%%capture
!uv pip install tbcontrol

import numpy as np
import sympy as sp
from sympy import symbols, apart, inverse_laplace_transform, simplify
from sympy import expand, apart, lcm, together, solve, Poly, Matrix
from sympy.abc import s, t
from scipy.signal import TransferFunction, step as sci_step, StateSpace
from scipy.optimize import fsolve
from tbcontrol.symbolic import routh

K_p = symbols("K_p")
K_d = symbols("K_d")
K_i = symbols("K_i")

# q1
K, K_v = symbols("K K_v")
G_s = (2 * K / s) * ((1/s) / (1-K_v/s))
# Define the equation for the poles
equation = s**2 - K_v*s

# Solve the equation for s
poles = solve(equation, s)
poles

[0, K_v]

OS = 0.2

# q2

G_s = (s-6)/((s+2)*(s+5)**2)
L_input = 1/s
Y_s = G_s*L_input
Y_s_apart = apart(Y_s)
output_time_domain = inverse_laplace_transform(Y_s_apart, s, t)
output_time_domain

$\displaystyle - \frac{11 t e^{- 5 t} \theta\left(t\right)}{15} - \frac{3 \theta\left(t\right)}{25} + \frac{4 e^{- 2 t} \theta\left(t\right)}{9} - \frac{73 e^{- 5 t} \theta\left(t\right)}{225}$

# q3

# q4

# q5

# q6
A = np.array([[1, 1, 1], [0, 1, 1], [3, -2, 4]])
B = np.array([[1], [0], [1]])
C = np.array([[4, 5, 2]])
D = np.array([[0]])

# Calculate poles and zeros
system = StateSpace(A, B, C, D)
poles = system.poles
zeros = system.zeros
zeros, poles

def q10_prac():
  K_p = symbols("K_p")
  G_s = 5/((s+1)**2)
  G_PID_s = K_p * (1 + 5/s + 0.2*s)
  open_loop = G_s * G_PID_s
  char_eq = 1 + open_loop
  expanded = char_eq.expand().collect(s)
  char_poly = Poly(expanded, s)
  coeffs = char_poly.all_coeffs()

  return [solve(it, K_p) for i, it in enumerate(coeffs)]