How do I implement LQR (Linear Quadratic Regulator) design script in my Piezo Bender Simulink Model ?

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Fajwa Noor
Fajwa Noor on 17 Jun 2022
Ran in:
Hi. Below is the LQR Design MATLAB code created to design a closed-loop controller of my Piezo Bender Energy Harvester in order to optimize the transfer of power and improve the efficiency of the energy harvester in different conditions. So, how do I implement this LQR script into my Simulink model ?
% Parameters
rho = 7804.89;
l = 3.175e-2;
w = 1.72e-2;
d = 5.084e-4;
E = 1.36628e11;
I = 1.3907e-14;
e31 = 7.5459;
epsl = 1.38965e-8;
bm = 0;
kb = 1e-5;
% Mass matrix
m = (rho*l*w*d)/420;
M1 = [156 22*l 54 -13*l;
22*l 4*l^2 13*l -3*l^2;
54 13*l 156 -22*l;
-13*l -3*l^2 -22*l 4*l^2];
M = m*M1;
% Stiffness matrix
K = [12*E*I/l^3 6*E*I/l^2 -12*E*I/l^3 6*E*I/l^2;
6*E*I/l^2 4*E*I/l -6*E*I/l^2 2*E*I/l;
-12*E*I/l^3 -6*E*I/l^2 12*E*I/l^3 -6*E*I/l^2;
6*E*I/l^2 2*E*I/l -6*E*I/l^2 4*E*I/l];
% Damping matrix
B = kb*K + bm*M;
%State-space
n = size(M, 1);
a11 = zeros(n);
a12 = eye(n);
a21 = - M\K;
a22 = - M\B;
b1 = zeros(n);
b2 = M\a12;
A = [a11 a12; a21 a22];
B = [b1; b2];
C = [eye(n) zeros(n)];
D = zeros(n);
sys1 = ss(A, B, C, D)
sys1 = A = x1 x2 x3 x4 x5 x6 x7 x8 x1 0 0 0 0 1 0 0 0 x2 0 0 0 0 0 1 0 0 x3 0 0 0 0 0 0 1 0 x4 0 0 0 0 0 0 0 1 x5 2.301e+07 4.175e+05 -2.301e+07 3.131e+05 230.1 4.175 -230.1 3.131 x6 -8.698e+09 -1.479e+08 8.698e+09 -1.282e+08 -8.698e+04 -1479 8.698e+04 -1282 x7 -2.301e+07 -3.131e+05 2.301e+07 -4.175e+05 -230.1 -3.131 230.1 -4.175 x8 -8.698e+09 -1.282e+08 8.698e+09 -1.479e+08 -8.698e+04 -1282 8.698e+04 -1479 B = u1 u2 u3 u4 x1 0 0 0 0 x2 0 0 0 0 x3 0 0 0 0 x4 0 0 0 0 x5 7384 -1.744e+06 -1846 -8.721e+05 x6 -1.744e+06 5.494e+08 8.721e+05 3.845e+08 x7 -1846 8.721e+05 7384 1.744e+06 x8 -8.721e+05 3.845e+08 1.744e+06 5.494e+08 C = x1 x2 x3 x4 x5 x6 x7 x8 y1 1 0 0 0 0 0 0 0 y2 0 1 0 0 0 0 0 0 y3 0 0 1 0 0 0 0 0 y4 0 0 0 1 0 0 0 0 D = u1 u2 u3 u4 y1 0 0 0 0 y2 0 0 0 0 y3 0 0 0 0 y4 0 0 0 0 Continuous-time state-space model.
step(sys1, 10)
%Optimal Gain matrix
q = 1.0e0; % to make the response faster, increase exponent of q with r fixed at 1
Q = q*eye(2*n);
r = 1.0e0; % to reduce the energy consumption, increase exponent of r with q fixed at 1
R = r*eye(n);
G = lqr(A, B, Q, R)
G = 4×8
0.5024 -0.0078 0.4976 -0.0078 0.9935 -0.0000 0.0076 -0.0000 -0.1397 0.9415 0.1397 0.0541 -0.0002 1.0000 0.0002 -0.0000 0.4976 0.0078 0.5024 0.0078 0.0076 0.0000 0.9935 0.0000 -0.1397 0.0541 0.1397 0.9415 -0.0002 -0.0000 0.0002 1.0000
H = A - B*G;
sys2 = ss(H, B, C, D)
sys2 = A = x1 x2 x3 x4 x5 x6 x7 x8 x1 0 0 0 0 1 0 0 0 x2 0 0 0 0 0 1 0 0 x3 0 0 0 0 0 0 1 0 x4 0 0 0 0 0 0 0 1 x5 2.264e+07 2.107e+06 -2.265e+07 1.229e+06 -7634 1.744e+06 2094 8.721e+05 x6 -8.567e+09 -6.86e+08 8.568e+09 -5.2e+08 1.833e+06 -5.493e+08 -9.609e+05 -3.845e+08 x7 -2.265e+07 -1.229e+06 2.264e+07 -2.107e+06 2094 -8.721e+05 -7634 -1.744e+06 x8 -8.568e+09 -5.2e+08 8.567e+09 -6.86e+08 9.609e+05 -3.845e+08 -1.833e+06 -5.493e+08 B = u1 u2 u3 u4 x1 0 0 0 0 x2 0 0 0 0 x3 0 0 0 0 x4 0 0 0 0 x5 7384 -1.744e+06 -1846 -8.721e+05 x6 -1.744e+06 5.494e+08 8.721e+05 3.845e+08 x7 -1846 8.721e+05 7384 1.744e+06 x8 -8.721e+05 3.845e+08 1.744e+06 5.494e+08 C = x1 x2 x3 x4 x5 x6 x7 x8 y1 1 0 0 0 0 0 0 0 y2 0 1 0 0 0 0 0 0 y3 0 0 1 0 0 0 0 0 y4 0 0 0 1 0 0 0 0 D = u1 u2 u3 u4 y1 0 0 0 0 y2 0 0 0 0 y3 0 0 0 0 y4 0 0 0 0 Continuous-time state-space model.
step(sys2, 10)
The Simulink Model.

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