Parameter Estimation of RIG Roll Experiments

Setup and descriptions

Without ACM model
Turn on wind tunnel
Only 1DoF for RIG roll movement
Use small-amplitude aileron command of CMP as inputs (in degrees) $$U = \delta_{a,cmp}(t)$$
Consider RIG roll angle and its derivative as States (in radians) $$X = \begin{pmatrix} \phi_{rig} \\ \dot{\phi}_{rig} \end{pmatrix}$$
Observe RIG roll angle and its derivative as Outputs (in degrees) $$Z = \begin{pmatrix} \phi_{rig} \\ \dot{\phi}_{rig} \end{pmatrix}$$
Use output error method based on most-likehood(ML) to estimate $$ \theta = \begin{pmatrix} C_{l,\delta_a,cmp} \\ C_{lp,cmp} \end{pmatrix} $$

Startup computation engines



In [ ]:

    
%run matt_startup
%run -i matt_utils

button_qtconsole()

#import other needed modules in all used engines
#with dview.sync_imports():
#   import os

Data preparation

Load raw data



In [2]:

    
filename = 'FIWT_Exp050_20150612160930.dat.npz'

def loadData():
    # Read and parse raw data
    global exp_data
    exp_data = np.load(filename)

    # Select colums
    global T_cmp, da1_cmp, da2_cmp, da3_cmp , da4_cmp
    T_cmp = exp_data['data33'][:,0]
    da1_cmp = exp_data['data33'][:,3]
    da2_cmp = exp_data['data33'][:,5]
    da3_cmp = exp_data['data33'][:,7]
    da4_cmp = exp_data['data33'][:,9]

    global T_rig, phi_rig
    T_rig = exp_data['data44'][:,0]
    phi_rig = exp_data['data44'][:,2]

loadData()



In [3]:

    
text_loadData()

Check time sequence and inputs/outputs

Click 'Check data' button to show the raw data.

Click on curves to select time point and push into queue; click 'T/s' text to pop up last point in the queue; and click 'Output' text to print time sequence table.



In [4]:

    
def checkInputOutputData():

    #check inputs/outputs
    fig, ax = plt.subplots(2,1,True)
    ax[0].plot(T_cmp,da1_cmp,'r', T_cmp,da2_cmp,'g',
               T_cmp,da3_cmp,'b', T_cmp,da4_cmp,'m',
               picker=1)
    ax[1].plot(T_rig,phi_rig, 'b', picker=2)
    ax[0].set_ylabel('$\delta \/ / \/ ^o$')
    ax[1].set_ylabel('$\phi \/ / \/ ^o/s$')
    ax[1].set_xlabel('$T \/ / \/ s$', picker=True)
    ax[0].set_title('Output', picker=True)
    

    fig.canvas.mpl_connect('pick_event', onPickTime)
    fig.show()
    display(fig)

button_CheckData()

Input $\delta_T$ and focused time ranges



In [5]:

    
# Pick up focused time ranges
time_marks = [
[28.4202657857,88.3682684612,"ramp cmp1 u"],
[90.4775063395,150.121612502,"ramp cmp1 d"],
[221.84785848,280.642700604,"ramp cmp2 u"],
[283.960460465,343.514106772,"ramp cmp2 d"],
[345.430891556,405.5707005,"ramp cmp3 u"],
[408.588176529,468.175897568,"ramp cmp3 d"],
[541.713519906,600.757388552,"ramp cmp4 u"],
[604.040332533,663.913079475,"ramp cmp4 d"],
[677.817484542,778.026124493,"ramp cmp1/3 d"],
[780.09665253,879.865190551,"ramp cmp1/3 u"],
[888.075513916,987.309008866,"ramp cmp2/4 d"],
[989.75879072,1088.99649253,"ramp cmp2/4 u"],
[1101.9320102,1221.45089264,"ramp cmpall u"],
[1240.32935658,1360.72513252,"ramp cmpall d"],
]

# Decide DT,U,Z and their processing method
DT=0.2
process_set = {
            'U':[(T_cmp, da1_cmp,0),
                (T_cmp, da2_cmp,0),
                (T_cmp, da3_cmp,0),
                (T_cmp, da4_cmp,0),],
            'Z':[(T_rig, phi_rig,1),],
            'cutoff_freq': 1 #Hz
           }

U_names = ['$\delta_{a1,cmp} \, / \, ^o$',
          '$\delta_{a2,cmp} \, / \, ^o$',
          '$\delta_{a3,cmp} \, / \, ^o$',
          '$\delta_{a4,cmp} \, / \, ^o$']
Y_names = Z_names = ['$\phi_{a,rig} \, / \, ^o$']

display_data_prepare()









    





$$\delta_T=200.0(ms)$$






    




Time ranges






    




Idx Start(s) Spam(s) Description
0 28.42 59.95 ramp cmp1 u
1 90.48 59.64 ramp cmp1 d
2 221.85 58.79 ramp cmp2 u
3 283.96 59.55 ramp cmp2 d
4 345.43 60.14 ramp cmp3 u
5 408.59 59.59 ramp cmp3 d
6 541.71 59.04 ramp cmp4 u
7 604.04 59.87 ramp cmp4 d
8 677.82 100.21 ramp cmp1/3 d
9 780.10 99.77 ramp cmp1/3 u
10 888.08 99.23 ramp cmp2/4 d
11 989.76 99.24 ramp cmp2/4 u
12 1101.93 119.52 ramp cmpall u
13 1240.33 120.40 ramp cmpall d






    




$U$
$\delta_{a1,cmp} \, / \, ^o$
$\delta_{a2,cmp} \, / \, ^o$
$\delta_{a3,cmp} \, / \, ^o$
$\delta_{a4,cmp} \, / \, ^o$






    




$Z$
$\phi_{a,rig} \, / \, ^o$

Resample and filter data in sections

For each section,

Select time range and shift it to start from zero;
Resample Time, Inputs, Outputs in unique $\delta_T$;
Smooth Input/Observe data if flag bit0 is set;
Take derivatives of observe data if flag bit1 is set.



