Inverted Pendulum

In this example, a mode predictive controller (MPC) is used to swing up and stabilize an inverted pendulum mounted on a moving cart.

Once Loop Reflect

Plot of the states and the control signal of the MPC solution.

The state vector consist of the angle of the pendulum \(\theta\), its angular velocity \(\omega\), the position of the cart \(x\), and its velocity \(v\). We use the following model:

\[\begin{split}\newcommand{\pend}[1]{#1_\mathrm{pend}} \newcommand{\cart}[1]{#1_\mathrm{cart}} \begin{aligned} \pend{a} &= \pend{l} \sin(\theta)\, \omega^2 - g \cos(\theta) \sin(\theta) \\ a &= \frac{F - \cart{b}\, v + \pend{m}\, \pend{a}}{\cart{m} + \pend{m}} \\ \alpha &= \frac{g \sin(\theta) - a \cos(\theta)}{\pend{l}} \\ \begin{pmatrix} \dot \theta \\ \dot \omega \\ \dot x \\ \dot v \end{pmatrix} &= \begin{pmatrix} \omega \\ \alpha \\ v \\ a \end{pmatrix} \end{aligned}\end{split}\]

A quadratic cost function is used, applied to the sine of half the pendulum angle, to make it periodic with period \(2\pi\):

\[\ell(x, v, \theta, \omega, F) = q_\theta \sin^2(\theta/2) + q_\omega\, \omega^2 + q_x\, x^2 + q_v\, v^2 + r_F\, F^2\]

The force \(F\) exerted on the cart is limited to \(\pm 5 \mathrm{N}\).

# %% Model

import casadi as cs
import numpy as np

# State vector
θ = cs.SX.sym("θ")  # Pendulum angle                     [rad]
ω = cs.SX.sym("ω")  # Pendulum angular velocity          [rad/s]
x = cs.SX.sym("x")  # Cart position                      [m]
v = cs.SX.sym("v")  # Cart velocity                      [m/s]
state = cs.vertcat(θ, ω, x, v)  # Full state vector
nx = state.shape[0]  # Number of states

# Input
F = cs.SX.sym("F")  # External force applied to the cart [N]
nu = F.shape[0]  # Number of inputs

# Parameters
F_max = 5  #        Maximum force applied to the cart    [N]
m_cart = 0.8  #     Mass of the cart                     [kg]
m_pend = 0.3  #     Mass of the pendulum                 [kg]
b_cart = 0.1  #     Friction coefficient of the cart     [N/m/s]
l_pend = 0.3  #     Length of the pendulum               [m]
g_gravity = 9.81  # Gravitational acceleration           [m/s²]
Ts = 0.025  #       Simulation sampling time             [s]
N_horiz = 50  #     MPC horizon                          [time steps]
N_sim = 300  #      Simulation length                    [time steps]

# Continous-time dynamics
a_pend = l_pend * cs.sin(θ) * ω**2 - g_gravity * cs.cos(θ) * cs.sin(θ)
a = (F - b_cart * v + m_pend * a_pend) / (m_cart + m_pend)
α = (g_gravity * cs.sin(θ) - a * cs.cos(θ)) / l_pend
f_c = cs.Function("f_c", [state, F], [cs.vertcat(ω, α, v, a)])

# 4th order Runge-Kutta integrator
k1 = f_c(state, F)
k2 = f_c(state + Ts * k1 / 2, F)
k3 = f_c(state + Ts * k2 / 2, F)
k4 = f_c(state + Ts * k3, F)

# Discrete-time dynamics
f_d_expr = state + (Ts / 6) * (k1 + 2 * k2 + 2 * k3 + k4)
f_d = cs.Function("f", [state, F], [f_d_expr])


# %% Model predictive control

# MPC inputs and states
mpc_x0 = cs.MX.sym("x0", nx)  # Initial state
mpc_u = cs.MX.sym("u", (1, N_horiz))  # Inputs
mpc_x = f_d.mapaccum(N_horiz)(mpc_x0, mpc_u)  # Simulated states

