Solver not applicable because of inverse?

[stevens]

Good morning,

I try to implement a Hinf Filter like in the book (theorem 2, p.293) by solving 2 LMIs (34 & 32)

Unfortunately I get this message:

 solvertime: 0

 info: 'Solver not applicable (sedumi)'

 problem: -4

 yalmiptime: 0.5470 

I think the reason of this problem is the inverse of the unknown Q1... I tried to implement Q1*Q1_inv = eye(n). Are there other ways to implement this?

[stevens]

% Hinf - Filter (Zhaojian Li et al.)

clear all

clc

% Parameter

m_c = 290;      % Aufbaumasse in kg

m_w = 60;       % Radmasse in kg

c_w = 19000;   % Radsteifigkeit in N/m

c_c = 16800;    % Aufbausteifigkeit in N/m

d_c = 200;     % Aufbaudaempfung in Ns/m

%% state space

% state 1: z_t

% state 2: z_w_dot

% state 3: z_cw

% state 4: z_c_dot 

alpha = 0.1;

A = [0 1 0 0;...

    -c_w/m_w -d_c/m_w c_c/m_w d_c/m_w;...

    0 -1 0 1;...

    0 d_c/m_c -c_c/m_c -d_c/m_c];

B = [0; 1/m_w; 0; -1/m_c];

B_r = [-1; 0; 0; 0];

B_w = [-alpha; 0; 0; 0];

C_0 = [0 1 0 0;...

       0 0 1 0;...

       0 0 0 1];

   

D_0 = [0.001; 0.001; 0.001];   

D_1 = 0.001;

D_r = 0.0001;

% augmented system

A_a = [A, B_r; zeros(1,4), zeros(1,1)];

B_a = [B_w; D_r];

C_a0 = [C_0, zeros(3,1); zeros(1,5)];

C_a1 = [zeros(3,5); zeros(1,4), 1];

D_a = [D_0; D_1];

E_a = [eye(4), zeros(4,1)];

%% LMI

yalmip('clear');

X = sdpvar(5,5,'symmetric');

Y = sdpvar(5,5,'symmetric');

AA = sdpvar(5,5,'full');

BB = sdpvar(5,4,'full');

CC = sdpvar(4,5,'full');

Q2 = sdpvar(1,1,'symmetric');

Q1 = sdpvar(5,5,'symmetric');

Q1inv = sdpvar(5,5,'symmetric');

gama = 0.5;

tauM = 0.5;

taum = 0.05;

% LMI (34)

T1 = [A_a*X + X*A_a', A_a + AA';...

      A_a' + AA, Y*A_a + A_a'*Y + BB*(C_a0+C_a1) + (C_a0+C_a1)'*BB'];

T2 = [B_a; Y*B_a + BB*D_a];

T3 = [X*E_a' - CC'; E_a'];

T4 = [zeros(5,5), zeros(5,1), X; BB*C_a1*A_a, BB*C_a1*B_a, eye(5,5)];

DD = [Q1, zeros(5,6); zeros(1,5),Q2,zeros(1,5); zeros(5,6), Q1inv];

L1 = [T1, T2, T3, T4;...

      T2', -gama^2+tauM*Q2, zeros(1,4), zeros(1,11);...

      T3', zeros(4,1), -eye(4,4), zeros(4,11);...

      T4', zeros(11,1), zeros(11,4), -(1/tauM)*DD]

% LMI (32)

L2 = [X, eye(5); eye(5), Y];

constraints = [L1<=0, -L2<=0, -X<=0, -Y<=0, -Q1<=0, -Q2<=0, Q1*Q1inv==eye(5)];

options = sdpsettings('verbose',1,'solver','sedumi');

solution = optimize(constraints,[],options)

X = double(X)

Y = double(Y)

AA = double(AA)

BB = double(BB)

CC = double(CC)

[Johan Löfberg]

A model involving both q and q^-1 is not convex and definitely not linear, hence not an lmi.

 

You’ve either misunderstood the paper, there is a typo in the paper, or they forgot to say something. Note that they say x,y,a,b,c are decision variables. They don’t list q1 and q2 as decision variables, so maybe they forgot to say they fix them, or they forgot to eliminate them using some transformation they’ve done previously.

 

You’re better off contacting the authors