Function Reference

care

Solve continuous-time algebraic Riccati equations (CARE)

Syntax

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

Description

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

such that has all its eigenvalues in the open left-half plane. The matrix is symmetric and called the stabilizing solution of . [X,L,G,rr] = care(A,B,Q) also returns:

• The eigenvalues L of
  
  
• The gain matrix
• The relative residual rr defined by
  
  

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

Here the gain matrix is and the "closed-loop" eigenvalues are L = eig(A-B*G,E).

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

[X,L,G,report] = care(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 Hamiltonian pencil has eigenvalues on or very near the imaginary axis (failure)
• -2 when there is no finite solution, i.e., with singular (failure)
• The relative residual defined above when the solution exists (success)

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

Note that this syntax returns report = 0 when successful.

Examples

Example 1

Given

you can solve the Riccati equation

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

To solve the -like Riccati equation

rewrite it in the care format as

You can now compute the stabilizing solution 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)
  

Algorithm

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

Limitations

The 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 detectable when and , or

See Also
dare        Solve discrete-time Riccati equations

lyap        Solve continuous-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.

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