How to plot a solution

In this tutorial, we explain the different options for plotting the solution of an optimal control problem using the plot and plot! functions, which are extensions of the Plots.jl package. Use plot to create a new plot object, and plot! to add to an existing one:

plot(args...; kw...)           # creates a new Plot, and set it to be the `current`
plot!(args...; kw...)          # modifies Plot `current()`
plot!(plt, args...; kw...)     # modifies Plot `plt`

More precisely, the signature of plot is as follows.

function plot(
    sol;            # optimal control solution
    layout,         # layout of the subplots
    control,        # plot the norm or components of the control
    time,           # normalise the time or not
    size,           # size of the figure
    state_style,    # style for the state trajectory
    costate_style,  # style for the costate trajectory
    control_style,  # style for the control trajectory
    kwargs...,      # attributes from Plots
)

In the following, we detail the roles of the arguments.

You can plot a solution obtained from the solve function, as well as from the flow computed using an optimal control problem and a control law. See, respectively, Basic Concepts and From Flow sections for more details.

You can add a plot to an existing one, thanks to the plot! function.

You can also retrieve the state, the costate and the control to create your own plots, see Custom plot section.

The problem and the solution

Let us start by importing the packages needed to define and solve the problem.

using OptimalControl
using NLPModelsIpopt

We consider the simple optimal control problem from the basic example tutorial.

const t0 = 0            # initial time
const tf = 1            # final time
const x0 = [ -1, 0 ]    # initial condition
const xf = [  0, 0 ]    # final condition

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)]

    ∫( 0.5u(t)^2 ) → min

end

sol = solve(ocp, display=false)

Basic concepts

The simplest way to plot the solution is to use the plot function with the solution as the only argument.

using Plots
plot(sol)

In the figure above, we have a grid of subplots: the left column displays the state component trajectories, the right column shows the costate component trajectories, and the bottom row contains the control component trajectories.

As in Plots, input data is passed positionally (for example, sol in plot(sol)), and attributes are passed as keyword arguments (for example, plot(sol; color = :blue)). After executing using Plots in the REPL, you can use the plotattr() function to print a list of all available attributes for series, plots, subplots, or axes.

# Valid Operations
plotattr(:Plot)
plotattr(:Series)
plotattr(:Subplot)
plotattr(:Axis)

Once you have the list of attributes, you can either use the aliases of a specific attribute or inspect a specific attribute to display its aliases and description.

julia> plotattr("color") # Specific Attribute Example:seriescolor

The base color for this series. `:auto` (the default) will select a color from the subplot's `color_palette`, based on the order it was added to the subplot. Also describes the colormap for surfaces.

Aliases: (:c, :cmap, :color, :colormap, :colour, :seriescolors).

Type: Union{Integer, Symbol, ColorSchemes.ColorScheme, ColorTypes.Colorant}.

`Series` attribute, defaults to `auto`.

You can also visit the Plot documentation online to get the descriptions of the attributes:

• To pass attributes to the plot, see the attributes plot documentation. For instance, you can specify the size of the figure.
• You can pass attributes to all subplots at once by referring to the attributes subplot documentation. For example, you can specify the location of the legends.
• Similarly, you can pass axis attributes to all subplots. See the attributes axis documentation. For example, you can remove the grid from every subplot.
• Finally, you can pass series attributes to all subplots. Refer to the attributes series documentation. For instance, you can set the width of the curves using linewidth.

plot(sol, size=(700, 450), legend=:bottomright, grid=false, linewidth=2)

To specify series attributes for a specific group of subplots (state, costate or control), you can use the optional keyword arguments state_style, costate_style, and control_style, which correspond to the state, costate, and control trajectories, respectively.

plot(sol;
     state_style   = (color=:blue,),                    # style of the state trajectory
     costate_style = (color=:black, linestyle=:dash),   # style of the costate trajectory
     control_style = (color=:red, linewidth=2))         # style of the control trajectory

From Flow

The previous solution of the optimal control problem was obtained using the solve function. If you prefer using an indirect shooting method and solving shooting equations, you may also want to plot the associated solution. To do this, you need to use the Flow function to reconstruct the solution. See the manual on how to compute flows for more details. In our case, you must provide the maximizing control $(x, p) \mapsto p_2$ along with the optimal control problem. For an introduction to simple indirect shooting, see the indirect simple shooting tutorial for an example.

