Issue in drawing the outer approximation

[DexterTsOH]

Thanks for sharing the toolbox. When I use it to draw the roa of a dynamic system of 4 state variables, the program failed to draw a outer approximation ,but successed in drawing an inner approximation. Is it normal? Thanks for your help!
Vdc=200;
rf=1.1;
Lf=39.5e-3;
C=501e-6;
pff=0.496;
f=2e-3;
J=3.2e-3;
Gamma0=1.6;
Gamma=1.6;
y=Gamma-Gamma0;
omega0=15002pi/60;
Iq0=(fomega0+Gamma0)/pff;
ps=pffIq0omega0;
Vs0=0.5(Vdc+sqrt(Vdc^2-4psrf));
Kpo=0.194;
Kio=0.121;

SYS4eq = [0; 0; 0; 0];
DeltaSYS4 = [20; 200; 80; 40];
SYS4= SOStab(SYS4eq, DeltaSYS4);
f1=1/Vs0-1/Vs0^2SYS4.x(2)+1/Vs0^3SYS4.x(2)^2;
f2=SYS4.x(3)+omega0;
SYS4.dynamics=[ -rf/LfSYS4.x(1)-1/LfSYS4.x(2);...
1/CSYS4.x(1)+pff/(CVs0)f1(omega0Iq0SYS4.x(2)+Vs0*(Kpof2-Iq0)SYS4.x(3)-Vs0Kiof2SYS4.x(4));...
-1/J(pffKpo+f)SYS4.x(3)+1/JpffKio*SYS4.x(4) ;...
-SYS4.x(3)];

T=10;
epsilon =200;
d=6;

[v1,vc1,wc1]=SYS4.SoS_out(d, T, epsilon);
SYS4.plot_roa(1,2,'outer',0);
[v2,vc2,wc2]=SYS4.SoS_in(d,T,epsilon);
SYS4.plot_roa(1,2,'inner',1);
% % figure
SYS4.plot_w(1,2,'outer')

[droste89]

Hello, thanks for your comment.
I'll just give you the short answer for the moment, as I unfortunately can't make the calculation on my side for the moment (I'm sorry I already took some days to answer you):

• it is "not normal" that the inner approximation is plotted and not the outer: the latter should be more accurately/more easily be calculated, so it seems strange that the only the former is plotted. Moreover, the fact that the inner region is inside the target set is also strange.
• I think the first thing to check is the value of the w polynomial: if it is too close to 1 (constant polynomial equals to 1, ie w as a vector looks like [1, 0.00001, 0.000, ...] it means that the polynomial degree is too low and the calculation result is not relevant. Then you could try to increase the degree to 10 and rerun the calculation.
• I see another parameter that can impact the quality of the result: the time horizon T. When it is "too long", the dynamics become more complex to compute, so you could also try to make a calculation with T=1 to see what happens. And then increase it if the results are more satisfying.