Exercise_06_08.py

# -*- coding: utf-8 -*-
"""
Created on Sat Jun 01 22:29:33 2013

@author: Irshad
#Exercise6.8
#Analyse the Controllability of G(s)
"""

import numpy as np
import matplotlib.pyplot as plt
import numpy.linalg as la

w = np.logspace(-2, 2, 1000)
s = 1j*w


def G(s):
    return (1/(s** 2 + 100))*np.matrix([[(1/(0.01*s + 1)), 1],
                                             [((s + 0.1)/(s + 1)), 1]])


def g11(s):
    return (1/(s**2 + 100))*(1/(0.01*s + 1))


def g12(s):
    return 1/(s**2 + 100)


def g21(s):
    return (1/(s**2 + 100))*((s + 0.1)/(s + 1))


def g22(s):
    return 1/(s**2 + 100)


def lambda11(s):
    return 1/(1 - ((g12(s)*g21(s))/(g11(s)*g22(s))))


# Lambda11=Lambda22
# Lambda12=Lambda21

freqresp = map(G, s)
l11 = np.array([lambda11(i) for i in s])
l21 = 1 - l11
sigmas = np.array([Sigma for U, Sigma, V in map(la.svd, freqresp)])
plt.figure(1)
plt.title('RGA Values over Frequency')
plt.xlabel(r'Frequency [rad/s]', fontsize=14)
plt.ylabel(r'Magnitude', fontsize=15)
plt.loglog(w, abs(l11))
plt.loglog(w, abs(l21))
plt.grid(b=None, which='both', axis='both')
plt.legend((r'$\lambda_{11}/\lambda_{22} $', r"$\lambda_{12}/\lambda_{21}$"), loc=2)
plt.figure(2)
plt.title('SVD Values over Frequency')
plt.xlabel(r'Frequency [rad/s]', fontsize=14)
plt.ylabel(r'Magnitude', fontsize=15)
plt.loglog(w, sigmas)
plt.grid(b=None, which='both', axis='both')
plt.show()