# Documentation

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# gdare

Generalized solver for discrete-time algebraic Riccati equation

## Syntax

`[X,L,report] = gdare(H,J,ns)[X1,X2,D,L] = gdare(H,J,NS,'factor')`

## Description

`[X,L,report] = gdare(H,J,ns)` computes the unique stabilizing solution `X` of the discrete-time algebraic Riccati equation associated with a Symplectic pencil of the form

`$H-tJ=\left[\begin{array}{ccc}A& F& B\\ -Q& E\prime & -S\\ S\prime & 0& R\end{array}\right]-\left[\begin{array}{ccc}E& 0& 0\\ 0& A\prime & 0\\ 0& B\prime & 0\end{array}\right]$`

The third input `ns` is the row size of the A matrix.

Optionally, `gdare` returns the vector `L` of closed-loop eigenvalues and a diagnosis `report` with value:

• -1 if the Symplectic pencil has eigenvalues on the unit circle

• -2 if there is no finite stabilizing solution `X`

• 0 if a finite stabilizing solution `X` exists

This syntax does not issue any error message when `X` fails to exist.

`[X1,X2,D,L] = gdare(H,J,NS,'factor')` returns two matrices `X1`, `X2` and a diagonal scaling matrix `D` such that `X = D*(X2/X1)*D`. The vector `L` contains the closed-loop eigenvalues. All outputs are empty when the Symplectic pencil has eigenvalues on the unit circle.