# dare

(Not recommended) Solve discrete-time algebraic Riccati equations

`dare`

is not recommended. Use `idare`

instead. For more information, see Version
History.

## Syntax

## Description

`[`

solves the more general discrete-time algebraic Riccati equation:`X`

,`L`

,`G`

] = dare(`A`

,`B`

,`Q`

,`R`

,`S`

,`E`

)

$${A}^{T}XA-{E}^{T}XE-({A}^{T}XB+S){({B}^{T}XB+R)}^{-1}({B}^{T}XA+{S}^{T})+Q=0$$

or, equivalently, if `R`

is nonsingular:

$${E}^{T}XE={F}^{T}XF-{F}^{T}XB{({B}^{T}XB+R)}^{-1}{B}^{T}XF+Q-S{R}^{-1}{S}^{T}$$

Here, $$F=A-B{R}^{-1}{S}^{T}$$.

## Examples

## Input Arguments

## Output Arguments

## Limitations

The (*A*, *B*) pair must be stabilizable (that is,
all eigenvalues of *A* outside the unit disk must be controllable). In
addition, the associated symplectic pencil must have no eigenvalue on the unit circle.
Sufficient conditions for this to hold are (*Q*, *A*)
detectable when *S* = *0* and *R*
> *0*, or

$$\left[\begin{array}{cc}Q& S\\ {S}^{T}& R\end{array}\right]>0$$

## Algorithms

`dare`

implements the algorithms described in [1]. It uses the QZ algorithm to deflate the extended symplectic pencil and compute its
stable invariant subspace.

## References

[1] Arnold, W.F., and A.J. Laub. “Generalized Eigenproblem
Algorithms and Software for Algebraic Riccati Equations.” *Proceedings of the IEEE* 72, no. 12 (1984): 1746–54.
https://doi.org/10.1109/PROC.1984.13083.

## Version History

**Introduced before R2006a**