LMI solving issue using MOSEK and GUROBI

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Md. Abdur Rahim

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Nov 2, 2023, 12:52:36 AM11/2/23

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I tried to solve the LMI using YALMIP with mosek as a solver. Unfortunately, it is showing "'Infeasible problem (learn to debug) (MOSEK-SDP)'". Also, I tried to use Gurobi, but it is not working and showing "Warning: Solver not applicable (gurobi does not support semidefinite constraints) Unrecognized field name "used_variables"." I am attaching my codes here. I would be grateful if you could help me with this code.

%System_parameters

m = 1800; % mass in kg

Iz= 2500; % Vehicle total inertia about the yaw axis in kg m2

lf= 1.03; % Distance of the front axle from CG in m

lr= 1.49; % Distance of the rear axle from CG in m

C_f= 40; % Nominal cornering stiffness of front tyres in KN/rad

C_r= 40; % Nominal cornering stiffness of rear tyres in KN/rad

D_l = 10; %look ahead distance in m

%for varying parameter vector p = (p1,...,ps)

v_x_min= 5;

v_x_max= 35;

%sys {v_x_min, 1/v_x_max}

A{1} = [-(C_r+C_f)./(m*v_x_max) ((lr*C_r)-(lf*C_f))./(m*v_x_max)-v_x_min 0 0; ((lr*C_r)-(lf*C_f))./(Iz*v_x_max) -((lr^2*C_r)+(lf^2*C_f))./(Iz*v_x_max) 0 0; -1 -D_l 0 v_x_min; 0 -1 0 0];

B{1} = [C_f./m; (lf*C_f)./Iz; 0; 0];

C{1} = [0 1 0 0; 0 0 1 0; 0 0 0 1];

E{1}=[0; 0; 0; v_x_min];

D{1} = zeros(3,1);

s1= ltisys(A{1},B{1},C{1},D{1});

%Sys {v_x_min, 1/v_x_min}

A{2} = [-(C_r+C_f)./(m*v_x_min) ((lr*C_r)-(lf*C_f))./(m*v_x_min)-v_x_min 0 0; ((lr*C_r)-(lf*C_f))./(Iz*v_x_min) -((lr^2*C_r)+(lf^2*C_f))./(Iz*v_x_min) 0 0; -1 -D_l 0 v_x_min; 0 -1 0 0];

B{2} = [C_f./m; (lf*C_f)./Iz; 0; 0];

C{2} = [0 1 0 0; 0 0 1 0; 0 0 0 1];

E{2}=[0; 0; 0; v_x_min];

D{2} = zeros(3,1);

s2= ltisys(A{2},B{2},C{2},D{2});

% Sys{v_x_max, 1/v_x_min}

A{3} = [-(C_r+C_f)./(m*v_x_min) ((lr*C_r)-(lf*C_f))./(m*v_x_min)-v_x_max 0 0; ((lr*C_r)-(lf*C_f))./(Iz*v_x_min) -((lr^2*C_r)+(lf^2*C_f))./(Iz*v_x_min) 0 0; -1 -D_l 0 v_x_max; 0 -1 0 0];

B{3} = [C_f./m; (lf*C_f)./Iz; 0; 0];

C{3} = [0 1 0 0; 0 0 1 0; 0 0 0 1];

E{3}=[0; 0; 0; v_x_max];

D{3} = zeros(3,1);

s3= ltisys(A{3},B{3},C{3},D{3});

% Sys {v_x_max, 1/v_x_max}

A{4} = [(C_r+C_f)./(m*v_x_max) ((lr*C_r)-(lf*C_f))./(m*v_x_max) - v_x_max 0 0; ((lr*C_r)-(lf*C_f))./(Iz*v_x_max) -((lr^2*C_r)+(lf^2*C_f))./(Iz*v_x_max) 0 0; -1 -D_l 0 v_x_max; 0 -1 0 0];

