'Infeasible problem (YALMIP)' problem : 1

219 views

Skip to first unread message

Ryad Ténor

unread,

Jun 17, 2015, 6:54:35 AM6/17/15

to yal...@googlegroups.com

Saisissez le code ici...


Saisissez le code ici...

Hello,

I want to solve an LMI constraint for getting a robust controller, i maked my augmented model by Schur complement, and use 'sdpt3' solver.

when i run, yalmip return that "the element wise enequality" is polynomial with error 'Infeasible problem (YALMIP)', are there a solution for this problem ?

With the lot of THANKS !

my code file :

close all

clear all

clc

%%% the differents constants

u=0.7;

Sf10=60712.7*u;

Sf20=4814*u;

Sr10=60088*u;

Sr20=3425*u;

m=1500;

Vx=20;

Iz=2208;

L2=1.4625;

L1=1.0065;

Is=5;

Iw=0.4;

Cx=0.4;%coefficient de trainé aérodynamique

rho=1.225;% densité volumique de l'air

Sx=3;% surface de la section exposée au vent

a11=(-(2*(Sf10+Sr10))/m*Vx);

a12=-1+((2*(L2*Sr10-L1*Sf10))/m*Vx^2);

a13=(2*(L2*Sr10-L1*Sr10))/Iz;

a14=(-2*(L2^2*Sr10+L1^2*Sf10))/Iz*Vx;

a21=(-(2*(Sf20+Sr20))/m*Vx);

a22=-1+((2*(L2*Sr20-L1*Sf20))/m*Vx^2);

a23=(2*(L2*Sr20-L1*Sr20))/Iz;

a24=(-2*(L2^2*Sr20+L1^2*Sf20))/Iz*Vx;

b11=2*Sf10/m*Vx;

b12=2*L1*Sf10/Iz;

b13=(rho*Cx*Sx)/m*Vx;

b14=(Iw*rho*Cx*Sx)/Iz;

b21=2*Sf20/m*Vx;

b22=2*L1*Sf20/Iz;

d1=0.12;

%%% the matrices of system

A1=[a11 a12 0 0;a13 a14 0 0;0 1 0 0;Vx Is Vx 0];

B1=[b11;b12;0;0];

A2=[a21 a22 0 0;a23 a24 0 0;0 1 0 0;Vx Is Vx 0];

B2=[b21;b22;0;0];

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

D=[[0;0;0;0],[0;0;0;0],[0;0;0;0]];

E11=[(-(2*(Sf10+Sr10))/m*Vx) ((2*(L2*Sr10-L1*Sf10))/m*Vx^2) 0 0;(2*(L2*Sr10-L1*Sr10))/Iz (-2*(L2^2*Sr10+L1^2*Sf10))/Iz*Vx 0 0;0 0 0 0;0 0 0 0];

E12=[(-(2*(Sf20+Sr20))/m*Vx) ((2*(L2*Sr20-L1*Sf20))/m*Vx^2) 0 0;(2*(L2*Sr20-L1*Sr20))/Iz (-2*(L2^2*Sr20+L1^2*Sf20))/Iz*Vx 0 0;0 0 0 0;0 0 0 0];

E21=[2*Sf10/m*Vx;2*L1*Sf10/Iz;0;0];

E22=[2*Sf20/m*Vx;2*L1*Sf20/Iz;0;0];

Bd=[(rho*Cx*Sx)/m*Vx 0;(Iw*rho*Cx*Sx)/Iz 0;0 -Vx;0 -Is*Vx];

Dd=[d1 0 0 0;0 d1 0 0;0 0 0 0;0 0 0 0];

%%% sdp variables

W2=sdpvar(4,4);

W1=sdpvar(4,4);

P22=sdpvar(4,4);

P21=sdpvar(4,4);

P2=sdpvar(4,4);

P1=sdpvar(4,4);

P3=sdpvar(4,4);

l1=sdpvar(1);l2=sdpvar(1);l3=sdpvar(1);l4=sdpvar(1);l5=sdpvar(1);l6=sdpvar(1);

l7=sdpvar(1);l8=sdpvar(1);l9=sdpvar(1);

K1=sdpvar(1,4);

K2=sdpvar(1,4);

eta=sdpvar(1);

%%%% The matrices of my augmented model (each one is with indices i and j)

