def solve_marathos_problem_with_setting(setting):
globalization = setting['globalization']
line_search_use_sufficient_descent = setting[
'line_search_use_sufficient_descent']
globalization_use_SOC = setting['globalization_use_SOC']
# create ocp object to formulate the OCP
ocp = AcadosOcp()
# set model
model = AcadosModel()
x1 = SX.sym('x1')
x2 = SX.sym('x2')
x = vertcat(x1, x2)
# dynamics: identity
model.disc_dyn_expr = x
model.x = x
model.u = SX.sym('u', 0, 0) # [] / None doesnt work
model.p = []
model.name = f'marathos_problem'
ocp.model = model
# discretization
Tf = 1
N = 1
ocp.dims.N = N
ocp.solver_options.tf = Tf
# cost
ocp.cost.cost_type_e = 'EXTERNAL'
ocp.model.cost_expr_ext_cost_e = x1
# constarints
ocp.model.con_h_expr = x1**2 + x2**2
ocp.constraints.lh = np.array([1.0])
ocp.constraints.uh = np.array([1.0])
# # soften
# ocp.constraints.idxsh = np.array([0])
# ocp.cost.zl = 1e5 * np.array([1])
# ocp.cost.zu = 1e5 * np.array([1])
# ocp.cost.Zl = 1e5 * np.array([1])
# ocp.cost.Zu = 1e5 * np.array([1])
# add bounds on x
# nx = 2
# ocp.constraints.idxbx_0 = np.array(range(nx))
# ocp.constraints.lbx_0 = -2 * np.ones((nx))
# ocp.constraints.ubx_0 = 2 * np.ones((nx))
# set options
ocp.solver_options.qp_solver = 'PARTIAL_CONDENSING_HPIPM' # FULL_CONDENSING_QPOASES
# PARTIAL_CONDENSING_HPIPM, FULL_CONDENSING_QPOASES, FULL_CONDENSING_HPIPM,
# PARTIAL_CONDENSING_QPDUNES, PARTIAL_CONDENSING_OSQP
ocp.solver_options.hessian_approx = 'EXACT'
ocp.solver_options.integrator_type = 'DISCRETE'
# ocp.solver_options.print_level = 1
ocp.solver_options.tol = TOL
ocp.solver_options.nlp_solver_type = 'SQP' # SQP_RTI, SQP
ocp.solver_options.globalization = globalization
ocp.solver_options.alpha_min = 1e-2
# ocp.solver_options.__initialize_t_slacks = 0
# ocp.solver_options.regularize_method = 'CONVEXIFY'
ocp.solver_options.levenberg_marquardt = 1e-1
# ocp.solver_options.print_level = 2
SQP_max_iter = 300
ocp.solver_options.qp_solver_iter_max = 400
ocp.solver_options.regularize_method = 'MIRROR'
# ocp.solver_options.exact_hess_constr = 0
ocp.solver_options.line_search_use_sufficient_descent = line_search_use_sufficient_descent
ocp.solver_options.globalization_use_SOC = globalization_use_SOC
ocp.solver_options.eps_sufficient_descent = 1e-1
ocp.solver_options.qp_tol = 5e-7
if FOR_LOOPING: # call solver in for loop to get all iterates
ocp.solver_options.nlp_solver_max_iter = 1
ocp_solver = AcadosOcpSolver(ocp, json_file=f'{model.name}.json')
else:
ocp.solver_options.nlp_solver_max_iter = SQP_max_iter
ocp_solver = AcadosOcpSolver(ocp, json_file=f'{model.name}.json')
# initialize solver
rad_init = 0.1 #0.1 #np.pi / 4
xinit = np.array([np.cos(rad_init), np.sin(rad_init)])
# xinit = np.array([0.82120912, 0.58406911])
[ocp_solver.set(i, "x", xinit) for i in range(N + 1)]
# solve
if FOR_LOOPING: # call solver in for loop to get all iterates
iterates = np.zeros((SQP_max_iter + 1, 2))
iterates[0, :] = xinit
alphas = np.zeros((SQP_max_iter, ))
qp_iters = np.zeros((SQP_max_iter, ))
iter = SQP_max_iter
residuals = np.zeros((4, SQP_max_iter))
# solve
for i in range(SQP_max_iter):
status = ocp_solver.solve()
ocp_solver.print_statistics(
) # encapsulates: stat = ocp_solver.get_stats("statistics")
# print(f'acados returned status {status}.')
iterates[i + 1, :] = ocp_solver.get(0, "x")
if status in [0, 4]:
iter = i
break
alphas[i] = ocp_solver.get_stats('alpha')[1]
qp_iters[i] = ocp_solver.get_stats('qp_iter')[1]
residuals[:, i] = ocp_solver.get_stats('residuals')
else:
ocp_solver.solve()
ocp_solver.print_statistics()
iter = ocp_solver.get_stats('sqp_iter')[0]
alphas = ocp_solver.get_stats('alpha')[1:]
qp_iters = ocp_solver.get_stats('qp_iter')
residuals = ocp_solver.get_stats('statistics')[1:5, 1:iter]
# get solution
solution = ocp_solver.get(0, "x")
# print summary
print(f"solved Marathos test problem with settings {setting}")
print(
f"cost function value = {ocp_solver.get_cost()} after {iter} SQP iterations"
)
print(f"alphas: {alphas[:iter]}")
print(f"total number of QP iterations: {sum(qp_iters[:iter])}")
max_infeasibility = np.max(residuals[1:3])
print(f"max infeasibility: {max_infeasibility}")
# compare to analytical solution
exact_solution = np.array([-1, 0])
sol_err = max(np.abs(solution - exact_solution))
# checks
if sol_err > TOL * 1e1:
raise Exception(
f"error of numerical solution wrt exact solution = {sol_err} > tol = {TOL*1e1}"
)
else:
print(f"matched analytical solution with tolerance {TOL}")
try:
if globalization == 'FIXED_STEP':
# import pdb; pdb.set_trace()
if max_infeasibility < 5.0:
raise Exception(
f"Expected max_infeasibility > 5.0 when using full step SQP on Marathos problem"
)
if iter != 10:
raise Exception(
f"Expected 10 SQP iterations when using full step SQP on Marathos problem, got {iter}"
)
if any(alphas[:iter] != 1.0):
raise Exception(
f"Expected all alphas = 1.0 when using full step SQP on Marathos problem"
)
elif globalization == 'MERIT_BACKTRACKING':
if max_infeasibility > 0.5:
raise Exception(
f"Expected max_infeasibility < 0.5 when using globalized SQP on Marathos problem"
)
if globalization_use_SOC == 0:
if FOR_LOOPING and iter != 57:
raise Exception(
f"Expected 57 SQP iterations when using globalized SQP without SOC on Marathos problem, got {iter}"
)
elif line_search_use_sufficient_descent == 1:
if iter not in range(29, 37):
