From 251fcd41d4b96ced8a0f934657435d9e18353ef6 Mon Sep 17 00:00:00 2001 From: Olivier Cots Date: Tue, 2 Sep 2025 17:00:37 +0200 Subject: [PATCH 01/22] beam ok --- docs/Project.toml | 2 + docs/make.jl | 2 +- docs/problems.jl | 429 +++++++++--------- docs/src/assets/Manifest.toml | 2 +- docs/src/assets/Project.toml | 2 + docs/src/problems-introduction.md | 1 - ext/Descriptions/beam.md | 45 +- ext/Descriptions/bioreactor.md | 2 +- ext/JuMPModels/ducted_fan.jl | 2 +- ext/OptimalControlModels/ducted_fan.jl | 2 +- test/Project.toml | 2 + test/extras/beam.jl | 28 ++ test/figures/init/beam.pdf | Bin 241996 -> 228976 bytes test/figures/init/bioreactor.pdf | Bin 284121 -> 284121 bytes test/figures/init/cart_pendulum.pdf | Bin 380215 -> 380215 bytes test/figures/init/chain.pdf | Bin 297428 -> 297428 bytes .../init/dielectrophoretic_particle.pdf | Bin 186083 -> 186083 bytes test/figures/init/double_oscillator.pdf | Bin 352998 -> 352998 bytes test/figures/init/ducted_fan.pdf | Bin 489602 -> 425107 bytes test/figures/init/electric_vehicle.pdf | Bin 229802 -> 229802 bytes test/figures/init/glider.pdf | Bin 383362 -> 383362 bytes test/figures/init/insurance.pdf | Bin 740103 -> 739789 bytes test/figures/init/jackson.pdf | Bin 251695 -> 251695 bytes test/figures/init/moonlander.pdf | Bin 538190 -> 538190 bytes test/figures/init/quadrotor.pdf | Bin 605623 -> 0 bytes test/figures/init/robbins.pdf | Bin 275604 -> 275604 bytes test/figures/init/robot.pdf | Bin 491930 -> 491930 bytes test/figures/init/rocket.pdf | Bin 287443 -> 287443 bytes test/figures/init/space_shuttle.pdf | Bin 591032 -> 591032 bytes test/figures/init/steering.pdf | Bin 365871 -> 365871 bytes test/figures/init/truck_trailer.pdf | Bin 617147 -> 617147 bytes test/figures/init/vanderpol.pdf | Bin 207452 -> 207452 bytes test/figures/solution/beam.pdf | Bin 219009 -> 219524 bytes test/figures/solution/bioreactor.pdf | Bin 289897 -> 289897 bytes test/figures/solution/cart_pendulum.pdf | Bin 324452 -> 324452 bytes test/figures/solution/chain.pdf | Bin 286444 -> 286444 bytes .../solution/dielectrophoretic_particle.pdf | Bin 179194 -> 179194 bytes test/figures/solution/double_oscillator.pdf | Bin 364926 -> 364926 bytes test/figures/solution/ducted_fan.pdf | Bin 418946 -> 399500 bytes test/figures/solution/electric_vehicle.pdf | Bin 208580 -> 208580 bytes test/figures/solution/glider.pdf | Bin 344505 -> 344505 bytes test/figures/solution/insurance.pdf | Bin 755442 -> 755128 bytes test/figures/solution/jackson.pdf | Bin 257574 -> 257574 bytes test/figures/solution/moonlander.pdf | Bin 571343 -> 571343 bytes test/figures/solution/quadrotor.pdf | Bin 525971 -> 0 bytes test/figures/solution/robbins.pdf | Bin 285603 -> 285603 bytes test/figures/solution/robot.pdf | Bin 390143 -> 390143 bytes test/figures/solution/rocket.pdf | Bin 298415 -> 298415 bytes test/figures/solution/space_shuttle.pdf | Bin 609738 -> 609738 bytes test/figures/solution/steering.pdf | Bin 412955 -> 412955 bytes test/figures/solution/truck_trailer.pdf | Bin 611565 -> 611565 bytes test/figures/solution/vanderpol.pdf | Bin 198020 -> 198020 bytes test/runtests.jl | 4 +- test/test_JuMP.jl | 10 +- test/test_OptimalControl.jl | 10 +- test/test_quick.jl | 8 +- test/utils.jl | 309 +++++-------- 57 files changed, 413 insertions(+), 447 deletions(-) create mode 100644 test/extras/beam.jl delete mode 100644 test/figures/init/quadrotor.pdf delete mode 100644 test/figures/solution/quadrotor.pdf diff --git a/docs/Project.toml b/docs/Project.toml index c2513dd1..04fca98e 100644 --- a/docs/Project.toml +++ b/docs/Project.toml @@ -11,6 +11,7 @@ NLPModels = "a4795742-8479-5a88-8948-cc11e1c8c1a6" NLPModelsIpopt = "f4238b75-b362-5c4c-b852-0801c9a21d71" OptimalControl = "5f98b655-cc9a-415a-b60e-744165666948" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +Printf = "de0858da-6303-5e67-8744-51eddeeeb8d7" TOML = "fa267f1f-6049-4f14-aa54-33bafae1ed76" [compat] @@ -26,4 +27,5 @@ NLPModels = "0.21" NLPModelsIpopt = "0.10" OptimalControl = "1" Plots = "1" +Printf = "1" TOML = "1" diff --git a/docs/make.jl b/docs/make.jl index e1ad9714..44aa315e 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -74,7 +74,7 @@ cp( repo_url = "github.com/control-toolbox/OptimalControlProblems.jl" # -draft = true +draft = false exclude_from_draft=Symbol[ # :beam ] diff --git a/docs/problems.jl b/docs/problems.jl index 476fea2c..bcf8a1a4 100644 --- a/docs/problems.jl +++ b/docs/problems.jl @@ -19,14 +19,18 @@ function generate_documentation( """ end + LEFT_MARGIN = get_left_margin(Symbol(PROBLEM)) + documentation=DRAFT * """ # $TITLE + ## Description of the problem + $DESCRIPTION - ## Packages + ## Numerical set-up - Import all necessary packages and define DataFrames to store information about the problem and resolution results. + In this section, we prepare the numerical environment required to study the problem. We begin by importing the relevant Julia packages and then initialise the data frames that will store the results of our simulations and computations. These structures provide the foundation for solving the problem and for comparing the different solution strategies in a consistent way. ```@example main using OptimalControlProblems # to access the Beam model @@ -38,6 +42,7 @@ function generate_documentation( using Plots.PlotMeasures # for leftmargin, bottommargin using JuMP # to import the JuMP model using Ipopt # to solve the JuMP model with Ipopt + using Printf # to print data_pb = DataFrame( # to store data about the problem Problem=Symbol[], @@ -55,95 +60,132 @@ function generate_documentation( nothing # hide ``` - ## initial guess + ## Initial guess - The initial guess (or first iterate) can be visualised by running the solver with `max_iter=0`. Here is the initial guess. + Before solving the problem, it is often useful to inspect the initial guess (sometimes called the first iterate). This guess is obtained by running the NLP solver with `max_iter = 0`, which evaluates the problem formulation without performing any optimisation steps. - ```@raw html -
Click to unfold and see the code for plotting the initial guess. - ``` + We plot the resulting trajectories for both the OptimalControl and JuMP models. Since both backends represent the same mathematical problem, their initial guesses should coincide, providing a useful consistency check before moving on to the optimised solution. - ```@example main - function plot_initial_guess(problem) - - # dimensions - x_vars = metadata[problem][:state_name] - u_vars = metadata[problem][:control_name] - n = length(x_vars) # number of states - m = length(u_vars) # number of controls - - # import OptimalControl model - docp = eval(problem)(OptimalControlBackend()) - nlp_oc = nlp_model(docp) - - # solve - nlp_oc_sol = NLPModelsIpopt.ipopt(nlp_oc; max_iter=0) - - # build an optimal control solution - ocp_sol = build_ocp_solution(docp, nlp_oc_sol) - - # plot the OptimalControl solution - plt = plot( - ocp_sol; - state_style=(color=1,), - costate_style=(color=1, legend=:none), - control_style=(color=1, legend=:none), - path_style=(color=1, legend=:none), - dual_style=(color=1, legend=:none), - size=(816, 220*(n+m)), - label="OptimalControl", - leftmargin=20mm, - ) - for i in 2:n - plot!(plt[i]; legend=:none) - end + !!! note "Code to plot the initial guess" - # import JuMP model - nlp_jp = eval(problem)(JuMPBackend()) + ```@raw html +
