ricc — Riccati equation
[X,RCOND,FERR]=ricc(A,B,C,"cont""method") [X,RCOND,FERR]=ricc(F,G,H,"disc","method")
real matrices of appropriate dimensions
real matrices of appropriate dimensions
real matrix
imposed string (flag for continuous or discrete)
'schr' or 'sign' for continuous-time systems and 'schr' or 'invf' for discrete-tyme systems
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.
//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)