Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
ZigangPan`RobustBackSteppingCancellation`
RobustBackSteppingArztan  |
| {V,α,x}=RobustBackSteppingArztan[Vo,αo,fo,ho,fa,ga,ha,xo,xa,xd,γ,Φfun,Δfun,Rfun,ϱ3,ϱ4,{ϱ1,ϱ2}] fo[xo,xa,xd] ho[xo,xa,xd] xa xo xa xd xo =fo[xo,xa,xd]+ho[xo,xa,xd]w xa =fa[xo,xa,xd]+ga[xo,xa,xd]u+ha[xo,xa,xd]w fo ho ga ha Vo xa=αo[xo] xo t ∫ 0 Vo w V α[xo,xa,xd] u Vo xo αo xa γ Φfun xo,xa,xd Δfun V xo,xa Rfun xo,xa,xd V=Vo+(xa-αo[xo])'Δfun(xa-αo[xo]) ϱ3 ϱ4 ϱ1,ϱ2 (1+ϱ2)ϱ1<1 t ∫ 0 V w l[xo,xa,xd]≥lo[xo,xd]+(xa-αo[xo])'Φfun[xo,xa,xd] V,α,Vo,αo,fo,ho,fa,ga,ha,Φfun,Δfun,Rfun | |
Examples
(1)
Basic Examples
(1)
In[1]:=
Needs["ZigangPan`Examples`"]
In[2]:=
Needs["ZigangPan`DifferentialEquationSolver`"]
In[3]:=
xc={x1,x2,x3,x4};$Assumptions=assumeReal[xc]
Out[3]=
x1∈&&x2∈&&x3∈&&x4∈
In[4]:=
fe[x1_,x2_,x3_,x4_]:={x3+x1^2,x1*x2+x4,x1*x3,x2*x4}
In[5]:=
ge[x1_,x2_,x3_,x4_]:={{0,0},{0,0},{1,0},{0,-1}}
In[6]:=
he[x1_,x2_,x3_,x4_]:={{1,0},{-1,x1},{0,0},{1,1}}
In[7]:=
V0=0;α0={0,0};γ=4;f=Apply[fe,xc];g=Apply[ge,xc];h=Apply[he,xc];rbar={{3,1},{1,2}};
In[8]:=
{V1,α1,x}=
[V0,α0,{},{},f〚1;;2〛/.{x30,x40},D[f〚1;;2〛,{{x3,x4}}],h〚1;;2,All〛,{},{x1,x2},{},γ,{x1,x2},IdentityMatrix[2]/2]
 | RobustBackSteppingCancellation |
Out[8]=
+,-x1-+(-x1+x2),-x2-x1x2+(x1-(1+)x2),{x1,x2}
2
x1
2
2
x2
2
2
x1
1
64
1
64
2
x1
In[9]:=
{V,α,x}=
[V1,α1,f〚1;;2〛,h〚1;;2,All〛,f〚3;;4〛,g〚3;;4,All〛,h〚3;;4,All〛,{x1,x2},{x3,x4},{},γ^2,{x1,x2}-α1,1/2*IdentityMatrix[2],rbar,0.7,0.8,{0.4,0.5}];
 | RobustBackSteppingArztan |
In[10]:=
closedloopdynamics=FormulaToFunction[xc,f+g.α+h.{0,0}];
In[11]:=
sol=RungeKutta45[closedloopdynamics,0,{-2,2.,1,0},20,1/10];
10% completed.
100% completed.
In[12]:=
Plot[sol[t],{t,0,20},PlotRangeAll]
Out[12]=
""