S11=[eye(4) zeros(4) zeros(4) (B1*K1)' zeros(4,1) zeros(4,2);

zeros(4) P21'*A1+P21*A1'+W1*C+W1'*C'+eye(4) (-P22*C*A1-W2*C-W1')' (-B1*K1+(A1-A1)'*P21)' ((B1-B1)'*P21)' (Bd'*P21)';

zeros(4) -P22*C*A1-W2*C-W1' W2+W2'+eye(4) ((C*(A1-A1)'*P22))' (-(C*(B1-B1))'*P22)' (-(C*Bd)'*P22)';

B1*K1 -B1*K1+(A1-A1)'*P21 -(C*(A1-A1)')*P22 A1+A1' B1 Bd;

zeros(1,4) (B1-B1)'*P21 -(C*(B1-B1))'*P22 B1' -eta^2*eye(1,1) zeros(1,2);

zeros(2,4) Bd'*P21 -(C*Bd)'*P22 Bd' zeros(2,1) eta^2*eye(2,2)];

S22=[eye(4) zeros(4) zeros(4) (B2*K2)' zeros(4,1) zeros(4,2);

zeros(4) P21'*A2+P21*A2'+W1*C+W1'*C'+eye(4) (-P22*C*A2-W2*C-W1')' (-B2*K2+(A2-A2)'*P21)' ((B2-B2)'*P21)' (Bd'*P21)';

zeros(4) -P22*C*A2-W2*C-W1' W2+W2'+eye(4) ((C*(A2-A2)'*P22))' (-(C*(B2-B2))'*P22)' (-(C*Bd)'*P22)';

B2*K2 -B2*K2+(A2-A2)'*P21 -(C*(A2-A2)')*P22 A2+A2' B1 Bd;

zeros(1,4) (B2-B2)'*P21 -(C*(B2-B2))'*P22 B2' -eta^2*eye(1,1) zeros(1,2);

zeros(2,4) Bd'*P21 -(C*Bd)'*P22 Bd' zeros(2,1) eta^2*eye(2,2)];

S12=[eye(4) zeros(4) zeros(4) (B1*K2)' zeros(4,1) zeros(4,2);

zeros(4) P21'*A2+P21*A2'+W1*C+W1'*C'+eye(4) (-P22*C*A2-W2*C-W1')' (-B1*K2+(A1-A2)'*P21)' ((B1-B2)'*P21)' (Bd'*P21)';

zeros(4) -P22*C*A2-W2*C-W1' W2+W2'+eye(4) ((C*(A1-A2)'*P22))' (-(C*(B1-B2))'*P22)' (-(C*Bd)'*P22)';

B1*K2 -B1*K2+(A1-A2)'*P21 -(C*(A1-A2)')*P22 A1+A1' B1 Bd;

zeros(1,4) (B1-B2)'*P21 -(C*(B1-B2))'*P22 B1' -eta^2*eye(1,1) zeros(1,2);

zeros(2,4) Bd'*P21 -(C*Bd)'*P22 Bd' zeros(2,1) eta^2*eye(2,2)];

S21=[eye(4) zeros(4) zeros(4) (B2*K1)' zeros(4,1) zeros(4,2);

zeros(4) P21'*A1+P21*A1'+W1*C+W1'*C'+eye(4) (-P22*C*A1-W2*C-W1')' (-B2*K1+(A2-A1)'*P21)' ((B2-B1)'*P21)' (Bd'*P21)';

zeros(4) -P22*C*A1-W2*C-W1' W2+W2'+eye(4) ((C*(A2-A1)'*P22))' (-(C*(B2-B1))'*P22)' (-(C*Bd)'*P22)';

B2*K1 -B2*K1+(A2-A1)'*P21 -(C*(A2-A1)')*P22 A2+A2' B2 Bd;

zeros(1,4) (B2-B1)'*P21 -(C*(B2-B1))'*P22 B2' -eta^2*eye(1,1) zeros(1,2);

zeros(2,4) Bd'*P21 -(C*Bd)'*P22 Bd' zeros(2,1) eta^2*eye(2,2)];

Sc11=[B1*K1 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

P1 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

B1*K1 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

C*(B1-B1)*K1 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) P21 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) (B1-B1)*K1 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) P21 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) C*(B1-B1)*K1 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) zeros(4) P22 zeros(4) zeros(4,1) zeros(4,2);

zeros(4) zeros(4) P22 zeros(4) zeros(4,1) zeros(4,2)];

Sc22=[B2*K2 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

P1 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

B2*K2 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

C*(B2-B2)*K2 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) P21 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) (B2-B2)*K1 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) P21 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) C*(B2-B2)*K2 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) zeros(4) P22 zeros(4) zeros(4,1) zeros(4,2);

zeros(4) zeros(4) P22 zeros(4) zeros(4,1) zeros(4,2)];