B{4} = [C_f./m; (lf*C_f)./Iz; 0; 0];

C{4} = [0 1 0 0; 0 0 1 0; 0 0 0 1];

E{4}=[0; 0; 0; v_x_max];

D{4} = zeros(3,1)

s4= ltisys(A{4},B{4},C{4},D{4});

pols = psys([s1,s2,s3,s4]);

[tmin,P] = quadstab(pols)

%% LMI Solution - Prepare Yalmip

yalmip('clear');

LMIs = [];

gamma2=sdpvar(1);

Z = sdpvar(4,4,'symmetric');

N = 4;

r = 40;

q = 40;

Z >= 0;

for k = 1:N % N is number of vertices

W{k} = sdpvar(1,4,'full');

%% Condition for Quadratic Stability

LMIs = [LMIs, A{k}*Z + Z*A{k}'+ B{k}*W{k} + W{k}'*B{k}'<= 0];

%% Condition for H_∞ Performance

LMIs = [LMIs, [A{k}*Z+Z'*A{k}'+B{k}*W{k}+W{k}'*B{k}' E{k} Z'*C{k}'; E{k}' -eye(1) D{k}'; C{k}*Z D{k} -gamma2*eye(3)] <= 0];

%% Condition for LMI Reion

LMIs = [LMIs, [-r*Z q*Z+A{k}*Z+B{k}*W{k}; q*Z+Z'*A{k}'+W{k}'*B{k}' -r*Z] <= 0];

end

%options = sdpsettings('solver','gurobi');

options = sdpsettings('solver','mosek');

solution1 = optimize(LMIs,Z,options)

solution2 = optimize(LMIs,gamma2,options)

%% Controller Gain at Individual Vertices

K1 = value(W{1})*inv(value(Z))

K2 = value(W{2})*inv(value(Z))

K3 = value(W{3})*inv(value(Z))

K4 = value(W{4})*inv(value(Z))

Johan Löfberg

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Nov 2, 2023, 3:05:47 AM11/2/23

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It makes no sense to have a matrix objective

>> Z
Linear matrix variable 4x4 (symmetric, real, 10 variables)

And yes, your model is infeasible, already the second and third constraint together form an ill-posed (most likely infeasible) model (i.e. H∞ Performance+LMI region for a single system fails) . Trying to use gurobi makes absolutely no sense as you have an SDP

>> optimize(LMIs(2:3))

MOSEK Version 10.0.16(BETA) (Build date: 2022-6-21 16:34:57)
Copyright (c) MOSEK ApS, Denmark WWW: mosek.com
Platform: Windows/64-X86

numbarvars = 2
numbarvars = 2
numbarvars = 2
Problem
Name :
Objective sense : minimize
Type : CONIC (conic optimization problem)
Constraints : 15
Affine conic cons. : 0
Disjunctive cons. : 0
Cones : 0
Scalar variables : 0
Matrix variables : 2
Integer variables : 0

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 1 time : 0.00
Lin. dep. - tries : 1 time : 0.02
Lin. dep. - number : 0
Presolve terminated. Time: 0.03
Problem
Name :
Objective sense : minimize
Type : CONIC (conic optimization problem)
Constraints : 15
Affine conic cons. : 0
Disjunctive cons. : 0
Cones : 0
Scalar variables : 0
Matrix variables : 2
Integer variables : 0

