 |
BUILT-IN MATHEMATICA SYMBOL
DiscreteRiccatiSolve
DiscreteRiccatiSolve[{a, b}, {q, r}]
gives the matrix 
that is the stabilizing solution of the discrete algebraic Riccati equation ![TemplateBox[{a}, ConjugateTranspose].x.a-x-TemplateBox[{a}, ConjugateTranspose].x.b.TemplateBox[{{(, {r, +, {TemplateBox[{b}, ConjugateTranspose], ., x, ., b}}, )}}, Inverse].TemplateBox[{b}, ConjugateTranspose].x.a+q=0](Files/DiscreteRiccatiSolve.en/2.png "TemplateBox[{a}, ConjugateTranspose].x.a-x-TemplateBox[{a}, ConjugateTranspose].x.b.TemplateBox[{{(, {r, +, {TemplateBox[{b}, ConjugateTranspose], ., x, ., b}}, )}}, Inverse].TemplateBox[{b}, ConjugateTranspose].x.a+q=0"){kind=small}
.
DiscreteRiccatiSolve[{a, b}, {q, r, p}]
solves ![TemplateBox[{a}, ConjugateTranspose].x.a-x-(TemplateBox[{a}, ConjugateTranspose].x.b+p).TemplateBox[{{(, {r, +, {TemplateBox[{b}, ConjugateTranspose], ., x, ., b}}, )}}, Inverse].(TemplateBox[{b}, ConjugateTranspose].x.a+TemplateBox[{p}, ConjugateTranspose])+q=0](Files/DiscreteRiccatiSolve.en/3.png "TemplateBox[{a}, ConjugateTranspose].x.a-x-(TemplateBox[{a}, ConjugateTranspose].x.b+p).TemplateBox[{{(, {r, +, {TemplateBox[{b}, ConjugateTranspose], ., x, ., b}}, )}}, Inverse].(TemplateBox[{b}, ConjugateTranspose].x.a+TemplateBox[{p}, ConjugateTranspose])+q=0"){kind=small}
.
Details and Options
- In
![TemplateBox[{a}, ConjugateTranspose].x.a-x-TemplateBox[{a}, ConjugateTranspose].x.b.TemplateBox[{{(, {r, +, {TemplateBox[{b}, ConjugateTranspose], ., x, ., b}}, )}}, Inverse].TemplateBox[{b}, ConjugateTranspose].x.a+q=0](Files/DiscreteRiccatiSolve.en/4.png "TemplateBox[{a}, ConjugateTranspose].x.a-x-TemplateBox[{a}, ConjugateTranspose].x.b.TemplateBox[{{(, {r, +, {TemplateBox[{b}, ConjugateTranspose], ., x, ., b}}, )}}, Inverse].TemplateBox[{b}, ConjugateTranspose].x.a+q=0")
, 
denotes the conjugate transpose. - The equation
![TemplateBox[{a}, ConjugateTranspose].x.a-x-TemplateBox[{a}, ConjugateTranspose].x.b.TemplateBox[{{(, {r, +, {TemplateBox[{b}, ConjugateTranspose], ., x, ., b}}, )}}, Inverse].TemplateBox[{b}, ConjugateTranspose].x.a+q=0](Files/DiscreteRiccatiSolve.en/6.png "TemplateBox[{a}, ConjugateTranspose].x.a-x-TemplateBox[{a}, ConjugateTranspose].x.b.TemplateBox[{{(, {r, +, {TemplateBox[{b}, ConjugateTranspose], ., x, ., b}}, )}}, Inverse].TemplateBox[{b}, ConjugateTranspose].x.a+q=0")
has a unique, symmetric, positive semidefinite solution only if 
is stabilizable, 
is detectable, 
, and 
. Consequently, all the eigenvalues of the matrix 
lie inside the unit circle, and the solution is stabilizing. - The solution is positive definite when
is controllable and 
is observable. - The eigenvalues of the symplectic matrix
![(a+b.r^(-1).b.TemplateBox[{{(, TemplateBox[{a}, ConjugateTranspose], )}}, Inverse].q -b .r^(-1).b.TemplateBox[{{(, TemplateBox[{a}, ConjugateTranspose], )}}, Inverse]; -TemplateBox[{{(, TemplateBox[{a}, ConjugateTranspose], )}}, Inverse].q TemplateBox[{{(, TemplateBox[{a}, ConjugateTranspose], )}}, Inverse])](Files/DiscreteRiccatiSolve.en/14.png "(a+b.r^(-1).b.TemplateBox[{{(, TemplateBox[{a}, ConjugateTranspose], )}}, Inverse].q -b .r^(-1).b.TemplateBox[{{(, TemplateBox[{a}, ConjugateTranspose], )}}, Inverse]; -TemplateBox[{{(, TemplateBox[{a}, ConjugateTranspose], )}}, Inverse].q TemplateBox[{{(, TemplateBox[{a}, ConjugateTranspose], )}}, Inverse])")
must not contain any symbolic expressions. - DiscreteRiccatiSolve supports a Method option. The following explicit settings can be specified:
-
"Eigensystem" uses eigenvalue decomposition "Schur" uses Schur decomposition - The default setting Method->Automatic selects
for exact matrices and 
as the primary method for real matrices. - Method->"Schur" only works with real matrices.
New in 8
