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# 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

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

The care function also returns the gain matrix, $G={R}^{-1}{B}^{T}XE$.

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

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

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

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

you can solve the Riccati equation

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

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 (${H}_{\infty }$)-like Riccati Equation

To solve the ${H}_{\infty }$-like Riccati equation

${A}^{T}X+XA+X\left({\gamma }^{-2}{B}_{1}{B}_{1}^{T}-{B}_{2}{B}_{2}^{T}\right)X+{C}^{T}C=0$

rewrite it in the care format as

${A}^{T}X+XA-X\text{\hspace{0.17em}}\underset{B}{\underbrace{\left[{B}_{1},{B}_{2}\right]}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\underset{R}{\underbrace{\left[\begin{array}{cc}-{\gamma }^{2}I& 0\\ 0& I\end{array}\right]}}}^{-1}\left[\begin{array}{c}{B}_{1}^{T}\\ {B}_{2}^{T}\end{array}\right]X+{C}^{T}C=0$

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 $\left(A,B\right)$ 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 $\left(Q,A\right)$ detectable when $S=0$ and $R>0$, or

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

expand all

### 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