置不为零，积分器就会一直累计，哪怕偏差很小也会把磁链估计带跑。

$$\Omega_m \neq \frac{d\Theta_m}{dt}\ \text{in practice, and}\ \hat{\psi}=\frac{1}{s}(u-Ri)\ \text{is not robust by itself.}$$

Engineering tradeoff工程上的权衡 Every observer is balancing two enemies: missing bandwidth and measurement defects. This is why Chapter 4 spends time on prediction and innovation instead of direct signal processing only. 每个观察器都在两类敌人之间做权衡：带宽不够 和 测量缺陷。所以 Chapter 4 才会强调预测与创新，而不是只做简单信号处理。

Unified Formulation / Active Field Model

A shared observer language for PMSM and IM用同一套语言描述 PMSM 和 IM

Section 4.4 rewrites the machine with the active-field-oriented model. The key modeling choice is to use the active-flux angle $\theta_d$ rather than raw rotor angle as a main state. Then PMSM and induction motor can be discussed inside one observer framework. 4.4 节把电机写成 active-field-oriented model。关键选择是把有功磁链角 $\theta_d$ 作为主要状态，而不是死盯机械转子角。这样 PMSM 和感应电机就能落在同一个观察器框架里讨论。

$$\begin{aligned} \dot{\theta}_d &= \omega_d = n_{pp}\Omega + \omega_{sl} \\ \dot{\Omega} &= \frac{1}{J_s}(T_{em}-T_L) = \frac{1}{J_s}(n_{pp}\psi_A i_{qs} - T_L) \end{aligned}$$

Why this modeling step is valuable为什么这一步重要

The observer can aim at one common state set.观察器可以围绕一组统一状态来设计。
Prediction terms can come from voltage equations rather than only mechanics.预测项可以来自电压方程，而不只是机械方程。
The role of back-EMF and flux information becomes explicit.反电势与磁链信息在估计中的作用会更清晰。

Specialization idea特化关系

PMSM: set $\omega_{sl}=0$.PMSM：令 $\omega_{sl}=0$。
IM: keep slip speed and rotor circuit relation.IM：保留 slip speed 和转子回路关系。
Same observer logic, different machine-specific parameters.同一观察器思路，不同电机对应不同参数化。

Observer Structure / Prediction + Innovation

Every useful observer has the same skeleton几乎所有实用观察器都有同一个骨架

$$\dot{\hat{x}} = f(\hat{x},u) + L\bigl(y-h(\hat{x})\bigr)$$

The first term says: “if my model were correct, how should the state evolve?” The second says: “how much should I correct because the predicted output does not match the measured one?” Chapter 4 repeatedly uses this structure in different disguises. 第一项回答“如果模型完全正确，状态应该怎样演化？”；第二项回答“如果预测输出和测量输出不一致，我该纠正多少？”Chapter 4 里的各种观察器，本质上都在重复这个结构。

Warm-up example from sensor fusion先用传感器融合热身

$$\dot{\hat{\Theta}} = \Omega_m + k_0(\Theta_m - \hat{\Theta})$$

Prediction comes from the gyroscope. Innovation comes from accelerometer angle mismatch. This is the cleanest prototype of “prediction + innovation.” 预测项来自陀螺仪，创新项来自加速度计角度误差。这就是“预测 + 创新”的最直观原型。

Encoderless speed observer idea in Section 4.54.5 节中的无编码器速度观察器思路

$$\dot{\hat{\theta}}_d = \hat{\omega}_d + k_0 e_{d,ss}$$

The prediction term $\hat{\omega}_d$ is reconstructed from the q-axis voltage equation; the innovation term $e_{d,ss}$ is built from d-axis mismatch. In words: use one model relation to predict, and another model relation to correct. 预测项 $\hat{\omega}_d$ 来自 q 轴电压方程，创新项 $e_{d,ss}$ 来自 d 轴不匹配。也就是：用一条模型关系做预测，用另一条模型关系做校正。

A powerful way to read Chapter 4读 Chapter 4 的高效方法 Whenever you see a new observer, ask only two questions first: what is the prediction term? and what output error is used as the innovation? 每看到一种新观察器，先只问两个问题：预测项来自哪里？ 以及 创新项到底用的是什么输出误差？

Observer Families / Typical Designs

Three observer families worth remembering至少记住这三类典型观察器

Flux estimator磁链估计器

Start with the open-loop stator-flux integrator, then add innovation. Section 4.7 shows that open-loop integration is informative but fragile, so it must be closed by a correction mechanism. 从开环定子磁链积分器出发，再补上创新项。4.7 节说明：开环积分有信息，但很脆弱，所以一定要闭环纠正。

Sliding-mode EMF observer滑模扰动反电势观察器

Use current error as the sliding surface. Once the current estimation error is driven to the manifold, the switching function reconstructs disturbance EMF. It is robust, but implementation details matter. 把电流误差作为滑模面。一旦电流估计误差被压到滑模面上，切换项就能重建扰动反电势。它很鲁棒，但实现细节很关键。

