Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
ZigangPan`NonlinearCholeskyFactorizationMIMO`
backsteppinglocaloptimalmatchingglobalinverseoptimalMIMO  |
| { V αn qcheck rcheck f g h qhat r γ xc m {n1,n2,...,nn} x1m x1M ϵ SC m u p w q t ∫ 0 f[ xiis n i 1≤ n 1 n 2 n n g[xc]= 0 n 1 0 n n-1 0 n ∑ j1 n j 0 n ∑ j1 n j pp ∘ r x1m,x1M n ∑ i=2 n i n 1 n 1 0 n 1 γ ϵ SC xc V αn qcheck rcheck SC αn qcheck rcheck SC | |
Examples
(1)
Basic Examples
(1)
In[1]:=
f={x2+x3+x4,-x1+x1^2-x3+x4,-x1-x3,-x2-x4};
In[2]:=
g={{0,0},{0,0},{1,x2},{-x2,1}};
In[3]:=
h={{1,0},{0,1},{0,0},{0,0}}
Out[3]=
{{1,0},{0,1},{0,0},{0,0}}
In[4]:=
qhat=x1^2+x2^2+x3^2+x4^2;r=IdentityMatrix[2]
Out[4]=
{{1,0},{0,1}}
In[5]:=
x1m={-Infinity,-Infinity};x1M={Infinity,Infinity};
In[6]:=
γ=2
Out[6]=
2
In[7]:=
xc={x1,x2,x3,x4}
Out[7]=
{x1,x2,x3,x4}
In[8]:=
nlsystem={xc,{u1,u2,w1,w2},Join[{y1,y2,y3,y4,z1,z2,z3,z4,z5,z6}],{FormulaToFunction[Join[xc,{u1,u2,w1,w2}],f+g.{u1,u2}+h.{w1,w2}],FormulaToFunction[Join[xc,{u1,u2,w1,w2}],{x1,x2,x3,x4,x1,x2,x3,x4,u1,u2}]},{1,2},{},{1,2},{3,4},{1,2,3,4},{5,6,7,8,9,10}}
Out[8]=
{{x1,x2,x3,x4},{u1,u2,w1,w2},{y1,y2,y3,y4,z1,z2,z3,z4,z5,z6},{Function[{x1,x2,x3,x4,u1,u2,w1,w2},{w1+x2+x3+x4,w2-x1+-x3+x4,u1-x1+u2x2-x3,u2-x2-u1x2-x4}],Function[{x1,x2,x3,x4,u1,u2,w1,w2},{x1,x2,x3,x4,x1,x2,x3,x4,u1,u2}]},{1,2},{},{1,2},{3,4},{1,2,3,4},{5,6,7,8,9,10}}
2
x1
In[9]:=
Va2=approximateHJIequation[f,(g.Inverse[r].Transpose[g]-h.Transpose[h]/γ^2)/4,qhat,xc,2]
Out[9]=
1.3279960796415705957959144606893692174354347621778+1.3279960796415705957959144606893692174354347621778+0.95864724879449000301768096905120855138966725905995x1x3-1.0049776339522068617657932214298995796894682578026x2x3+1.0209370152900697609038233237234845982591297973183+1.0049776339522068617657932214298995796894682578026x1x4+0.95864724879449000301768096905120855138966725905995x2x4+1.0209370152900697609038233237234845982591297973183
2
x1
2
x2
2
x3
2
x4
In[10]:=
approximatenonlinearCholeskyfactorization[Va2,xc〚4;;1;;-1〛,2]
Out[10]=
{{Function[{zl1,zl2,zl3,zl4},1.0209370152900697609038233237234845982591297973183],Function[{zl2,zl3,zl4},1.0209370152900697609038233237234845982591297973183],Function[{zl3,zl4},0.855639665025396783256209836754720413904946876272],Function[{zl4},0.855639665025396783256209836754720413904946876272]},{0.49218395400556198760587093525286486621433053068883x1+0.46949382500453178293663344349754520365791471426778x2+x4,0.46949382500453178293663344349754520365791471426778x1-0.49218395400556198760587093525286486621433053068883x2+x3,x2,x1}}
In[11]:=
m=1;
In[12]:=
nvec={2,2}
Out[12]=
{2,2}
In[13]:=
ϵ=m;
In[14]:=
{V,controllaw,qcheck,rcheck}=
[f,g,h,qhat,r,γ,xc,m,nvec,x1m,x1M,ϵ,False];
 | backsteppinglocaloptimalmatchingglobalinverseoptimalMIMO |
Please check Assumption 8.29! This program will assume that Assumption 8.29 holds!
In[15]:=
controller={{},{y1,y2,y3,y4},{u1,u2},{FormulaToFunction[{y1,y2,y3,y4},{}],FormulaToFunction[xc,controllaw]},{1,2,3,4},{1,2},{1,2,3,4},{},{1,2},{}};
In[16]:=
disturbancesystem={{},{z1,z2,z3,z4,z5,z6},{w1,w2},{FormulaToFunction[{x1,x2,x3,x4,u1,u2},{}],FormulaToFunction[{x1,x2,x3,x4,u1,u2},{0,0}]},{1,2,3,4,5,6},{1,2},{1,2,3,4,5,6},{},{1,2},{}};
In[17]:=
{
systemstates
,controllerstates
,disturbancestates
,controlinput
,disturbanceinput
,measurementoutput
,controlledoutput
,controllersignals
}=simulationNLsystem[nlsystem
,{0.7,0.6,0.5,0.4},controller
,{},disturbancesystem
,{},0,10,Command"RungeKutta45",StartingStepSize1/100];No direct feedthrough in first system!
No direct feedthrough in first system!
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In[18]:=
Plot[{systemstates[t]〚1〛,systemstates[t]〚2〛,systemstates[t]〚3〛,systemstates[t]〚4〛},{t,0,10},PlotRange{-0.7,1.1}]
Out[18]=
In[19]:=
Plot[{controlinput[t]〚1〛,controlinput[t]〚2〛},{t,0,10},PlotRange{-2.8,2}]
Out[19]=