# NOTE: got 29 locally and 36 on Github actions.
# On Github actions the inequality constraint was numerically violated in the beginning.
# This leads to very different behavior, since the merit gradient is so different.
# Github actions: merit_grad = -1.669330e+00, merit_grad_cost = -1.737950e-01, merit_grad_dyn = 0.000000e+00, merit_grad_ineq = -1.495535e+00
# Jonathan Laptop: merit_grad = -1.737950e-01, merit_grad_cost = -1.737950e-01, merit_grad_dyn = 0.000000e+00, merit_grad_ineq = 0.000000e+00
raise Exception(
f"Expected SQP iterations in range(29, 37) when using globalized SQP with SOC on Marathos problem, got {iter}"
)
else:
if iter != 12:
raise Exception(
f"Expected 12 SQP iterations when using globalized SQP with SOC on Marathos problem, got {iter}"
)
except Exception as inst:
if FOR_LOOPING and globalization == "MERIT_BACKTRACKING":
print(
"\nAcados globalized OCP solver behaves different when for looping due to different merit function weights.",
"Following exception is not raised\n")
print(inst, "\n")
else:
raise (inst)
if PLOT:
plt.figure()
axs = plt.plot(solution[0], solution[1], 'x', label='solution')
if FOR_LOOPING: # call solver in for loop to get all iterates
cm = plt.cm.get_cmap('RdYlBu')
axs = plt.scatter(iterates[:iter + 1, 0],
iterates[:iter + 1, 1],
c=range(iter + 1),
s=35,
cmap=cm,
label='iterates')
plt.colorbar(axs)
ts = np.linspace(0, 2 * np.pi, 100)
plt.plot(1 * np.cos(ts) + 0, 1 * np.sin(ts) - 0, 'r')
plt.axis('square')
plt.legend()
plt.title(
f"Marathos problem with N = {N}, x formulation, SOC {globalization_use_SOC}"
)
plt.show()
print(f"\n\n----------------------\n")
simX[N, :] = ocp_solver.get(N, "x")
print("inequality multipliers at stage 1")
print(ocp_solver.get(1, "lam")) # inequality multipliers at stage 1
print("slack values at stage 1")
print(ocp_solver.get(1, "t")) # slack values at stage 1
print("multipliers of dynamic conditions between stage 1 and 2")
print(ocp_solver.get(
1, "pi")) # multipliers of dynamic conditions between stage 1 and 2
# initialize ineq multipliers and slacks at stage 1
ocp_solver.set(1, "lam", np.zeros(2, ))
ocp_solver.set(1, "t", np.zeros(2, ))
ocp_solver.print_statistics(
) # encapsulates: stat = ocp_solver.get_stats("statistics")
# timings
time_tot = ocp_solver.get_stats("time_tot")
time_lin = ocp_solver.get_stats("time_lin")
time_sim = ocp_solver.get_stats("time_sim")
time_qp = ocp_solver.get_stats("time_qp")
print(
f"timings OCP solver: total: {1e3*time_tot}ms, lin: {1e3*time_lin}ms, sim: {1e3*time_sim}ms, qp: {1e3*time_qp}ms"
)
# print("simU", simU)
# print("simX", simX)
plot_pendulum(shooting_nodes, Fmax, simU, simX, latexify=False)
def solve_marathos_ocp(setting):
globalization = setting['globalization']
line_search_use_sufficient_descent = setting[
'line_search_use_sufficient_descent']
globalization_use_SOC = setting['globalization_use_SOC']
qp_solver = setting['qp_solver']
# create ocp object to formulate the OCP
ocp = AcadosOcp()
# set model
model = export_linear_mass_model()
ocp.model = model
nx = model.x.size()[0]
nu = model.u.size()[0]
ny = nu
# discretization
Tf = 2
N = 20
shooting_nodes = np.linspace(0, Tf, N + 1)
ocp.dims.N = N
# set cost
Q = 2 * np.diag([])
R = 2 * np.diag([1e1, 1e1])
ocp.cost.W_e = Q
ocp.cost.W = scipy.linalg.block_diag(Q, R)
ocp.cost.cost_type = 'LINEAR_LS'
ocp.cost.cost_type_e = 'LINEAR_LS'
ocp.cost.Vx = np.zeros((ny, nx))
Vu = np.eye((nu))
ocp.cost.Vu = Vu
ocp.cost.yref = np.zeros((ny, ))
# set constraints
Fmax = 5
ocp.constraints.lbu = -Fmax * np.ones((nu, ))
ocp.constraints.ubu = +Fmax * np.ones((nu, ))
ocp.constraints.idxbu = np.array(range(nu))
x0 = np.array([1e-1, 1.1, 0, 0])
ocp.constraints.x0 = x0
# terminal constraint
x_goal = np.array([0, -1.1, 0, 0])
ocp.constraints.idxbx_e = np.array(range(nx))
ocp.constraints.lbx_e = x_goal
ocp.constraints.ubx_e = x_goal
if SOFTEN_TERMINAL:
ocp.constraints.idxsbx_e = np.array(range(nx))
ocp.cost.zl_e = 1e4 * np.ones(nx)
ocp.cost.zu_e = 1e4 * np.ones(nx)
ocp.cost.Zl_e = 1e6 * np.ones(nx)
ocp.cost.Zu_e = 1e6 * np.ones(nx)
# add obstacle
if OBSTACLE:
obs_rad = 1.0
obs_x = 0.0
obs_y = 0.0
circle = (obs_x, obs_y, obs_rad)
ocp.constraints.uh = np.array([100.0]) # doenst matter
ocp.constraints.lh = np.array([obs_rad**2])
x_square = model.x[0]**OBSTACLE_POWER + model.x[1]**OBSTACLE_POWER
ocp.model.con_h_expr = x_square
# copy for terminal
ocp.constraints.uh_e = ocp.constraints.uh
ocp.constraints.lh_e = ocp.constraints.lh
ocp.model.con_h_expr_e = ocp.model.con_h_expr
else:
circle = None
# soften
if OBSTACLE and SOFTEN_OBSTACLE:
ocp.constraints.idxsh = np.array([0])
ocp.constraints.idxsh_e = np.array([0])
Zh = 1e6 * np.ones(1)
zh = 1e4 * np.ones(1)
ocp.cost.zl = zh
ocp.cost.zu = zh
ocp.cost.Zl = Zh
ocp.cost.Zu = Zh
ocp.cost.zl_e = np.concatenate((ocp.cost.zl_e, zh))
ocp.cost.zu_e = np.concatenate((ocp.cost.zu_e, zh))
ocp.cost.Zl_e = np.concatenate((ocp.cost.Zl_e, Zh))
ocp.cost.Zu_e = np.concatenate((ocp.cost.Zu_e, Zh))
# set options
ocp.solver_options.qp_solver = qp_solver # FULL_CONDENSING_QPOASES
# PARTIAL_CONDENSING_HPIPM, FULL_CONDENSING_QPOASES, FULL_CONDENSING_HPIPM,
# PARTIAL_CONDENSING_QPDUNES, PARTIAL_CONDENSING_OSQP
ocp.solver_options.hessian_approx = 'GAUSS_NEWTON'
ocp.solver_options.integrator_type = 'ERK'
# ocp.solver_options.print_level = 1
ocp.solver_options.nlp_solver_type = 'SQP' # SQP_RTI, SQP
ocp.solver_options.globalization = globalization
ocp.solver_options.alpha_min = 0.01
# ocp.solver_options.__initialize_t_slacks = 0
# ocp.solver_options.levenberg_marquardt = 1e-2
ocp.solver_options.qp_solver_cond_N = 0
ocp.solver_options.print_level = 1
ocp.solver_options.nlp_solver_max_iter = 200
ocp.solver_options.qp_solver_iter_max = 400
# NOTE: this is needed for PARTIAL_CONDENSING_HPIPM to get expected behavior
qp_tol = 5e-7
ocp.solver_options.qp_solver_tol_stat = qp_tol
ocp.solver_options.qp_solver_tol_eq = qp_tol
ocp.solver_options.qp_solver_tol_ineq = qp_tol
ocp.solver_options.qp_solver_tol_comp = qp_tol
ocp.solver_options.qp_solver_ric_alg = 1
# ocp.solver_options.qp_solver_cond_ric_alg = 1
# set prediction horizon
ocp.solver_options.tf = Tf
ocp_solver = AcadosOcpSolver(ocp, json_file=f'{model.name}_ocp.json')
ocp_solver.options_set('line_search_use_sufficient_descent',
line_search_use_sufficient_descent)
ocp_solver.options_set('globalization_use_SOC', globalization_use_SOC)
ocp_solver.options_set('full_step_dual', 1)
if INITIALIZE: # initialize solver
# [ocp_solver.set(i, "x", x0 + (i/N) * (x_goal-x0)) for i in range(N+1)]
[ocp_solver.set(i, "x", x0) for i in range(N + 1)]
# [ocp_solver.set(i, "u", 2*(np.random.rand(2) - 0.5)) for i in range(N)]
# solve
status = ocp_solver.solve()
ocp_solver.print_statistics(
) # encapsulates: stat = ocp_solver.get_stats("statistics")
sqp_iter = ocp_solver.get_stats('sqp_iter')[0]
print(f'acados returned status {status}.')
# ocp_solver.store_iterate(f'it{ocp.solver_options.nlp_solver_max_iter}_{model.name}.json')
# get solution
simX = np.array([ocp_solver.get(i, "x") for i in range(N + 1)])
simU = np.array([ocp_solver.get(i, "u") for i in range(N)])
pi_multiplier = [ocp_solver.get(i, "pi") for i in range(N)]
print(f"cost function value = {ocp_solver.get_cost()}")
# print summary
print(f"solved Marathos test problem with settings {setting}")
print(
f"cost function value = {ocp_solver.get_cost()} after {sqp_iter} SQP iterations"
)
# print(f"alphas: {alphas[:iter]}")
# print(f"total number of QP iterations: {sum(qp_iters[:iter])}")
# max_infeasibility = np.max(residuals[1:3])
# print(f"max infeasibility: {max_infeasibility}")
# checks
if status != 0:
raise Exception(f"acados solver returned status {status} != 0.")
if globalization == "FIXED_STEP":
if sqp_iter != 18:
raise Exception(
f"acados solver took {sqp_iter} iterations, expected 18.")
elif globalization == "MERIT_BACKTRACKING":
if globalization_use_SOC == 1 and line_search_use_sufficient_descent == 0 and sqp_iter not in range(
21, 23):
raise Exception(
f"acados solver took {sqp_iter} iterations, expected range(21, 23)."
)
elif globalization_use_SOC == 1 and line_search_use_sufficient_descent == 1 and sqp_iter not in range(
21, 24):
raise Exception(
f"acados solver took {sqp_iter} iterations, expected range(21, 24)."
)
elif globalization_use_SOC == 0 and line_search_use_sufficient_descent == 0 and sqp_iter not in range(
155, 165):
raise Exception(
f"acados solver took {sqp_iter} iterations, expected range(155, 165)."
)
elif globalization_use_SOC == 0 and line_search_use_sufficient_descent == 1 and sqp_iter not in range(
160, 175):
raise Exception(
f"acados solver took {sqp_iter} iterations, expected range(160, 175)."
