icare function doesn't return the value for X_lqr, K_lqr.

I using icare function for the lqr controller.

Alift is 9x9 matrix, Blift is 9x1 matrix, Q is 9x9 identity matrix, and R is 1.

Alift, Blift matrix is work well.

But it only return the L_lqr(9x1) matrix.

What is the problem?

Below is my code.

x_ref = [5 5]';
x_ref_lift = liftFtn(x_ref);

Unrecognized function or variable 'liftFtn'.

[X_lqr, K_lqr, L_lqr] = icare(Alift, Blift, Q, R);
for m = 1 : Nsim - 1
    u = -K_lqr * (xlift2(:, end) - x_ref_lift);
    xlift2 = [xlift2, Alift * xlift2(:, end) + Blift * u]; % Lifted dynamics
end

[규연]

I don't know why, but L_lqr(9x1) matrix's components are all complex number.

Answer 1

[Sam Chak]

OP: I don't know why, but L_lqr(9x1) matrix's components are all complex number.

The description in icare() documentation says that the syntax "[X,K,L] = icare(A,B,Q,R,S,E,G)" computes the unique stabilizing solution X, state-feedback gain K, and the closed-loop eigenvalues L of the following continuous-time algebraic Riccati equation.

[Sam Chak]

%% Double Integrator system, x" = u
A    = [0 1;
        0 0];
B    = [0;
        1];

%% Settings to force the Controller for a critically-damped response, u = – 2·x' – x
Q    = diag([1 2]);
R    = 1;

%% Tuning by LQR method
[K, X, L] = lqr(A, B, Q, R)

K = 1×2

1.0000 2.0000

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

X = 2×2

2.0000 1.0000 1.0000 2.0000

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

L = 2×1

-1.0000 -1.0000

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

%% Tuning by solving Continuous-time Algebraic Riccati Equations
[X, K, L] = icare(A, B, Q, R)

X = 2×2

2.0000 1.0000 1.0000 2.0000

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

K = 1×2

1.0000 2.0000

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

L = 2×1

-1.0000 -1.0000

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

[규연]

i think the system has the problem.

i'll try lqr controller for another system.

thank you!