State-space model, algebraic degeneration example


tableau model, equations

(%i6) e1 : i1 + i2 = 0 ;
e2 : u1 v1 = 0 ;
e3 : u2 v1 = 0 ;
e4 : u2 = eg ;
e5 : C· Du2 i2 = 0 ;
T :[ e1, e2, e3, e4, e5] ;
(e1) i2 + i1 = 0 (e2) u1 v1 = 0 (e3) u2 v1 = 0 (e4) u2 = eg (e5) C Du2 i2 = 0 (T) [ i2 + i1 = 0 , u1 v1 = 0 , u2 v1 = 0 , u2 = eg , C Du2 i2 = 0 ]


special vectors

(%i8) Dx :[ Du2] ;
y :[ i1, i2, u1, v1] ;
(Dx) [ Du2 ] (y) [ i1 , i2 , u1 , v1 ]


state equations

(%i9) se : eliminate( T, y) ;
(se) [ u2 eg ]

algebraic degeneration!

solve for the degenerate variable

(%i10) se : linsolve( se, u2)[ 1] ;
(se) u2 = eg

algebraic relation!

subsitute for the degenerated state variable derivative

(%i11) de : Du2 = Deg ;
(de) Du2 = Deg
(%i12) Tmod : ev( T, de) ;
(Tmod) [ i2 + i1 = 0 , u1 v1 = 0 , u2 v1 = 0 , u2 = eg , C Deg i2 = 0 ]


there are no state equations; only output equations remain

u2 is now an output variable, let's change the output vector

(%i13) ynew :[ i1, i2, u1, u2, v1] ;
(ynew) [ i1 , i2 , u1 , u2 , v1 ]
(%i14) oe : linsolve( Tmod, ynew) ;
(oe) [ i1 = C Deg , i2 = C Deg , u1 = eg , u2 = eg , v1 = eg ]

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