)
if PLOT:
plot_linear_mass_system_X_state_space(simX,
circle=circle,
x_goal=x_goal)
plot_linear_mass_system_U(shooting_nodes, simU)
# plot_linear_mass_system_X(shooting_nodes, simX)
# import pdb; pdb.set_trace()
print(f"\n\n----------------------\n")
def main(discretization='shooting_nodes'):
# create ocp object to formulate the OCP
ocp = AcadosOcp()
# set model
model = export_pendulum_ode_model()
ocp.model = model
integrator_type = 'LIFTED_IRK' # ERK, IRK, GNSF, LIFTED_IRK
if integrator_type == 'GNSF':
acados_dae_model_json_dump(model)
# structure detection in Matlab/Octave -> produces 'pendulum_ode_gnsf_functions.json'
status = os.system('octave detect_gnsf_from_json.m')
# load gnsf from json
with open(model.name + '_gnsf_functions.json', 'r') as f:
gnsf_dict = json.load(f)
ocp.gnsf_model = gnsf_dict
Tf = 1.0
nx = model.x.size()[0]
nu = model.u.size()[0]
ny = nx + nu
ny_e = nx
N = 15
# discretization
ocp.dims.N = N
# shooting_nodes = np.linspace(0, Tf, N+1)
time_steps = np.linspace(0, 1, N)
time_steps = Tf * time_steps / sum(time_steps)
shooting_nodes = np.zeros((N + 1, ))
for i in range(len(time_steps)):
shooting_nodes[i + 1] = shooting_nodes[i] + time_steps[i]
# nonuniform discretizations can be defined either by shooting_nodes or time_steps:
if discretization == 'shooting_nodes':
ocp.solver_options.shooting_nodes = shooting_nodes
elif discretization == 'time_steps':
ocp.solver_options.time_steps = time_steps
else:
raise NotImplementedError(
f"discretization type {discretization} not supported.")
# set num_steps
ocp.solver_options.sim_method_num_steps = 2 * np.ones((N, ))
ocp.solver_options.sim_method_num_steps[0] = 3
# set num_stages
ocp.solver_options.sim_method_num_stages = 2 * np.ones((N, ))
ocp.solver_options.sim_method_num_stages[0] = 4
# set cost
Q = 2 * np.diag([1e3, 1e3, 1e-2, 1e-2])
R = 2 * np.diag([1e-2])
ocp.cost.W_e = Q
ocp.cost.W = scipy.linalg.block_diag(Q, R)
ocp.cost.cost_type = 'LINEAR_LS'
ocp.cost.cost_type_e = 'LINEAR_LS'
ocp.cost.Vx = np.zeros((ny, nx))
ocp.cost.Vx[:nx, :nx] = np.eye(nx)
Vu = np.zeros((ny, nu))
Vu[4, 0] = 1.0
ocp.cost.Vu = Vu
ocp.cost.Vx_e = np.eye(nx)
ocp.cost.yref = np.zeros((ny, ))
ocp.cost.yref_e = np.zeros((ny_e, ))
# set constraints
Fmax = 80
ocp.constraints.lbu = np.array([-Fmax])
ocp.constraints.ubu = np.array([+Fmax])
x0 = np.array([0.0, np.pi, 0.0, 0.0])
ocp.constraints.x0 = x0
ocp.constraints.idxbu = np.array([0])
ocp.solver_options.qp_solver = 'PARTIAL_CONDENSING_HPIPM' # FULL_CONDENSING_QPOASES
ocp.solver_options.hessian_approx = 'GAUSS_NEWTON'
ocp.solver_options.integrator_type = integrator_type
ocp.solver_options.print_level = 0
ocp.solver_options.nlp_solver_type = 'SQP' # SQP_RTI, SQP
# set prediction horizon
ocp.solver_options.tf = Tf
ocp.solver_options.initialize_t_slacks = 1
# Set additional options for Simulink interface:
acados_path = get_acados_path()
json_path = os.path.join(acados_path,
'interfaces/acados_template/acados_template')
with open(json_path + '/simulink_default_opts.json', 'r') as f:
simulink_opts = json.load(f)
ocp_solver = AcadosOcpSolver(ocp,
json_file='acados_ocp.json',
simulink_opts=simulink_opts)
# ocp_solver = AcadosOcpSolver(ocp, json_file = 'acados_ocp.json')
simX = np.ndarray((N + 1, nx))
simU = np.ndarray((N, nu))
# change options after creating ocp_solver
ocp_solver.options_set("step_length", 0.99999)
ocp_solver.options_set("globalization",
"fixed_step") # fixed_step, merit_backtracking
ocp_solver.options_set("tol_eq", TOL)
ocp_solver.options_set("tol_stat", TOL)
ocp_solver.options_set("tol_ineq", TOL)
ocp_solver.options_set("tol_comp", TOL)
# initialize solver
for i in range(N):
ocp_solver.set(i, "x", x0)
status = ocp_solver.solve()
if status not in [0, 2]:
raise Exception('acados returned status {}. Exiting.'.format(status))
# get primal solution
for i in range(N):
simX[i, :] = ocp_solver.get(i, "x")
simU[i, :] = ocp_solver.get(i, "u")
simX[N, :] = ocp_solver.get(N, "x")
print("inequality multipliers at stage 1")
print(ocp_solver.get(1, "lam")) # inequality multipliers at stage 1
print("slack values at stage 1")
print(ocp_solver.get(1, "t")) # slack values at stage 1
print("multipliers of dynamic conditions between stage 1 and 2")
print(ocp_solver.get(
1, "pi")) # multipliers of dynamic conditions between stage 1 and 2
# initialize ineq multipliers and slacks at stage 1
ocp_solver.set(1, "lam", np.zeros(2, ))
ocp_solver.set(1, "t", np.zeros(2, ))
ocp_solver.print_statistics(
) # encapsulates: stat = ocp_solver.get_stats("statistics")
# timings
time_tot = ocp_solver.get_stats("time_tot")
time_lin = ocp_solver.get_stats("time_lin")
time_sim = ocp_solver.get_stats("time_sim")
time_qp = ocp_solver.get_stats("time_qp")
print(
f"timings OCP solver: total: {1e3*time_tot}ms, lin: {1e3*time_lin}ms, sim: {1e3*time_sim}ms, qp: {1e3*time_qp}ms"
)
# print("simU", simU)
# print("simX", simX)
iterate_filename = f'final_iterate_{discretization}.json'
ocp_solver.store_iterate(filename=iterate_filename, overwrite=True)
plot_pendulum(shooting_nodes, Fmax, simU, simX, latexify=False)
del ocp_solver
def solve_armijo_problem_with_setting(setting):
globalization = setting['globalization']
line_search_use_sufficient_descent = setting[
'line_search_use_sufficient_descent']
globalization_use_SOC = setting['globalization_use_SOC']
# create ocp object to formulate the OCP
ocp = AcadosOcp()
# set model
model = AcadosModel()
x = SX.sym('x')
# dynamics: identity
model.disc_dyn_expr = x
model.x = x
model.u = SX.sym('u', 0, 0) # [] / None doesnt work
model.p = []
model.name = f'armijo_problem'
ocp.model = model
# discretization
Tf = 1
N = 1
ocp.dims.N = N
ocp.solver_options.tf = Tf
# cost
ocp.cost.cost_type_e = 'EXTERNAL'
ocp.model.cost_expr_ext_cost_e = x @ x
ocp.model.cost_expr_ext_cost_custom_hess_e = 1.0 # 2.0 is the actual hessian
# constarints
ocp.constraints.idxbx = np.array([0])
ocp.constraints.lbx = np.array([-10.0])
ocp.constraints.ubx = np.array([10.0])
# options
ocp.solver_options.qp_solver = 'FULL_CONDENSING_QPOASES' # 'PARTIAL_CONDENSING_HPIPM' # FULL_CONDENSING_QPOASES
ocp.solver_options.hessian_approx = 'EXACT'
ocp.solver_options.integrator_type = 'DISCRETE'
ocp.solver_options.print_level = 0
ocp.solver_options.tol = TOL
ocp.solver_options.nlp_solver_type = 'SQP' # SQP_RTI, SQP
ocp.solver_options.globalization = globalization
ocp.solver_options.alpha_reduction = 0.9
ocp.solver_options.line_search_use_sufficient_descent = line_search_use_sufficient_descent
ocp.solver_options.globalization_use_SOC = globalization_use_SOC
ocp.solver_options.eps_sufficient_descent = 5e-1
SQP_max_iter = 200
ocp.solver_options.qp_solver_iter_max = 400
ocp.solver_options.nlp_solver_max_iter = SQP_max_iter
# create solver
ocp_solver = AcadosOcpSolver(ocp, json_file=f'{model.name}.json')
# initialize solver
xinit = np.array([1.0])
[ocp_solver.set(i, "x", xinit) for i in range(N + 1)]
# get stats
status = ocp_solver.solve()
ocp_solver.print_statistics()
iter = ocp_solver.get_stats('sqp_iter')[0]
alphas = ocp_solver.get_stats('alpha')[1:]
qp_iters = ocp_solver.get_stats('qp_iter')
print(f"acados ocp solver returned status {status}")
# get solution
solution = ocp_solver.get(0, "x")
print(f"found solution {solution}")
# print summary
print(f"solved Armijo test problem with settings {setting}")
print(
f"cost function value = {ocp_solver.get_cost()} after {iter} SQP iterations"
)
print(f"alphas: {alphas[:iter]}")
print(f"total number of QP iterations: {sum(qp_iters[:iter])}")
# compare to analytical solution
exact_solution = np.array([0.0])
sol_err = max(np.abs(solution - exact_solution))
print(f"error wrt analytical solution {sol_err}")
# checks
if ocp.model.cost_expr_ext_cost_custom_hess_e == 1.0:
if globalization == 'MERIT_BACKTRACKING':
if sol_err > TOL * 1e1:
raise Exception(
f"error of numerical solution wrt exact solution = {sol_err} > tol = {TOL*1e1}"
)
else:
print(f"matched analytical solution with tolerance {TOL}")
if status != 0:
raise Exception(
f"acados solver returned status {status} != 0.")
if line_search_use_sufficient_descent == 1:
if iter > 22:
raise Exception(f"acados ocp solver took {iter} iterations." + \
"Expected <= 22 iterations for globalized SQP method with aggressive eps_sufficient_descent condition on Armijo test problem.")
else:
if iter < 64:
raise Exception(f"acados ocp solver took {iter} iterations." + \
"Expected > 64 iterations for globalized SQP method without sufficient descent condition on Armijo test problem.")
elif globalization == 'FIXED_STEP':
if status != 2:
raise Exception(
f"acados solver returned status {status} != 2. Expected maximum iterations for full-step SQP on Armijo test problem."
