ricc

ricc — Riccati equation

Calling sequence

[X,RCOND,FERR]=ricc(A,B,C,"cont""method")  
[X,RCOND,FERR]=ricc(F,G,H,"disc","method")

Parameters

`A,B,C`	: real matrices of appropriate dimensions
`F,G,H`	: real matrices of appropriate dimensions
`X`	: real matrix
`"cont","disc"'`	: imposed string (flag for continuous or discrete)
`method`	: 'schr' or 'sign' for continuous-time systems and 'schr' or 'invf' for discrete-tyme systems

Description

Riccati solver.

Continuous time:



  X=ricc(A,B,C,'cont')

gives a solution to the continuous time ARE



  A'*X+X*A-X*B*X+C=0 .

B and C are assumed to be nonnegative definite. (A,G) is assumed to be stabilizable with G*G' a full rank factorization of B.

(A,H) is assumed to be detectable with H*H' a full rank factorization of C.

Discrete time:



  X=ricc(F,G,H,'disc')

gives a solution to the discrete time ARE



  X=F'*X*F-F'*X*G1*((G2+G1'*X*G1)^-1)*G1'*X*F+H

F is assumed invertible and G = G1*inv(G2)*G1'.

One assumes (F,G1) stabilizable and (C,F) detectable with C'*C full rank factorization of H. Use preferably ric_desc.

C, D are symmetric .It is assumed that the matrices A, C and D are such that the corresponding matrix pencil has N eigenvalues with moduli less than one.

Error bound on the solution and a condition estimate are also provided. It is assumed that the matrices A, C and D are such that the corresponding Hamiltonian matrix has N eigenvalues with negative real parts.

Examples



//Standard formulas to compute Riccati solutions
A=rand(3,3);B=rand(3,2);C=rand(3,3);C=C*C';R=rand(2,2);R=R*R'+eye();
B=B*inv(R)*B';
X=ricc(A,B,C,'cont');
norm(A'*X+X*A-X*B*X+C,1)
H=[A -B;-C -A'];
[T,d]=schur(eye(H),H,'cont');T=T(:,1:d);
X1=T(4:6,:)/T(1:3,:);
norm(X1-X,1)
[T,d]=schur(H,'cont');T=T(:,1:d);
X2=T(4:6,:)/T(1:3,:);
norm(X2-X,1)
//       Discrete time case
F=A;B=rand(3,2);G1=B;G2=R;G=G1/G2*G1';H=C;
X=ricc(F,G,H,'disc');
norm(F'*X*F-(F'*X*G1/(G2+G1'*X*G1))*(G1'*X*F)+H-X)
H1=[eye(3,3) G;zeros(3,3) F'];
H2=[F zeros(3,3);-H eye(3,3)];
[T,d]=schur(H2,H1,'disc');T=T(:,1:d);X1=T(4:6,:)/T(1:3,:);
norm(X1-X,1)
Fi=inv(F);
Hami=[Fi Fi*G;H*Fi F'+H*Fi*G];
[T,d]=schur(Hami,'d');T=T(:,1:d);
Fit=inv(F');
Ham=[F+G*Fit*H -G*Fit;-Fit*H Fit];
[T,d]=schur(Ham,'d');T=T(:,1:d);X2=T(4:6,:)/T(1:3,:);
norm(X2-X,1)

Author

P. Petkov