dare (Control System Toolbox)

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