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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 System Identification
Step1:  Control System Model
Step2:  $ y[n] = k u[n - d] $
$ y[n] = k z^{-d} u[n] $
$ G = \frac{y[n]}{u[n]} = \frac{k}{z^{d}} $
Step3:  The integrator transfer function
Step4:  The derivative transfer function
Step5:  The combined PID transfer function
Step6:  Roll
Step7:  Continuous Time Optimization
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Python Code:
def model_to_acc_tf(m):
    # open loop, rate output, mix input plant
    acc_tf = m['gain'] * control.tf(*control.pade(m['delay'], 1))  # order 1 approx
    tf_integrator = control.tf((1), (1, 0))
    return acc_tf * tf_integrator
def pid_design(G_rate_ol, d_tc=1.0/125, K0=[0.01, 0.01, 0.01, 5]):
    K_rate, G_cl_rate, G_ol_rate_comp = ut.logsysid.pid_design(
        G_rate_ol, K0[:3], d_tc)
    tf_integrator = control.tf((1), (1, 0))
    K, G_cl, G_ol_comp = ut.logsysid.pid_design(
        G_cl_rate*tf_integrator, K0[3:], d_tc,
        use_I=False, use_D=False)
    return np.vstack([K_rate, K])
def pid_tf(use_P=True, use_I=True, use_D=True, d_tc=0.1):
    H = []
    if use_P:
        H += [control.tf(1, 1)]
    if use_I:
        H += [control.tf((1), (1, 0))]
    if use_D:
        H += [control.tf((1, 0), (d_tc, 1))]
    H = np.array([H]).T
    H_num = [[H[i][j].num[0][0] for i in range(H.shape[0])] for j in range(H.shape[1])]
    H_den = [[H[i][j].den[0][0] for i in range(H.shape[0])] for j in range(H.shape[1])]
    H = control.tf(H_num, H_den)
    return H
log_file = ut.ulog.download_log('http://review.px4.io/download?log=35b27fdb-6a93-427a-b634-72ab45b9609e', '/tmp')
data = ut.sysid.prepare_data(log_file)
res = ut.sysid.attitude_sysid(data)
res
res = ut.sysid.attitude_sysid(data)
res
Explanation: System Identification
End of explanation
fs = ut.ulog.sample_frequency(data)
fs
Explanation: Control System Model
End of explanation
roll_tf = ut.sysid.delay_gain_model_to_tf(res['roll']['model'])
roll_tf
pitch_tf = ut.sysid.delay_gain_model_to_tf(res['pitch']['model'])
pitch_tf
yaw_tf = ut.sysid.delay_gain_model_to_tf(res['yaw']['model'])
yaw_tf
## Proportional
P_tf = control.tf(1, 1, 1.0/fs)
P_tf
Explanation: $ y[n] = k u[n - d] $
$ y[n] = k z^{-d} u[n] $
$ G = \frac{y[n]}{u[n]} = \frac{k}{z^{d}} $
End of explanation
z, f_s = sympy.symbols('z, f_s')
G_I = z / (f_s*(z-1))
G_I
I_tf = ut.control.sympy_to_tf(G_I, {'f_s': fs, 'dt': 1.0/fs})
I_tf
Explanation: The integrator transfer function:
\begin{align}
y[n] &= y[n-1] + \frac{u[n]}{f_s} \
Y[n] &= z^{-1}Y[n] + \frac{U[n]}{f_s} \
Y[n] &(1 - z^{-1}) = \frac{U[n]}{f_s} \
\frac{Y[n]}{U[n]} &= \frac{z}{fs (z - 1)} \
\end{align}
End of explanation
G_D = f_s * (z-1) / z
G_D
D_tf = ut.control.sympy_to_tf(G_D, {'f_s': fs, 'dt': 1.0/fs})
D_tf
Explanation: The derivative transfer function:
\begin{align}
y[n] &= f_s(u[n] - u[n-1]) \
Y[n] &= f_s(U[n] - z^{-1}U[n]) \
\frac{Y[n]}{U[n]} &= f_s(1 - z^{-1}) \
\frac{Y[n]}{U[n]} &= \frac{f_s(z - 1)}{z}
\end{align}
End of explanation
pid_tf = ut.control.tf_vstack([P_tf, I_tf, D_tf])
pid_tf
Explanation: The combined PID transfer function:
End of explanation
roll_tf_comp = roll_tf*pid_tf
roll_tf_comp
roll_ss_comp = control.tf2ss(roll_tf_comp);
Explanation: Roll
End of explanation
def attitude_loop_design(m, name):
    d_tc = 1.0/10
    G_rate_ol = model_to_acc_tf(m)
    K0=[0.182, 1.9, 0.01]
    K_rate, G_rate_ol, G_rate_comp_cl = ut.logsysid.pid_design(
        G_rate_ol, [0.2, 0.2, 0], d_tc)
    tf_integrator = control.tf([1], [1, 0])
    G_ol = G_rate_comp_cl*tf_integrator
    K, G_ol, G_comp_cl = ut.logsysid.pid_design(
        G_ol, [1], d_tc,
        use_I=False, use_D=False)
    K_rate, K
    return {
        'MC_{:s}RATE_P'.format(name): K_rate[0, 0],
        'MC_{:s}RATE_I'.format(name): K_rate[1, 0],
        'MC_{:s}RATE_D'.format(name): K_rate[2, 0],
        'MC_{:s}_P'.format(name): K[0, 0],
    }
log_file = ut.ulog.download_log('http://review.px4.io/download?log=35b27fdb-6a93-427a-b634-72ab45b9609e', '/tmp')
data = ut.sysid.prepare_data(log_file)
res = ut.sysid.attitude_sysid(data)
res
attitude_loop_design(res['roll']['model'], 'ROLL')
attitude_loop_design(res['pitch']['model'], 'PITCH')
attitude_loop_design(res['yaw']['model'], 'YAW')
Explanation: Continuous Time Optimization
End of explanation
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