Hi

I am trying to use the LINEAR_LS as cost function for the MPC. However, it does not get the result I would expect. I want to formulate the cost (x-1)^2 for the lterm and mterm. However, the solution is not u = 1 which should be optimal. Here, is a minimal code example:

```
import casadi as cd
import numpy as np
from acados_template import AcadosModel, AcadosOcp, AcadosOcpSolver
def export_simple_dae_model() -> AcadosModel:
model_name = 'minimal_example'
# set up states & controls
x = cd.SX.sym('x')
u = cd.SX.sym('u')
p = cd.SX.sym('p')
# xdot
x_dot = cd.SX.sym('x_dot')
# algebraic variables
z = cd.SX.sym('z')
# dynamics
f_expl = u - 1
f_impl = x_dot - f_expl
alg = z - 4 + u
f_impl = cd.vertcat(f_impl, alg)
model = AcadosModel()
model.f_impl_expr = f_impl
model.f_expl_expr = f_expl
model.x = x
model.xdot = x_dot
model.z = z
model.u = u
model.p = p
# set model_name
model.name = model_name
return model
def test_minimal_mpc_example_with_linear_ls() -> None:
# create ocp object to formulate the OCP
ocp = AcadosOcp()
# set model
model = export_simple_dae_model()
ocp.model = model
Tf = 1.0
N = 2
# set dimensions
ocp.dims.N = N
ocp.cost.cost_type = 'LINEAR_LS'
ocp.cost.cost_type_e = 'LINEAR_LS'
# The cost function should be (x - 1)**2
# Because x starts at 1, the optimal solution should be u=1 which results
# in a cost value of 0.
ocp.cost.Vx = np.array([[1]])
ocp.cost.Vu = np.array([[0]])
ocp.cost.Vz = np.array([[0]])
ocp.cost.yref = np.array([[1]])
ocp.cost.Vx_e = np.array([[1]])
ocp.cost.yref_e = np.array([[1]])
ocp.cost.W = np.eye(1)
ocp.cost.W_e = np.eye(1)
# set constraints
ocp.constraints.x0 = np.array([1.0])
# set options
ocp.solver_options.qp_solver = 'PARTIAL_CONDENSING_HPIPM'
ocp.solver_options.hessian_approx = 'EXACT' # 'GAUSS_NEWTON', 'EXACT'
ocp.solver_options.integrator_type = 'IRK'
ocp.solver_options.nlp_solver_type = 'SQP' # SQP_RTI, SQP
# set prediction horizon
ocp.solver_options.tf = Tf
p_init = np.array([2])
ocp.parameter_values = p_init
ocp_solver = AcadosOcpSolver(ocp, json_file='acados_ocp.json')
ocp_solver.solve()
cost_value = ocp_solver.get_cost()
n_horizon: int = ocp_solver.acados_ocp.dims.N # type: ignore
result_x = []
for stage_index in range(n_horizon + 1):
stage_x = ocp_solver.get(stage_index, 'x')
result_x.append(stage_x)
print(result_x)
result_u = []
for stage_index in range(n_horizon):
stage_u = ocp_solver.get(stage_index, 'u')
result_u.append(stage_u)
print(result_u)
assert cost_value == 0, 'Something is wrong.'
if __name__ == '__main__':
test_minimal_mpc_example_with_linear_ls()
```

Thanks,

David