Discret time LQR R matrix must be symmetric positive definite with as many columns as B error

Hello!

I'm trying to make a discret time control with LQR and I have this error:

Error using dlqr (line 31)

In the "lqr(A,B,Q,R,...)" command, the

R matrix must be symmetric positive

definite with as many columns as B.

Error in LQR_model2 (line 82)

K_d = dlqr(Ad,Bd,Q,R,N);

Matlab Code:

clc 
clear all
close all
%% parametrii modelelului 2
Mp = 0.272;                     
dM = 0.071;                                              
g = 9.81;                       
J = 0.002;                       
Jw = 3.941*10^-5;              
b_theta = 0.82*10^-3;           
b_altha_kmke_Ra = 1.202*10^-4; 
km_Ra = 1.837*10^-4;

%% matricea modelului spatial
A = [0 1 0 0;
    (Mp*dM*g)/J -(b_theta)/J 0 1/J*b_altha_kmke_Ra;
    0 0 0 1;
    -(Mp*g*dM)/J b_theta/J 0 -(J+Jw)/(J*Jw)*b_altha_kmke_Ra];
B = [0 0;
    -1/J*km_Ra 1/J;
    0 0;
    (J+Jw)/(J*Jw)*km_Ra -1/J];
C = [1 0 0 0;
    0 1 0 0;
    0 0 1 0;
    0 0 0 1];
D = [0 0; 0 0;0 0;0 0];
%% construirea sistemului
sys = ss(A,B,C,0); %construirea sistemului
xd = [50;0;200;0]; %conditii dorite
x0 = 1*randn(4,1);

%% simulare 
Q = [100 0 0 0;
    0 100 0 0;
    0 0 10 0; 
    0 0 0 1];
R=5*[1 0.1;
   0.1 1];
N = zeros(4,2);

[K,S,e] = lqr(sys,Q,R,N);
K_lqr = lqr(A,B,Q,R);

eig(A-B*K)

eig(A)

Acl=A-B*K
Bcl  = -Acl;
sys_cl = ss(Acl,Bcl,C,0)


t_f = 30;
t = 0:0.001:t_f;

cl_intrari = [xd(1)*ones(size(t));xd(2)*ones(size(t));xd(3)*ones(size(t));xd(4)*ones(size(t))]

[iesire_cl,t_cl,s_cl] = lsim(sys_cl,cl_intrari,t,x0);

% figure;
% plot(t,s_cl(:,1),'k')
% hold on
% plot(t,s_cl(:,2),'r')
% hold on
% plot(t,s_cl(:,3),'g')
% hold on
% plot(t,s_cl(:,4),'m')
% legend('\theta','\theta dot','\alpha','\alpha dot')
% grid on

%% Discretizare 
Ts = 0.1;
sys_d = c2d(sys,Ts);
Ad = sys_d.a;
Bd = sys_d.b;
Cd = sys_d.c;
Dd = sys_d.d;

R=eye(4)

K_d = dlqr(Ad,Bd,Q,R,N);
Ob = dlqr(Ad',Cd',Q,R,N);

And I tried to make R symmetric positiv and it's not working. What to do?

Answer 1

[Sam Chak]

Hi @Andrei Rotaru

Fixing the code as shown will eliminate the error message.

R   = eye(size(B, 2));
K_d = dlqr(Ad, Bd, Q, R, N)

K_d = 2x4

0.6249 0.8269 2.4984 0.8009 0.2062 0.0264 0.0001 0.0001

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R   = eye(size(Cd));
N   = zeros(size(Ad));
Ob  = dlqr(Ad', Cd', Q, R, N)

Ob = 4x4

1.4894 10.6811 -0.4535 -9.2995 0.1132 1.4441 -0.0141 -0.4215 -0.0000 -0.0001 0.9165 0.0030 -0.0032 -0.0258 0.0887 0.7332

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[Andrei Rotaru]

Thanks!

[Sam Chak]

You are welcome, @Andrei Rotaru. 👍