question on computing mrpi

[lweidsab]

Hi,

I have a 4-order linear discreted system with A, B, Q, R, when I use the function compute_mrpi_set() to solve the maximum robust invariant set, the result of Miniski addition is not convergered. I mean the alpha is becoming larger with the increasing iterative steps, while the correct alpha should become smaller. I want to know what's the reason, it is caused by the feedback maricx Ak or disturbance polyhedron W？

Your answer would be appreciated.

Yongxiang

[wtllll]

 您好，已收到您的邮件，谢谢！

[KKaiWang]

just check if eigenvalues of A_K is strictly less than 1 and if W is bounded.

[lweidsab]

Thanks for your suggestions. I tried to use the continuous state-space function, it's converged. While when I discretize the state-space with a zero-order holder, it is not convergent. The eigenvalues of A_K is almost 1 and system is borderline stable.

clear all; clc;

% fix random seed
rng(0);

A = [0 1 0 0;
-60.723 -1.0795 -3.1167 -0.32884;
0 8 -13.59 -13.35;
0 0 8 0];

B = [0;
0.6577;
0;
0];

C = [1 0 0 0;
0 1 0 0];

D = [0;0];

sys = ss(A,B,C,D);
sysd = c2d(sys,0.01,'zoh');

Ad = sysd.A;
Bd = sysd.B;
Cd = sysd.C;
Dd = sysd.D;

Q = diag([1,1,1,1]);
R = 0.1;

w_min = 1e-3*[-0.1; -0.1; -0.1; -0.1];
w_max = -w_min;

D = normalized_inequality_constraint(w_min, w_max);
qw = length(D);
W = Polyhedron('A',D,'b',ones(qw,1));

disturbance_system = DisturbanceLinearSystem(Ad, Bd, Q, R, W);

x_min = [-0.2; -1; -1e3; -1e3];
x_max = [0.2; 1; 1e3; 1e3];

u_min = -12;
u_max = -u_min;

% constraints on state Xc and input Uc
Uc_vertex = [12; -12];
Xc =Polyhedron( 'A', [eye(4,4);(-1)*eye(4,4)], 'b',[0.0174;0.15;0.15;0.15;0.0174;0.15;0.15;0.15]);
Uc = Polyhedron(Uc_vertex);

%% State Constraints
G = normalized_inequality_constraint(x_min, x_max);
qx = length(G);
X = Polyhedron('A',G,'b',ones(qx,1));

%% Input Constraints
H = normalized_inequality_constraint(u_min, u_max);
qu = length(H);
U = Polyhedron('A',H,'b',ones(qu,1));

N_horizon = 10;
mpc = TubeModelPredictiveControl(disturbance_system, Xc, Uc, N_horizon);

just check if eigenvalues of A_K is strictly less than 1 and if W is bounded.