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z-Kfib0qImEa*9k!!TkY?Fpu(Hky!MEkq&{J4M%ND3?#BBu#Ig);v-CRWK$;@jGNh`pv(PcIu zei3c|32bQVTmfD-4y(<>i|AdWllNh^Rp3kJxUxrP32pe+X9lUR0!dQJdDp8;h+5ac zb?L?`(JwOeOO0DXyt7G%-9lGEn(&N$>DUr#%#BW#Dp>^|G~SVsy^8L(UHz}uhS>3%Y?WRfT}3|)jf1sbt^+%Y zKxy59H5B~vXp_&6b@(}=d%CZenZX9op1U~jAG(gz1UMA2%LbrV@fY=WXM+Y&y-JL& z9PnIMYTf3O3{af*F#P)`Y+RedlDp1;vz==r$={s&>c Bbz}el diff --git a/examples-gallery/plot_non_contiguous_parameter_identification_bias.py b/examples-gallery/plot_non_contiguous_parameter_identification_bias.py index 4c61325..a442363 100644 --- a/examples-gallery/plot_non_contiguous_parameter_identification_bias.py +++ b/examples-gallery/plot_non_contiguous_parameter_identification_bias.py @@ -18,15 +18,15 @@ If the measurements are biased in addition to being noisy, one way to handle this is to double the set of differential equations created above, where the -second set is a shifted copy of the first set, shifted by twice the expected -bias. +second set is a shifted copy of the first set. -Mass-spring-damper Example -========================== +Mass-spring-damper-friction Example +=================================== The position of a simple system consisting of a mass connected to a fixed point -by a spring and a damper is simulated and recorded as noisy measurements with -bias. The spring constant and the damping coefficient will be identified. +by a spring and a damper and subject to friction is simulated and recorded as +noisy measurements with bias. The spring constant, the damping coefficient +and the coefficient of friction will be identified. **State Variables** @@ -39,6 +39,7 @@ - :math:`c`: linear damping coefficient [Ns/m] - :math:`k`: linear spring constant [N/m] - :math:`l_0`: natural length of the spring [m] +- :math:`friction`: friction coefficient [N] """ import sympy as sm @@ -46,6 +47,7 @@ import numpy as np from scipy.integrate import solve_ivp from opty import Problem +from opty.utils import parse_free import matplotlib.pyplot as plt # %% @@ -60,7 +62,7 @@ t0, tf = 0.0, 10.0 num_nodes = 500 -m, c, k, l0 = sm.symbols('m, c, k, l0') +m, c, k, l0, friction = sm.symbols('m, c, k, l0, friction') bias_ident = sm.symbols(f'bias:{number_of_measurements}') t = me.dynamicsymbols._t @@ -76,8 +78,13 @@ for i in range(2*number_of_measurements): Points[i].set_pos(O, x[i]*N.x) Points[i].set_vel(N, u[i]*N.x) -bodies = [me.Particle('part_' + str(i), Points[i], m) for i in range(2*number_of_measurements)] -forces = [(Points[i], -c*u[i]*N.x - k*(x[i]-l0)*N.x) for i in range(2*number_of_measurements)] +bodies = [me.Particle('part_' + str(i), Points[i], m) + for i in range(2*number_of_measurements)] +# %% +# :math:`\tanh(\alpha \cdot x) \approx sign(x)` for large :math:`\alpha`, and +# is differentiale everywhere. +forces = [(Points[i], -c*u[i]*N.x - k*(x[i]-l0)*N.x - + friction * sm.tanh(30*u[i])*N.x) for i in range(2*number_of_measurements)] q_ind = x[: number_of_measurements] q_dep = x[number_of_measurements :] @@ -86,7 +93,7 @@ kd = sm.Matrix([u[i] - x[i].diff(t) for i in range(2*number_of_measurements)]) -config_constr = sm.Matrix([x[i] - x[number_of_measurements + i] + 2*bias_ident[i] +config_constr = sm.Matrix([x[i] - x[number_of_measurements + i] + 1*bias_ident[i] for i in range(number_of_measurements)]) speed_constr = config_constr.diff(t) @@ -104,7 +111,18 @@ fr, frstar = KM.kanes_equations(bodies, forces) eom =kd.col_join(fr+frstar) eom = eom.col_join(config_constr) -sm.pprint(eom) + +for i in range(2): + sm.pprint(eom[i]) +for i in range(2): + print('.........') +for i in range(2): + sm.pprint(eom[2*number_of_measurements+i]) +for i in range(2): + print('.........') +for i in range(2): + sm.pprint(eom[-(2-i)]) + # %% # Generate