def test_balreal():
isys = Lowpass(0.05)
noise = 0.5 * Lowpass(0.01) + 0.5 * Alpha(0.005)
p = 0.8
sys = p * isys + (1 - p) * noise
T, Tinv, S = balanced_transformation(sys)
assert np.allclose(inv(T), Tinv)
assert np.allclose(S, hsvd(sys))
balsys = sys.transform(T, Tinv)
assert balsys == sys
assert np.all(S >= 0)
assert np.all(S[0] > 0.3)
assert np.all(S[1:] < 0.05)
assert np.allclose(sorted(S, reverse=True), S)
P = control_gram(balsys)
Q = observe_gram(balsys)
diag = np.diag_indices(len(P))
offdiag = np.ones_like(P, dtype=bool)
offdiag[diag] = False
offdiag = np.where(offdiag)
assert np.allclose(P[diag], S)
assert np.allclose(P[offdiag], 0)
assert np.allclose(Q[diag], S)
assert np.allclose(Q[offdiag], 0)
def balreal(sys):
"""Computes the balanced realization of sys and returns its eigenvalues.
References:
[1] http://www.mathworks.com/help/control/ref/balreal.html
[2] Laub, A.J., M.T. Heath, C.C. Paige, and R.C. Ward, "Computation of
System Balancing Transformations and Other Applications of
Simultaneous Diagonalization Algorithms," *IEEE Trans. Automatic
Control*, AC-32 (1987), pp. 115-122.
"""
sys = LinearSystem(sys) # cast first to memoize sys2ss
if not sys.analog:
raise NotImplementedError("balanced digital filters not supported")
R = control_gram(sys)
O = observe_gram(sys)
LR = cholesky(R, lower=True)
LO = cholesky(O, lower=True)
U, S, V = svd(np.dot(LO.T, LR))
T = np.dot(LR, V.T) * S ** (-1. / 2)
Tinv = (S ** (-1. / 2))[:, None] * np.dot(U.T, LO.T)
return similarity_transform(sys, T, Tinv), S
def test_grams():
sys = 0.6*Alpha(0.01) + 0.4*Lowpass(0.05)
A, B, C, D = sys2ss(sys)
P = control_gram(sys)
assert np.allclose(np.dot(A, P) + np.dot(P, A.T), -np.dot(B, B.T))
assert np.linalg.matrix_rank(P) == len(P) # controllable
Q = observe_gram(sys)
assert np.allclose(np.dot(A.T, Q) + np.dot(Q, A), -np.dot(C.T, C))
assert np.linalg.matrix_rank(Q) == len(Q) # observable
def test_grams():
sys = 0.6 * Alpha(0.01) + 0.4 * Lowpass(0.05)
A, B, C, D = sys2ss(sys)
P = control_gram(sys)
assert np.allclose(np.dot(A, P) + np.dot(P, A.T), -np.dot(B, B.T))
assert matrix_rank(P) == len(P) # controllable
Q = observe_gram(sys)
assert np.allclose(np.dot(A.T, Q) + np.dot(Q, A), -np.dot(C.T, C))
assert matrix_rank(Q) == len(Q) # observable
def hankel(sys):
"""Compute Hankel singular values of a linear system.
References:
[1] Glover, Keith, and Jonathan R. Partington. "Bounds on the
achievable accuracy in model reduction." Modelling, robustness and
sensitivity reduction in control systems. Springer Berlin
Heidelberg, 1987. 95-118.
"""
sys = LinearSystem(sys)
R, O = control_gram(sys), observe_gram(sys)
# sort needed to be consistent across different versions
return np.sort(np.sqrt(abs(eig(np.dot(O, R))[0])))[::-1]