Overview / 总述
作业目标 / Objective
Lecture 4 的主线是：当控制器需要速度、角度、磁链、扰动这些传感器不能直接提供的信息时，必须依靠 observer 去重建缺失状态。后面两部分的主线则是：即便控制律在 $\alpha\beta$ 或 $dq$ 坐标系里已经写好，真正能施加到电机上的仍然是逆变器所允许的开关电压，因此控制器必须继续向下走一层，完成从电压参考到占空比的映射。
$$\dot{\hat{x}} = f(\hat{x},u) + L\bigl(y-h(\hat{x})\bigr), \qquad
u_{xG}=S_xV_{dc},\; x\in\{a,b,c\}$$
本次作业要求你把这两条线接起来： 先用 Lecture 4 的语言解释 sensorless observer 的预测项、创新项、扩展状态和低速可观测性；再用后两部分的语言说明三相三线逆变器、SPWM 的利用率限制，以及为什么零序注入会导向 SVPWM。
建议完成方式：
Part 1 不要只背公式，要明确“哪条物理关系做预测，哪种输出误差做校正”；Part 2 和 Part 3 不要只写调制结果，要明确“为什么三相三线系统里真正被约束的是线电压与 span，而不是某一个相对中性点的任意电压”。
Part 1 / 第一部分 (30 分)
Sensorless Observer / 无传感器观察器
1.1 预测 + 创新的观察器骨架 / The Prediction + Innovation Skeleton
$$\dot{\hat{x}} = f(\hat{x},u) + L\bigl(y-h(\hat{x})\bigr), \qquad
\dot{\hat{\theta}}_d = \hat{\omega}_d + k_0 e_{d,ss}$$
Task 1.1
解释为什么 sensorless control 不是“没有信息”，而是“可测信息不足以直接形成闭环”的问题。请至少列出 3 个通常可测信号，以及 3 个需要观察器重建的量。
结合通式 $\dot{\hat{x}} = f(\hat{x},u) + L(y-h(\hat{x}))$，解释预测项 $f(\hat{x},u)$ 与创新项 $L(y-h(\hat{x}))$ 的物理意义，并说明为什么二者缺一不可。
对于 Lecture 4 给出的无编码器速度观察器想法
$$\dot{\hat{\theta}}_d=\hat{\omega}_d+k_0e_{d,ss},$$
请明确指出：哪个量是预测项，哪个量是创新项；$\hat{\omega}_d$ 主要来自哪条模型关系，$e_{d,ss}$ 又来自哪种模型不匹配。
1.2 观察器扩展状态、可观测性与低速困难 / Extended States, Observability, and Low-Speed Difficulty
$$\begin{aligned}
\dot{\hat{\Theta}} &= \hat{\Omega} + k_0(\Theta-\hat{\Theta}) \\
\dot{\hat{\Omega}} &= T_{em} + k_1(\Theta-\hat{\Theta})
\end{aligned}$$
Task 1.2
Lecture 4 提到三类典型观察器：flux estimator、sliding-mode EMF observer、position-output speed/load observer。请分别说明：它们使用什么测量输出、主要重建什么隐藏量、以及它们各自最容易暴露的实现风险是什么。
从上面的二阶位置输出观察器出发，增加一个扰动状态 $\hat d$，使速度方程能够表示负载转矩或建模误差。写出你设计的三状态观察器，并说明如果扰动更接近“斜坡型”而不是“常值型”，为什么还需要再增加一个扩展状态。
解释为什么低速 sensorless control 的困难，本质上有一部分是“信息/可观测性问题”；并结合“直接对角度求导得到速度”与 observer bandwidth 的讨论，说明低速时为什么更容易暴露噪声、滞后与闭环耦合问题。
关键结论：
Part 1 不是“多学几种观测器公式”这么简单。你真正要掌握的是：什么信息来自模型预测，什么信息来自测量误差校正，以及为什么低速时可用于校正的信息会天然变少。
Part 2 / 第二部分 (30 分)
Part 2 — Inverter / 三相三线逆变器
2.1 从占空比到端电压 / From Duty Ratio to Terminal Voltage
$$u_{aG}=S_aV_{dc}, \qquad u_{bG}=S_bV_{dc}, \qquad u_{cG}=S_cV_{dc}, \qquad S_x\in[0,1]$$
Task 2.1
定义 $S_a,S_b,S_c$、$V_{dc}$、下标 $G$ 的物理意义，并解释教材中 “volt-second equivalence” 在这里到底意味着什么。
说明为什么在三相三线、星形连接且中性点开路的系统中，端电压 $u_{xG}$ 与相电压 $u_{xn}$ 不是同一个概念。这里的“中性点漂移 / common-mode” 是如何出现的？
从 $u_{xG}=S_xV_{dc}$ 出发，推导线电压 $u_{ab},u_{bc},u_{ca}$ 与占空比的关系，并证明 $u_{ab}+u_{bc}+u_{ca}=0$。
2.2 三相三线约束、控制桥接与带宽限制 / Constraints, Control Bridging, and Bandwidth Limits