Sc12=[B1*K2 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

P1 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

B1*K2 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

C*(B1-B2)*K2 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) P21 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) (B1-B2)*K2 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) P21 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) C*(B1-B2)*K2 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) zeros(4) P22 zeros(4) zeros(4,1) zeros(4,2);

zeros(4) zeros(4) P22 zeros(4) zeros(4,1) zeros(4,2)];

Sc21=[B2*K1 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

P1 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

B1*K2 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

C*(B2-B1)*K1 zeros(4) zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) P21 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) (B2-B1)*K1 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) P21 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) C*(B2-B1)*K1 zeros(4) zeros(4) zeros(4,1) zeros(4,2);

zeros(4) zeros(4) P22 zeros(4) zeros(4,1) zeros(4,2);

zeros(4) zeros(4) P22 zeros(4) zeros(4,1) zeros(4,2)];

So11=[Dd'*P1 zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

Dd'*P1 zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

E21*K1 zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) Dd'*P21 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) Dd'*P2 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) E21*K1 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) E11*P21 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) Dd'*P21 zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) Dd'*C'*P2 zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) Dd' zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) Dd'*C'*P22 zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1)];

So22=[Dd'*P1 zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

Dd'*P1 zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

E22*K2 zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) Dd'*P21 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) Dd'*P2 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) E22*K2 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) E12*P21 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) Dd'*P21 zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) Dd'*C'*P2 zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) Dd' zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) Dd'*C'*P22 zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1)];

So12=[Dd'*P1 zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

Dd'*P1 zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

E21*K2 zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) Dd'*P21 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) Dd'*P2 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) E21*K2 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) E11*P21 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) Dd'*P21 zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) Dd'*C'*P2 zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) Dd' zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) Dd'*C'*P22 zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1)];

So21=[Dd'*P1 zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

Dd'*P1 zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

E22*K1 zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) Dd'*P21 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) Dd'*P2 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) E22*K1 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) E12*P21 zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) Dd'*P21 zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) Dd'*C'*P2 zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) Dd' zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) Dd'*C'*P22 zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1)];

Su11=[l6*E11'*E11+l1*E21*E21' zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) E21*E21' zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) l8*E11'*E11+l6*E21*E21' zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) l9*E11'*E11+l7*Dd*Dd' zeros(4,2) zeros(4,1);

zeros(2,4) zeros(2,4) zeros(2,4) zeros(2,4) zeros(2,2) zeros(2,1);

zeros(1,4) zeros(1,4) zeros(1,4) zeros(1,4) zeros(1,2) zeros(1,1)];

Su22=[l6*E12'*E12+l1*E22*E22' zeros(4) zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) E22*E22' zeros(4) zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) l8*E12'*E12+l6*E22*E22' zeros(4) zeros(4,2) zeros(4,1);

zeros(4) zeros(4) zeros(4) l9*E12'*E12+l7*Dd*Dd' zeros(4,2) zeros(4,1);

zeros(2,4) zeros(2,4) zeros(2,4) zeros(2,4) zeros(2,2) zeros(2,1);

zeros(1,4) zeros(1,4) zeros(1,4) zeros(1,4) zeros(1,2) zeros(1,1)];

Sk=[16*eye(4) zeros(4,50);zeros(4) l1*eye(4) zeros(4,46);zeros(4,8) (l1+l4+l5)*eye(4) zeros(4,42);zeros(4,12) eye(4) zeros(4,38);zeros(4,16) l2*(l3+l5)*eye(4) zeros(4,34);

zeros(1,20) l3 zeros(1,33);zeros(4,21) l4*eye(4) zeros(4,29);zeros(4,25) eye(4) zeros(4,25);zeros(4,29) l4*eye(4) zeros(4,21);zeros(4,33) l5*eye(4) zeros(4,17);

zeros(4,37) l3*l1*eye(4) zeros(4,13);zeros(4,41) eye(4) zeros(4,9);zeros(4,45) l6*eye(4) zeros(4,5);zeros(1,49) l5 zeros(1,4);zeros(4,50) eye(4)];

% Sys=[S11 So11' Sc11';So11 Sk zeros(54);Sc11 zeros(54) eye(54)]

%%% the constraints

Sys=[((S11+Su11)-[So11;Sc11]'*([-Sk zeros(54);zeros(54) eye(54)])*[So11;Sc11])<=0];

Sys=[Sys, ((S22+Su22)-[So22;Sc22]'*([-Sk zeros(54);zeros(54) eye(54)])*[So22;Sc22])<=0];