In [6]:

    
resample(True);

Define dynamic model to be estimated

$$\left\{\begin{matrix}\begin{align} M_{x,rig} &= M_{x,a} + M_{x,f} + M_{x,cg} = 0 \\ M_{x,a} &= \frac{1}{2} \rho V^2S_cb_c C_{la,cmp}\delta_{a,cmp} \\ M_{x,f} &= -F_c \, sign(\dot{\phi}_{rig}) \\ M_{x,cg} &= -m_T g l_{zT} \sin \left ( \phi - \phi_0 \right ) \end{align}\end{matrix}\right.$$



In [14]:

    
%%px --local
#update common const parameters in all engines

angles = range(-40,41,5)
angles[0] -= 1
angles[-1] += 1
del angles[angles.index(0)]
angles_num = len(angles)

#problem size
Nx = 0
Nu = 4
Ny = 1
Npar = 4*angles_num+1

#reference
S_c = 0.1254   #S_c(m2) 
b_c = 0.7      #b_c(m)
g = 9.81       #g(m/s2)

#static measurement
m_T = 9.585   #m_T(kg)
l_z_T = 0.0416 #l_z_T(m)
V = 25         #V(m/s)

#previous estimations
F_c = 0.06 #F_c(N*m)
Clda_cmp = -0.2966 #Clda_cmp(1/rad)
kFbrk = 1.01
shuffle = 0.01

#other parameters

qbarSb = 0.5*1.225*V*V*S_c*b_c
_m_T_l_z_T_g = -(m_T*l_z_T)*g


def obs(Z,T,U,params):
    s = T.size
    
    k1 = np.array(params[0:angles_num])
    k2 = np.array(params[angles_num:angles_num*2])
    k3 = np.array(params[angles_num*2:angles_num*3])
    k4 = np.array(params[angles_num*3:angles_num*4])
    phi0 = params[-1]

    Clda1 = scipy.interpolate.interp1d(angles, Clda_cmp*0.00436332313*k1*angles,assume_sorted=True)
    Clda2 = scipy.interpolate.interp1d(angles, Clda_cmp*0.00436332313*k2*angles,assume_sorted=True)
    Clda3 = scipy.interpolate.interp1d(angles, Clda_cmp*0.00436332313*k3*angles,assume_sorted=True)
    Clda4 = scipy.interpolate.interp1d(angles, Clda_cmp*0.00436332313*k4*angles,assume_sorted=True)
    
    moments_a = qbarSb*(Clda1(U[1:s,0])+Clda2(U[1:s,1])
                       +Clda3(U[1:s,2])+Clda4(U[1:s,3]))

    phi = Z[0,0]*0.0174533
    phi_rslt = [phi]
    for t,m_a in itertools.izip(T[1:],moments_a):
        moments_cg = _m_T_l_z_T_g*math.sin(phi-phi0)
        if moments_cg + m_a > F_c*kFbrk:
            k = ( F_c-m_a)/_m_T_l_z_T_g
            if k > 1:
                k = 1
            elif k < -1:
                k = -1
            phi = phi0+math.asin(k)
        elif moments_cg + m_a < -F_c*kFbrk:
            k = (-F_c-m_a)/_m_T_l_z_T_g
            if k > 1:
                k = 1
            elif k < -1:
                k = -1
            phi = phi0+math.asin(k)
        else :
            phi += (moments_cg + m_a)/(F_c*kFbrk)*shuffle/57.3
        phi_rslt.append(phi)
    
    return (np.array(phi_rslt)*57.3).reshape((-1,1))



In [15]:

    
display(HTML('<b>Constant Parameters</b>'))
table = ListTable()
table.append(['Name','Value','unit'])
table.append(['$S_c$',S_c,'$m^2$'])
table.append(['$b_c$',b_c,'$m$'])
table.append(['$g$',g,'$m/s^2$'])
table.append(['$m_T$',m_T,'$kg$'])
table.append(['$l_{zT}$',l_z_T,'$m$'])
table.append(['$V$',V,'$m/s$'])
table.append(['$F_c$',F_c,'$Nm$'])
table.append(['$C_{l \delta a,cmp}$',Clda_cmp,'$rad^{-1}$'])
display(table)









    




Constant Parameters






    




Name Value unit
$S_c$ 0.1254 $m^2$
$b_c$ 0.7 $m$
$g$ 9.81 $m/s^2$
$m_T$ 9.585 $kg$
$l_{zT}$ 0.0416 $m$
$V$ 25 $m/s$
$F_c$ 0.06 $Nm$
$C_{l \delta a,cmp}$ -0.2966 $rad^{-1}$

Initial guess

Input default values and ranges for parameters
Select sections for trainning
Adjust parameters based on simulation results
Decide start values of parameters for optimization



In [16]:

    
#initial guess
param0 = [1]*(4*angles_num)+[0]

param_name = ['k_{}_{}'.format(i/angles_num+1, angles[i%angles_num]) for i in range(4*angles_num)] + ['$phi_0$']
param_unit = ['1']*(4*angles_num) + ['$rad$']

NparID = Npar
opt_idx = range(Npar)
opt_param0 = [param0[i] for i in opt_idx]
par_del = [0.001]*(4*angles_num) + [0.0001]
bounds =  [(0,1.5)]*(4*angles_num) +[(-0.1, 0.1)]
display_default_params()

#select sections for training
section_idx = range(8)
del section_idx[3]
display_data_for_train()

#push parameters to engines
push_opt_param()









    




Default Parameters for optimization






    




