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# ricc

Riccati equation

### Syntax

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

### Arguments

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 `riccati()`.

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)```

• riccati — Solves the matricial Riccati equation (continuous | discrete time domain)
• schur — decomposição (ordenada) de Schur de matrizes e feixes

### Used Functions

See SCI/modules/cacsd/src/slicot/riccpack.f