)
else:
print(
f"Sucess: Expected maximum iterations for full-step SQP on Armijo test problem."
)
print(f"\n\n----------------------\n")
class Pmpc(object):
def __init__(self,
N,
sys,
cost,
wref=None,
tuning=None,
lam_g_ref=None,
sensitivities=None,
options={}):
""" Constructor
"""
# store construction data
self.__N = N
self.__vars = sys['vars']
self.__nx = sys['vars']['x'].shape[0]
self.__nu = sys['vars']['u'].shape[0]
# nonlinear inequalities slacks
if 'us' in sys['vars']:
self.__ns = sys['vars']['us'].shape[0]
else:
self.__ns = 0
# mpc slacks
if 'usc' in sys['vars']:
self.__nsc = sys['vars']['usc'].shape[0]
self.__scost = sys['scost']
else:
self.__nsc = 0
# store system dynamics
self.__F = sys['f']
# store path constraints
if 'h' in sys:
self.__h = sys['h']
h_lin = self.__h(*self.__vars.values())
self.__h_x_idx = [
idx for idx in range(h_lin.shape[0])
if not True in ca.which_depends(
h_lin[idx], ct.vertcat(*list(self.__vars.values())[1:]))
]
else:
self.__h = None
# store slacked nonlinear inequality constraints
if 'g' in sys:
self.__gnl = sys['g']
self.__detect_state_dependent_constraints()
else:
self.__gnl = None
self.__h_us_idx = [] # no nonlinear state-dependent constraints
# store system sensitivities around steady state
self.__S = sys['S']
self.__cost = cost
# set options
self.__options = self.__default_options()
for option in options:
if option in self.__options:
self.__options[option] = options[option]
else:
raise ValueError(
'Unknown option for Pmpc class instance: "{}"'.format(
option))
# detect cost-type
if self.__cost.n_in() == 2:
# cost function of the form: l(x,u)
self.__type = 'economic'
# no tuning required
tuning = None
if self.__options['hessian_approximation'] == 'gauss_newton':
self.__options['hessian_approximation'] = 'exact'
Logger.logger.warning(
'Gauss-Newton Hessian approximation cannot be applied for economic MPC problem. Switched to exact Hessian.'
)
else:
# cost function of the form: (w-wref)'*H*(w-wref) + q'w
self.__type = 'tracking'
# check if tuning matrices are provided
assert tuning != None, 'Provide tuning matrices for tracking MPC!'
# periodicity operator
self.__p_operator = self.__options['p_operator']
self.__jac_p_operator = ca.Function('jac_p', [sys['vars']['x']], [
ca.jacobian(self.__p_operator(sys['vars']['x']), sys['vars']['x'])
])
self.__S = sensitivities
# construct MPC solver
self.__construct_solver()
# periodic indexing
self.__index = 0
self.__index_acados = 0
# create periodic reference
assert wref != None, 'Provide reference trajectory!'
self.__create_reference(wref, tuning, lam_g_ref)
# initialize log
self.__initialize_log()
# initialize acados solvers
self.__acados_ocp_solver = None
self.__acados_integrator = None
# solver initial guess
self.__set_initial_guess()
return None
def __default_options(self):
# default options
opts = {
'hessian_approximation':
'exact',
'ipopt_presolve':
False,
'max_iter':
2000,
'p_operator':
ca.Function('p_operator', [self.__vars['x']], [self.__vars['x']]),
'slack_flag':
'none'
}
return opts
def __construct_solver(self):
""" Construct periodic MPC solver
"""
# system variables and dimensions
x = self.__vars['x']
u = self.__vars['u']
# NLP parameters
if self.__type == 'economic':
# parameters
self.__p = ct.struct_symMX([
ct.entry('x0', shape=(self.__nx, 1)),
ct.entry('xN', shape=(self.__nx, 1))
])
# reassign for brevity
x0 = self.__p['x0']
xN = self.__p['xN']
if self.__type == 'tracking':
ref_vars = (ct.entry('x', shape=(self.__nx, ),
repeat=self.__N + 1),
ct.entry('u', shape=(self.__nu, ), repeat=self.__N))
if 'us' in self.__vars:
ref_vars += (ct.entry('us',
shape=(self.__ns, ),
repeat=self.__N), )
# reference trajectory
wref = ct.struct_symMX([ref_vars])
nw = self.__nx + self.__nu + self.__ns
tuning = ct.struct_symMX([ # tracking tuning
ct.entry('H', shape=(nw, nw), repeat=self.__N),
ct.entry('q', shape=(nw, 1), repeat=self.__N)
])
# parameters
self.__p = ct.struct_symMX([
ct.entry('x0', shape=(self.__nx, )),
ct.entry('wref', struct=wref),
ct.entry('tuning', struct=tuning)
])
# reassign for brevity
x0 = self.__p['x0']
wref = self.__p.prefix['wref']
tuning = self.__p.prefix['tuning']
xN = wref['x', -1]
# NLP variables
variables_entry = (ct.entry('x',
shape=(self.__nx, ),
repeat=self.__N + 1),
ct.entry('u', shape=(self.__nu, ), repeat=self.__N))
if 'us' in self.__vars:
variables_entry += (ct.entry('us',
shape=(self.__ns, ),
repeat=self.__N), )
self.__wref = ct.struct_symMX([variables_entry
]) # structure of reference
if 'usc' in self.__vars:
variables_entry += (ct.entry('usc',
shape=(self.__nsc, ),
repeat=self.__N), )
# nlp variables + bounds
w = ct.struct_symMX([variables_entry])
# variable bounds are implemented as inequalities
self.__lbw = w(-np.inf)
self.__ubw = w(np.inf)
# prepare dynamics and path constraints entry
constraints_entry = (ct.entry('dyn',
shape=(self.__nx, ),
repeat=self.__N), )
if self.__gnl is not None:
constraints_entry += (ct.entry('g',
shape=self.__gnl.size1_out(0),
repeat=self.__N), )
if self.__h is not None:
constraints_entry += (ct.entry('h',
shape=self.__h.size1_out(0),
repeat=self.__N), )
# terminal constraint
self.__nx_term = self.__p_operator.size1_out(0)
# create general constraints structure
g_struct = ct.struct_symMX([
ct.entry('init', shape=(self.__nx, 1)), constraints_entry,
ct.entry('term', shape=(self.__nx_term, 1))
])
# create symbolic constraint expressions
map_args = collections.OrderedDict()
map_args['x0'] = ct.horzcat(*w['x', :-1])
map_args['p'] = ct.horzcat(*w['u'])
F_constr = ct.horzsplit(self.__F.map(self.__N)(**map_args)['xf'])
# generate constraints
constr = collections.OrderedDict()
constr['dyn'] = [a - b for a, b in zip(F_constr, w['x', 1:])]
if 'us' in self.__vars:
map_args['us'] = ct.horzcat(*w['us'])
if self.__gnl is not None:
constr['g'] = ct.horzsplit(
self.__gnl.map(self.__N)(*map_args.values()))
if 'usc' in self.__vars:
map_args['usc'] = ct.horzcat(*w['usc'])
if self.__h is not None:
constr['h'] = ct.horzsplit(
self.__h.map(self.__N)(*map_args.values()))
repeated_constr = list(
itertools.chain.from_iterable(zip(*constr.values())))
term_constraint = self.__p_operator(w['x', -1] - xN)
self.__g = g_struct(
ca.vertcat(w['x', 0] - x0, *repeated_constr, term_constraint))
self.__lbg = g_struct(np.zeros(self.__g.shape))
self.__ubg = g_struct(np.zeros(self.__g.shape))
if self.__h is not None:
self.__ubg['h', :] = np.inf
for i in self.__h_us_idx + self.__h_x_idx: # rm constraints the only depend on x at k = 0
self.__lbg['h', 0, i] = -np.inf
# nlp cost
cost_map = self.__cost.map(self.__N)
if self.__type == 'economic':
cost_args = [ct.horzcat(*w['x', :-1]), ct.horzcat(*w['u'])]
elif self.__type == 'tracking':
if self.__ns != 0:
cost_args_w = ct.horzcat(*[
ct.vertcat(w['x', k], w['u', k], w['us', k])
for k in range(self.__N)
])
cost_args_w_ref = ct.horzcat(*[
ct.vertcat(wref['x', k], wref['u', k], wref['us', k])
for k in range(self.__N)
])
else:
cost_args_w = ct.horzcat(*[
ct.vertcat(w['x', k], w['u', k]) for k in range(self.__N)
])
cost_args_w_ref = ct.horzcat(*[
ct.vertcat(wref['x', k], wref['u', k])
for k in range(self.__N)
])
cost_args = [
cost_args_w, cost_args_w_ref,
ct.horzcat(*tuning['H']),
ct.horzcat(*tuning['q'])
]
if self.__options['hessian_approximation'] == 'gauss_newton':
if 'usc' not in self.__vars:
hess_gn = ct.diagcat(*tuning['H'],
ca.DM.zeros(self.__nx, self.__nx))
else:
hess_block = list(
itertools.chain.from_iterable(
zip(tuning['H'],
[ca.DM.zeros(self.__nsc, self.__nsc)] *
self.__N)))
hess_gn = ct.diagcat(*hess_block,
ca.DM.zeros(self.__nx, self.__nx))
J = ca.sum2(cost_map(*cost_args))
# add cost on slacks
if 'usc' in self.__vars:
J += ca.sum2(ct.mtimes(self.__scost.T, ct.horzcat(*w['usc'])))
# create solver
prob = {'f': J, 'g': self.__g, 'x': w, 'p': self.__p}
self.__w = w
self.__g_fun = ca.Function('g_fun', [self.__w, self.__p], [self.__g])
# create IPOPT-solver instance if needed
if self.__options['ipopt_presolve']:
opts = {
'ipopt': {
'linear_solver': 'ma57',
'print_level': 0
},
'expand': False
}
if Logger.logger.getEffectiveLevel() > 10:
opts['ipopt']['print_level'] = 0
opts['print_time'] = 0
opts['ipopt']['sb'] = 'yes'
self.__solver = ca.nlpsol('solver', 'ipopt', prob, opts)
# create hessian approximation function
if self.__options['hessian_approximation'] == 'gauss_newton':
lam_g = ca.MX.sym('lam_g', self.__g.shape) # will not be used
hess_approx = ca.Function('hess_approx',
[self.__w, self.__p, lam_g], [hess_gn])
elif self.__options['hessian_approximation'] == 'exact':
hess_approx = 'exact'
# create sqp solver
prob['lbg'] = self.__lbg
prob['ubg'] = self.__ubg
sqp_opts = {
'hessian_approximation': hess_approx,
'max_iter': self.__options['max_iter']
}
self.__sqp_solver = sqp_method.Sqp(prob, sqp_opts)
def step(self, x0):
""" Compute periodic MPC feedback control for given initial condition.