$$u_{xn}=u_{xG}+u_{Gn}, \qquad
u = Ri + L\frac{di}{dt} + e, \qquad x\in\{a,b,c\}$$
Task 2.2
前面部分的控制律通常输出的是 $\alpha\beta$ 或 $dq$ 坐标系里的电压参考。请解释为什么 Part 2 必须继续往下走一层，建立从 $u^*_{\alpha\beta}$ 到 $S_a,S_b,S_c$ 的桥接关系；并证明若三相相电压命令同时加上同一个零序 / 中性轴电压分量，所有线电压保持不变。
解释为什么“三相三线 + 中性点开路”的约束意味着三相相电压并非三个独立自由度，而只能等效为两个自由度加一个公共偏置。
根据 5.1.6 的结论，结合
$$u = Ri + L\frac{di}{dt} + e$$
说明为什么最高电流带宽由 dc-bus voltage 决定；并进一步讨论当转速升高、反电势变大时，为什么更容易出现饱和、anti-windup 问题和动态性能恶化。
本部分的主线：
逆变器不是“把参考电压原样搬到电机端子”的理想执行器。它只提供由开关、母线电压和拓扑决定的有限电压集合，因此控制器真正面对的是一个 受限输入系统。
Part 3 / 第三部分 (40 分)
Part 3 — SVPWM / SPWM、零序注入与电压利用率
3.1 SPWM 的利用率上限 / The Utilization Limit of SPWM
$$\max(u_{ab})=\frac{\sqrt{3}}{2}V_{dc}=0.866\,V_{dc}, \qquad
\eta_{Vdc}=\frac{\max(u_{an},u_{bn},u_{cn})-\min(u_{an},u_{bn},u_{cn})}{2}$$
Task 3.1
解释为什么直接使用正弦参考的 SPWM，虽然实现自然，但并没有把三相三线逆变器的 dc bus capability 用满。
从
$$\max(u_{ab})=\frac{\sqrt{3}}{2}V_{dc}$$
出发，说明 SPWM 的电压利用率上限代表什么物理意义，并解释为什么它低于线电压所允许的理论上限 $V_{dc}$。
教材用 span
$$\eta_{Vdc}=\frac{\max(u_{an},u_{bn},u_{cn})-\min(u_{an},u_{bn},u_{cn})}{2}$$
来可视化直流母线利用率。请解释为什么“同一时刻三相相电压的最大值与最小值之间的距离”能够准确反映利用率问题。
3.2 中点、零序注入与 SVPWM / Midpoint, Zero-Sequence Injection, and SVPWM
$$u_{\text{mid}}=\frac{\max(u_{an},u_{bn},u_{cn})+\min(u_{an},u_{bn},u_{cn})}{2}, \qquad
u_{ZSM}=-u_{\text{mid}}$$
Task 3.2
用教材中 Fig. 27 / Fig. 28 的语言解释 “black dots” 与 “red dots” 分别表示什么，以及为什么把 red dots 压到 0 V 会为 span 腾出更多空间。随后推导中点公式
$$u_{\text{mid}}=\frac{\max(u_{an},u_{bn},u_{cn})+\min(u_{an},u_{bn},u_{cn})}{2}$$
以及相应的零序注入
$$u_{ZSM}=-u_{\text{mid}}.$$
说明这个注入项本质上是在做什么几何移动。
解释为什么零序注入会改变相对中性点的波形，却不会改变任何线电压波形。你的回答必须同时包含“公式证明”与“物理解释”两部分。
根据教材给出的结果，推导
$$\frac{V_{SVPWM}}{V_{SPWM}}=\frac{2}{\sqrt{3}}\approx1.1547,$$
并说明这意味着相对于 SPWM，SVPWM 的电压可用幅值提高了多少。接着写出一个从 $u^*_{\alpha\beta}$ 到 $S_a^*,S_b^*,S_c^*$ 的步骤链，至少包含：inverse Clarke、phase command、zero-sequence injection、dc-bus normalization、duty saturation 这 5 个环节。
本部分真正要理解的不是“再记一个调制公式”，而是：
SVPWM 的核心并不是神秘的空间矢量名词，而是一个非常直接的操作：利用零序自由度重新摆放三相相电压，使三相 span 在不改变线电压目标的前提下尽可能贴近 dc bus 极限。
English Version
Homework 3 in English
Lecture 4 asks one central question: when the controller needs speed, angle, flux, or disturbance information that sensors do not provide directly, how can that missing feedback information be reconstructed online? The later parts ask another: once the control law has already produced a voltage reference in $\alpha\beta$ or $dq$, how is that reference turned into something the three-phase inverter can actually apply?