Click to unfold and see the code for plotting the initial guess. + ``` - # solve - set_optimizer(nlp_jp, Ipopt.Optimizer) - set_optimizer_attribute(nlp_jp, "max_iter", 0) - optimize!(nlp_jp) + ```@example main + function plot_initial_guess(problem) + + # ----------------------------- + # Extract dimensions from metadata + # ----------------------------- + x_vars = metadata[problem][:state_name] + u_vars = metadata[problem][:control_name] + n_states = length(x_vars) + n_controls = length(u_vars) + + # ----------------------------- + # Build OptimalControl problem + # ----------------------------- + ocp_model = eval(problem)(OptimalControlBackend()) + nlp_oc = nlp_model(ocp_model) + + # Solve NLP with zero iterations (initial guess) + nlp_oc_sol = NLPModelsIpopt.ipopt(nlp_oc; max_iter=0) + + # Build OptimalControl solution + ocp_sol = build_ocp_solution(ocp_model, nlp_oc_sol) + + # ----------------------------- + # Plot OptimalControl solution + # ----------------------------- + plt = plot( + ocp_sol; + state_style=(color=1,), + costate_style=(color=1, legend=:none), + control_style=(color=1, legend=:none), + path_style=(color=1, legend=:none), + dual_style=(color=1, legend=:none), + size=(816, 220*(n_states+n_controls)), + label="OptimalControl", + leftmargin=20mm, + ) + + # Hide legend for additional state plots + for i in 2:n_states + plot!(plt[i]; legend=:none) + end - # plot - t = time_grid(problem, nlp_jp) # t0, ..., tN = tf - x = state(problem, nlp_jp) # function of time - u = control(problem, nlp_jp) # function of time - p = costate(problem, nlp_jp) # function of time + # ----------------------------- + # Build JuMP model + # ----------------------------- + nlp_jp = eval(problem)(JuMPBackend()) + + # Solve NLP with zero iterations (initial guess) + set_optimizer(nlp_jp, Ipopt.Optimizer) + set_optimizer_attribute(nlp_jp, "max_iter", 0) + optimize!(nlp_jp) + + # Extract trajectories + t_grid = time_grid(problem, nlp_jp) + x_fun = state(problem, nlp_jp) + u_fun = control(problem, nlp_jp) + p_fun = costate(problem, nlp_jp) + + # ----------------------------- + # Plot JuMP solution on top + # ----------------------------- + # States + for i in 1:n_states + label = i == 1 ? "JuMP" : :none + plot!(plt[i], t_grid, t -> x_fun(t)[i]; color=2, linestyle=:dash, label=label) + end - for i in 1:n # state - label = i == 1 ? "JuMP" : :none - plot!(plt[i], t, t -> x(t)[i]; color=2, linestyle=:dash, label=label) - end + # Costates + for i in 1:n_states + plot!(plt[n_states+i], t_grid, t -> -p_fun(t)[i]; color=2, linestyle=:dash, label=:none) + end - for i in 1:n # costate - plot!(plt[n+i], t, t -> -p(t)[i]; color=2, linestyle=:dash, label=:none) - end + # Controls + for i in 1:n_controls + plot!(plt[2*n_states+i], t_grid, t -> u_fun(t)[i]; color=2, linestyle=:dash, label=:none) + end - for i in 1:m # control - plot!(plt[2n+i], t, t -> u(t)[i]; color=2, linestyle=:dash, label=:none) + return plt end + nothing # hide + ``` - return plt - end - nothing # hide - ``` - - ```@raw html -
-
- ``` + ```@raw html +