Position-output speed/load observer位置输出型速度/负载观察器

Use measured angle as output, then reconstruct speed and disturbance by a Luenberger-style observer. This route connects directly to our simulation code and Coding Project 4. 用角度测量作为输出，再用 Luenberger 风格的观察器重建速度和扰动。这一路线与课程仿真代码和 Coding Project 4 直接对应。

The position-output observer viewpoint位置输出观察器的直观视角

$$\begin{aligned} \dot{\hat{\Theta}} &= \hat{\Omega} + k_0(\Theta - \hat{\Theta}) \\ \dot{\hat{\Omega}} &= T_{em} + 0 + k_1(\Theta - \hat{\Theta}) \end{aligned}$$

Even before estimating load torque explicitly, this structure already tells you something important: angle error contains information about speed error, and speed mismatch contains information about disturbance. 哪怕还没有显式引入负载转矩估计，这个结构已经说明了一件事：角度误差里含有速度误差信息，而速度不匹配里又含有扰动信息。

Observability / Low-Speed Difficulty

Why low-speed sensorless control is fundamentally hard为什么低速 sensorless control 在根子上就更难

One of the most important teaching points of Chapter 4 is that low-speed difficulty is not only an algorithm problem. It is often an information problem. When speed is low, back-EMF and speed-related excitation are weak, so the measured voltages and currents carry less usable information about angle and speed. Chapter 4 最重要的教学点之一是：低速困难 不只是算法问题，很多时候更是一个 信息量问题。速度低时，反电势和与速度有关的激励都变弱，电压电流里能反映角度和速度的信息自然就少了。

What becomes weak at low speed低速时变弱的东西

Back-EMF magnitude反电势幅值
Current and voltage mismatch sensitivity to angle error电流电压不匹配对角度误差的敏感度
Signal-to-noise ratio of speed-related information速度相关信息的信噪比

What this means for design这对设计意味着什么

Observer bandwidth cannot be increased without limit.观察器带宽不能无脑提高。
Noise, model error, and controller interaction become more visible.噪声、模型误差和控制器耦合会更明显。
Near zero speed, many methods need extra saliency or signal injection ideas.接近零速时，很多方法需要额外依赖凸极性或注入信号等机制。

Observer-controller coupling观察器与控制器耦合 The observer is inside the control loop, not beside it. Changing observer bandwidth changes estimation lag and noise; that then changes current command, torque ripple, and closed-loop behavior. So “more aggressive observer” does not automatically mean “better control.” 观察器不是摆在控制器旁边的装饰，它就在闭环里。改变观察器带宽会改变估计延迟和噪声，而这些又会反馈到电流指令、转矩波动和整体闭环表现上。因此“观察器更激进”并不自动等于“控制更好”。

Discussion / Think in Class

Questions students should be able to reason through学生应该能够独立思考的问题

Q1

If current is measurable but speed is not, can the speed loop still be closed? What must be added to the control architecture?如果电流可测但转速不可测，速度环还能闭起来吗？控制结构里必须多出什么？

Q2

Why is “differentiate encoder angle to get speed” often unacceptable at high precision or low speed?为什么“对编码器角度求导得到速度”在高精度或低速下常常不够好？

Q3

Why does open-loop flux integration drift even if the formula is mathematically correct?为什么开环磁链积分即使公式本身没错，也依然会漂移？

Q4

In an observer, which physical relation is used for prediction and which mismatch is used for innovation?在一个观察器里，哪条物理关系被拿来做预测，哪种不匹配被拿来做创新？

Q5

Why is low-speed sensorless difficulty partly an observability issue rather than only a tuning issue?为什么低速 sensorless 的困难有一部分是可观测性问题，而不只是调参问题？

Q6

Why can two observers with similar equations behave very differently once they are embedded in a closed-loop controller?为什么两个形式相近的观察器，一旦放进闭环控制器里，效果可能差很多？

Q7

What extra quantity are you really trying to estimate when a position-output observer starts to reconstruct load torque?当位置输出观察器进一步重建负载转矩时，本质上增加了什么估计量？

Q8

What experimental evidence would convince you that a speed estimate is “usable for control” rather than merely “looks close”?怎样的实验现象才能证明某个速度估计“不只是看起来接近”，而是真的“可用于控制”？

Read Section 4.5 and identify the prediction term and innovation term explicitly.读 4.5 节，明确指出预测项和创新项各是什么。

Read Section 4.7 and explain why open-loop integration is informative but unsafe.读 4.7 节，解释为什么开环积分“有信息但不安全”。

Carry these questions into Coding Project 4 and answer them with figures, not only words.把这些问题带到 Coding Project 4 中，用图而不只是用文字回答。