dare

(Not recommended) Solve discrete-time algebraic Riccati equations

dare is not recommended. Use idare instead. For more information, see Version History.

Syntax

[X,L,G] = dare(A,B,Q,R)

[X,L,G] = dare(A,B,Q,R,S,E)

[X,L,G,report] = dare(___)

[X1,X2,D,L] = dare(___,'factor')

Description

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

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

The dare function also returns a gain matrix G and a vector L of closed-loop eigenvalues.

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

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

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

[X,L,G,report] = dare(___) also returns a diagnosis report.

[X1,X2,D,L] = dare(___,'factor') returns a factorized solution of the Riccati equation.

Examples

Solve Discrete-Time Algebraic Riccati Equation

Solve the discrete-time algebraic Riccati equation considering the following set of matrices:

A = [-0.9,-0.3;0.7,0.1];
B = [1;1];
Q = [1,0;0,3];
R = 0.1;

Find the stabilizing solution using dare to solve for the above matrices with default values for S and E.

[X,L,G,report] = dare(A,B,Q,R)

X =

    4.7687    0.9438
    0.9438    3.2369


L =

   -0.4460
   -0.0027


G =

   -0.2216   -0.1297


report =

   9.4192e-16

Input Arguments

A, B, Q, R, S, E — Input matrices

matrices

Input matrices, specified as matrices. When you omit R, S, and E, the function uses the default values R = I, S = 0, and E = I.

Output Arguments

X — Riccati equation solution

matrix

Solution to the discrete-time algebraic Riccati equation, returned as a matrix.

dare returns [] for X when the associated Symplectic matrix has eigenvalues on the unit circle.

L — Closed-loop eigenvalues

vector

Closed-loop eigenvalues, returned as a matrix.

The closed-loop eigenvalues L is computed as:

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

G — State-feedback gain

matrix

State-feedback gain, returned as a matrix.

The state-feedback gain G is computed as:

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

dare returns [] for G when the associated Symplectic matrix has eigenvalues on the unit circle.

report — Diagnosis report

scalar

Diagnosis report, returned as a scalar with one of these values:

• -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, D — Factorized solution

matrices

Factorized solution matrices, returned as matrices. The function returns X₁, X₂, and a diagonal scaling matrix D such that X = D(X₁/X₂)D.

dare returns X1, X2, and D as 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 [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

R2019a: dare not recommended

Starting in R2019a, use the idare command to solve discrete-time Riccati equations. This approach has improved accuracy through better scaling and the computation of K is more accurate when R is ill-conditioned relative to dare. Furthermore, idare includes an optional info structure to gather the implicit solution data of the Riccati equation.

The following table shows some typical uses of dare and how to update your code to use idare instead.

Not Recommended	Recommended
[X,L,G] = dare(A,B,Q,R,S,E)	[X,K,L] = idare(A,B,Q,R,S,E) computes the stabilizing solution X, the state-feedback gain K and the closed-loop eigenvalues L of the discrete-time algebraic Riccati equation. For more information, see idare.
[X,L,G,report] = dare(A,B,Q,R,S,E)	[X,K,L,info] = idare(A,B,Q,R,S,E) computes the stabilizing solution X, the state-feedback gain K, the closed-loop eigenvalues L of the discrete-time algebraic Riccati equation. The info structure contains the implicit solution data. For more information, see idare.

There are no plans to remove dare at this time.