dare

(Not recommended) Solve discrete-time algebraic Riccati equations (DAREs)

Syntax

[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')

Description

[X,L,G] = dare(A,B,Q,R) computes the unique stabilizing solution X of the discrete-time algebraic Riccati equation

${A}^{T}XA-X-{A}^{T}XB{\left({B}^{T}XB+R\right)}^{-1}{B}^{T}XA+Q=0$

The dare function also returns the gain matrix, $G={\left({B}^{T}XB+R\right)}^{-1}{B}^{T}XA$, and the vector L of closed loop eigenvalues, where

L=eig(A-B*G,E)

[X,L,G] = dare(A,B,Q,R,S,E) solves the more general discrete-time algebraic Riccati equation,

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

or, equivalently, if R is nonsingular,

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

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 $G={\left({B}^{T}XB+R\right)}^{-1}\left({B}^{T}XA+{S}^{T}\right)$

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.

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 . It uses the QZ algorithm to deflate the extended symplectic pencil and compute its stable invariant subspace.

Compatibility Considerations

expand all

Not recommended starting in R2019a

References

 Arnold, W.F., III and A.J. Laub, "Generalized Eigenproblem Algorithms and Software for Algebraic Riccati Equations," Proc. IEEE®, 72 (1984), pp. 1746-1754.