Noisy Measurement Data with Bias @@ -116,26 +134,28 @@ # rhs = KM.rhs() states = x + u -parameters = [m, c, k, l0] + list(bias_ident) -par_vals = [1.0, 0.75, 2.0, 1.0] + [0] * number_of_measurements +parameters = [m, c, k, l0, friction] + list(bias_ident) +par_vals = [1.0, 0.25, 2.0, 1.0, 0.5] + [0] * number_of_measurements eval_rhs = sm.lambdify(states + parameters, rhs) times = np.linspace(t0, tf, num=num_nodes) measurements = [] -np.random.seed(12345) +np.random.seed(123) # %% # As the second half of x, u are a shifted copy of the first half, they must get # identical initial conditions. +start1_list, start2_list = [], [] for i in range(number_of_measurements): start1 = 4.0*np.random.randn(number_of_measurements) start2 = 4.0*np.random.randn(number_of_measurements) + x0 = np.hstack((start1, start1, start2, start2)) sol = solve_ivp(lambda t, x, p: eval_rhs(*x, *p).squeeze(), (t0, tf), x0, t_eval=times, args=(par_vals,)) - measurements.append(sol.y[0, :] + 1.0*np.random.randn(num_nodes)) + measurements.append(sol.y[1, :] + 1.0*np.random.randn(num_nodes)) # %% # Add bias to the measurements bias = 4.0 *np.random.uniform(0., 1., size=number_of_measurements) @@ -147,8 +167,9 @@ # Setup the Identification Problem # -------------------------------- # -# The goal is to identify the damping coefficient :math:`c` and the spring -# constant :math:`k`. The objective :math:`J` is to minimize the least square +# The goal is to identify the damping coefficient :math:`c`, the spring +# constant :math:`k` and the coefficient of friction :math:`friction`. +# The objective :math:`J` is to minimize the least square # difference in the optimal simulation as compared to the measurements. If # some measurement is considered more reliable, its weight :math:`w` may be # increased relative to the other measurements. @@ -177,14 +198,15 @@ def obj_grad(free): # %% -# By not including :math:`c`, :math:`k`, :math:`bias_i` in the parameter map, -# they will be treated as unknown parameters. +# By not including :math:`c`, :math:`k`, :math:`friction`, :math:`bias_i` +# in the parameter map, they will be treated as unknown parameters. par_map = {m: par_vals[0], l0: par_vals[3]} # %% bounds = { c: (0.01, 2.0), k: (0.1, 10.0), + friction: (0.1, 10.0), } | {u[i]: (-10, 10) for i in range(2*number_of_measurements)} problem = Problem( @@ -215,7 +237,7 @@ def obj_grad(free): np.array(measurements).flatten(), # x1 np.zeros(2*number_of_measurements*num_nodes), # u [0.5 for _ in range(number_of_measurements)], # bias - [0.1, 3.0], # c, k + [0.1, 3.0, 2.0], # c, k, friction )) @@ -223,6 +245,8 @@ def obj_grad(free): # Solve the Optimization Problem # ------------------------------ # +# Here, the initial guess is the result of some previous run, stored in +# 'non_contiguous_parameter_identification_bias_solution.npy'. initial_guess = np.load('non_contiguous_parameter_identification_bias_solution.npy') solution, info = problem.solve(initial_guess) # %% @@ -243,10 +267,9 @@ def obj_grad(free): # The identified parameters are: # ------------------------------ # -print(f'Estimate of damping coefficient is {solution[-2]: 1.2f}') -print(f'Estimate of the spring constant is {solution[-1]: 1.2f} \n') -for i in range(number_of_measurements): - print(f'estimated {i}-th bias is {solution[-8+i]:.2f}, true bias is {bias[i]:.2f}') +print(f'Estimate of damping coefficient is {solution[-3]: 1.2f}') +print(f'Estimate of the spring constant is {solution[-1]: 1.2f}') +print(f'Estimate of the friction coefficient is {solution[-2]: 1.2f}') # %% # Plot the measurements and the trajectories calculated. @@ -264,4 +287,5 @@ def obj_grad(free): # %% # # sphinx_gallery_thumbnail_number = 3 + plt.show()