```
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
system description: A cartpole system is considered, which consists of a wagon with the mass M,
a rope with the constant length l, which is attached to the wagon, and a load,
which is located at the free end of the rope. The force that can be applied to the wagon
is available as a manipulated variable.
problem specification for control problem: design of a full state feedback controller to control
the x-position of the load to 1.5m.
"""
import numpy as np
import sympy as sp
from sympy import cos, sin
from math import pi
from ackrep_core import ResultContainer
from system_models.cartpole_system.system_model import Model
class ProblemSpecification(object):
# system symbols for setting up the equation of motion
model = Model()
x1, x2, x3, x4 = model.xx_symb
xx = sp.Matrix(model.xx_symb) # states of system
u = [model.uu_symb[0]] # input of system
# equilibrium point for linearization of the nonlinear system
eqrt = [(x1, 0), (x2, 0), (x3, 0), (x4, 0), (u, 0)]
xx0 = np.array([0.2, pi / 6, 1, 0.2]) # initial condition for simulation
tt = np.linspace(0, 5, 1000) # vector for the time axis for simulating
poles_cl = [-3, -3, -3, -3] # desired poles of closed loop
yr = 1.5 # reference output
# plotting parameters
titles_state = ["x1", "x2", "x1_dot", "x2_dot"]
titles_output = ["y"]
x_label = "time [s]"
y_label_state = ["position [m]", "angular position [rad]", "velocity [m/s]", "angular velocity [rad/s]"]
y_label_output = ["x-position of pendulum m"]
graph_color = "r"
row_number = 2 # the number of images in each row
@classmethod
def rhs(cls):
"""Right hand side of the equation of motion in nonlinear state space form
:return: nonlinear state space
"""
return sp.Matrix(cls.model.get_rhs_symbolic_num_params())
@classmethod
def output_func(cls):
"""output equation of the system: x-position of the load
:return: output equation y = x1
"""
x1, x2, x3, x4 = cls.xx
u = cls.u
l = cls.model.pp_str_dict["l"] # geometry constant
return sp.Matrix([x1 + l * sin(x2)])
def evaluate_solution(solution_data):
"""
Condition: the x-position of the load reaches 1.5m after at least 4 seconds
:param solution_data: solution data of problem of solution
:return:
"""
P = ProblemSpecification
success = all(abs(solution_data.yy[800:] - [P.yr] * 200) < 1e-2)
return ResultContainer(success=success, score=1.0)
```

controller design via full state feedback

cartpole system