# MPC cost
Q = cs.MX.sym("Q", nx)  # Stage state cost
Qf = cs.MX.sym("Qf", nx)  # Terminal state cost
R = cs.MX.sym("R", nu)  # Stage input cost
s, u = cs.MX.sym("s", nx), cs.MX.sym("u", nu)
sin_s = cs.vertcat(cs.sin(s[0] / 2), s[1:])  # Penalize sin(θ/2), not θ
stage_cost_x = cs.Function("lx", [s, Q], [cs.dot(sin_s, cs.diag(Q) @ sin_s)])
terminal_cost_x = cs.Function("lf", [s, Qf], [cs.dot(sin_s, cs.diag(Qf) @ sin_s)])
stage_cost_u = cs.Function("lu", [u, R], [cs.dot(u, cs.diag(R) @ u)])

mpc_param = cs.vertcat(mpc_x0, Q, Qf, R)
mpc_y_cost = cs.sum2(stage_cost_x.map(N_horiz - 1)(mpc_x[:, :-1], Q))
mpc_u_cost = cs.sum2(stage_cost_u.map(N_horiz)(mpc_u, R))
mpc_terminal_cost = terminal_cost_x(mpc_x[:, -1], Qf)
mpc_cost = mpc_y_cost + mpc_u_cost + mpc_terminal_cost

# Box constraints on the actuator force:
C = -F_max * np.ones(N_horiz), +F_max * np.ones(N_horiz)

# Compile into an alpaqa problem
import alpaqa

# Generate C code for the cost function, compile it, and load it as an
# alpaqa problem description:
problem = (
    alpaqa.minimize(mpc_cost, cs.vec(mpc_u))  # objective and variables
    .subject_to_box(C)  #                       box constraints on the variables
    .with_param(mpc_param)  #                   parameters to be changed later
).compile(sym=cs.MX.sym)

from datetime import timedelta

# Configure an alpaqa solver:
Solver = alpaqa.PANOCSolver
inner_solver = Solver(
    panoc_params={
        "max_time": timedelta(seconds=Ts),
        "max_iter": 200,
        "stop_crit": alpaqa.PANOCStopCrit.ProjGradUnitNorm2,
    },
    direction=alpaqa.StructuredLBFGSDirection(
        lbfgs_params={"memory": 15},
        direction_params={"hessian_vec_factor": 0},
    ),
)
alm_params = {
    "tolerance": 1e-4,
    "initial_tolerance": 1e-4,
}
solver = alpaqa.ALMSolver(
    alm_params=alm_params,
    inner_solver=inner_solver,
)


# %% Controller class


# Wrap the solver in a class that solves the optimal control problem at each
# time step, implementing warm starting:
class MPCController:
    def __init__(self, problem: alpaqa.CasADiProblem):
        self.problem = problem
        self.tot_it = 0
        self.tot_time = timedelta()
        self.max_time = timedelta()
        self.failures = 0
        self.u = np.ones(problem.n)

    def __call__(self, state_n):
        state_n = np.array(state_n).ravel()
        # Set the current state as the initial state
        self.problem.param[:nx] = state_n
        # Shift over the previous solution by one time step
        self.u = np.concatenate((self.u[nu:], self.u[-nu:]))
        # Solve the optimal control problem
        # (warm start using the shifted previous solution)
        self.u, _, stats = solver(self.problem, self.u)
        # Print some solver statistics
        print(
            f"{stats['status']!s:24} iter={stats['inner']['iterations']:4} "
            f"time={stats['elapsed_time']} "
            f"failures={stats['inner_convergence_failures']} "
            f"tol={stats['ε']:1.3e} stepsize={stats['inner']['final_γ']:1.3e}"
        )
        self.tot_it += stats["inner"]["iterations"]
        self.failures += stats["status"] != alpaqa.SolverStatus.Converged
        self.tot_time += stats["elapsed_time"]
        self.max_time = max(self.max_time, stats["elapsed_time"])
        # Return the optimal control signal for the first time step
        return self.u[:nu]