$$\dot{\hat{x}} = f(\hat{x},u) + L\bigl(y-h(\hat{x})\bigr), \qquad
u_{xG}=S_xV_{dc},\; x\in\{a,b,c\}$$
This homework asks you to connect those two stories. Part 1 focuses on the observer viewpoint of Lecture 4: prediction, innovation, extended states, and low-speed observability. Part 2 focuses on the three-phase three-wire inverter. Part 3 focuses on why zero-sequence injection leads to SVPWM and better dc-bus utilization.
How to approach this homework:
In Part 1, always ask what physical relation provides the prediction term and what output mismatch provides the innovation term. In Parts 2 and 3, always ask what is truly constrained in a three-wire inverter: the available switching states, the line voltages, and the span of the three phase commands.
Part 1. Sensorless Observer
1.1 The Prediction + Innovation Skeleton
$$\dot{\hat{x}} = f(\hat{x},u) + L\bigl(y-h(\hat{x})\bigr), \qquad
\dot{\hat{\theta}}_d = \hat{\omega}_d + k_0 e_{d,ss}$$
Task 1.1
Explain why sensorless control is not a problem of “having no information,” but rather a problem of insufficient directly measured feedback information. List at least three signals that are usually measured and three quantities that must be reconstructed by an observer.
Using the generic observer form $\dot{\hat{x}} = f(\hat{x},u) + L(y-h(\hat{x}))$, explain the physical meaning of the prediction term and the innovation term, and explain why both are necessary.
For the encoderless observer idea
$$\dot{\hat{\theta}}_d=\hat{\omega}_d+k_0e_{d,ss},$$
identify explicitly which term is the prediction term and which is the innovation term. State what model relation mainly generates $\hat{\omega}_d$ and what mismatch generates $e_{d,ss}$.
1.2 Extended States, Observability, and Low-Speed Difficulty
$$\begin{aligned}
\dot{\hat{\Theta}} &= \hat{\Omega} + k_0(\Theta-\hat{\Theta}) \\
\dot{\hat{\Omega}} &= T_{em} + k_1(\Theta-\hat{\Theta})
\end{aligned}$$
Task 1.2
Lecture 4 highlights three typical observer families: flux estimator, sliding-mode EMF observer, and position-output speed/load observer. For each one, state what measured output it uses, what hidden quantity it reconstructs, and what practical implementation risk is most visible.
Starting from the second-order position-output observer above, add one disturbance state $\hat d$ so that the speed equation can represent load torque or modeling error. Write the resulting three-state observer, and explain why one more state such as $\hat p_d$ becomes useful when the disturbance is closer to a ramp than a constant.
Explain why the low-speed difficulty of sensorless control is partly an information/observability problem. Then connect this to the “differentiate angle to get speed” idea and to aggressive observer bandwidth: why do low-speed operation, noise, lag, and closed-loop coupling become more problematic together?
Main takeaway:
Part 1 is not mainly about memorizing observer formulas. It is about understanding what information comes from model prediction, what information comes from mismatch-based correction, and why low-speed operation naturally reduces the information available for correction.
Part 2. Inverter
2.1 From Duty Ratio to Terminal Voltage
$$u_{aG}=S_aV_{dc}, \qquad u_{bG}=S_bV_{dc}, \qquad u_{cG}=S_cV_{dc}, \qquad S_x\in[0,1]$$
Task 2.1
Define the physical meaning of $S_a,S_b,S_c$, $V_{dc}$, the subscript $G$, and the phrase “volt-second equivalence.”