Idx Description $x_0$ unit opt $\delta_{par}$ min max
0 k_1_-41 1 1 * 0.001 0 1.5
1 k_1_-35 1 1 * 0.001 0 1.5
2 k_1_-30 1 1 * 0.001 0 1.5
3 k_1_-25 1 1 * 0.001 0 1.5
4 k_1_-20 1 1 * 0.001 0 1.5
5 k_1_-15 1 1 * 0.001 0 1.5
6 k_1_-10 1 1 * 0.001 0 1.5
7 k_1_-5 1 1 * 0.001 0 1.5
8 k_1_5 1 1 * 0.001 0 1.5
9 k_1_10 1 1 * 0.001 0 1.5
10 k_1_15 1 1 * 0.001 0 1.5
11 k_1_20 1 1 * 0.001 0 1.5
12 k_1_25 1 1 * 0.001 0 1.5
13 k_1_30 1 1 * 0.001 0 1.5
14 k_1_35 1 1 * 0.001 0 1.5
15 k_1_41 1 1 * 0.001 0 1.5
16 k_2_-41 1 1 * 0.001 0 1.5
17 k_2_-35 1 1 * 0.001 0 1.5
18 k_2_-30 1 1 * 0.001 0 1.5
19 k_2_-25 1 1 * 0.001 0 1.5
20 k_2_-20 1 1 * 0.001 0 1.5
21 k_2_-15 1 1 * 0.001 0 1.5
22 k_2_-10 1 1 * 0.001 0 1.5
23 k_2_-5 1 1 * 0.001 0 1.5
24 k_2_5 1 1 * 0.001 0 1.5
25 k_2_10 1 1 * 0.001 0 1.5
26 k_2_15 1 1 * 0.001 0 1.5
27 k_2_20 1 1 * 0.001 0 1.5
28 k_2_25 1 1 * 0.001 0 1.5
29 k_2_30 1 1 * 0.001 0 1.5
30 k_2_35 1 1 * 0.001 0 1.5
31 k_2_41 1 1 * 0.001 0 1.5
32 k_3_-41 1 1 * 0.001 0 1.5
33 k_3_-35 1 1 * 0.001 0 1.5
34 k_3_-30 1 1 * 0.001 0 1.5
35 k_3_-25 1 1 * 0.001 0 1.5
36 k_3_-20 1 1 * 0.001 0 1.5
37 k_3_-15 1 1 * 0.001 0 1.5
38 k_3_-10 1 1 * 0.001 0 1.5
39 k_3_-5 1 1 * 0.001 0 1.5
40 k_3_5 1 1 * 0.001 0 1.5
41 k_3_10 1 1 * 0.001 0 1.5
42 k_3_15 1 1 * 0.001 0 1.5
43 k_3_20 1 1 * 0.001 0 1.5
44 k_3_25 1 1 * 0.001 0 1.5
45 k_3_30 1 1 * 0.001 0 1.5
46 k_3_35 1 1 * 0.001 0 1.5
47 k_3_41 1 1 * 0.001 0 1.5
48 k_4_-41 1 1 * 0.001 0 1.5
49 k_4_-35 1 1 * 0.001 0 1.5
50 k_4_-30 1 1 * 0.001 0 1.5
51 k_4_-25 1 1 * 0.001 0 1.5
52 k_4_-20 1 1 * 0.001 0 1.5
53 k_4_-15 1 1 * 0.001 0 1.5
54 k_4_-10 1 1 * 0.001 0 1.5
55 k_4_-5 1 1 * 0.001 0 1.5
56 k_4_5 1 1 * 0.001 0 1.5
57 k_4_10 1 1 * 0.001 0 1.5
58 k_4_15 1 1 * 0.001 0 1.5
59 k_4_20 1 1 * 0.001 0 1.5
60 k_4_25 1 1 * 0.001 0 1.5
61 k_4_30 1 1 * 0.001 0 1.5
62 k_4_35 1 1 * 0.001 0 1.5
63 k_4_41 1 1 * 0.001 0 1.5
64 $phi_0$ 0 $rad$ * 0.0001 -0.1 0.1






    




Training Data






    




Idx Start(s) Spam(s) Description
0 28.42 59.95 ramp cmp1 u
1 90.48 59.64 ramp cmp1 d
2 221.85 58.79 ramp cmp2 u
4 345.43 60.14 ramp cmp3 u
5 408.59 59.59 ramp cmp3 d
6 541.71 59.04 ramp cmp4 u
7 604.04 59.87 ramp cmp4 d



In [17]:

    
# select 4 section from training data
#idx = random.sample(section_idx, 4)
idx = section_idx[:]

interact_guess();

Optimize using ML



In [18]:

    
display_preopt_params()

if False:
    InfoMat = None
    method = 'trust-ncg'
    def hessian(opt_params, index):
        global InfoMat
        return InfoMat
    dview['enable_infomat']=True
    options={'gtol':1}
    opt_bounds = None
else:
    method = 'L-BFGS-B'
    hessian = None
    dview['enable_infomat']=False
    options={'ftol':1e-6,'maxfun':400}
    opt_bounds = bounds

cnt = 0
tmp_rslt = None
T0 = time.time()
print('#cnt,     Time,      |R|')

%time res =  sp.optimize.minimize(fun=costfunc, x0=opt_param0, \
        args=(opt_idx,), method=method, jac=True, hess=hessian, \
        bounds=opt_bounds, options=options)









    




Parameters for optimization






    




