Example: Gol, Lazar and Belta (2013) solved by Bemporad Morari.

This example reproduces parts of the numerical results of [3]. A similar example reproducing all results of [3] is available as a codeocean capsule in [4].

This example was borrowed from [1, Example VIII.A] and tackles an optimal control for the hybrid system with state evolution governed by

\[x(k+1) = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}x(k) + \begin{bmatrix} 0.5 \\ 1.0 \end{bmatrix} u(k)\]

The goal is to take the state vector toward a target set XT by visiting one of the squares A or B and avoiding the obstacles O1 and O2

First, let us import CDDLib, GLPK, OSQP, JuMP, Pavito and Ipopt

import CDDLib
import GLPK
import OSQP
using JuMP
import Pavito
import HiGHS
import Ipopt

At this point we import Dionysos

using Dionysos
const DI = Dionysos
const UT = DI.Utils
const ST = DI.System
const OP = DI.Optim

Dionysos.Optim

And the file defining the hybrid system for this problem

include(joinpath(dirname(dirname(pathof(Dionysos))), "problems", "gol_lazar_belta.jl"))

Main.var"__atexample__named__Gol,_Lazar_&_Belta_(2013)".GolLazarBelta

Now we instantiate our optimal control problem using the function provided by GolLazarBelta.jl

problem = GolLazarBelta.problem(CDDLib.Library(), Float64);

Finally, we select the method presented in [2] as our optimizer

qp_solver = optimizer_with_attributes(
    OSQP.Optimizer,
    "eps_abs" => 1e-8,
    "eps_rel" => 1e-8,
    "max_iter" => 100000,
    MOI.Silent() => true,
);

mip_solver = optimizer_with_attributes(HiGHS.Optimizer, MOI.Silent() => true);

cont_solver = optimizer_with_attributes(Ipopt.Optimizer, MOI.Silent() => true);

miqp_solver = optimizer_with_attributes(
    Pavito.Optimizer,
    "mip_solver" => mip_solver,
    "cont_solver" => cont_solver,
    MOI.Silent() => true,
);

algo = optimizer_with_attributes(
    OP.BemporadMorari.Optimizer{Float64},
    "continuous_solver" => qp_solver,
    "mixed_integer_solver" => miqp_solver,
    "indicator" => false,
    "log_level" => 0,
);

and use it to solve the given problem, with the help of the abstraction layer MathOptInterface provided by JuMP

optimizer = MOI.instantiate(algo)
MOI.set(optimizer, MOI.RawOptimizerAttribute("problem"), problem)
MOI.optimize!(optimizer)

4.102313995361328

We check the solver time

MOI.get(optimizer, MOI.SolveTimeSec())

4.102313995361328

the termination status

termination = MOI.get(optimizer, MOI.TerminationStatus())

LOCALLY_SOLVED::TerminationStatusCode = 4

the objective value

objective_value = MOI.get(optimizer, MOI.ObjectiveValue())

11.385062952226303

and recover the corresponding continuous trajectory

xu = MOI.get(optimizer, ST.ContinuousTrajectoryAttribute());

A little bit of data visualization now:

using Plots
using Polyhedra
using HybridSystems
using Suppressor

#Initialize our canvas
fig = plot(;
    aspect_ratio = :equal,
    xtickfontsize = 10,
    ytickfontsize = 10,
    guidefontsize = 16,
    titlefontsize = 14,
);
xlims!(-10.5, 3.0)
ylims!(-10.5, 3.0)

#Plot the discrete modes
for mode in states(problem.system)
    t =
        (problem.system.ext[:q_T] in [mode, mode + 11]) ? "XT" :
        (
            mode == problem.system.ext[:q_A] ? "A" :
            (
                mode == problem.system.ext[:q_B] ? "B" :
                mode <= 11 ? string(mode) : string(mode - 11)
            )
        )
    set = stateset(problem.system, mode)
    plot!(set; color = :white)
    UT.text_in_set_plot!(fig, set, t)
end

#Plot obstacles
for i in eachindex(problem.system.ext[:obstacles])
    set = problem.system.ext[:obstacles][i]
    plot!(set; color = :black, opacity = 0.5)
    UT.text_in_set_plot!(fig, set, "O$i")
end

#Plot trajectory
x0 = problem.initial_set[2]
x_traj = [x0, xu.x...]
plot!(fig, UT.DrawTrajectory(x_traj));

#Plot initial point
plot!(fig, UT.DrawPoint(x0); color = :blue)
annotate!(fig, x0[1], x0[2] - 0.5, "x0")

References

1. Gol, E. A., Lazar, M., & Belta, C. (2013). Language-guided controller synthesis for linear systems. IEEE Transactions on Automatic Control, 59(5), 1163-1176.
2. Bemporad, A., & Morari, M. (1999). Control of systems integrating logic, dynamics, and constraints. Automatica, 35(3), 407-427.
3. Legat B., Bouchat J., Jungers R. M. (2021). Abstraction-based branch and bound approach to Q-learning for hybrid optimal control. 3rd Annual Learning for Dynamics & Control Conference, 2021.
4. Legat B., Bouchat J., Jungers R. M. (2021). Abstraction-based branch and bound approach to Q-learning for hybrid optimal control. https://www.codeocean.com/. https://doi.org/10.24433/CO.6650697.v1.

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