How to use pwm with lqr controller?

Question

Mikail 2025-3-25

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2175634-how-to-use-pwm-with-lqr-controller

评论： Sam Chak 2025-4-28

ode45_1.m

I want see Fx,Fy and Fz in discrete form using pwm. But ı have an lqr controller and Fx, Fy and Fz depending on u . Please help me with it.

clear all;

clc;

close all;

% **Sistem Parametreleri**

m_val = 25.0;

Ix_val = 0.25;

Iy_val = 5.28;

Iz_val = 5.28;

theta_val=0;

phi_val=0;

psi_val=0;

g_val = 9.81;

rho_val = 1.225;

Sx_val = 0.00;% x eksenindeki tahmini alan

Sy_val = 0.0; % y eksenindeki tahmini alan

Sz_val = 0.0; % z eksenindeki tahmini alan

Cx_val = 0.0; % Aerodinamik katsayı

Cy_val = 0.0; % Aerodinamik katsayı

Cz_val = 0; % Aerodinamik katsayı

Cl_val = 0.0; Cm_val = 0.0; Cn_val = 0.0;

d=0.75;

kp=70.779169670332;

ki=40.999999999919;

kd=40.3888999977265;

c=29.33685249238;

wn_close = 400/2.5;

wn = 400/2.5;

dr = 1;

% **Sembolik Değişkenler**

syms u v w p q r phi theta psi Fx Fy Fz L M N t1 t2 t3 p_x p_y p_z real

syms m Ix Iy Iz g real

syms rho Sx Sy Sz Cx Cy Cz Cl Cn Cm real

% **Durum Değişkenleri ve Girdi Vektörü**

x = [u; v; w; p; q; r; phi; theta; psi; p_x; p_y; p_z];

u_vec = [Fx; Fy; Fz; L; M; N];

% **Dinamik Denklem**

f = [

-g*cos(theta)*cos(psi)+Fx/m-0.5*rho*Sx*Cx*u^2/m-q*w+r*v;

-g*(sin(phi)*sin(theta)*cos(psi)-cos(phi)*sin(psi))+(Fy/m)-0.5*rho*Sy*Cy*v^2/m-r*u+ p*w;

-g*(cos(phi)*sin(theta)*cos(psi)+sin(phi)*sin(psi))+(Fz/m)-0.5*rho*Sz*Cz*w^2/m-p*v + q*u;

(0.5*rho*Sx*d*Cl*L^2-L)/Ix;

(0.5*rho*Sy*d*Cm*q^2-M)/Iy;

(0.5*rho*Sz*d*Cn*r^2-N)/Iz;

p + (q*sin(phi) + r*cos(phi)) * tan(theta);

q*cos(phi) - r*sin(phi);

(q*sin(phi) + r*cos(phi)) / cos(theta);

u*cos(theta)*cos(psi)+u*cos(theta)*sin(psi)-u*sin(theta);

v*(cos(phi)*cos(psi)-cos(phi)*sin(psi)-sin(psi));

w*(cos(phi)*cos(theta)+cos(phi)*sin(theta)+sin(phi))

];

% **Jacobian Matrisleri**

A_matrix = jacobian(f, x);

B_matrix = jacobian(f, u_vec);

% **Denge Noktaları**

x_equilibrium = zeros(12,1);

u_equilibrium = zeros(6,1);

% **A ve B Matrislerinin Değeri**

A_eq = subs(A_matrix, x, x_equilibrium);

A_eq = subs(A_eq, u_vec, u_equilibrium);

A_eq = subs(A_eq, [m, Ix, Iy, Iz, g, phi, theta, psi], ...

[m_val, Ix_val, Iy_val, Iz_val, g_val, phi_val, theta_val, psi_val]);

A = double(A_eq);

B_eq = subs(B_matrix, [x; u_vec], [x_equilibrium; u_equilibrium]);

B_eq = subs(B_eq, [m, Ix, Iy, Iz, g, phi, theta, psi], ...

[m_val, Ix_val, Iy_val, Iz_val, g_val, phi_val, theta_val, psi_val]);

B = double(B_eq);

% **Kontrol Edilebilirlik**

controllability = ctrb(A, B);

rank_controllability = rank(controllability);

% **LQR Kazanç Matrisi**

Q = diag([280,14,14,1,6,6,1,1,1,50,115,115]);

R = diag([0.052,0.06,0.06,2,2,2]);

K = lqr(A, B, Q, R);

% **Başlangıç Koşulları ve Referans**

xref = [-0.5; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0]; % Hedef durum

x_init = [0; 0; 0; 0; 0.7; 0.5; 0; 0; 0; 9; 0; 0]; % Başlangıç durumu

% **ODE Çözümü**

tspan = 0:0.01:20;

u_history = zeros(6, length(tspan));