using OrdinaryDiffEq

p  = costate(sol)                # costate as a function of time
p0 = p(t0)                       # costate solution at the initial time
f  = Flow(ocp, (x, p) -> p[2])   # flow from an ocp and a control law

sol_flow = f( (t0, tf), x0, p0 ) # compute the solution
plot(sol_flow)                   # plot the solution from a flow

We may notice that the time grid contains very few points. This is evident from the subplot of $x_2$, or by retrieving the time grid directly from the solution.

time_grid(sol_flow)

6-element Vector{Float64}:
 0.0
 0.002765360243588406
 0.023631266188700685
 0.12877324251759437
 0.5279162676838509
 1.0

To improve visualization (without changing the accuracy), you can provide a finer grid.

fine_grid = range(t0, tf, 100)
sol_flow = f( (t0, tf), x0, p0; saveat=fine_grid )
plot(sol_flow)

Split versus group layout

If you prefer to get a more compact figure, you can use the layout optional keyword argument with :group value. It will group the state, costate and control trajectories in one subplot for each.

plot(sol; layout=:group)

The default layout value is :split which corresponds to the grid of subplots presented above.

plot(sol; layout=:split)

Add a plot

You can plot the solution of a second optimal control problem on the same figure if it has the same number of states, costates and controls. For instance, consider the same optimal control problem but with a different initial condition.

ocp = @def begin

    t ∈ [t0, tf], time
    x ∈ R², state
    u ∈ R, control

    x(t0) == [-0.5, -0.5]
    x(tf) == xf

    ẋ(t) == [x₂(t), u(t)]

    ∫( 0.5u(t)^2 ) → min

end
sol2 = solve(ocp; display=false)

We first plot the solution of the first optimal control problem, then, we plot the solution of the second optimal control problem on the same figure, but with dashed lines.

plt = plot(sol; label="sol1", size=(700, 500))
plot!(plt, sol2; label="sol2", linestyle=:dash)

Plot the norm of the control

For some problem, it is interesting to plot the (Euclidean) norm of the control. You can do it by using the control optional keyword argument with :norm value.

plot(sol; control=:norm, size=(800, 300), layout=:group)

The default value is :components.

plot(sol; control=:components, size=(800, 300), layout=:group)

Custom plot

You can, of course, create your own plots by extracting the state, costate, and control from the optimal control solution. For instance, let us plot the norm of the control.

using LinearAlgebra
t = time_grid(sol)
u = control(sol)
plot(t, norm∘u; label="‖u‖", xlabel="t")

Normalised time

We consider a LQR example and solve the problem for different values of the final time tf. Then, we plot the solutions on the same figure using a normalized time $s = (t - t_0) / (t_f - t_0)$, enabled by the keyword argument time = :normalize (or :normalise) in the plot function.

# definition of the problem, parameterised by the final time
function lqr(tf)

    ocp = @def begin
        t ∈ [0, tf], time
        x ∈ R², state
        u ∈ R, control
        x(0) == [0, 1]
        ẋ(t) == [x₂(t), - x₁(t) + u(t)]
        ∫( 0.5(x₁(t)^2 + x₂(t)^2 + u(t)^2) ) → min
    end

    return ocp
end;

# solve the problems and store them
solutions = []
tfs = [3, 5, 30]
for tf ∈ tfs
    solution = solve(lqr(tf); display=false)
    push!(solutions, solution)
end

# create plots
plt = plot(solutions[1]; time=:normalize)
for sol ∈ solutions[2:end]
    plot!(plt, sol; time=:normalize)
end

# make a custom plot: keep only state and control
N = length(tfs)
px1 = plot(plt[1]; legend=false, xlabel="s", ylabel="x₁")
px2 = plot(plt[2]; label=reshape(["tf = $tf" for tf ∈ tfs], (1, N)), xlabel="s", ylabel="x₂")
pu  = plot(plt[5]; legend=false, xlabel="s", ylabel="u")

using Plots.PlotMeasures # for leftmargin, bottommargin
plot(px1, px2, pu; layout=(1, 3), size=(800, 300), leftmargin=5mm, bottommargin=5mm)