"""
# reset periodic indexing if necessary
self.__index = self.__index % len(self.__ref)
# update nlp parameters
p0 = self.__p(0.0)
p0['x0'] = x0
if self.__type == 'economic':
p0['xN'] = self.__ref[self.__index][-x0.shape[0]:]
elif self.__type == 'tracking':
p0['wref'] = self.__ref[self.__index]
p0['tuning', 'H'] = self.__Href[self.__index]
p0['tuning', 'q'] = self.__qref[self.__index]
# pre-solve NLP with IPOPT for globalization
if self.__options['ipopt_presolve']:
ipopt_sol = self.__solver(x0=self.__w0,
lbg=self.__lbg,
ubg=self.__ubg,
p=p0)
self.__w0 = self.__w(ipopt_sol['x'])
self.__lam_g0 = self.__g(ipopt_sol['lam_g'])
# solve NLP
sol = self.__sqp_solver.solve(self.__w0.cat, p0.cat, self.__lam_g0.cat)
# store solution
self.__g_sol = self.__g(self.__g_fun(sol['x'], p0))
self.__w_sol = self.__w(sol['x'])
self.__extract_solver_stats()
# shift reference
self.__index += 1
# update initial guess
self.__w0, self.__lam_g0 = self.__shift_initial_guess(
self.__w_sol, self.__g(sol['lam_g']))
return self.__w_sol['u', 0]
def step_acados(self, x0):
# reset periodic indexing if necessary
self.__index_acados = self.__index_acados % self.__Nref
# format x0
x0 = np.squeeze(x0.full())
# update NLP parameters
self.__acados_ocp_solver.set(0, "lbx", x0)
self.__acados_ocp_solver.set(0, "ubx", x0)
# update reference and tuning matrices
self.__set_acados_reference()
# solve
status = self.__acados_ocp_solver.solve()
# timings
# np.append(self.__acados_times, self.__acados_ocp_solver.get_stats("time_tot"))
print("acados timings: total: ", self.__acados_ocp_solver.get_stats("time_tot"), \
" lin: ", self.__acados_ocp_solver.get_stats("time_lin"), \
" sim: ", self.__acados_ocp_solver.get_stats("time_sim"), " qp: ", \
self.__acados_ocp_solver.get_stats("time_qp"))
# if status != 0:
# raise Exception('acados solver returned status {}. Exiting.'.format(status))
# save solution
self.__w_sol_acados = self.__w(0.0)
for i in range(self.__N):
self.__w_sol_acados['x', i] = self.__acados_ocp_solver.get(i, "x")
self.__w_sol_acados['u', i] = self.__acados_ocp_solver.get(
i, "u")[:self.__nu]
if 'us' in self.__vars:
self.__w_sol_acados['us', i] = self.__acados_ocp_solver.get(
i, "u")[self.__nu:]
self.__w_sol_acados['x', self.__N] = self.__acados_ocp_solver.get(
self.__N, "x")
self.__extract_acados_solver_stats()
# feedback policy
u0 = self.__acados_ocp_solver.get(0, "u")[:self.__nu]
# update initial guess
self.__shift_initial_guess_acados()
# shift index
self.__index_acados += 1
return u0
def generate(self, dae=None, quad=None, name='tunempc', opts={}):
""" Create embeddable NLP solver
"""
from acados_template import AcadosModel, AcadosOcp, AcadosOcpSolver, AcadosSimSolver
# extract dimensions
nx = self.__nx
nu = self.__nu + self.__ns # treat slacks as pseudo-controls
# extract reference
ref = self.__ref
xref = np.squeeze(self.__ref[0][:nx], axis=1)
uref = np.squeeze(self.__ref[0][nx:nx + nu], axis=1)
# sampling time
self.__ts = opts['tf'] / self.__N
# create acados model
model = AcadosModel()
model.x = ca.MX.sym('x', nx)
model.u = ca.MX.sym('u', nu)
model.p = []
model.name = name
# detect input type
if dae is None:
model.f_expl_expr = self.__F(x0=model.x,
p=model.u)['xf'] / self.__ts
opts['integrator_type'] = 'ERK'
opts['sim_method_num_stages'] = 1
opts['sim_method_num_steps'] = 1
else:
n_in = dae.n_in()
if n_in == 2:
# xdot = f(x, u)
if 'integrator_type' in opts:
if opts['integrator_type'] in ['IRK', 'GNSF']:
xdot = ca.MX.sym('xdot', nx)
model.xdot = xdot
model.f_impl_expr = xdot - dae(model.x,
model.u[:self.__nu])
model.f_expl_expr = xdot
elif opts['integrator_type'] == 'ERK':
model.f_expl_expr = dae(model.x, model.u[:self.__nu])
else:
raise ValueError('Provide numerical integrator type!')
else:
xdot = ca.MX.sym('xdot', nx)
model.xdot = xdot
model.f_expl_expr = xdot
if n_in == 3:
# f(xdot, x, u) = 0
model.f_impl_expr = dae(xdot, model.x, model.u[:self.__nu])
elif n_in == 4:
# f(xdot, x, u, z) = 0
nz = dae.size1_in(3)
z = ca.MX.sym('z', nz)
model.z = z
model.f_impl_expr = dae(xdot, model.x, model.u[:self.__nu],
z)
else:
raise ValueError(
'Invalid number of inputs for system dynamics function.'
