Računanje $H_\infty$ norme pomoću LNM
import control as ct
import numpy as np
import scipy as sc
import pymor
import pymor.discretizations.iosys as iosys
from pymor.basic import *
from pymor.core.config import config
set_log_levels({'pymor.algorithms.gram_schmidt.gram_schmidt': 'WARNING'})
p = InstationaryProblem(
StationaryProblem(
domain=LineDomain([0.,1.], left='robin', right='robin'),
diffusion=ConstantFunction(1., 1),
robin_data=(ConstantFunction(1., 1), ExpressionFunction('(x[...,0] < 1e-10) * 1.', 1)),
functionals={'output': ('l2_boundary', ExpressionFunction('(x[...,0] > (1 - 1e-10)) * 1.', 1))}
),
ConstantFunction(0., 1),
T=3.
)
d, _ = discretize_instationary_cg(p, diameter=1/50, nt=50)
00:39 DiffusionOperatorP1: Calulate gradients of shape functions transformed by reference map ... 00:39 DiffusionOperatorP1: Calculate all local scalar products beween gradients ... 00:39 DiffusionOperatorP1: Determine global dofs ... 00:39 DiffusionOperatorP1: Boundary treatment ... 00:39 DiffusionOperatorP1: Assemble system matrix ... 00:39 DiffusionOperatorP1: Calulate gradients of shape functions transformed by reference map ... 00:39 DiffusionOperatorP1: Calculate all local scalar products beween gradients ... 00:39 DiffusionOperatorP1: Determine global dofs ... 00:39 DiffusionOperatorP1: Boundary treatment ... 00:39 DiffusionOperatorP1: Assemble system matrix ... 00:39 L2ProductP1: Integrate the products of the shape functions on each element 00:39 L2ProductP1: Determine global dofs ... 00:39 L2ProductP1: Boundary treatment ... 00:39 L2ProductP1: Assemble system matrix ... 00:39 DiffusionOperatorP1: Calulate gradients of shape functions transformed by reference map ... 00:39 DiffusionOperatorP1: Calculate all local scalar products beween gradients ... 00:39 DiffusionOperatorP1: Determine global dofs ... 00:39 DiffusionOperatorP1: Boundary treatment ... 00:39 DiffusionOperatorP1: Assemble system matrix ... 00:39 L2ProductP1: Integrate the products of the shape functions on each element 00:39 L2ProductP1: Determine global dofs ... 00:39 L2ProductP1: Boundary treatment ... 00:39 L2ProductP1: Assemble system matrix ... 00:39 DiffusionOperatorP1: Calulate gradients of shape functions transformed by reference map ... 00:39 DiffusionOperatorP1: Calculate all local scalar products beween gradients ... 00:39 DiffusionOperatorP1: Determine global dofs ... 00:39 DiffusionOperatorP1: Boundary treatment ... 00:39 DiffusionOperatorP1: Assemble system matrix ...
lti = d.to_lti()
from pymor.algorithms.to_matrix import to_matrix
A = to_matrix(lti.A,format='dense')
B = to_matrix(lti.B,format='dense')
C = to_matrix(lti.C,format='dense')
D = to_matrix(lti.D,format='dense')
%%time
lti.hinf_norm()
CPU times: user 418 µs, sys: 13 µs, total: 431 µs Wall time: 452 µs
0.3333333333332594
import picos as pic
import cvxopt as cvx
AA = pic.new_param('AA',cvx.matrix(A))
BB = pic.new_param('BB',cvx.matrix(B))
CC = pic.new_param('CC',cvx.matrix(C))
DD = pic.new_param('DD',cvx.matrix(D))
n = np.size(AA)[0]
H = pic.Problem()
X = H.add_variable('X',(n,n),vtype='symmetric')
gamma = H.add_variable('gamma',1)
H.set_objective('min',gamma)
H.add_constraint(X>>0)
H.add_constraint( ((AA.T*X + X*AA & X*BB & CC.T) // (BB.T*X & -gamma & DD.T) // (CC & DD & -gamma )) << 0 )
<53×53 LMI Constraint: [AAᵀ·X + X·AA, X·BB, CCᵀ; BBᵀ·X, -gamma, DDᵀ; CC, DD, -gamma] ≼ 0>
%%time
H.solve(verbose = 0, solver = 'cvxopt');
CPU times: user 34.7 s, sys: 5.12 s, total: 39.8 s Wall time: 10.8 s
{'cvxopt_sol': {'x': <1327x1 matrix, tc='d'>, 'y': <0x1 matrix, tc='d'>, 's': <5410x1 matrix, tc='d'>, 'z': <5410x1 matrix, tc='d'>, 'status': 'optimal', 'gap': 8.928032640284613e-10, 'relative gap': 2.6784097923146256e-09, 'primal objective': 0.33333333329947223, 'dual objective': 0.33333333330480375, 'primal infeasibility': 1.265575137583725e-10, 'dual infeasibility': 4.384820382091425e-09, 'primal slack': 5.054705081915405e-13, 'dual slack': 5.236947309112654e-13, 'residual as primal infeasibility certificate': None, 'residual as dual infeasibility certificate': None, 'iterations': 19}, 'status': 'optimal', 'time': 10.806421041488647, 'primals': {'X': [0.31825297959015725, 0.37502238667454174, 0.27821577599148356, 0.3100239882446223, 0.3263285425172467, 0.24897637608706874, 0.2537184986779456, 0.2680774308536679, 0.29116473932122217, 0.22812587567901035, 0.204961540352629, 0.21755212810969493, 0.23815693630295615, 0.26644998583437146, 0.21372574091139807, 0.1627827250749125, 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print ('Norma je {0:.8f}'.format(H.obj_value()))
Norma je 0.33333333