Optimizer - threads : 2
Optimizer - solved problem : the primal
Optimizer - Constraints : 15
Optimizer - Cones : 0
Optimizer - Scalar variables : 0 conic : 0
Optimizer - Semi-definite variables: 2 scalarized : 72
Factor - setup time : 0.00 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 120 after factor : 120
Factor - dense dim. : 0 flops : 1.08e+04
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 3.0e+00 5.0e+00 2.0e+00 0.00e+00 1.000000000e+00 0.000000000e+00 1.0e+00 0.05
1 4.9e-01 8.1e-01 7.9e-01 -9.23e-01 -3.930233281e+00 0.000000000e+00 1.6e-01 0.19
2 7.9e-02 1.3e-01 3.2e-01 -9.89e-01 -3.419807106e+01 0.000000000e+00 2.6e-02 0.19
3 1.6e-02 2.7e-02 1.4e-01 -9.95e-01 -1.749981358e+02 0.000000000e+00 5.3e-03 0.19
4 2.7e-03 4.5e-03 5.8e-02 -1.01e+00 -1.065013674e+03 0.000000000e+00 8.9e-04 0.20
5 2.2e-04 3.7e-04 1.7e-02 -1.00e+00 -1.266120992e+04 0.000000000e+00 7.4e-05 0.20
6 7.2e-05 1.2e-04 9.5e-03 -1.00e+00 -3.926472578e+04 0.000000000e+00 2.4e-05 0.20
7 1.2e-05 2.0e-05 3.9e-03 -1.00e+00 -2.379410259e+05 0.000000000e+00 4.0e-06 0.22
8 2.3e-06 3.8e-06 1.7e-03 -1.00e+00 -1.240872769e+06 0.000000000e+00 7.6e-07 0.22
9 4.5e-07 7.4e-07 7.6e-04 -1.01e+00 -6.510249064e+06 0.000000000e+00 1.5e-07 0.23
10 1.6e-07 2.6e-07 4.5e-04 -1.01e+00 -1.839285562e+07 0.000000000e+00 5.2e-08 0.23
11 5.2e-08 8.6e-08 2.5e-04 -9.81e-01 -5.362656782e+07 0.000000000e+00 1.7e-08 0.23
12 2.6e-09 4.3e-09 4.3e-05 -9.37e-01 -6.190881577e+08 0.000000000e+00 8.5e-10 0.25
13 4.5e-10 7.5e-10 5.7e-06 -5.56e-02 -3.596613787e+08 0.000000000e+00 1.5e-10 0.25
14 1.5e-11 2.5e-11 2.0e-08 8.72e-01 -3.866489600e+06 0.000000000e+00 5.1e-12 0.27
15 1.0e-15 7.5e-12 8.2e-15 9.96e-01 -1.323069083e+02 0.000000000e+00 3.5e-16 0.27
16 8.8e-23 1.9e-12 5.0e-25 1.00e+00 -1.001184592e-05 0.000000000e+00 2.3e-23 0.27
17 8.2e-23 2.3e-12 4.6e-25 1.00e+00 -9.386105549e-06 0.000000000e+00 2.1e-23 0.28
18 6.2e-23 1.1e-11 3.0e-25 1.00e+00 -7.039579165e-06 0.000000000e+00 1.6e-23 0.28
19 9.1e-30 3.5e-11 2.4e-37 1.00e+00 -2.858439529e-15 0.000000000e+00 1.0e-32 0.30
20 1.3e-38 5.8e-11 3.4e-50 1.00e+00 -1.752358982e-24 0.000000000e+00 7.0e-42 0.30
21 1.2e-38 4.4e-11 2.8e-50 1.00e+00 -1.533314110e-24 0.000000000e+00 6.1e-42 0.31
22 1.2e-38 4.4e-11 2.8e-50 1.00e+00 -1.533314110e-24 0.000000000e+00 6.1e-42 0.31
23 1.2e-38 4.4e-11 2.8e-50 1.00e+00 -1.533314110e-24 0.000000000e+00 6.1e-42 0.33
Optimizer terminated. Time: 0.41

Interior-point solution summary
Problem status : ILL_POSED
Solution status : PRIMAL_ILLPOSED_CER
Dual. obj: 0.0000000000e+00 nrm: 2e+01 Viol. con: 0e+00 barvar: 4e-09

BTW, this line

Z >= 0;

does not do anything (and if you added it to your model it would be redundant as you already impose that in your "LMI Reion" constraint

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