care

Continuous-time algebraic Riccati equation solution

Syntax

[X,L,G] = care(A,B,Q)
[X,L,G] = care(A,B,Q,R,S,E)
[X,L,G,report] = care(A,B,Q,...)
[X1,X2,D,L] = care(A,B,Q,...,'factor')

Description

[X,L,G] = care(A,B,Q) computes the unique solution X of the continuous-time algebraic Riccati equation

ATX+XAXBBTX+Q=0

The care function also returns the gain matrix, G=R1BTXE.

[X,L,G] = care(A,B,Q,R,S,E) solves the more general Riccati equation

ATXE+ETXA(ETXB+S)R1(BTXE+ST)+Q=0

When omitted, R, S, and E are set to the default values R=I, S=0, and E=I. Along with the solution X, care returns the gain matrix G=R1(BTXE+ST) and a vector L of closed-loop eigenvalues, where

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

[X,L,G,report] = care(A,B,Q,...) returns a diagnosis report with:

  • -1 when the associated Hamiltonian pencil has eigenvalues on or very near the imaginary axis (failure)

  • -2 when there is no finite stabilizing solution X

  • The Frobenius norm of the relative residual if X exists and is finite.

This syntax does not issue any error message when X fails to exist.

[X1,X2,D,L] = care(A,B,Q,...,'factor') returns two matrices X1, 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 Hamiltonian matrix has eigenvalues on the imaginary axis.

Examples

Example 1

Solve Algebraic Riccati Equation

Given

A=[3211]B=[01]C=[11]R=3

you can solve the Riccati equation

ATX+XAXBR1BTX+CTC=0

by

a = [-3 2;1 1]
b = [0 ; 1]
c = [1 -1]
r = 3
[x,l,g] = care(a,b,c'*c,r)

This yields the solution

x

x =
    0.5895    1.8216
    1.8216    8.8188

You can verify that this solution is indeed stabilizing by comparing the eigenvalues of a and a-b*g.

[eig(a)   eig(a-b*g)]
 
ans =
   -3.4495   -3.5026
    1.4495   -1.4370

Finally, note that the variable l contains the closed-loop eigenvalues eig(a-b*g).

l

l =
   -3.5026
   -1.4370

Example 2

Solve H-infinity (H)-like Riccati Equation

To solve the H-like Riccati equation

ATX+XA+X(γ2B1B1TB2B2T)X+CTC=0

rewrite it in the care format as

ATX+XAX[B1,B2]B[γ2I00I]R1[B1TB2T]X+CTC=0

You can now compute the stabilizing solution X by

B = [B1 , B2]
m1 = size(B1,2)
m2 = size(B2,2)
R = [-g^2*eye(m1) zeros(m1,m2) ; zeros(m2,m1) eye(m2)]

X = care(A,B,C'*C,R)

Limitations

The (A,B) pair must be stabilizable (that is, all unstable modes are controllable). In addition, the associated Hamiltonian matrix or pencil must have no eigenvalue on the imaginary axis. Sufficient conditions for this to hold are (Q,A) detectable when S=0 and R>0, or

[QSSTR]>0

More About

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Algorithms

care implements the algorithms described in [1]. It works with the Hamiltonian matrix when R is well-conditioned and E=I; otherwise it uses the extended Hamiltonian pencil and QZ algorithm.

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

See Also

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