Idx Description $x_0$ unit opt $\delta_{par}$ min max
0 k_1_-41 1.0 1 * 0.001 0 1.5
1 k_1_-35 1.0 1 * 0.001 0 1.5
2 k_1_-30 1.0 1 * 0.001 0 1.5
3 k_1_-25 1.0 1 * 0.001 0 1.5
4 k_1_-20 1.0 1 * 0.001 0 1.5
5 k_1_-15 1.0 1 * 0.001 0 1.5
6 k_1_-10 1.0 1 * 0.001 0 1.5
7 k_1_-5 1.0 1 * 0.001 0 1.5
8 k_1_5 1.0 1 * 0.001 0 1.5
9 k_1_10 1.0 1 * 0.001 0 1.5
10 k_1_15 1.0 1 * 0.001 0 1.5
11 k_1_20 1.0 1 * 0.001 0 1.5
12 k_1_25 1.0 1 * 0.001 0 1.5
13 k_1_30 1.0 1 * 0.001 0 1.5
14 k_1_35 1.0 1 * 0.001 0 1.5
15 k_1_41 1.0 1 * 0.001 0 1.5
16 k_2_-41 1.0 1 * 0.001 0 1.5
17 k_2_-35 1.0 1 * 0.001 0 1.5
18 k_2_-30 1.0 1 * 0.001 0 1.5
19 k_2_-25 1.0 1 * 0.001 0 1.5
20 k_2_-20 1.0 1 * 0.001 0 1.5
21 k_2_-15 1.0 1 * 0.001 0 1.5
22 k_2_-10 1.0 1 * 0.001 0 1.5
23 k_2_-5 1.0 1 * 0.001 0 1.5
24 k_2_5 1.0 1 * 0.001 0 1.5
25 k_2_10 1.0 1 * 0.001 0 1.5
26 k_2_15 1.0 1 * 0.001 0 1.5
27 k_2_20 1.0 1 * 0.001 0 1.5
28 k_2_25 1.0 1 * 0.001 0 1.5
29 k_2_30 1.0 1 * 0.001 0 1.5
30 k_2_35 1.0 1 * 0.001 0 1.5
31 k_2_41 1.0 1 * 0.001 0 1.5
32 k_3_-41 1.0 1 * 0.001 0 1.5
33 k_3_-35 1.0 1 * 0.001 0 1.5
34 k_3_-30 1.0 1 * 0.001 0 1.5
35 k_3_-25 1.0 1 * 0.001 0 1.5
36 k_3_-20 1.0 1 * 0.001 0 1.5
37 k_3_-15 1.0 1 * 0.001 0 1.5
38 k_3_-10 1.0 1 * 0.001 0 1.5
39 k_3_-5 1.0 1 * 0.001 0 1.5
40 k_3_5 1.0 1 * 0.001 0 1.5
41 k_3_10 1.0 1 * 0.001 0 1.5
42 k_3_15 1.0 1 * 0.001 0 1.5
43 k_3_20 1.0 1 * 0.001 0 1.5
44 k_3_25 1.0 1 * 0.001 0 1.5
45 k_3_30 1.0 1 * 0.001 0 1.5
46 k_3_35 1.0 1 * 0.001 0 1.5
47 k_3_41 1.0 1 * 0.001 0 1.5
48 k_4_-41 1.0 1 * 0.001 0 1.5
49 k_4_-35 1.0 1 * 0.001 0 1.5
50 k_4_-30 1.0 1 * 0.001 0 1.5
51 k_4_-25 1.0 1 * 0.001 0 1.5
52 k_4_-20 1.0 1 * 0.001 0 1.5
53 k_4_-15 1.0 1 * 0.001 0 1.5
54 k_4_-10 1.0 1 * 0.001 0 1.5
55 k_4_-5 1.0 1 * 0.001 0 1.5
56 k_4_5 1.0 1 * 0.001 0 1.5
57 k_4_10 1.0 1 * 0.001 0 1.5
58 k_4_15 1.0 1 * 0.001 0 1.5
59 k_4_20 1.0 1 * 0.001 0 1.5
60 k_4_25 1.0 1 * 0.001 0 1.5
61 k_4_30 1.0 1 * 0.001 0 1.5
62 k_4_35 1.0 1 * 0.001 0 1.5
63 k_4_41 1.0 1 * 0.001 0 1.5
64 $phi_0$ 0.0 $rad$ * 0.0001 -0.1 0.1






    



