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 2020-6-17

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/549765-change-continuous-state-space-mimo-model-to-discrete-state-space-mimo-model-without-change-to-transf

编辑： Ivan Dwi Putra 2020-6-17

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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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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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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