Function Reference

dare

Solve discrete-time algebraic Riccati equations (DARE)

Syntax

• [X,L,G,rr] = dare(A,B,Q,R)
  [X,L,G,rr] = dare(A,B,Q,R,S,E)
  
  [X,L,G,report] = dare(A,B,Q,...,'report')
  [X1,X2,L,report] = dare(A,B,Q,...,'implicit')
  

Description

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

such that the "closed-loop" matrix

has all its eigenvalues inside the unit disk. The matrix is symmetric and called the stabilizing solution of . [X,L,G,rr] = dare(A,B,Q,R) also returns:

• The eigenvalues L of
• The gain matrix

• The relative residual rr defined by

[X,L,G,rr] = dare(A,B,Q,R,S,E) solves the more general DARE:

The corresponding gain matrix and closed-loop eigenvalues are

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

Two additional syntaxes are provided to help develop applications such as -optimal control design.

[X,L,G,report] = dare(A,B,Q,...,'report') turns off the error messages when the solution fails to exist and returns a failure report instead. The value of report is:

• -1 when the associated symplectic pencil has eigenvalues on or very near the unit circle (failure)
• -2 when there is no finite solution, that is, with singular (failure)
• The relative residual defined above when the solution exists (success)

Alternatively, [X1,X2,L,report] = dare(A,B,Q,...,'implicit') also turns off error messages but now returns in implicit form as

Note that this syntax returns report = 0 when successful.

Algorithm

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

Limitations

The pair must be stabilizable (that is, all eigenvalues of 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 detectable when and , or

See Also
care        Solve continuous-time Riccati equations

dlyap       Solve discrete-time Lyapunov equations

References

[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.

damp

dcgain