#cnt,     Time,      |R|
0001,    0.152,       13.8
0002,    0.294,      400.9
0003,    0.438,       12.5
0004,    0.578,        5.5
0005,    0.725,        5.3
0006,    0.867,        7.1
0007,    1.003,        5.3
0008,    1.142,        5.0
0009,    1.282,        4.4
0010,    1.424,        4.3
0011,    1.567,        3.8
0012,    1.719,        3.8
0013,    1.875,        3.8
0014,    2.021,        3.8
0015,    2.173,        3.8
0016,    2.317,        3.8
0017,    2.470,        3.8
0018,    2.623,        3.8
0019,    2.764,        3.8
0020,    2.905,        3.8
0021,    3.048,        3.8
0022,    3.198,        3.8
0023,    3.353,        3.8
0024,    3.508,        3.8
0025,    3.650,        3.8
0026,    3.789,        3.8
0027,    3.927,        3.8
0028,    4.064,        3.8
0029,    4.206,        3.8
0030,    4.347,        3.8
0031,    4.490,        3.8
0032,    4.628,      249.1
0033,    4.767,       19.4
0034,    4.909,        4.9
0035,    5.065,        3.5
0036,    5.205,        2.9
0037,    5.348,        3.8
0038,    5.485,        2.3
0039,    5.634,        2.4
0040,    5.794,        2.3
0041,    5.938,        2.3
0042,    6.076,        2.2
0043,    6.218,        2.0
0044,    6.360,        1.8
0045,    6.509,        1.6
0046,    6.650,        1.7
0047,    6.793,        1.7
0048,    6.935,        1.6
0049,    7.076,        1.6
0050,    7.214,        1.6
0051,    7.354,        1.6
0052,    7.493,        1.6
0053,    7.636,        1.6
0054,    7.789,        1.6
0055,    7.955,        1.6
0056,    8.100,        1.6
0057,    8.238,        1.6
0058,    8.373,        1.6
0059,    8.513,        1.6
0060,    8.653,        1.6
0061,    8.789,        1.6
0062,    8.933,        1.6
0063,    9.073,        1.6
0064,    9.212,        1.6
0065,    9.357,        1.6
0066,    9.504,      208.0
0067,    9.642,       16.3
0068,    9.781,        2.6
0069,    9.926,        1.3
0070,   10.071,        1.3
0071,   10.210,        1.3
0072,   10.346,        1.3
0073,   10.484,        1.3
0074,   10.621,        1.3
0075,   10.757,        1.3
0076,   10.891,        1.3
0077,   11.030,        1.3
0078,   11.166,        1.3
0079,   11.307,        1.3
0080,   11.451,        1.3
0081,   11.593,        1.3
0082,   11.729,        1.3
0083,   11.868,        1.3
0084,   12.026,        1.3
0085,   12.189,        1.3
0086,   12.345,        1.3
0087,   12.501,        1.3
0088,   12.642,        1.3
0089,   12.782,        1.3
0090,   12.923,        1.3
0091,   13.059,        1.3
0092,   13.198,        1.3
0093,   13.333,      325.0
0094,   13.469,       25.3
0095,   13.605,        4.1
0096,   13.746,        1.7
0097,   13.881,        1.3
0098,   14.020,        1.2
0099,   14.158,        1.2
0100,   14.298,        1.2
0101,   14.436,        1.2
0102,   14.572,        1.2
0103,   14.710,        1.1
0104,   14.848,        1.1
0105,   14.995,        1.1
0106,   15.130,        1.0
0107,   15.273,        1.0
0108,   15.420,        1.0
0109,   15.574,        1.0
0110,   15.749,        1.0
0111,   15.939,        1.0
0112,   16.116,        1.0
0113,   16.314,        1.0
0114,   16.457,        1.0
0115,   16.598,        1.0
0116,   16.735,        1.0
0117,   16.878,        1.0
0118,   17.019,        1.0
0119,   17.167,        1.0
0120,   17.327,        1.0
0121,   17.481,        1.0
0122,   17.616,        1.0
0123,   17.748,        1.0
0124,   17.891,        1.0
0125,   18.027,        1.0
0126,   18.166,        1.0
0127,   18.307,        1.0
0128,   18.454,      165.3
0129,   18.592,       15.4
0130,   18.734,        3.0
0131,   18.874,        1.3
0132,   19.020,        1.0
0133,   19.161,        1.0
0134,   19.300,        1.0
0135,   19.446,        1.0
0136,   19.586,        1.0
0137,   19.726,        1.0
0138,   19.859,        1.0
0139,   20.004,        1.0
0140,   20.141,        1.0
0141,   20.278,        1.0
0142,   20.419,        1.0
0143,   20.562,        1.0
0144,   20.698,        1.0
0145,   20.835,        1.0
0146,   20.968,        1.0
0147,   21.098,        1.0
0148,   21.241,        1.0
0149,   21.386,        1.0
0150,   21.531,        1.0
0151,   21.671,        1.0
0152,   21.815,        1.0
0153,   21.949,        1.0
0154,   22.092,        1.0
0155,   22.230,        1.0
0156,   22.373,        1.0
0157,   22.515,        1.0
0158,   22.662,        1.0
0159,   22.823,        1.0
0160,   22.982,        1.0
0161,   23.142,        1.0
0162,   23.277,        1.0
0163,   23.416,        1.0
0164,   23.553,        1.0
0165,   23.686,        1.0
0166,   23.823,        1.0
0167,   23.969,        1.0
0168,   24.107,        1.0
0169,   24.245,        1.0
0170,   24.397,        1.0
0171,   24.537,        1.0
0172,   24.676,        1.0
0173,   24.816,        1.0
0174,   24.956,        1.0
0175,   25.093,        1.0
0176,   25.231,        1.0
0177,   25.377,        1.0
0178,   25.517,        1.0
0179,   25.659,        1.0
0180,   25.803,        1.0
0181,   25.938,        1.0
0182,   26.073,        1.0
0183,   26.206,        1.0
0184,   26.339,        1.0
0185,   26.475,        1.0
0186,   26.619,        1.0
0187,   26.755,        1.0
0188,   26.898,        1.0
0189,   27.035,        1.0
0190,   27.172,        1.0
0191,   27.321,        1.0
0192,   27.476,        1.0
0193,   27.618,        1.0
0194,   27.754,        1.0
0195,   27.894,        1.0
0196,   28.035,        1.0
0197,   28.175,        1.0
0198,   28.315,        1.0
0199,   28.455,        1.0
0200,   28.602,        1.0
0201,   28.742,        1.0
0202,   28.876,        1.0
0203,   29.016,        1.0
0204,   29.155,        1.0
0205,   29.292,        1.0
0206,   29.440,        1.0
0207,   29.577,        1.0
0208,   29.714,        1.0
0209,   29.857,        1.0
0210,   29.997,        1.0
0211,   30.138,        1.0
0212,   30.272,        1.0
0213,   30.417,        1.0
0214,   30.564,        1.0
0215,   30.708,        1.0
0216,   30.846,        1.0
0217,   30.980,        1.0
0218,   31.117,        1.0
0219,   31.256,        1.0
0220,   31.391,        1.0
0221,   31.539,        1.0
0222,   31.678,        1.0
0223,   31.825,        1.0
0224,   31.989,        1.0
0225,   32.157,        1.0
0226,   32.298,        1.0
0227,   32.436,        1.0
0228,   32.573,        1.0
0229,   32.705,        1.0
0230,   32.848,        1.0
0231,   32.991,        1.0
0232,   33.128,        1.0
0233,   33.271,        1.0
0234,   33.428,        1.0
0235,   33.566,        1.0
0236,   33.710,        1.0
0237,   33.850,        1.0
0238,   33.987,        1.0
0239,   34.133,        1.0
0240,   34.272,        1.0
0241,   34.420,        1.0
0242,   34.555,        1.0
0243,   34.697,        1.0
0244,   34.839,        1.0
0245,   34.979,        1.0
0246,   35.120,        1.0
0247,   35.258,        1.0
0248,   35.397,        1.0
0249,   35.537,        1.0
0250,   35.677,        1.0
0251,   35.817,        1.0
0252,   35.957,        1.0
0253,   36.097,        1.0
0254,   36.234,        1.0
0255,   36.375,        1.0
0256,   36.521,        1.0
0257,   36.660,        1.0
0258,   36.800,        1.0
0259,   36.942,        1.0
0260,   37.077,        1.0
0261,   37.216,        1.0
0262,   37.354,        1.0
0263,   37.501,        1.0
0264,   37.645,        1.0
0265,   37.787,        1.0
0266,   37.928,        1.0
0267,   38.067,        1.0
0268,   38.205,        1.0
0269,   38.349,        1.0
0270,   38.491,        1.0
0271,   38.633,        1.0
0272,   38.768,        1.0
0273,   38.909,        1.0
0274,   39.049,        1.0
0275,   39.185,        1.0
0276,   39.327,        1.0
0277,   39.486,        1.0
0278,   39.625,        1.0
0279,   39.771,        1.0
0280,   39.912,        1.0
0281,   40.048,        1.0
0282,   40.185,        1.0
0283,   40.325,        1.0
0284,   40.513,        1.0
0285,   40.656,        1.0
0286,   40.796,        1.0
0287,   40.939,        1.0
0288,   41.077,        1.0
0289,   41.225,        1.0
0290,   41.363,        1.0
0291,   41.507,        1.0
0292,   41.650,        1.0
0293,   41.786,        1.0
0294,   41.926,        0.9
0295,   42.066,        0.9
0296,   42.207,        0.9
0297,   42.346,        0.9
0298,   42.495,        0.9
0299,   42.637,        0.9
0300,   42.773,        0.8
0301,   42.912,        0.8
0302,   43.053,        0.7
0303,   43.192,        0.7
0304,   43.333,        0.7
0305,   43.476,        0.7
0306,   43.620,        0.7
0307,   43.758,        0.7
0308,   43.895,        0.7
0309,   44.027,        0.7
0310,   44.166,        0.7
0311,   44.310,        0.7
0312,   44.453,        0.7
0313,   44.594,        0.7
0314,   44.733,        0.7
0315,   44.871,        0.7
0316,   45.012,        0.7
0317,   45.145,        0.7
0318,   45.291,        0.7
0319,   45.441,        0.7
0320,   45.588,        0.7
0321,   45.730,        0.7
0322,   45.868,        0.7
0323,   46.004,        0.7
0324,   46.143,      330.3
0325,   46.284,       22.4
0326,   46.423,        4.0
0327,   46.561,        1.5
0328,   46.704,        0.9
0329,   46.841,        0.7
0330,   46.979,        0.7
0331,   47.123,        0.7
0332,   47.267,        0.7
0333,   47.412,        0.7
0334,   47.569,        0.7
0335,   47.725,        0.7
0336,   47.880,        0.7
0337,   48.036,        0.7
0338,   48.188,        0.7
0339,   48.351,        0.7
0340,   48.519,        0.7
0341,   48.685,        0.7
0342,   48.831,        0.7
0343,   48.985,        0.7
CPU times: user 12.5 s, sys: 856 ms, total: 13.4 s
Wall time: 49 s

