Solve discrete-time algebraic Riccati equations (DAREs)
[X,L,G] = dare(A,B,Q,R)
[X,L,G] = dare(A,B,Q,R,S,E)
[X,L,G,report] = dare(A,B,Q,...)
[X1,X2,L,report] = dare(A,B,Q,...,'factor')
computes the unique stabilizing solution [X,L,G] = dare(A,B,Q,R)
X
of the
discrete-time algebraic Riccati equation
$${A}^{T}XA-X-{A}^{T}XB{({B}^{T}XB+R)}^{-1}{B}^{T}XA+Q=0$$
The dare
function also returns the gain matrix, $$G={({B}^{T}XB+R)}^{-1}{B}^{T}XA$$, and the vector L
of
closed loop eigenvalues, where
L=eig(A-B*G,E)
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,
$${E}^{T}XE={F}^{T}XF-{F}^{T}XB{({B}^{T}XB+R)}^{-1}{B}^{T}XF+Q-S{R}^{-1}{S}^{T}$$
where $$F=A-B{R}^{-1}{S}^{T}$$. When omitted, R
, S
,
and E
are set to the default values R=I
, S=0
,
and E=I
.
The dare
function returns the corresponding
gain matrix $$G={({B}^{T}XB+R)}^{-1}({B}^{T}XA+{S}^{T})$$
and a vector L
of closed-loop eigenvalues,
where
L= eig(A-B*G,E)
[X,L,G,report] = dare(A,B,Q,...)
returns
a diagnosis report
with value:
-1
when the associated symplectic
pencil has eigenvalues on or very near the unit circle
-2
when there is no finite stabilizing
solution X
The Frobenius norm if X
exists
and is finite
[X1,X2,L,report] = dare(A,B,Q,...,'factor')
returns
two matrices, X1
and 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 associated Symplectic matrix has eigenvalues on the
unit circle.
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
$$\left[\begin{array}{cc}Q& S\\ {S}^{T}& R\end{array}\right]>0$$
dare
implements the algorithms described
in [1]. It uses the QZ algorithm
to deflate the extended symplectic pencil and compute its stable invariant subspace.
[1] Arnold, W.F., III and A.J. Laub, "Generalized Eigenproblem Algorithms and Software for Algebraic Riccati Equations," Proc. IEEE^{®}, 72 (1984), pp. 1746-1754.