care

(Not recommended) Solve continuous-time algebraic Riccati equation

care is not recommended. Use icare instead. For more information, see Version History.

Syntax

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

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

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

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

Description

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

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

Along with the solution X, care returns the gain matrix G and a vector L of closed-loop eigenvalues.

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

$$A^{T}X+XA-XB{B}^{T}X+Q=0$$

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

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

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

Examples

Solve Algebraic Riccati Equation

This example show how to solve the following Riccati equation:

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

Given

$$A=\left[\begin{array}{cc}-3& 2\\ 1& 1\end{array}\right]B=\left[\begin{array}{c}0\\ 1\end{array}\right]C=\left[\begin{array}{cc}1& -1\end{array}\right]R=3$$

Define the matrices and solve the equation.

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

x =

    0.5895    1.8216
    1.8216    8.8188


l =

   -1.4370
   -3.5026


g =

    0.6072    2.9396

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).

Solve H-infinity-like Riccati Equation

This example show how to solve the $$H_{\infty }$$-like Riccati equation

$$A^{T}X+XA+X({\gamma }^{-2}{B}_1{B}_1^T-B_2{B}_2^T)X+C^{T}C=0$$

You can rewrite the equation in the format supported by care as follows:

$$A^{T}X+XA-X\underset{B}{\underbrace{[B_1,B_2]}}{\underset{R}{\underbrace{\left[\begin{array}{cc}-{\gamma }^{2}I& 0\\ 0& I\end{array}\right]}}}^{-1}\left[\begin{array}{c}{B}_1^T\\ B_2^T\end{array}\right]X+C^{T}C=0$$

For any given input matrices, 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)

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 continuous-time algebraic Riccati equation, returned as a matrix.

care returns [] for X when the associated Hamiltonian matrix has eigenvalues on the imaginary axis.

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:

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

care returns [] for G when the associated Hamiltonian matrix has eigenvalues on the imaginary axis.

report — Diagnosis report

scalar

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

• -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.

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.

care returns X1, X2, and D as empty when the associated Hamiltonian matrix has eigenvalues on the imaginary axis.

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

$$\left[\begin{array}{cc}Q& S\\ S^T& R\end{array}\right]>0$$

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., 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: care is not recommended

Starting in R2019a, use the icare command to solve continuous-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 care. Furthermore, icare includes an optional info structure to gather the implicit solution data of the Riccati equation.

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

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

There are no plans to remove care at this time.