Double integrator: energy minimisation

Let us consider a wagon moving along a rail, whose acceleration can be controlled by a force $u$. We denote by $x = (x_1, x_2)$ the state of the wagon, where $x_1$ is the position and $x_2$ the velocity.

We assume that the mass is constant and equal to one, and that there is no friction. The dynamics are given by

\[ \dot x_1(t) = x_2(t), \quad \dot x_2(t) = u(t),\quad u(t) \in \R,\]

which is simply the double integrator system. Let us consider a transfer starting at time $t_0 = 0$ and ending at time $t_f = 1$, for which we want to minimise the transfer energy

\[ \frac{1}{2}\int_{0}^{1} u^2(t) \, \mathrm{d}t\]

starting from $x(0) = (-1, 0)$ and aiming to reach the target $x(1) = (0, 0)$.

First, we need to import the OptimalControl.jl package to define the optimal control problem, NLPModelsIpopt.jl to solve it, and Plots.jl to visualise the solution.

using OptimalControl
using NLPModelsIpopt
using Plots

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Optimal control problem

Let us define the problem with the @def macro:

t0 = 0
tf = 1
x0 = [-1, 0]
xf = [0, 0]
ocp = @def begin
    t ∈ [t0, tf], time
    x ∈ R², state
    u ∈ R, control
    x(t0) == x0
    x(tf) == xf
    ẋ(t) == [x₂(t), u(t)]
    0.5∫( u(t)^2 ) → min
end
nothing # hide

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Solve and plot

Direct method

We can solve it simply with:

sol = solve(ocp)
nothing # hide

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And plot the solution with:

plot(sol)

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Indirect method

The first solution was obtained using the so-called direct method.^[1] Another approach is to use an indirect simple shooting method. We begin by importing the necessary packages.

using OrdinaryDiffEq # Ordinary Differential Equations (ODE) solver
using NonlinearSolve # Nonlinear Equations (NLE) solver

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To define the shooting function, we must provide the maximising control in feedback form:

# maximising control, H(x, p, u) = p₁x₂ + p₂u - u²/2
u(x, p) = p[2]

# Hamiltonian flow
f = Flow(ocp, u)

# state projection, p being the costate
π((x, p)) = x

# shooting function
S(p0) = π( f(t0, x0, p0, tf) ) - xf
nothing # hide

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We are now ready to solve the shooting equations.

# auxiliary in-place NLE function
nle!(s, p0, λ) = s[:] = S(p0)

# initial guess for the Newton solver
p0_guess = [1, 1]

# NLE problem with initial guess
prob = NonlinearProblem(nle!, p0_guess)

# resolution of S(p0) = 0
sol = solve(prob; show_trace=Val(true))
p0_sol = sol.u # costate solution

# print the costate solution and the shooting function evaluation
println("\ncostate: p0 = ", p0_sol)
println("shoot: S(p0) = ", S(p0_sol), "\n")

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To plot the solution obtained by the indirect method, we need to build the solution of the optimal control problem. This is done using the costate solution and the flow function.

sol = f((t0, tf), x0, p0_sol; saveat=range(t0, tf, 100))
plot(sol)

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State constraint

Direct method: constrained case

We add the path constraint

\[ x_2(t) \le 1.2.\]

Let us model, solve and plot the optimal control problem with this constraint.

# the upper bound for x₂
a = 1.2

# the optimal control problem
ocp = @def begin
    t ∈ [t0, tf], time
    x ∈ R², state
    u ∈ R, control
    x₂(t) ≤ a
    x(t0) == x0
    x(tf) == xf
    ẋ(t) == [x₂(t), u(t)]
    0.5∫( u(t)^2 ) → min
end

# solve with a direct method using default settings
sol = solve(ocp)

# plot the solution
plt = plot(sol; label="Direct", size=(800, 600))

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Indirect method: constrained case

The pseudo-Hamiltonian is (considering the normal case):

\[H(x, p, u, \mu) = p_1 x_2 + p_2 u - \frac{u^2}{2} + \mu\, c(x),\]

with $c(x) = x_2 - a$. Along a boundary arc we have $c(x(t)) = 0$. Differentiating, we obtain:

\[ \frac{\mathrm{d}}{\mathrm{d}t}c(x(t)) = \dot{x}_2(t) = u(t) = 0.\]

The zero control is maximising; hence, $p_2(t) = 0$ along the boundary arc.

\[ \dot{p}_2(t) = -p_1(t) - \mu(t) \quad \Rightarrow \mu(t) = -p_1(t).\]

Since the adjoint vector is continuous at the entry time $t_1$ and the exit time $t_2$, we have four unknowns: the initial costate $p_0 \in \mathbb{R}^2$ and the times $t_1$ and $t_2$. We need four equations: the target condition provides two, reaching the constraint at time $t_1$ gives $c(x(t_1)) = 0$, and finally $p_2(t_1) = 0$.

# flow for unconstrained extremals
f = Flow(ocp, (x, p) -> p[2])

ub = 0          # boundary control
c(x) = x[2]-a   # constraint: c(x) ≥ 0
μ(p) = -p[1]    # dual variable

# flow for boundary extremals
g = Flow(ocp, (x, p) -> ub, (x, u) -> c(x), (x, p) -> μ(p))

# shooting function
function shoot!(s, p0, t1, t2)
    x_t0, p_t0 = x0, p0
    x_t1, p_t1 = f(t0, x_t0, p_t0, t1)
    x_t2, p_t2 = g(t1, x_t1, p_t1, t2)
    x_tf, p_tf = f(t2, x_t2, p_t2, tf)
    s[1:2] = x_tf - xf
    s[3] = c(x_t1)
    s[4] = p_t1[2]
end
nothing # hide

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We are now ready to solve the shooting equations.

# auxiliary in-place NLE function
nle!(s, ξ, λ) = shoot!(s, ξ[1:2], ξ[3], ξ[4])

# initial guess for the Newton solver
ξ_guess = [40, 10, 0.25, 0.75]

# NLE problem with initial guess
prob = NonlinearProblem(nle!, ξ_guess)

# resolution of the shooting equations
sol = solve(prob; show_trace=Val(true))
p0, t1, t2 = sol.u[1:2], sol.u[3], sol.u[4]

# print the costate solution and the entry and exit times
println("\np0 = ", p0, "\nt1 = ", t1, "\nt2 = ", t2)

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To reconstruct the trajectory obtained with the state constraint, we concatenate the flows: one unconstrained arc up to the entry time $t_1$, a boundary arc between $t_1$ and $t_2$, and finally another unconstrained arc up to $t_f$. This concatenation allows us to compute the complete solution — state, costate, and control — which we can then plot together with the direct solution for comparison.

# concatenation of the flows
φ = f * (t1, g) * (t2, f)

# compute the solution: state, costate, control...
flow_sol = φ((t0, tf), x0, p0; saveat=range(t0, tf, 100))      

# plot the solution on the previous plot
plot!(plt, flow_sol; label="Indirect", color=2, linestyle=:dash)

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