Explain why terminal voltage $u_{xG}$ and phase voltage $u_{xn}$ are not the same quantity in a three-phase three-wire star-connected system with open neutral. How does neutral/common-mode motion appear?
Starting from $u_{xG}=S_xV_{dc}$, derive the line voltages $u_{ab},u_{bc},u_{ca}$ in terms of duty ratios, and show that $u_{ab}+u_{bc}+u_{ca}=0$.
2.2 Constraints, Control Bridging, and Bandwidth Limits
$$u_{xn}=u_{xG}+u_{Gn}, \qquad
u = Ri + L\frac{di}{dt} + e, \qquad x\in\{a,b,c\}$$
Task 2.2
Earlier parts usually produce a voltage reference in $\alpha\beta$ or $dq$. Explain why Part 2 must continue one layer lower and build a bridge from $u^*_{\alpha\beta}$ to switching variables $S_a,S_b,S_c$. Then prove algebraically that adding the same zero-sequence / common-mode voltage to all three phase commands does not change any line voltage.
Explain why the three-phase three-wire open-neutral constraint means that the three phase voltages do not form three independent degrees of freedom, but only two degrees of freedom plus one common offset.
Using the conclusion of Section 5.1.6 together with
$$u = Ri + L\frac{di}{dt} + e,$$
explain why maximum current-loop bandwidth is limited by dc-bus voltage. Then discuss why higher speed and larger back-EMF make saturation, anti-windup problems, and dynamic-performance degradation more likely.
Main takeaway:
The inverter is not an ideal actuator that simply copies your reference voltage. It only offers a limited switching-based voltage set determined by topology and dc-bus voltage, so the controller is fundamentally dealing with a constrained-input system.
Part 3. SVPWM
3.1 The Utilization Limit of SPWM
$$\max(u_{ab})=\frac{\sqrt{3}}{2}V_{dc}=0.866\,V_{dc}, \qquad
\eta_{Vdc}=\frac{\max(u_{an},u_{bn},u_{cn})-\min(u_{an},u_{bn},u_{cn})}{2}$$
Task 3.1
Explain why straightforward SPWM based on sinusoidal phase commands does not fully use the dc-bus capability of the three-phase three-wire inverter.
Starting from
$$\max(u_{ab})=\frac{\sqrt{3}}{2}V_{dc},$$
explain the physical meaning of the SPWM utilization limit and why it is below the theoretical line-voltage ceiling $V_{dc}$.
The textbook visualizes dc-bus utilization using the span
$$\eta_{Vdc}=\frac{\max(u_{an},u_{bn},u_{cn})-\min(u_{an},u_{bn},u_{cn})}{2}.$$
Explain why the distance between the largest and smallest phase commands at the same instant is the right quantity to examine.
3.2 Midpoint, Zero-Sequence Injection, and SVPWM
$$u_{\text{mid}}=\frac{\max(u_{an},u_{bn},u_{cn})+\min(u_{an},u_{bn},u_{cn})}{2}, \qquad
u_{ZSM}=-u_{\text{mid}}$$
Task 3.2
Using the language of Fig. 27 and Fig. 28, explain what the “black dots” and “red dots” represent, and why clamping the red dots to 0 V creates more room for the span. Then derive the midpoint expression
$$u_{\text{mid}}=\frac{\max(u_{an},u_{bn},u_{cn})+\min(u_{an},u_{bn},u_{cn})}{2}$$
and the injected zero-sequence voltage
$$u_{ZSM}=-u_{\text{mid}}.$$
Explain what geometric shift is being applied to the three phase commands.
Explain why zero-sequence injection changes phase-to-neutral waveforms but does not change any line-voltage waveform. Your answer must contain both an algebraic proof and a physical explanation.
Derive
$$\frac{V_{SVPWM}}{V_{SPWM}}=\frac{2}{\sqrt{3}}\approx1.1547,$$
and explain how much usable voltage amplitude is gained by SVPWM relative to SPWM. Then write a step-by-step signal path from $u^*_{\alpha\beta}$ to $S_a^*,S_b^*,S_c^*$, including inverse Clarke, phase commands, zero-sequence injection, dc-bus normalization, and duty saturation.
Main takeaway:
The core of SVPWM here is not the label “space vector” by itself. The real idea is very concrete: use the zero-sequence degree of freedom to reposition the three phase commands so that their span gets as close as possible to the dc-bus limits without changing the desired line voltage.