% ODE Çözümü ve Kontrol Girdilerini Kaydetme

[t, x] = ode45(@(t, x) ode_with_control(t, x, A, B, K, xref), tspan, x_init);

for i = 1:length(t)

[~, u_history(:, i)] = ode_with_control(t(i), x(i, :)', A, B, K, xref);

end

% **Sonuçları Çizdirme**

figure;

subplot(5,1,1);

plot(t, x(:,1:3), 'LineWidth', 1.5);

legend('x_1', 'x_2', 'x_3');

xlabel('Zaman (s)');

ylabel('Değerler');

title('Durum Değişkenleri 1-3');

grid on;

subplot(5,1,2);

plot(t, x(:,4:6), 'LineWidth', 1.5);

legend('x_4', 'x_5', 'x_6');

xlabel('Zaman (s)');

ylabel('Değerler');

title('Durum Değişkenleri 4-6');

grid on;

subplot(5,1,3);

plot(t, x(:,7:9), 'LineWidth', 1.5);

legend('x_7', 'x_8', 'x_9');

xlabel('Zaman (s)');

ylabel('Değerler');

title('Durum Değişkenleri 7-9');

grid on;

subplot(5,1,4);

plot(t, x(:,10:12), 'LineWidth', 1.5);

legend('x_{10}', 'x_{11}', 'x_{12}');

xlabel('Zaman (s)');

ylabel('Değerler');

title('Durum Değişkenleri 10-12');

grid on;

subplot(5,1,5);

plot(t, u_history(1,:), 'r', 'LineWidth', 1.5);

hold on;

plot(t, u_history(2,:), 'g', 'LineWidth', 1.5);

plot(t, u_history(3,:), 'b', 'LineWidth', 1.5);

legend('F_x', 'F_y', 'F_z');

xlabel('Zaman (s)');

ylabel('Kuvvet (N)');

title('Uygulanan Kuvvetler');

grid on;

hold off;

●

% **ODE Fonksiyonu**

function [dx, u] = ode_with_control(t, x, A, B, K, xref)

u = -K * (x); % LQR ile kontrol

dx = A * x + B * u; % Sistem dinamikleri

end

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Mathieu NOE 2025-3-27

hello

just to be sure , you mean pwm = pulse width modulation ? like in doc : PWM Generator ?

you need to create a carrier (triangular waveform) at a much higher frequency then your signal frequency content , and make a logical comparison between the signal and the carrier (as explained in the link)

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Answer 1

Mathieu NOE 2025-3-31

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2175634-how-to-use-pwm-with-lqr-controller#answer_1562855

在 MATLAB Online 中打开

hello again

I am not sure what it brings here to convert your linear outputs to pwm , but a simple demo below if you want to try it and adapt it to your code :

%% demo pwm output

tf = 0.5; % final time

dt = 1e-2;% signal sampling rate

t = (0:dt:tf);

freq = 2; % signal frequency

signal = 0.95*sin(2*pi*freq*t); % !! signal amplitude must be normalized to +/- 1 range !!

% pwm signal

factor = 100; % upsample factor (between incoming signal sampling rate and PWM carrier sampling rate

dtpwm = dt/factor; % much faster sampling rate

tpwm = (0:dtpwm:tf);

% carrier signal = triangular waveform

signal_carrier = sawtooth(2*pi*freq*factor*tpwm, 0.5); % 0.5 specifies a triangular waveform

%% compare signal to carrier

% first you have to resample the signal to the carrier sampling freq

signal2 = interp1(t,signal,tpwm);

% apply logical conditions to out_pwm

ind_pos = (signal2>signal_carrier);

ind_neg = (signal2<signal_carrier);

out_pwm = zeros(size(signal_carrier)); % initialize out_pwm to zero

out_pwm(ind_pos) = +1;

out_pwm(ind_neg) = -1;

% low pass filter the PWM output => should reproduce input waveform

[b,a] = butter(2,freq*dt/2);

out_pwm_filtered = filtfilt(b,a,out_pwm);

% plot

subplot(2,1,1),plot(t,signal,tpwm,signal_carrier);

legend('signal','pwm carrier')

xlabel('time(s)');

ylim([-1.2 1.2])

subplot(2,1,2),plot(tpwm,out_pwm,tpwm,out_pwm_filtered);

ylim([-1.2 1.2])

legend('pwm signal','lowpass filtered pwm signal')

xlabel('time(s)');

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Mikail 2025-4-16

Thanks for your reply . I consider again and decide that I won't use lqr controller and pwm because of active control limitation so ı didn't use your code but it will be very helpful on my future project.

Sam Chak 2025-4-28

@Mikail, After making a decision, have you considered the alternatives?

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How to use pwm with lqr controller?

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How to use pwm with lqr controller?

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