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Change Continuous State Space MIMO Model to Discrete State Space MIMO Model without Change to Transfer Function and Using idare Function to find S matrix, K matrix and eigen value S Matrix

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Ivan Dwi Putra
Ivan Dwi Putra on 17 Jun 2020
Edited: Ivan Dwi Putra on 17 Jun 2020
this is my code for my system model
clear; close; clc;
syms a1_head a2_head b hstar
%Parameter Massa
m1 = 8095; % massa train set 1 dalam kg
m2 = 8500; % massa train set 2 dalam kg
g = 10;
c_0_1 = 0.01176;
c_1_1 = 0.00077616;
c_2_1 = 4.48 ;
c_0_2 = 0.01176 ;
c_1_2 = 0.00077616;
c_2_2 = 4.48;
v_0 = 300;
hstar = 120;
a_1 = -1./m1.*(c_1_1 + 2.*c_2_1.*v_0);
a_2 = -1./m2.*(c_1_2 + 2.*c_2_2.*v_0);
a_1_head = 1-(a_1.*hstar);
a_2_head = 1-(a_2.*hstar);
b = 1;
% Model data
A = sym(zeros(4,4));
A(1,2) = a_1_head;
A(3,2) = (a_2_head) - 1;
A(3,4) = a_2_head;
display(A);
B = sym(zeros(4,2));
B(1,1) = -b*hstar;
B(2,1) = b;
B(3,2) = -b*hstar;
B(4,1) = -b;
B(4,2) = b;
display(B);
C = [1 0 0 0;
0 1 0 0;
0 0 1 0;
0 0 0 1];
p_1 = -1./m1.*(c_0_1 - c_2_1.*(v_0).^2);
p_2 = -1./m2.*(c_0_2 - c_2_2.*(v_0).^2);
W = [((a_1 - 1).*v_0) - (p_1.*hstar);
0;
((a_2 - 1).*v_0) - (p_2.*hstar);
((a_1 - 1).*v_0) - (p_1.*hstar);
];
This is Q and R matrics
% Q and R matrices for ARE
Q = sym(eye(4)); display(Q);
R = sym(zeros(2,2)); R(1,:) = [1 2]; R(2,:) = [2 3]; display(R);
This is S matrix I want to find
% Matrix S to find
svar = sym('s',[1 16]);
S = [svar(1:4); svar(5:8); svar(9:12); svar(13:16)];
display(S);
This is Algebra Ricatti Equation
% LHS of ARE: A'*S + S*A' - S*B*Rinv*B'*S
left_ARE = transpose(A)*S + S*A - S*B*inv(R)*transpose(B)*S;
display(left_ARE);
% RHS of ARE: -Q
right_ARE = -Q;
display(right_ARE);
I want to change to discrete state space model
ss_model = ss(double(A),double(B(:,1)),C,W(:,1));
dt = 1 ;
d_sys = c2d(ss_model,dt);
is that true use this code?
from d_sys code i get
d_sys =
A =
x1 x2 x3 x4
x1 1 40.85 0 0
x2 0 1 0 0
x3 0 37.95 1 38.95
x4 0 0 0 1
B =
u1
x1 -99.58
x2 1
x3 -0.5
x4 -1
C =
x1 x2 x3 x4
y1 1 0 0 0
y2 0 1 0 0
y3 0 0 1 0
y4 0 0 0 1
D =
u1
y1 -6377
y2 0
y3 -6087
y4 -6377
Sample time: 1 seconds
Discrete-time state-space model.
now i wan to use idare to find X(S matrix),K matrix in lqr,L(Eigen value S matrix) using idare function
how i use idare function using A,B,C,D in discrete state space model?
from the d_sys my B matrix is weird because my B matrix become 4x1 different from my model 4x2

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