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无速度传感器交流电机的扩展Luenberger观测器

news 2025/9/16 11:25:40

扩展Luenberger观测器是一种用于无速度传感器交流电机控制的重要技术，它能够估计电机的内部状态（如转子磁链）和转速。

理论背景

对于感应电机，在静止α-β坐标系下的模型可以表示为：

状态方程：
dx/dt = A(ω)x + Bu
y = Cx其中：
x = [isα, isβ, ψrα, ψrβ]^T
u = [usα, usβ]^T
y = [isα, isβ]^T

扩展Luenberger观测器通过构建一个状态观测器来估计这些状态变量，包括转速ω。

MATLAB实现

%% 感应电机扩展Luenberger观测器仿真
clc; clear; close all;%% 电机参数设置
Rs = 1.115;     % 定子电阻 (Ω)
Rr = 1.083;     % 转子电阻 (Ω)
Ls = 0.145;     % 定子自感 (H)
Lr = 0.145;     % 转子自感 (H)
Lm = 0.139;     % 互感 (H)
J = 0.02;       % 转动惯量 (kg·m²)
P = 4;          % 极对数
sigma = 1 - Lm^2/(Ls*Lr); % 漏感系数%% 观测器参数
Tr = Lr/Rr;     % 转子时间常数
gamma = (Ls*Lr - Lm^2)/(Lr); % 计算常数% 观测器增益矩阵设计
lambda = 100;   % 观测器极点
K1 = lambda;
K2 = lambda^2;
K3 = lambda^3;%% 仿真参数
dt = 1e-5;      % 采样时间 (s)
T = 0.5;        % 总仿真时间 (s)
N = T/dt;       % 总步数%% 初始状态
% 真实状态: [isα, isβ, ψrα, ψrβ, ωr]
x_true = [0; 0; 0; 0; 0];
% 估计状态: [isα, isβ, ψrα, ψrβ, ωr]
x_est = [0; 0; 0.1; 0.1; 0.1];
% 估计误差协方差
P = eye(5);%% 参考信号和负载转矩
f_ref = 50;     % 参考频率 (Hz)
omega_ref = 2*pi*f_ref; % 参考电角速度 (rad/s)
T_load = 5;     % 负载转矩 (N·m)%% 控制参数 (简单的V/f控制)
V_base = 220;   % 基波电压 (V)
f_base = 50;    % 基波频率 (Hz)%% 数据存储
time = zeros(1, N);
speed_true = zeros(1, N);
speed_est = zeros(1, N);
current_true = zeros(2, N);
current_est = zeros(2, N);
flux_true = zeros(2, N);
flux_est = zeros(2, N);%% 主仿真循环
for k = 1:Nt = k * dt;time(k) = t;% 生成参考电压信号 (V/f控制)if t < 0.1freq = 0.5 * f_base * (t/0.1);amp = 0.5 * V_base * (t/0.1);elsefreq = f_base;amp = V_base;endus_alpha = amp * cos(2*pi*freq*t);us_beta = amp * sin(2*pi*freq*t);u = [us_alpha; us_beta];% 真实系统动态x_true = induction_motor_dynamics(x_true, u, Rs, Rr, Ls, Lr, Lm, J, P, T_load, dt);% 测量输出 (带噪声)y_true = [x_true(1); x_true(2)] + 0.1 * randn(2, 1);% 扩展Luenberger观测器[x_est, omega_est] = extended_luenberger_observer(x_est, u, y_true, Rs, Rr, Ls, Lr, Lm, K1, K2, K3, dt);% 存储数据speed_true(k) = x_true(5) * 60 / (2 * pi * P); % 转换为RPMspeed_est(k) = omega_est * 60 / (2 * pi * P); % 转换为RPMcurrent_true(:, k) = x_true(1:2);current_est(:, k) = x_est(1:2);flux_true(:, k) = x_true(3:4);flux_est(:, k) = x_est(3:4);
end%% 绘制结果
figure('Position', [100, 100, 1200, 800]);% 速度估计
subplot(2, 2, 1);
plot(time, speed_true, 'LineWidth', 2); hold on;
plot(time, speed_est, '--', 'LineWidth', 2);
xlabel('时间 (s)');
ylabel('转速 (RPM)');
title('转速估计');
legend('真实转速', '估计转速');
grid on;% 电流估计 (α轴)
subplot(2, 2, 2);
plot(time, current_true(1, :), 'LineWidth', 2); hold on;
plot(time, current_est(1, :), '--', 'LineWidth', 2);
xlabel('时间 (s)');
ylabel('电流 (A)');
title('α轴电流估计');
legend('真实电流', '估计电流');
grid on;% 磁链估计 (α轴)
subplot(2, 2, 3);
plot(time, flux_true(1, :), 'LineWidth', 2); hold on;
plot(time, flux_est(1, :), '--', 'LineWidth', 2);
xlabel('时间 (s)');
ylabel('磁链 (Wb)');
