Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
ZigangPan`NonlinearCholeskyFactorization`
backsteppinglocaloptimalmatchingglobalinverseoptimalN  |
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Examples
(1)
Basic Examples
(1)
In[1]:=
xc={x1,x2};x1m=-2;x1M=2;
In[2]:=
$Assumptions=((x1|x2)∈Reals);
In[3]:=
fe[x1_,x2_]:={x2+x1^2,0}
In[4]:=
ge[x1_,x2_]:={{0},{1}}
In[5]:=
he[x1_,x2_]:={{1},{0}}
In[6]:=
qe[x1_,x2_]:=x1^2/(x1-x1m)/(x1M-x1)+x2^2
In[7]:=
γ=4;
In[8]:=
f=Apply[fe,xc];
In[9]:=
g=Apply[ge,xc];
In[10]:=
h=Apply[he,xc];
In[11]:=
r={{1}};
In[12]:=
q=Apply[qe,xc];
In[13]:=
s=(g.Inverse[r].Transpose[g]-h.Transpose[h]/γ^2)/4;
In[14]:=
Va2=
[f,s,q,xc,2]
 | approximateHJIequation |
Out[14]=
0.74566078353098269398742583173524076969324162745495+1.0672406012816462175243920149575540149877246679402x1x2+1.4439658982677348429739207447791524129284836319254
2
x1
2
x2
In[15]:=
Va3=
[f,s,q,xc,3]
| approximateHJIequation |
Out[15]=
0.74566078353098269398742583173524076969324162745495+1.7999503770403896165422532400554460798401124030983+1.0672406012816462175243920149575540149877246679402x1x2+3.266320905623785474113185224022831513184551539736x2+1.4439658982677348429739207447791524129284836319254+2.0943483047946629482898669389150365008839723100133x1+0.4995957038976601482760750899761287023393977940874
2
x1
3
x1
2
x1
2
x2
2
x2
3
x2
In[16]:=
ans=
[f,g,h,q,r,γ,xc,3,x1m,x1M,3];
 | backsteppinglocaloptimalmatchingglobalinverseoptimalN |
In[17]:=
Vx=D[ans〚1〛,{xc}];
In[18]:=
Plot3D[ans〚1〛,{x1,-2,2},{x2,-5,5},PlotRange{0,3}]
Out[18]=
In[19]:=
Plot3D[ans〚4〛,{x1,-2,2},{x2,-5,5},PlotRange{0,2}]
Out[19]=
In[20]:=
Plot3D[ans〚3〛,{x1,-2,2},{x2,-5,5},PlotRange{0,20}]
Out[20]=
In[21]:=
Plot3D[ans〚2〛,{x1,-2,2},{x2,-5,5},PlotRange{-60,10}]
Out[21]=
In[22]:=
Plot3D[Vx.f-Vx.(g.Inverse[ans〚4〛].Transpose[g]-1/γ^2h.Transpose[h]).Vx/4+ans〚3〛,{x1,-2,2},{x2,-5,5},PlotRange{-0.1,0.1}]
Out[22]=
In[23]:=
closedloopdynamics=FormulaToFunction[xc,f+g.ans〚2〛+h.{0}];
In[24]:=
sol=RungeKutta45[closedloopdynamics,0,{0.4,0.5},20,1/100];
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