# %% Simulate the system using the MPC controller

state_0 = np.array([-np.pi, 0, 0.2, 0])  # Initial state of the system

# Parameters
Q = [10, 1e-2, 1, 1e-2]
Qf = np.array([10, 1e-1, 1000, 1e-1])
R = [1e-1]
problem.param = np.concatenate((state_0, Q, Qf, R))

# Simulation
mpc_states = np.empty((nx, N_sim + 1))
mpc_inputs = np.empty((nu, N_sim))
mpc_states[:, 0] = state_0
controller = MPCController(problem)
for n in range(N_sim):
    # Solve the optimal control problem:
    mpc_inputs[:, n] = controller(mpc_states[:, n])
    # Apply the first optimal control input to the system and simulate for
    # one time step, then update the state:
    mpc_states[:, n + 1] = f_d(mpc_states[:, n], mpc_inputs[:, n]).T

print(f"{controller.tot_it} iterations, {controller.failures} failures")
print(
    f"time: {controller.tot_time} (total), {controller.max_time} (max), "
    f"{controller.tot_time / N_sim} (avg)"
)


# %% Visualize the results

import matplotlib.pyplot as plt
import matplotlib as mpl
from matplotlib import animation

mpl.rcParams["animation.frame_format"] = "svg"

# Plot the cart and pendulum
fig, ax = plt.subplots()
h = 0.04
x = [state_0[2], state_0[2] + l_pend * np.sin(state_0[0])]
y = [h, h + l_pend * np.cos(state_0[0])]
(target_pend,) = ax.plot(x, y, "--", color="tab:blue", label="Initial state")
state_N = [0, 0, 0, 0]
x = [state_N[2], state_N[2] + l_pend * np.sin(state_N[0])]
y = [h, h + l_pend * np.cos(state_N[0])]
(target_pend,) = ax.plot(x, y, "--", color="tab:green", label="Target state")
(pend,) = ax.plot(x, y, "-o", color="tab:orange", alpha=0.9, label="MPC")
cart = plt.Rectangle((-2 * h, 0), 4 * h, h, color="tab:orange")
ax.add_patch(cart)
plt.legend()
plt.ylim([-l_pend + h, l_pend + 2 * h])
plt.xlim([-1.5 * l_pend, +1.5 * l_pend])
plt.gca().set_aspect("equal", "box")
plt.tight_layout()


class Animation:
    def __call__(self, n):
        state_n = mpc_states[:, n]
        x = [state_n[2], state_n[2] + l_pend * np.sin(state_n[0])]
        y = [h, h + l_pend * np.cos(state_n[0])]
        pend.set_xdata(x)
        pend.set_ydata(y)
        cart.set_x(state_n[2] - 2 * h)
        return [pend, cart]


ani = animation.FuncAnimation(
    fig, Animation(), interval=1000 * Ts, blit=True, repeat=True, frames=N_sim
)

fig, axs = plt.subplots(5, sharex=True, figsize=(6, 9))
ts = np.arange(N_sim + 1) * Ts
labels = [
    "Angle $\\theta$ [rad]",
    "Angular velocity $\\omega$ [rad/s]",
    "Position $x$ [m]",
    "Velocity $v$ [m/s]",
]
for i, (ax, lbl) in enumerate(zip(axs[:-1], labels)):
    ax.plot(ts, mpc_states[i, :])
    ax.set_title(lbl)
ax = axs[-1]
ax.plot(ts[:N_sim], mpc_inputs.T)
ax.set_title("Control input $F$ [N]")
ax.set_xlabel("Simulation time $t$ [s]")
ax.set_xlim(0, N_sim * Ts)
plt.tight_layout()

# Show the animation
plt.show()