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riccati
Solves the matricial Riccati equation (continuous | discrete time domain)
Syntax
X = riccati(H) X = riccati(H, E) X = riccati(A, B, C, dom) X = riccati(A, B, C, dom, method) [X1, X2, residual] = riccati(...)
Arguments
- A, B, C
- Square matrices of real numbers, of size n x n: Matricial coefficients of the equation.
- H, E
- Square matrices of real numbers, of size 2n x 2n: Hamiltonian matrices | pencil of the equation.
- dom
- Type / time domain of the Riccati equation:
"c"
or"continuous"
, or"d"
or"discrete"
. - method
- string :
"eigen"
for block diagonalization (default), or"schur"
for Schur method. - X1, X2, X
- square matrices of real numbers (X2 invertible). X is symetric when B and C are so.
- residual
- real number: norm1 of the actual Equation(X) residue.
Description
Continuous time domain
X = riccati(A, B, C, "c") solves the matricial Riccati equation:
A'*X + X*A - X*B*X + C = 0
X = riccati(H)
where H = [A, -B ; -C, -A']
is the Hamiltonian matrix,
does the same, using the 'eigen' method, but is more stable.
Discrete time domain
X = riccati(A, B, C, "d") solves the Riccati equation:
A'*X*A - (A'*X*B1) / (B2 +
B1'*X*B1) *
(B1'*X*A) + C - X = 0
with B = B1 / B2 *
B1'
.
X = riccati(H, E) does the same, where
H = [A, zeros(n,n) ; -C, eye(n,n)]
and
E = [eye(n,n), B ; zeros(n,n), A']
define the Hamiltonian pencil (E,H)
.
Output options
[X1, X2] = riccati(..) provides
X1
and X2
, with X2
invertible,
such that X = X1 / X2
.
residual
is the L1-norm of the actual equation's result. If
X
is an actual solution, residual
should be 0.
Most often its value is close to %eps*norm(X)
.
In the discrete case, sometimes
B1 and
B2 can't be retrieved from B .
Then residual can't be assessed and is set to %nan . |
When no solution is found, X=[] or X1=[] is
returned, and a warning is yielded. |
Examples
Continuous time domainn = 3; // [A, B, C] = (grand(n,n,"uin",-9,9), grand(n,n,"uin",-9,9), grand(n,n,"uin",-9,9)) A = [ -62. 91. 57. -43. -45. -19. 58. 83. -62. ]; B = [ 75. -31. -10. -79. 70. 68. -72. -5. 32. ]; C = [ -56. 70. 58. -41. 54. 50. 90. 2. -40. ]; // With A, B, C // ------------ X = riccati(A, B, C, "c") clean(A'*X + X*A - X*B*X + C) [x1, x2, res] = riccati(A,B,C, "c"); x = x1 / x2; and(x==X) res // With the Hamiltonian // -------------------- H = [A, -B; -C, -A']; X = riccati(H) clean(A'*X + X*A - X*B*X + C) [x1, x2] = riccati(H); x = x1 / x2; and(x==X)
--> X = riccati(A, B, C, "c") X = -0.1790367 0.4166084 0.2319152 -0.4977093 0.7993495 0.3086213 0.5595286 0.3202094 -0.103394 --> clean(A'*X + X*A - X*B*X + C) ans = 0. 0. 0. 0. 0. 0. 0. 0. 0. --> [x1, x2, res] = riccati(A,B,C, "c"); --> x = x1 / x2; --> and(x==X) ans = T --> res res = 3.340D-13 --> // With the Hamiltonian --> // -------------------- --> H = [A, -B; -C, -A']; --> X = riccati(H); X = -0.1790367 0.4166084 0.2319152 -0.4977093 0.7993495 0.3086213 0.5595286 0.3202094 -0.103394 --> clean(A'*X + X*A - X*B*X + C) ans = 0. 0. 0. 0. 0. 0. 0. 0. 0. --> [x1, x2] = riccati(H); --> x = x1 / x2; --> and(x==X) ans = T
// Eq: A'*X*A - (A'*X*B1) / (B2 + B1'*X*B1) * (B1'*X*A) + C - X = 0 n = 4; [A, B1, B2, C] = (rand(n,n), rand(n,n), rand(n,n), rand(n,n)); B = B1 / B2 * B1'; X = riccati(A, B, C, 'd', 'schur') if X <> [] then clean(A'*X*A - (A'*X*B1) / (B2 + B1'*X*B1) * (B1'*X*A) + C - X) [x1, x2, res] = riccati(A, B, C, 'd', 'schur'); x = x1 / x2; and(x==X) res else disp("Riccati: No solution found") end
History
Version | Description |
6.1.0 |
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