)
if self.__gnl is not None:
model.con_h_expr = self.__gnl(model.x, model.u[:self.__nu],
model.u[self.__nu:])
if self.__type == 'economic':
if quad is None:
model.cost_expr_ext_cost = self.__cost(
model.x, model.u[:self.__nu]) / self.__ts
else:
model.cost_expr_ext_cost = self.__cost(model.x,
model.u[:self.__nu])
# create acados ocp
ocp = AcadosOcp()
ocp.model = model
ny = nx + nu
ny_e = nx
if 'integrator_type' in opts and opts['integrator_type'] == 'GNSF':
from acados_template import acados_dae_model_json_dump
import os
acados_dae_model_json_dump(model)
# Set up Octave to be able to run the following:
## if using a virtual python env, the following lines can be added to the env/bin/activate script:
# export OCTAVE_PATH=$OCTAVE_PATH:$ACADOS_INSTALL_DIR/external/casadi-octave
# export OCTAVE_PATH=$OCTAVE_PATH:$ACADOS_INSTALL_DIR/interfaces/acados_matlab_octave/
# export OCTAVE_PATH=$OCTAVE_PATH:$ACADOS_INSTALL_DIR/interfaces/acados_matlab_octave/acados_template_mex/
# echo
# echo "OCTAVE_PATH=$OCTAVE_PATH"
status = os.system(
"octave --eval \"convert({})\"".format("\'" + model.name +
"_acados_dae.json\'"))
# load gnsf from json
with open(model.name + '_gnsf_functions.json', 'r') as f:
import json
gnsf_dict = json.load(f)
ocp.gnsf_model = gnsf_dict
# set horizon length
ocp.dims.N = self.__N
# set cost module
if self.__type == 'economic':
# set cost function type to external (provided in model)
ocp.cost.cost_type = 'EXTERNAL'
if quad is not None:
ocp.solver_options.cost_discretization = 'INTEGRATOR'
elif self.__type == 'tracking':
# set weighting matrices
ocp.cost.W = self.__Href[0][0]
# set-up linear least squares cost
ocp.cost.cost_type = 'LINEAR_LS'
ocp.cost.W_e = np.zeros((nx, nx))
ocp.cost.Vx = np.zeros((ny, nx))
ocp.cost.Vx[:nx, :nx] = np.eye(nx)
Vu = np.zeros((ny, nu))
Vu[nx:, :] = np.eye(nu)
ocp.cost.Vu = Vu
ocp.cost.Vx_e = np.eye(nx)
ocp.cost.yref = np.squeeze(
ca.vertcat(xref,uref).full() - \
ct.mtimes(np.linalg.inv(ocp.cost.W),self.__qref[0][0].T).full(), # gradient term
axis = 1
)
ocp.cost.yref_e = np.zeros((ny_e, ))
if n_in == 4: # DAE flag
ocp.cost.Vz = np.zeros((ny, nz))
# if 'custom_hessian' in opts:
# self.__custom_hessian = opts['custom_hessian']
# initial condition
ocp.constraints.x0 = xref
# set inequality constraints
ocp.constraints.constr_type = 'BGH'
if self.__S['C'] is not None:
C = self.__S['C'][0][:, :nx]
D = self.__S['C'][0][:, nx:]
lg = -self.__S['e'][0] + ct.mtimes(C, xref).full() + ct.mtimes(
D, uref).full()
ug = 1e8 - self.__S['e'][0] + ct.mtimes(
C, xref).full() + ct.mtimes(D, uref).full()
ocp.constraints.lg = np.squeeze(lg, axis=1)
ocp.constraints.ug = np.squeeze(ug, axis=1)
ocp.constraints.C = C
ocp.constraints.D = D
if 'usc' in self.__vars:
if 'us' in self.__vars:
arg = [
self.__vars['x'], self.__vars['u'], self.__vars['us'],
self.__vars['usc']
]
else:
arg = [
self.__vars['x'], self.__vars['u'], self.__vars['usc']
]
Jsg = ca.Function(
'Jsg', [self.__vars['usc']],
[ca.jacobian(self.__h(*arg), self.__vars['usc'])])(0.0)
self.__Jsg = Jsg.full()[:-self.__nsc, :]
ocp.constraints.Jsg = self.__Jsg
ocp.cost.Zl = np.zeros((self.__nsc, ))
ocp.cost.Zu = np.zeros((self.__nsc, ))
ocp.cost.zl = np.squeeze(self.__scost.full(),
axis=1) / self.__ts
ocp.cost.zu = np.squeeze(self.__scost.full(),
axis=1) / self.__ts
# set nonlinear equality constraints
if self.__gnl is not None:
ocp.constraints.lh = np.zeros(self.__ns, )
ocp.constraints.uh = np.zeros(self.__ns, )
# terminal constraint:
x_term = self.__p_operator(model.x)
Jbx = ca.Function('Jbx', [model.x],
[ca.jacobian(x_term, model.x)])(0.0)
ocp.constraints.Jbx_e = Jbx.full()
ocp.constraints.lbx_e = np.squeeze(self.__p_operator(xref).full(),
axis=1)
ocp.constraints.ubx_e = np.squeeze(self.__p_operator(xref).full(),
axis=1)
for option in list(opts.keys()):
if hasattr(ocp.solver_options, option):
setattr(ocp.solver_options, option, opts[option])
self.__acados_ocp_solver = AcadosOcpSolver(ocp,
json_file='acados_ocp_' +
model.name + '.json')
self.__acados_integrator = AcadosSimSolver(ocp,
json_file='acados_ocp_' +
model.name + '.json')
# set initial guess
self.__set_acados_initial_guess()
return self.__acados_ocp_solver, self.__acados_integrator
def __create_reference(self, wref, tuning, lam_g_ref):
""" Create periodic reference and tuning data.
"""
# period of reference
self.__Nref = len(wref['u'])
# create reference and tuning sequence
# for each starting point in period
ref_pr = []
ref_du = []
ref_du_struct = []
H = []
q = []
for k in range(self.__Nref):
# reference primal solution
refk = []
for j in range(self.__N):
refk += [
wref['x', (k + j) % self.__Nref],
wref['u', (k + j) % self.__Nref]
]
if 'us' in self.__vars:
refk += [wref['us', (k + j) % self.__Nref]]
refk.append(wref['x', (k + self.__N) % self.__Nref])
# reference dual solution
lamgk = self.__g(0.0)
lamgk['init'] = -lam_g_ref['dyn', (k - 1) % self.__Nref]
for j in range(self.__N):
lamgk['dyn', j] = lam_g_ref['dyn', (k + j) % self.__Nref]
if 'g' in list(lamgk.keys()):
lamgk['g', j] = lam_g_ref['g', (k + j) % self.__Nref]
if 'h' in list(lamgk.keys()):
lam_h = [lam_g_ref['h', (k + j) % self.__Nref]]
if 'usc' in self.__vars:
lam_h += [-self.__scost] # TODO not entirely correct
lamgk['h', j] = ct.vertcat(*lam_h)
lamgk['term'] = self.__p_operator(
lam_g_ref['dyn', (k + self.__N - 1) % self.__Nref])
# adjust dual solution of terminal constraint is projected
if self.__nx_term != self.__nx:
# find new terminal multiplier
A_m = []
b_m = []
A_factor = ca.DM.eye(self.__nx)
for j in range(self.__N):
A_m.append(
ct.mtimes(
ct.mtimes(
self.__S['B'][(self.__N - j - 1) %
self.__Nref].T, A_factor),
self.__jac_p_operator(ca.DM.ones(self.__nx, 1)).T))
b_m.append(
ct.mtimes(
ct.mtimes(
self.__S['B'][(self.__N - j - 1) %
self.__Nref].T, A_factor),
lam_g_ref['dyn',
(k + self.__N - 1) % self.__Nref]))
A_factor = ct.mtimes(
self.__S['A'][(self.__N - j - 1) % self.__Nref].T,
A_factor)
A_m = ct.vertcat(*A_m)
b_m = ct.vertcat(*b_m)
LI_indeces = [
] # indeces of first full rank number linearly independent rows
R0 = 0
for i in range(A_m.shape[0]):
R = np.linalg.matrix_rank(A_m[LI_indeces + [i], :])
if R > R0:
LI_indeces.append(i)
R0 = R
lamgk['term'] = ca.solve(A_m[LI_indeces, :],
b_m[LI_indeces, :])
# recursively update dynamics multipliers
delta_lam = -lam_g_ref['dyn', (k + self.__N - 1) %
self.__Nref] + ct.mtimes(
self.__jac_p_operator(
ca.DM.ones(self.__nx, 1)).T,
lamgk['term'])
lamgk['dyn', self.__N - 1] += delta_lam
for j in range(1, self.__N + 1):
delta_lam = ct.mtimes(
self.__S['A'][(self.__N - j) % self.__Nref].T,
delta_lam)
if j < self.__N:
lamgk['dyn', self.__N - 1 - j] += delta_lam
else:
lamgk['init'] += -delta_lam
ref_pr.append(ct.vertcat(*refk))
ref_du.append(lamgk.cat)
ref_du_struct.append(lamgk)
if tuning is not None:
H.append([
tuning['H'][(k + j) % self.__Nref] for j in range(self.__N)
])
q.append([
tuning['q'][(k + j) % self.__Nref] for j in range(self.__N)
])
self.__ref = ref_pr
self.__ref_du = ref_du
self.__ref_du_struct = ref_du_struct
self.__Href = H
self.__qref = q
return None
def __initialize_log(self):
self.__log = {
'cpu': [],
'iter': [],
'f': [],
'status': [],
'sol_x': [],
'lam_x': [],
'lam_g': [],
'u0': [],
'nACtot': [],
'nAC': [],
'idx_AC': [],
'nAS': []
}
self.__log_acados = {
'time_tot': [],
'time_lin': [],
'time_sim': [],
'time_qp': [],
'sqp_iter': [],
'time_reg': [],
'time_qp_xcond': [],
'time_qp_solver_call': [],
}
return None
def __extract_solver_stats(self):
info = self.__sqp_solver.stats
self.__log['cpu'].append(info['t_wall_total'])
self.__log['iter'].append(info['iter_count'])
self.__log['status'].append(info['return_status'])
self.__log['sol_x'].append(info['x'])
self.__log['lam_g'].append(info['lam_g'])
self.__log['f'].append(info['f'])
self.__log['u0'].append(self.__w(info['x'])['u', 0])
self.__log['nACtot'].append(info['nAC'])
nAC, idx_AC = self.__detect_AC(self.__g(info['lam_g']))
self.__log['nAC'].append(nAC)
self.__log['idx_AC'].append(nAC)
self.__log['nAS'].append(info['nAS'])
return None
def __extract_acados_solver_stats(self):
for key in list(self.__log_acados.keys()):
self.__log_acados[key].append(
self.__acados_ocp_solver.get_stats(key))
return None
def __detect_AC(self, lam_g_opt):
# optimal active set
if 'h' in lam_g_opt.keys():
idx_opt = [