Show and test results



In [19]:

    
display_opt_params()

# show result
idx = range(len(time_marks))
display_data_for_test();

update_guess();









    




Parameters from optimization






    




Idx Description $x$
0 k_1_-41 1.01114537558
1 k_1_-35 1.06964977738
2 k_1_-30 1.13337426265
3 k_1_-25 1.19943077017
4 k_1_-20 1.22000196622
5 k_1_-15 1.24991354029
6 k_1_-10 1.23143272329
7 k_1_-5 1.03755684016
8 k_1_5 0.846035509447
9 k_1_10 1.2617931813
10 k_1_15 1.27809150051
11 k_1_20 1.22433259622
12 k_1_25 1.0465672512
13 k_1_30 1.0286947112
14 k_1_35 0.987233221447
15 k_1_41 0.937897430281
16 k_2_-41 1.0
17 k_2_-35 1.0
18 k_2_-30 1.0
19 k_2_-25 1.0
20 k_2_-20 1.0
21 k_2_-15 1.0
22 k_2_-10 1.0
23 k_2_-5 1.11896655811
24 k_2_5 0.753403281578
25 k_2_10 1.02926687063
26 k_2_15 0.983482028172
27 k_2_20 0.923433988277
28 k_2_25 0.85778075496
29 k_2_30 0.845995462606
30 k_2_35 0.848370297607
31 k_2_41 0.862036781638
32 k_3_-41 0.734766321056
33 k_3_-35 0.736067709239
34 k_3_-30 0.736804555394
35 k_3_-25 0.737841666536
36 k_3_-20 0.751930160755
37 k_3_-15 0.817344411153
38 k_3_-10 0.995254801914
39 k_3_-5 1.06511394712
40 k_3_5 0.878411999056
41 k_3_10 0.854901612376
42 k_3_15 0.9423031392
43 k_3_20 0.954849872397
44 k_3_25 0.924917778453
45 k_3_30 0.880464821699
46 k_3_35 0.830803608297
47 k_3_41 0.83830255326
48 k_4_-41 0.752025117385
49 k_4_-35 0.697429879246
50 k_4_-30 0.755344949468
51 k_4_-25 0.780372556987
52 k_4_-20 0.785149396505
53 k_4_-15 0.905903979292
54 k_4_-10 0.920501474895
55 k_4_-5 0.957986995052
56 k_4_5 0.875386593958
57 k_4_10 0.813355757391
58 k_4_15 0.765194724818
59 k_4_20 0.623408591035
60 k_4_25 0.638072460684
61 k_4_30 0.650309417787
62 k_4_35 0.651176135704
63 k_4_41 0.669445414955
64 $phi_0$ -0.00227919058251