title('α轴转子磁链估计');
legend('真实磁链', '估计磁链');
grid on;% 估计误差
speed_error = speed_est - speed_true;
subplot(2, 2, 4);
plot(time, speed_error, 'LineWidth', 2);
xlabel('时间 (s)');
ylabel('转速误差 (RPM)');
title('转速估计误差');
grid on;% 计算性能指标
rmse_speed = sqrt(mean(speed_error.^2));
fprintf('转速估计RMSE: %.2f RPM\n', rmse_speed);
fprintf('最大转速误差: %.2f RPM\n', max(abs(speed_error)));%% 感应电机动力学模型
function x_next = induction_motor_dynamics(x, u, Rs, Rr, Ls, Lr, Lm, J, P, T_load, dt)% 状态: x = [isα, isβ, ψrα, ψrβ, ωr]% 输入: u = [usα, usβ]% 输出: y = [isα, isβ]is_alpha = x(1);is_beta = x(2);psi_r_alpha = x(3);psi_r_beta = x(4);omega_r = x(5);us_alpha = u(1);us_beta = u(2);% 计算常数sigma = 1 - Lm^2/(Ls*Lr);Tr = Lr/Rr;K = Lm/(sigma * Ls * Lr);% 定子电流微分dis_alpha_dt = (-(Rs/(sigma*Ls) + (1-sigma)/(sigma*Tr))*is_alpha ...+ K*(1/Tr * psi_r_alpha + omega_r * psi_r_beta) ...+ 1/(sigma*Ls) * us_alpha);dis_beta_dt = (-(Rs/(sigma*Ls) + (1-sigma)/(sigma*Tr))*is_beta ...+ K*(1/Tr * psi_r_beta - omega_r * psi_r_alpha) ...+ 1/(sigma*Ls) * us_beta);% 转子磁链微分dpsi_r_alpha_dt = (Lm/Tr * is_alpha - 1/Tr * psi_r_alpha - omega_r * psi_r_beta);dpsi_r_beta_dt = (Lm/Tr * is_beta - 1/Tr * psi_r_beta + omega_r * psi_r_alpha);% 电磁转矩Te = (3/2) * P * (Lm/Lr) * (psi_r_alpha * is_beta - psi_r_beta * is_alpha);% 机械运动方程domega_r_dt = (P/J) * (Te - T_load);% 欧拉积分x_dot = [dis_alpha_dt; dis_beta_dt; dpsi_r_alpha_dt; dpsi_r_beta_dt; domega_r_dt];x_next = x + x_dot * dt;
end%% 扩展Luenberger观测器
function [x_est_next, omega_est] = extended_luenberger_observer(x_est, u, y, Rs, Rr, Ls, Lr, Lm, K1, K2, K3, dt)% 状态估计: x_est = [isα, isβ, ψrα, ψrβ, ωr]% 输入: u = [usα, usβ]% 测量: y = [isα, isβ]is_alpha_est = x_est(1);is_beta_est = x_est(2);psi_r_alpha_est = x_est(3);psi_r_beta_est = x_est(4);omega_est = x_est(5);us_alpha = u(1);us_beta = u(2);% 计算常数sigma = 1 - Lm^2/(Ls*Lr);Tr = Lr/Rr;K = Lm/(sigma * Ls * Lr);% 观测器模型dis_alpha_est_dt = (-(Rs/(sigma*Ls) + (1-sigma)/(sigma*Tr))*is_alpha_est ...+ K*(1/Tr * psi_r_alpha_est + omega_est * psi_r_beta_est) ...+ 1/(sigma*Ls) * us_alpha ...+ K1 * (y(1) - is_alpha_est));dis_beta_est_dt = (-(Rs/(sigma*Ls) + (1-sigma)/(sigma*Tr))*is_beta_est ...+ K*(1/Tr * psi_r_beta_est - omega_est * psi_r_alpha_est) ...+ 1/(sigma*Ls) * us_beta ...+ K1 * (y(2) - is_beta_est));dpsi_r_alpha_est_dt = (Lm/Tr * is_alpha_est - 1/Tr * psi_r_alpha_est - omega_est * psi_r_beta_est) ...+ K2 * (y(1) - is_alpha_est);dpsi_r_beta_est_dt = (Lm/Tr * is_beta_est - 1/Tr * psi_r_beta_est + omega_est * psi_r_alpha_est) ...+ K2 * (y(2) - is_beta_est);% 转速自适应律epsilon = (psi_r_alpha_est * (y(2) - is_beta_est) - psi_r_beta_est * (y(1) - is_alpha_est));domega_est_dt = K3 * epsilon;% 欧拉积分x_dot_est = [dis_alpha_est_dt; dis_beta_est_dt; dpsi_r_alpha_est_dt; dpsi_r_beta_est_dt; domega_est_dt];x_est_next = x_est + x_dot_est * dt;
end