k for k in range(self.__h.size1_out(0) - self.__nsc)
if lam_g_opt['h', 0][k] != 0
]
lam_g_ref = self.__g(self.__ref_du[self.__index])
idx_ref = [
k for k in range(self.__h.size1_out(0) - self.__nsc)
if lam_g_ref['h', 0][k] != 0
]
else:
idx_opt = []
idx_ref = []
# get number of active set changes
nAC = len([k for k in idx_opt if k not in idx_ref])
nAC += len([k for k in idx_ref if k not in idx_opt])
return nAC, idx_opt
def reset(self):
self.__index = 0
self.__index_acados = 0
self.__initialize_log()
self.__set_initial_guess()
return None
def __shift_initial_guess(self, w0, lam_g0):
w_shifted = self.__w(0.0)
lam_g_shifted = self.__g(0.0)
lam_g_shifted['init'] = lam_g0['dyn', 0]
# shift states and controls
for i in range(self.__N):
# shift primal solution
w_shifted['x', i] = w0['x', i + 1]
if i < self.__N - 1:
w_shifted['u', i] = w0['u', i + 1]
if 'us' in self.__vars:
w_shifted['us', i] = w0['us', i + 1]
if 'usc' in self.__vars:
w_shifted['usc', i] = w0['usc', i + 1]
# shift dual solution
lam_g_shifted['dyn', i] = lam_g0['dyn', i + 1]
for constr in ['g', 'h']:
if constr in lam_g0.keys():
lam_g_shifted[constr, i] = lam_g0[constr, i + 1]
# copy final interval
w_shifted['x', self.__N] = w_shifted['x', self.__N - 1]
w_shifted['u', self.__N - 1] = w_shifted['u', self.__N - 2]
if 'us' in self.__vars:
w_shifted['us', self.__N - 1] = w_shifted['us', self.__N - 2]
if 'usc' in self.__vars:
w_shifted['usc', self.__N - 1] = w_shifted['usc', self.__N - 2]
lam_g_shifted['dyn', self.__N - 1] = lam_g_shifted['dyn', self.__N - 2]
for constr in ['g', 'h']:
if constr in lam_g0.keys():
lam_g_shifted[constr,
self.__N - 1] = lam_g_shifted[constr,
self.__N - 2]
lam_g_shifted['term'] = lam_g0['term']
return w_shifted, lam_g_shifted
def __shift_initial_guess_acados(self):
for i in range(self.__N):
x_prev = np.squeeze(self.__w_sol_acados['x', i + 1].full(), axis=1)
self.__acados_ocp_solver.set(i, "x", x_prev)
if i < self.__N - 1:
u_prev = np.squeeze(self.__w_sol_acados['u', i + 1].full(),
axis=1)
if 'us' in self.__vars:
u_prev = np.squeeze(ct.vertcat(
u_prev, self.__w_sol_acados['us', i + 1]).full(),
axis=1)
self.__acados_ocp_solver.set(i, "u", u_prev)
# initial guess in terminal stage on periodic trajectory
idx = (self.__index_acados + self.__N) % self.__Nref
# reference
xref = np.squeeze(self.__ref[(idx + 1) % self.__Nref][:self.__nx],
axis=1)
uref = np.squeeze(self.__ref[idx][self.__nx:self.__nx + self.__nu +
self.__ns],
axis=1)
self.__acados_ocp_solver.set(self.__N, "x", xref)
self.__acados_ocp_solver.set(self.__N - 1, "u", uref)
return None
def __set_initial_guess(self):
# create initial guess at steady state
wref = self.__wref(self.__ref[self.__index])
w0 = self.__w(0.0)
w0['x'] = wref['x']
w0['u'] = wref['u']
if 'us' in self.__vars:
w0['us'] = wref['us']
self.__w0 = w0
# initial guess for multipliers
self.__lam_g0 = self.__g(self.__ref_du[self.__index])
# acados solver initialization at reference
if self.__acados_ocp_solver is not None:
self.__set_acados_initial_guess()
return None
def __set_acados_reference(self):
for i in range(self.__N):
# periodic index
idx = (self.__index_acados + i) % self.__Nref
# reference
xref = np.squeeze(self.__ref[idx][:self.__nx], axis=1)
uref = np.squeeze(self.__ref[idx][self.__nx:self.__nx + self.__nu +
self.__ns],
axis=1)
if self.__type == 'tracking':
# construct output reference with gradient term
yref = np.squeeze(
ca.vertcat(xref,uref).full() - \
ct.mtimes(
np.linalg.inv(self.__Href[idx][0]/self.__ts), # inverse of weighting matrix
self.__qref[idx][0].T).full()/self.__ts, # gradient term
axis = 1
)
self.__acados_ocp_solver.set(i, 'yref', yref)
# update tuning matrix
self.__acados_ocp_solver.cost_set(
i, 'W', self.__Href[idx][0] / self.__ts)
# set custom hessians if applicable
# if self.__acados_ocp_solver.acados_ocp.solver_options.ext_cost_custom_hessian:
# self.__acados_ocp_solver.cost_set(i, "cost_custom_hess", self.__custom_hessian[idx])
# update constraint bounds
if self.__h is not None:
C = self.__S['C'][idx][:, :self.__nx]
D = self.__S['C'][idx][:, self.__nx:]
lg = -self.__S['e'][idx] + ct.mtimes(
C, xref).full() + ct.mtimes(D, uref).full()
ug = 1e8 - self.__S['e'][idx] + ct.mtimes(
C, xref).full() + ct.mtimes(D, uref).full()
# remove constraints that depend on states only from first shooting node
if i == 0:
for k in range(D.shape[0]):
if k in self.__h_us_idx + self.__h_x_idx:
lg[k] += -1e8
self.__acados_ocp_solver.constraints_set(
i, 'lg', np.squeeze(lg, axis=1))
self.__acados_ocp_solver.constraints_set(
i, 'ug', np.squeeze(ug, axis=1))
# update terminal constraint
idx = (self.__index_acados + self.__N) % self.__Nref
x_term = np.squeeze(self.__p_operator(self.__ref[idx][:self.__nx]),
axis=1)
self.__acados_ocp_solver.set(self.__N, 'lbx', x_term)
self.__acados_ocp_solver.set(self.__N, 'ubx', x_term)
return None
def __set_acados_initial_guess(self):
# dual reference solution
ref_dual = self.__ref_du_struct[self.__index_acados % self.__Nref]
for i in range(self.__N):
# periodic index
idx = (self.__index_acados + i) % self.__Nref
# initialize at reference
xref = np.squeeze(self.__ref[idx][:self.__nx], axis=1)
uref = np.squeeze(self.__ref[idx][self.__nx:self.__nx + self.__nu +
self.__ns],
axis=1)
# set initial guess
self.__acados_ocp_solver.set(i, "x", xref)
self.__acados_ocp_solver.set(i, "u", uref)
# set dual initial guess
self.__acados_ocp_solver.set(i, "pi",
np.squeeze(ref_dual['dyn', i].full()))
# the inequalities are internally organized in the following order:
# [ lbu lbx lg lh ubu ubx ug uh ]
lam_h = []
t = []
if i == 0:
lam_x0 = copy.deepcopy(ref_dual['init'])
if 'h' in list(ref_dual.keys()):
lam_lh0 = -ref_dual['h', i][:ref_dual['h', i].shape[0] -
self.__nsc]
t_lh0 = copy.deepcopy(self.__S['e'][idx % self.__Nref])
if i == 0:
# set unused constraints at i=0 to be inactive
C = self.__S['C'][idx][:, :self.__nx]
D = self.__S['C'][idx][:, self.__nx:]
for k in range(D.shape[0]):
if k in self.__h_us_idx + self.__h_x_idx:
lam_x0 += -ct.mtimes(lam_lh0[k], C[k, :])
lam_lh0[k] = 0.0
t_lh0[k] += 1e8
lam_lx0 = -copy.deepcopy(lam_x0)
for k in range(self.__nx):
if lam_lx0[k] < 0.0:
lam_lx0[k] = 0.0 # assign multiplier to upper bound
lam_h.append(lam_lx0) # lbx_0
t.append(np.zeros((self.__nx, )))
if 'h' in list(ref_dual.keys()):
if i == 0:
lam_lh = lam_lh0
t_lh = t_lh0
else:
lam_lh = -ref_dual['h', i][:ref_dual['h', i].shape[0] -
self.__nsc]
t_lh = copy.deepcopy(self.__S['e'][idx % self.__Nref])
lam_h.append(lam_lh) # lg
t.append(t_lh)
if 'g' in list(ref_dual.keys()):
lam_lg0 = -ref_dual['g', i]
lam_ug0 = np.zeros(lam_lg0.shape)
for k in range(lam_lg0.shape[0]):
if lam_lg0[k] < 0.0:
lam_ug0[k] = -lam_lg0[k]
lam_lg0[k] = 0.0
lam_h.append(lam_lg0) # lh
t.append(np.zeros((ref_dual['g', i].shape[0], )))
if i == 0:
lam_ux0 = copy.deepcopy(lam_x0)
for k in range(self.__nx):
if lam_ux0[k] < 0.0:
lam_ux0[k] = 0.0 # assign multiplier to lower bound
lam_h.append(lam_ux0) # ubx_0
t.append(np.zeros((self.__nx, )))
if 'h' in list(ref_dual.keys()):
lam_h.append(
np.zeros((ref_dual['h', i].shape[0] - self.__nsc, ))) # ug
t.append(1e8 *
np.ones((ref_dual['h', i].shape[0] - self.__nsc, 1)) -
self.__S['e'][idx])
if 'g' in list(ref_dual.keys()):
lam_h.append(lam_ug0) # uh
t.append(np.zeros((ref_dual['g', i].shape[0], )))
if self.__nsc > 0:
lam_sl = self.__scost - ct.mtimes(lam_lh.T, self.__Jsg).T
lam_h.append(lam_sl) # ls
lam_h.append(self.__scost) # us
t.append(np.zeros((self.__nsc, ))) # slg > 0
t.append(np.zeros((self.__nsc, ))) # sug > 0
if len(lam_h) != 0:
self.__acados_ocp_solver.set(
i, "lam", np.squeeze(ct.vertcat(*lam_h).full()))
self.__acados_ocp_solver.set(i, "t",
np.squeeze(ct.vertcat(*t).full()))
# terminal state
idx = (self.__index_acados + self.__N) % self.__Nref
xref = np.squeeze(self.__ref[idx][:self.__nx], axis=1)
self.__acados_ocp_solver.set(self.__N, "x", xref)
# terminal multipliers
lam_lterm = -ref_dual['term']
lam_uterm = np.zeros((ref_dual['term'].shape[0], ))
for k in range(lam_lterm.shape[0]):
if lam_lterm[k] < 0.0:
lam_uterm[k] = -lam_lterm[k]
lam_lterm[k] = 0.0
lam_term = np.squeeze(ct.vertcat(lam_lterm, lam_uterm).full())
self.__acados_ocp_solver.set(self.__N, "lam", lam_term)
return None
def __detect_state_dependent_constraints(self):
""" Detect which nonlinear equalities depend on states but not on controls.