    




Test Data






    




Idx Start(s) Spam(s) Description
0 28.42 59.95 ramp cmp1 u
1 90.48 59.64 ramp cmp1 d
2 221.85 58.79 ramp cmp2 u
3 283.96 59.55 ramp cmp2 d
4 345.43 60.14 ramp cmp3 u
5 408.59 59.59 ramp cmp3 d
6 541.71 59.04 ramp cmp4 u
7 604.04 59.87 ramp cmp4 d
8 677.82 100.21 ramp cmp1/3 d
9 780.10 99.77 ramp cmp1/3 u
10 888.08 99.23 ramp cmp2/4 d
11 989.76 99.24 ramp cmp2/4 u
12 1101.93 119.52 ramp cmpall u
13 1240.33 120.40 ramp cmpall d



In [20]:

    
res_params = res['x']
params = param0[:]
for i,j in enumerate(opt_idx):
    params[j] = res_params[i]

k1 = np.array(params[0:angles_num])
k2 = np.array(params[angles_num:angles_num*2])
k3 = np.array(params[angles_num*2:angles_num*3])
k4 = np.array(params[angles_num*3:angles_num*4])

Clda_cmp1 = Clda_cmp*0.00436332313*k1*angles
Clda_cmp2 = Clda_cmp*0.00436332313*k2*angles
Clda_cmp3 = Clda_cmp*0.00436332313*k3*angles
Clda_cmp4 = Clda_cmp*0.00436332313*k4*angles

print('angeles = ')
print(angles)
print('Clda_cmpx = ')
print(np.vstack((Clda_cmp1,Clda_cmp2,Clda_cmp3,Clda_cmp4)))

%matplotlib inline
plt.figure(figsize=(12,8),dpi=300)
plt.plot(angles,  Clda_cmp1, 'r')
plt.plot(angles,  Clda_cmp2, 'g')
plt.plot(angles,  Clda_cmp3, 'b')
plt.plot(angles,  Clda_cmp4, 'm')
plt.xlabel('$\delta_{a,cmp}$')
plt.ylabel('$C_{l \delta a,cmp}$')
plt.show()









    



angeles = 
[-41, -35, -30, -25, -20, -15, -10, -5, 5, 10, 15, 20, 25, 30, 35, 41]
Clda_cmpx = 
[[ 0.05365201  0.04845049  0.04400308  0.03880643  0.03157759  0.02426385
   0.01593673  0.00671383 -0.00547453 -0.01632964 -0.02481085 -0.03168969
  -0.03386068 -0.03993892 -0.04471738 -0.04976543]
 [ 0.05306063  0.04529566  0.03882485  0.03235404  0.02588323  0.01941242
   0.01294162  0.00724062 -0.00487513 -0.01332038 -0.01909177 -0.02390146
  -0.02775267 -0.03284565 -0.03842749 -0.04574021]
 [ 0.03898716  0.03334067  0.02860633  0.02387216  0.01946238  0.01586664
   0.01288021  0.00689215 -0.00568404 -0.01106381 -0.01829239 -0.0247146
  -0.02992483 -0.03418391 -0.0376318  -0.04448086]
 [ 0.03990292  0.03159054  0.02932615  0.02524821  0.0203222   0.01758579
   0.01191278  0.00619895 -0.00566446 -0.01052614 -0.01485428 -0.01613583
  -0.02064422 -0.02524817 -0.02949545 -0.03552119]]



In [ ]:

    
toggle_inputs()
button_qtconsole()

Idx	Start(s)	Spam(s)	Description
0	28.42	59.95	ramp cmp1 u
1	90.48	59.64	ramp cmp1 d
2	221.85	58.79	ramp cmp2 u
3	283.96	59.55	ramp cmp2 d
4	345.43	60.14	ramp cmp3 u
5	408.59	59.59	ramp cmp3 d
6	541.71	59.04	ramp cmp4 u
7	604.04	59.87	ramp cmp4 d
8	677.82	100.21	ramp cmp1/3 d
9	780.10	99.77	ramp cmp1/3 u
10	888.08	99.23	ramp cmp2/4 d
11	989.76	99.24	ramp cmp2/4 u
12	1101.93	119.52	ramp cmpall u
13	1240.33	120.40	ramp cmpall d

Name	Value	unit
$S_c$	0.1254	$m^2$
$b_c$	0.7	$m$
$g$	9.81	$m/s^2$
$m_T$	9.585	$kg$
$l_{zT}$	0.0416	$m$
$V$	25	$m/s$
$F_c$	0.06	$Nm$
$C_{l \delta a,cmp}$	-0.2966	$rad^{-1}$

