Can a higher-order observer predict time-varying disturbances? 高阶观测器能否预测时变扰动？

From constant disturbance to time-varying disturbance 从恒定扰动到时变扰动

The mechanical equation机械方程

Define the total disturbance $d_{to}$ as everything the controller does not model: 定义总扰动 $d_{to}$ 为控制器未建模的所有内容：

An observer that estimates $d_{to}$ can feed it forward to cancel the disturbance before the PI regulator reacts. The question is: how well can the observer track $d_{to}$ when it changes over time? 估计 $d_{to}$ 的观测器可以将其前馈，在 PI 调节器反应之前就抵消扰动。问题是：当 $d_{to}$ 随时间变化时，观测器能跟踪得多好？

3rd-order observer: assumes constant disturbance三阶观测器：假设扰动恒定

States: $[\hat{\theta}_d,\ \hat{\omega}_r,\ \hat{d}_{to}]$. The internal model assumes $\dot{d}_{to} = 0$.状态：$[\hat{\theta}_d,\ \hat{\omega}_r,\ \hat{d}_{to}]$。内部模型假设 $\dot{d}_{to} = 0$。

Gains placed at $(s + \omega_{ob})^3$:增益配置在 $(s + \omega_{ob})^3$：

4th-order observer: models ramp disturbance四阶观测器：建模斜坡扰动

States: $[\hat{\theta}_d,\ \hat{\omega}_r,\ \hat{d}_{to},\ \hat{p}_{to}]$. The internal model assumes $\dot{d}_{to} = p_{to}$, $\dot{p}_{to} = 0$.状态：$[\hat{\theta}_d,\ \hat{\omega}_r,\ \hat{d}_{to},\ \hat{p}_{to}]$。内部模型假设 $\dot{d}_{to} = p_{to}$，$\dot{p}_{to} = 0$。

Gains placed at $(s + \omega_{ob})^4$:增益配置在 $(s + \omega_{ob})^4$：

Where to find and modify the observer观测器代码位置与修改点

1. Observer gains initialization1. 观测器增益初始化

In The_Motor_Controller.__init__ (around line 165), the observer gains are selected by an if/elif/else chain. Currently the 3rd-order branch is active:在 The_Motor_Controller.__init__（约第 165 行），观测器增益通过 if/elif/else 链选择。当前三阶分支生效：

omega_ob = 100  # [rad/s]
if False:    # 2nd-order speed observer
    ...
elif False:  # 2nd-order position observer
    ...
elif True:   # 3rd-order  <--- CURRENTLY ACTIVE
    self.ell1 = 3 * omega_ob
    self.ell2 = 3 * omega_ob**2
    self.ell3 = omega_ob**3 * init_Js/init_npp
else:        # 4th-order  <--- YOU NEED TO ACTIVATE THIS
    self.ell1 = 4 * omega_ob
    self.ell2 = 6 * omega_ob**2
    self.ell3 = 4 * omega_ob**3 * init_Js/init_npp
    self.ell4 = omega_ob**4

2. Observer dynamics (already supports 4th-order)2. 观测器动态方程（已支持四阶）

def DYNAMICS_SpeedObserver(x, CTRL):
    fx = np.zeros(6)
    output_error = angle_diff(CTRL.theta_d, x[0])
    fx[0] = CTRL.ell1*output_error + x[1]           # theta
    fx[1] = CTRL.ell2*output_error + (CTRL.Tem + x[2]) * CTRL.npp/CTRL.Js  # omega
    fx[2] = CTRL.ell3*output_error + x[3]           # d_to
    fx[3] = CTRL.ell4*output_error + 0.0             # p_to (rate)
    return fx

When ell4 = 0 (3rd-order), x[3] stays zero. When ell4 > 0 (4th-order), x[3] estimates the rate of disturbance change.当 ell4 = 0（三阶）时，x[3] 保持为零。当 ell4 > 0（四阶）时，x[3] 估计扰动变化率。

3. The missing link: feedforward is not wired in!3. 缺失的环节：前馈未接入！

In FOC() (line 638), the speed controller output goes directly to cmd_idq[1]:在 FOC()（第 638 行），速度控制器输出直接赋给 cmd_idq[1]：

# Current code (line 638-639):
if CTRL.bool_apply_speed_closed_loop_control == True:
    CTRL.cmd_idq[1] = reg_speed.Out    # feedforward NOT added!

The observer computes total_disrubance_feedforward (line 757-761) but it is never used. You must wire it in.观测器计算了 total_disrubance_feedforward（第 757-761 行），但从未使用。你需要将其接入。

4. Watch data for plotting4. 画图用的观测数据

sim.gdd['CTRL.xS[2]']       # estimated disturbance (d_to)
sim.gdd['CTRL.xS[3]']       # estimated rate (p_to)
sim.gdd['ACM.TLoad']         # actual load torque
sim.gdd['CTRL.omega_r_mech'] # measured speed [rpm]
sim.gdd['CTRL.cmd_rpm']      # speed command [rpm]

Switch from 3rd-order to 4th-order observer 从三阶切换到四阶观测器

You need a way to select observer order from the d configuration dictionary, without hard-coding if True/if False in source code. 你需要一种方法，从 d 配置字典选择观测器阶数，而不是在源代码中硬编码 if True/if False。

Connect the observer output to the speed controller 将观测器输出连接到速度控制器

The observer estimates disturbance $\hat{d}_{to}$ in torque-like units. To use it as a feedforward, convert to $i_q$ current: 观测器估计的扰动 $\hat{d}_{to}$ 单位类似转矩。要用作前馈，需转换为 $i_q$ 电流：

# BEFORE:
CTRL.cmd_idq[1] = reg_speed.Out

# AFTER:
CTRL.cmd_idq[1] = reg_speed.Out
if CTRL.use_disturbance_feedforward_rejection > 0:
    if CTRL.KA > 0:
        CTRL.cmd_idq[1] += CTRL.total_disrubance_feedforward / (1.5 * CTRL.npp * CTRL.KA)

Prove the 4th-order observer can track ramp disturbance 证明四阶观测器能跟踪斜坡扰动

Ramp load torque setup斜坡负载转矩设置

# In user_system_input_code:
"CTRL.cmd_rpm = 50\n"
"if ii >= 2: ACM.TLoad = 0.05 * (ii - 2) * d['TIME_SLICE']"

This creates a linearly increasing load starting at time slice 2. Adjust the slope (0.05) so the motor doesn't stall.这将从时间片 2 开始创建线性增加的负载。调整斜率（0.05）使电机不会堵转。

What to plot for each scenario每个场景需要画什么

• Subplot 1: Speed command CTRL.cmd_rpm vs actual speed CTRL.omega_r_mech [rpm]子图 1：速度给定 CTRL.cmd_rpm vs 实际速度 CTRL.omega_r_mech [rpm]
• Subplot 2: Actual load ACM.TLoad vs estimated disturbance CTRL.xS[2] (note: opposite sign)子图 2：实际负载 ACM.TLoad vs 估计扰动 CTRL.xS[2]（注意：符号相反）
• Subplot 3: Estimated rate CTRL.xS[3] — should converge to the ramp slope for 4th-order子图 3：估计变化率 CTRL.xS[3]——四阶时应收敛到斜坡斜率

Tracking, disturbance rejection, and noise: the three-way tradeoff 跟踪、抗扰与噪声：三方权衡

A closed-loop system's output can be decomposed into three paths: 闭环系统的输出可以分解为三条通道：

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