"""
g_nl = self.__gnl(self.__vars['x'], self.__vars['u'],
self.__vars['us'])
self.__gnl_x_idx = []
for i in range(g_nl.shape[0]):
if not True in ca.which_depends(g_nl[i], self.__vars['u'], 1):
self.__gnl_x_idx.append(i)
self.__h_us_idx = [
idx + self.__h.size1_out(0) - self.__ns for idx in self.__gnl_x_idx
]
return None
@property
def w(self):
return self.__w
@property
def g_sol(self):
return self.__g_sol
@property
def w_sol(self):
return self.__w_sol
@property
def log(self):
return self.__log
@property
def log_acados(self):
return self.__log_acados
@property
def index(self):
return self.__index
@property
def acados_ocp_solver(self):
return self.__acados_ocp_solver
@property
def acados_integrator(self):
return self.__acados_integrator
@property
def w_sol_acados(self):
return self.__w_sol_acados
def run_nominal_control(chain_params):
# create ocp object to formulate the OCP
ocp = AcadosOcp()
# chain parameters
n_mass = chain_params["n_mass"]
M = chain_params["n_mass"] - 2 # number of intermediate masses
Ts = chain_params["Ts"]
Tsim = chain_params["Tsim"]
N = chain_params["N"]
u_init = chain_params["u_init"]
with_wall = chain_params["with_wall"]
yPosWall = chain_params["yPosWall"]
m = chain_params["m"]
D = chain_params["D"]
L = chain_params["L"]
perturb_scale = chain_params["perturb_scale"]
nlp_iter = chain_params["nlp_iter"]
nlp_tol = chain_params["nlp_tol"]
save_results = chain_params["save_results"]
show_plots = chain_params["show_plots"]
seed = chain_params["seed"]
np.random.seed(seed)
nparam = 3*M
W = perturb_scale * np.eye(nparam)
# export model
model = export_disturbed_chain_mass_model(n_mass, m, D, L)
# set model
ocp.model = model
nx = model.x.size()[0]
nu = model.u.size()[0]
ny = nx + nu
ny_e = nx
Tf = N * Ts
# initial state
xPosFirstMass = np.zeros((3,1))
xEndRef = np.zeros((3,1))
xEndRef[0] = L * (M+1) * 6
pos0_x = np.linspace(xPosFirstMass[0], xEndRef[0], n_mass)
xrest = compute_steady_state(n_mass, m, D, L, xPosFirstMass, xEndRef)
x0 = xrest
# set dimensions
ocp.dims.N = N
# set cost module
ocp.cost.cost_type = 'LINEAR_LS'
ocp.cost.cost_type_e = 'LINEAR_LS'
Q = 2*np.diagflat( np.ones((nx, 1)) )
q_diag = np.ones((nx,1))
strong_penalty = M+1
q_diag[3*M] = strong_penalty
q_diag[3*M+1] = strong_penalty
q_diag[3*M+2] = strong_penalty
Q = 2*np.diagflat( q_diag )
R = 2*np.diagflat( 1e-2 * np.ones((nu, 1)) )
ocp.cost.W = scipy.linalg.block_diag(Q, R)
ocp.cost.W_e = Q
ocp.cost.Vx = np.zeros((ny, nx))
ocp.cost.Vx[:nx,:nx] = np.eye(nx)
Vu = np.zeros((ny, nu))
Vu[nx:nx+nu, :] = np.eye(nu)
ocp.cost.Vu = Vu
ocp.cost.Vx_e = np.eye(nx)
# import pdb; pdb.set_trace()
yref = np.vstack((xrest, np.zeros((nu,1)))).flatten()
ocp.cost.yref = yref
ocp.cost.yref_e = xrest.flatten()
# set constraints
umax = 1*np.ones((nu,))
ocp.constraints.constr_type = 'BGH'
ocp.constraints.lbu = -umax
ocp.constraints.ubu = umax
ocp.constraints.x0 = x0.reshape((nx,))
ocp.constraints.idxbu = np.array(range(nu))
# disturbances
nparam = 3*M
ocp.parameter_values = np.zeros((nparam,))
# wall constraint
if with_wall:
nbx = M + 1
Jbx = np.zeros((nbx,nx))
for i in range(nbx):
Jbx[i, 3*i+1] = 1.0
ocp.constraints.Jbx = Jbx
ocp.constraints.lbx = yPosWall * np.ones((nbx,))
ocp.constraints.ubx = 1e9 * np.ones((nbx,))
# slacks
ocp.constraints.Jsbx = np.eye(nbx)
L2_pen = 1e3
L1_pen = 1
ocp.cost.Zl = L2_pen * np.ones((nbx,))
ocp.cost.Zu = L2_pen * np.ones((nbx,))
ocp.cost.zl = L1_pen * np.ones((nbx,))
ocp.cost.zu = L1_pen * np.ones((nbx,))
# solver options
ocp.solver_options.qp_solver = 'PARTIAL_CONDENSING_HPIPM' # FULL_CONDENSING_QPOASES
ocp.solver_options.hessian_approx = 'GAUSS_NEWTON'
ocp.solver_options.integrator_type = 'IRK'
ocp.solver_options.nlp_solver_type = 'SQP' # SQP_RTI
ocp.solver_options.nlp_solver_max_iter = nlp_iter
ocp.solver_options.sim_method_num_stages = 2
ocp.solver_options.sim_method_num_steps = 2
ocp.solver_options.qp_solver_cond_N = N
ocp.solver_options.qp_tol = nlp_tol
ocp.solver_options.tol = nlp_tol
# ocp.solver_options.nlp_solver_tol_eq = 1e-9
# set prediction horizon
ocp.solver_options.tf = Tf
acados_ocp_solver = AcadosOcpSolver(ocp, json_file = 'acados_ocp_' + model.name + '.json')
# acados_integrator = AcadosSimSolver(ocp, json_file = 'acados_ocp_' + model.name + '.json')
acados_integrator = export_chain_mass_integrator(n_mass, m, D, L)
#%% get initial state from xrest
xcurrent = x0.reshape((nx,))
for i in range(5):
acados_integrator.set("x", xcurrent)
acados_integrator.set("u", u_init)
status = acados_integrator.solve()
if status != 0:
raise Exception('acados integrator returned status {}. Exiting.'.format(status))
# update state
xcurrent = acados_integrator.get("x")
#%% actual simulation
N_sim = int(np.floor(Tsim/Ts))
simX = np.ndarray((N_sim+1, nx))
simU = np.ndarray((N_sim, nu))
wall_dist = np.zeros((N_sim,))
timings = np.zeros((N_sim,))
simX[0,:] = xcurrent
# closed loop
for i in range(N_sim):
# solve ocp
acados_ocp_solver.set(0, "lbx", xcurrent)
acados_ocp_solver.set(0, "ubx", xcurrent)
status = acados_ocp_solver.solve()
timings[i] = acados_ocp_solver.get_stats("time_tot")[0]
if status != 0:
raise Exception('acados acados_ocp_solver returned status {} in time step {}. Exiting.'.format(status, i))
simU[i,:] = acados_ocp_solver.get(0, "u")
print("control at time", i, ":", simU[i,:])
# simulate system
acados_integrator.set("x", xcurrent)
acados_integrator.set("u", simU[i,:])
pertubation = sampleFromEllipsoid(np.zeros((nparam,)), W)
acados_integrator.set("p", pertubation)
status = acados_integrator.solve()
if status != 0:
raise Exception('acados integrator returned status {}. Exiting.'.format(status))
# update state
xcurrent = acados_integrator.get("x")
simX[i+1,:] = xcurrent
# xOcpPredict = acados_ocp_solver.get(1, "x")
# print("model mismatch = ", str(np.max(xOcpPredict - xcurrent)))
yPos = xcurrent[range(1,3*M+1,3)]
wall_dist[i] = np.min(yPos - yPosWall)
print("time i = ", str(i), " dist2wall ", str(wall_dist[i]))
print("dist2wall (minimum over simulation) ", str(np.min(wall_dist)))
#%% plot results
if os.environ.get('ACADOS_ON_TRAVIS') is None and show_plots:
plot_chain_control_traj(simU)
plot_chain_position_traj(simX, yPosWall=yPosWall)
plot_chain_velocity_traj(simX)
animate_chain_position(simX, xPosFirstMass, yPosWall=yPosWall)
# animate_chain_position_3D(simX, xPosFirstMass)
plt.show()