Idx	Description	$x_0$	unit	opt	$\delta_{par}$	min	max
0	k_1_-41	1	1	*	0.001	0	1.5
1	k_1_-35	1	1	*	0.001	0	1.5
2	k_1_-30	1	1	*	0.001	0	1.5
3	k_1_-25	1	1	*	0.001	0	1.5
4	k_1_-20	1	1	*	0.001	0	1.5
5	k_1_-15	1	1	*	0.001	0	1.5
6	k_1_-10	1	1	*	0.001	0	1.5
7	k_1_-5	1	1	*	0.001	0	1.5
8	k_1_5	1	1	*	0.001	0	1.5
9	k_1_10	1	1	*	0.001	0	1.5
10	k_1_15	1	1	*	0.001	0	1.5
11	k_1_20	1	1	*	0.001	0	1.5
12	k_1_25	1	1	*	0.001	0	1.5
13	k_1_30	1	1	*	0.001	0	1.5
14	k_1_35	1	1	*	0.001	0	1.5
15	k_1_41	1	1	*	0.001	0	1.5
16	k_2_-41	1	1	*	0.001	0	1.5
17	k_2_-35	1	1	*	0.001	0	1.5
18	k_2_-30	1	1	*	0.001	0	1.5
19	k_2_-25	1	1	*	0.001	0	1.5
20	k_2_-20	1	1	*	0.001	0	1.5
21	k_2_-15	1	1	*	0.001	0	1.5
22	k_2_-10	1	1	*	0.001	0	1.5
23	k_2_-5	1	1	*	0.001	0	1.5
24	k_2_5	1	1	*	0.001	0	1.5
25	k_2_10	1	1	*	0.001	0	1.5
26	k_2_15	1	1	*	0.001	0	1.5
27	k_2_20	1	1	*	0.001	0	1.5
28	k_2_25	1	1	*	0.001	0	1.5
29	k_2_30	1	1	*	0.001	0	1.5
30	k_2_35	1	1	*	0.001	0	1.5
31	k_2_41	1	1	*	0.001	0	1.5
32	k_3_-41	1	1	*	0.001	0	1.5
33	k_3_-35	1	1	*	0.001	0	1.5
34	k_3_-30	1	1	*	0.001	0	1.5
35	k_3_-25	1	1	*	0.001	0	1.5
36	k_3_-20	1	1	*	0.001	0	1.5
37	k_3_-15	1	1	*	0.001	0	1.5
38	k_3_-10	1	1	*	0.001	0	1.5
39	k_3_-5	1	1	*	0.001	0	1.5
40	k_3_5	1	1	*	0.001	0	1.5
41	k_3_10	1	1	*	0.001	0	1.5
42	k_3_15	1	1	*	0.001	0	1.5
43	k_3_20	1	1	*	0.001	0	1.5
44	k_3_25	1	1	*	0.001	0	1.5
45	k_3_30	1	1	*	0.001	0	1.5
46	k_3_35	1	1	*	0.001	0	1.5
47	k_3_41	1	1	*	0.001	0	1.5
48	k_4_-41	1	1	*	0.001	0	1.5
49	k_4_-35	1	1	*	0.001	0	1.5
50	k_4_-30	1	1	*	0.001	0	1.5
51	k_4_-25	1	1	*	0.001	0	1.5
52	k_4_-20	1	1	*	0.001	0	1.5
53	k_4_-15	1	1	*	0.001	0	1.5
54	k_4_-10	1	1	*	0.001	0	1.5
55	k_4_-5	1	1	*	0.001	0	1.5
56	k_4_5	1	1	*	0.001	0	1.5
57	k_4_10	1	1	*	0.001	0	1.5
58	k_4_15	1	1	*	0.001	0	1.5
59	k_4_20	1	1	*	0.001	0	1.5
60	k_4_25	1	1	*	0.001	0	1.5
61	k_4_30	1	1	*	0.001	0	1.5
62	k_4_35	1	1	*	0.001	0	1.5
63	k_4_41	1	1	*	0.001	0	1.5
64	$phi_0$	0	$rad$	*	0.0001	-0.1	0.1

Idx	Description	$x$
0	k_1_-41	1.01114537558
1	k_1_-35	1.06964977738
2	k_1_-30	1.13337426265
3	k_1_-25	1.19943077017
4	k_1_-20	1.22000196622
5	k_1_-15	1.24991354029
6	k_1_-10	1.23143272329
7	k_1_-5	1.03755684016
8	k_1_5	0.846035509447
9	k_1_10	1.2617931813
10	k_1_15	1.27809150051
11	k_1_20	1.22433259622
12	k_1_25	1.0465672512
13	k_1_30	1.0286947112
14	k_1_35	0.987233221447
15	k_1_41	0.937897430281
16	k_2_-41	1.0
17	k_2_-35	1.0
18	k_2_-30	1.0
19	k_2_-25	1.0
20	k_2_-20	1.0
21	k_2_-15	1.0
22	k_2_-10	1.0
23	k_2_-5	1.11896655811
24	k_2_5	0.753403281578
25	k_2_10	1.02926687063
26	k_2_15	0.983482028172
27	k_2_20	0.923433988277
28	k_2_25	0.85778075496
29	k_2_30	0.845995462606
30	k_2_35	0.848370297607
31	k_2_41	0.862036781638
32	k_3_-41	0.734766321056
33	k_3_-35	0.736067709239
34	k_3_-30	0.736804555394
35	k_3_-25	0.737841666536
36	k_3_-20	0.751930160755
37	k_3_-15	0.817344411153
38	k_3_-10	0.995254801914
39	k_3_-5	1.06511394712
40	k_3_5	0.878411999056
41	k_3_10	0.854901612376
42	k_3_15	0.9423031392
43	k_3_20	0.954849872397
44	k_3_25	0.924917778453
45	k_3_30	0.880464821699
46	k_3_35	0.830803608297
47	k_3_41	0.83830255326
48	k_4_-41	0.752025117385
49	k_4_-35	0.697429879246
50	k_4_-30	0.755344949468
51	k_4_-25	0.780372556987
52	k_4_-20	0.785149396505
53	k_4_-15	0.905903979292
54	k_4_-10	0.920501474895
55	k_4_-5	0.957986995052
56	k_4_5	0.875386593958
57	k_4_10	0.813355757391
58	k_4_15	0.765194724818
59	k_4_20	0.623408591035
60	k_4_25	0.638072460684
61	k_4_30	0.650309417787
62	k_4_35	0.651176135704
63	k_4_41	0.669445414955
64	$phi_0$	-0.00227919058251