fail to estimate a simple linear system identification (the result is sensitive to discretization freq)

[utotch]

I tried to solve a simplest (linear grey-box system) problem to check nlgreyfast but it produced an incorrect result, which was unexpected.

Problem setting:
linear continuous-time grey-box system
#parameter = 1
#state = 2
#input = 1

I expected nlgreyfast solve this easily if I discretized the system with enough frequency, but it estimated the wrong value even though I chose enough freq.

I'm not sure it's my setting problem or inherent difficulty, but I'd be glad if you comment something to this.

code

% nlgreyfast sample
% dx/dt = [0 1; 0 theta]*[x1; x2] + [0; 1]*u1
% estimate theta
x1 = casadi.MX.sym('x1');
x2 = casadi.MX.sym('x2');
u1 = casadi.MX.sym('u1');
theta = casadi.MX.sym('theta');
n_param = 1;
Ac = [0 1; 0 theta];
Bc = [0; 1];
rhs = Ac*[x1; x2] + Bc*u1;
ode = casadi.Function('ode', {[x1; x2], u1, theta}, {rhs});
sample_freq = 20.0; % [Hz]: estimate value is very sensitive to this
num_sample_sim = 100;
num_steps_per_sample = 10;
[model, all_samples] = nlsim(ode, sample_freq, num_sample_sim, num_steps_per_sample);
fopts = struct;
fopts.scale_v       = ones(n_param,1);
fopts.param_guess_v = 1.0;
fopts.param_lb_v    = -20;
fopts.param_ub_v    =  20;
fopts.x0_v          = [0; 0];
fopts.x0_free_v     = [false; false];
fopts.y_states_v    = [1];
fopts.X_guess_H     = 'guess_mechatronic';
fopts.N_group_size  = 1;

% dataset(train)
dt = 0.1;
ts = (0:dt:100)';
us = sawtooth(ts);
theta_true = -2.0;
sysc = ss([0 1; 0 theta_true], Bc, [1 0], 0);
x0 = [0;0];
[ys, ts_, xs] = lsim(sysc, us, ts, x0);
data_train = iddata(ys, us, dt);
plot(data_train);
model_est = nlgreyfast(data_train, model, fopts);
fprintf(2,'[theta(true)    ]%f\n', theta_true);
fprintf(2,'[theta(estimate)]%f\n', model_est.param_est_v);
fprintf(2,'[x0(true)       ][%f;%f]\n',  x0(1), x0(2));
fprintf(2,'[x0(estimate)   ][%f;%f]\n',  model_est.x0_est_v(1), model_est.x0_est_v(2));

result

[theta(true)    ]-2.000000
[theta(estimate)]-1.000812
[x0(true)       ][0.000000;0.000000]
[x0(estimate)   ][0.000000;0.000000]