MATLAB code error, LQR

Hello: for the follwoing code:

% Part 1: Define the system parameters and initial conditions

% Define the system parameters

eta = ones(4,1); N = 2*ones(4,1); A = ones(4,1); Z_m = 10*ones(4,1); m = ones(4,1); H = 0.5*ones(4,1);

% Define the torque disturbance

T_d = @(t) [0.05*sin(t); 0.05*cos(t); 0.05*sin(1.3*t); 0.1*sin(t)];

% Define the communication topology

L = [1 0 0 -1;

-1 1 0 0;

0 -1 1 0;

0 0 -1 1];

% Define the initial conditions

x0 = [0; 1; 0.5;

1; 1; 0;

2; 0.1; 1;

3; 0; 1];

% Part 2: Define the LQR controller and simulate the system

% Define the state-space matrices for the system

A_sys = zeros(12,12);

B_sys = zeros(12,4);

C_sys = eye(4,12);

D_sys = zeros(4,4);

for i = 1:4

% Define the state matrices for the ith link

A_i = [0 1 0; -(N(i)/eta(i))*sin(x0(i,1)) -(A(i)/eta(i)) 0; 0 0 0];

B_i = [0; 1/eta(i); 1/(m(i)*eta(i))];

% Update the system matrices with the ith link matrices

A_sys((3*i-2):(3*i),(3*i-2):(3*i)) = A_i;

B_sys((3*i-2):(3*i),i) = B_i;

end

% Define the LQR controller

Q = eye(12);

R = 0.1*eye(4);

[K,~,~] = lqr(A_sys,B_sys,Q,R);

% Define the reference signal

yd = 2;

% Define the simulation time span and time step

tspan = [0 15];

dt = 0.01;

% Define the simulation function

f = @(t,x) multi_agent_system(t,x,K,L,N,eta,A,Z_m,m,H,T_d,yd);

% Simulate the system using theode45 solver and plot the results

% Simulate the system using ode45

[t, x] = ode45(f, tspan, x0);

% Plot the results

figure;

for i = 1:4

subplot(4,1,i)

plot(t, x(:,(3i-2)), 'r-', 'LineWidth', 2); hold on;

plot(t, ydones(size(t)), 'b--', 'LineWidth', 2); hold off;

xlabel('Time (s)');

ylabel(['Position of Link ' num2str(i)]);

legend(['Link ' num2str(i) ' Position'], 'Reference Signal');

end

% Part 3: Define the system dynamics function**

function dxdt = multi_agent_system(t,x,K,L,N,eta,A,Z_m,m,H,T_d,yd)

% Define the state variables

w = x(:,1:3:end);

dotw = x(:,2:3:end);

T = x(:,3:3:end);

% Define the control input

u = -K*x';

% Define the time derivatives of the state variables

d_w = dotw;

d_dotw = -((N./eta).*sin(w) + (A./eta).*dotw - T_d(t)./eta) + u;

d_T = (1./(m.*eta)).*u - (Z_m./(m.*eta)).*dotw -(H./m).*T;

% Define the consensus term

consensus_term = L*x;

% Define the time derivatives of the state variables for each agent

dxdt = zeros(size(x));

for i = 1:4

dxdt(:,(3i-2):(3i)) = [d_w(:,i), d_dotw(:,i), d_T(:,i)] + consensus_term(:,(3i-2):(3i));

end

% Define the reference signal tracking error

error = x(:,1:3:end) - yd;

% Add the reference signal tracking error to the first agent's torque

dxdt(:,3) = dxdt(:,3) - error./eta;

end

it give me the following error:

Error using lqr (line 42)

Cannot compute the stabilizing Riccati solution S for the LQR design.

This could be because:

* R is singular,

* [Q N;N' R] needs to be positive definite,

* The E matrix in the state equation is singular.

Error in New_Model7 (line 44)

[K,~,~] = lqr(A_sys,B_sys,Q,R);

if anyone can help me?

Answer 1

[Nathan Hardenberg]

I do not know the exact working behind the "Riccati Solution", but I have had this error before. For me it helped to adjust Q and/or R values slightly.

In your example you could for example try this:

R = 0.1*eye(4);
R(1,1) = 0.09; R(2,2) = 0.99; R(3,3) = 0.101; R(4,4) = 0.102

R = 4×4

0.0900 0 0 0 0 0.9900 0 0 0 0 0.1010 0 0 0 0 0.1020

This works for you example. If you have more Info abour you system you can also adjust Q and R a bit more informed. But for a first shot this should be good enough

Also: There is another error in your code afterwards, but this does not seem to have anything to do with the lqr method