Sys=[Sys, ((S12+Su11)-[So12;Sc12]'*([-Sk zeros(54);zeros(54) eye(54)])*[So12;Sc12])<=0];

Sys=[Sys, ((S21+Su22)-[So21;Sc21]'*([-Sk zeros(54);zeros(54) eye(54)])*[So21;Sc21])<=0];

Sys=[Sys, ((S11+Su11+S12+Su11+S21+Su22)-[So11+So12+So21;Sc11+Sc12+Sc21]'*3*([-Sk zeros(54);zeros(54) eye(54)])*[So11+So12+So21;Sc11+Sc12+Sc21])<=0];

Sys=[Sys, ((S22+Su22+S12+Su11+S21+Su22)-[So22+So21+So21;Sc11+Sc12+Sc21]'*3*([-Sk zeros(54);zeros(54) eye(54)])*[So22+So21+So21;Sc11+Sc12+Sc21])<=0];

Sys=[Sys, P1>0, P21>0, P22>0, eta>0, P3>0, l1>0, l2>0, l3>0, l4>0, l5>0, l6>0, l7>0, l8>0, l9>0]

opt=sdpsettings('solver','sdpt3');

yalmipdiagnostics=solvesdp(Sys)

Johan Löfberg

unread,

Jun 17, 2015, 7:05:26 AM6/17/15

to yal...@googlegroups.com

Several issues with your code.

I would assume you want to have semidefinite constraints. However, the stuff you generate isn't symmetric. This is the term in the first constraint

>> ((S11+Su11)-[So11;Sc11]'*([-Sk zeros(54);zeros(54) eye(54)])*[So11;Sc11])

Polynomial matrix variable 19x19 (full, real, 71 variables)

and then, even if you got your transposes right etc and managed to create a symmetric expression, you are creating quartic stuff here, very far from LINEAR matrix inequalities, hence completely intractable to solve.

An obvious start is to note that you only use eta^2, hence you should replace eta^2 with a new linear variable, say, alpha.

However, you have a bunch of other nonlinear stuff, such as l3*l1, not to speak of the final construction of the constraints where you multiply several matrices all involving decision variables.

BTW, strict inequalities is not allowed, and solvesdp is called optimize now.

Ryad Ténor

unread,

Jun 18, 2015, 6:09:39 AM6/18/15

to yal...@googlegroups.com

Hello,

I have tried to change someting in my code.

Firstly I understand that non- linearities are caused by the product of variables SDP, for that i arranged to make variables changes (replacing eta^2 by eta, l1*l3=l3 etc...), i also see the problem of constraint wich isn't symetric, well i have change ((S11+Su11)-[So11;Sc11]'*([-Sk zeros(54);zeros(54) eye(54)])*[So11;Sc11]) by ((S11+Su11)-[So11;Sc11]'*([-Sk zeros(68,40);zeros(40,68) eye(40)])*[So11;Sc11]) giving : Polynomial matrix variable 19x19 (full, real, 60 variables) the number of variables diminished, and here i would ask a question, if i do M-M'

where M=((S11+Su11)-[So11;Sc11]'*([-Sk zeros(68,40);zeros(40,68) eye(40)])*[So11;Sc11]) do - I must have the result equal to zero to know that the constraint is symetric ??

In another register, I understood that the product of SDP matrices in the formulation of the model increases the number of non linearities, from this perspective I searched a way for linearizing and I found a command "linearize" in "http://www.mathworks.com/matlabcentral/fx_files/21936/1/content/ellipsoids/yalmip/htmldata/reference.htm", i use also "dualize" to convert a SDP problem given in primal form to the corresponding dual problem, when i run the file i have this error "NaN's cannot be converted to logicals."

Is it true what i have do ?

With my THANKS !

Johan Löfberg

unread,

Jun 18, 2015, 6:35:19 AM6/18/15

to yal...@googlegroups.com

You expression is still non-symmetric, and the reason is, e.g., that S11 is unsymmetric

You cannot use linearize to avoid your current issues. You have a bad polynomial model at the moment, you have to come up with a better model. linearize is used for obtaining a Taylor expansion around a given point. Dualize cannot be used either, it is for something very different. You have to fix the problem using pen-and-paper first, it is not something the modelling language can solve for you

Don't look in some old file mathworks refuses to remove from the matlab central. Use the YALMIP Wiki

http://users.isy.liu.se/johanl/yalmip/

Ryad Ténor

unread,

Jun 18, 2015, 6:38:22 AM6/18/15

to yal...@googlegroups.com

Ok i see, i'll begin from zero.

THANKS

Reply all

Reply to author

Forward

0 new messages