+ ``` ```@example main plot_initial_guess(:$PROBLEM) ``` - ## Solve the problem + ## Solving the problem + + To solve an optimal control problem, we can rely on two complementary formulations: the `OptimalControl` backend, which works directly with the discretised control problem, and the `JuMP` backend, which leverages JuMP’s flexible modelling framework. + + Both approaches generate equivalent NLPs that can be solved with Ipopt, and comparing them ensures consistency between the two formulations. + + Before solving, we can inspect the discretisation details of the problem. The table below reports the number of grid points, decision variables, and constraints associated with the chosen formulation. + + ```@example main + push!(data_pb, + ( + Problem=:$PROBLEM, + Grid_Size=metadata[:$PROBLEM][:N], + Variables=get_nvar(nlp_model($PROBLEM(OptimalControlBackend()))), + Constraints=get_ncon(nlp_model($PROBLEM(OptimalControlBackend()))), + ) + ) + data_pb # hide + ``` ### OptimalControl model - Import the OptimalControl model and solve it. + We first solve the problem using the `OptimalControl` backend. The process begins by importing the problem definition and constructing the associated nonlinear programming (NLP) model. This NLP is then passed to the Ipopt solver, with standard options for tolerance and barrier parameter strategy. ```@example main # import DOCP model @@ -163,29 +205,15 @@ function generate_documentation( nothing # hide ``` - The problem has the following numbers of steps, variables and constraints. - - ```@example main - push!(data_pb, - ( - Problem=:$PROBLEM, - Grid_Size=metadata[:$PROBLEM][:N], - Variables=get_nvar(nlp_oc), - Constraints=get_ncon(nlp_oc), - ) - ) - data_pb # hide - ``` - ### JuMP model - Import the JuMP model and solve it. + We now repeat the procedure using the `JuMP` backend. Here, the problem is reformulated as a JuMP model, which offers a flexible and widely used framework for nonlinear optimisation in Julia. The solver settings are chosen to mirror those used previously, so that the results can be compared on an equal footing. ```@example main # import model nlp_jp = $PROBLEM(JuMPBackend()) - # solve + # solve with Ipopt set_optimizer(nlp_jp, Ipopt.Optimizer) set_optimizer_attribute(nlp_jp, "print_level", 4) set_optimizer_attribute(nlp_jp, "tol", 1e-8) @@ -197,7 +225,7 @@ function generate_documentation( ## Numerical comparisons - Let's get the flag, the number of iterations and the objective value from the resolutions. + In this section, we examine the results of the problem resolutions. We extract the solver status (flag), the number of iterations, and the objective value for each model. This provides a first overview of how each approach performs and sets the stage for a more detailed comparison of the solution trajectories. ```@example main # from OptimalControl model @@ -222,139 +250,108 @@ function generate_documentation( data_re # hide ``` - We compare the OptimalControl and JuMP solutions in terms of the number of iterations, the \$L^2\$-norm of the differences in the state, control, and variable, as well as the objective values. Both absolute and relative errors are reported. + We compare the solutions obtained from the OptimalControl and JuMP models by examining the number of iterations required for convergence, the \$L^2\$-norms of the differences in states, controls, and additional variables, and the corresponding objective values. Both absolute and relative errors are reported, providing a clear quantitative measure of the agreement between the two approaches. - ```@raw html -
Click to unfold and get the code of the numerical comparison. - ``` + !!! note "Code to print the numerical comparisons" - ```@example main - function L2_norm(T, X) - # T and X are supposed to be one dimensional - s = 0.0 - for i in 1:(length(T) - 1) - s += 0.5 * (X[i]^2 + X[i + 1]^2) * (T[i + 1]-T[i]) - end - return √(s) - end + ```@raw html +
Click to unfold and get the code of the numerical comparisons. + ``` - function numerical_comparison(problem, docp, nlp_oc_sol, nlp_jp) - - # get relevant data from OptimalControl model - ocp_sol = build_ocp_solution(docp, nlp_oc_sol) # build an ocp solution - t_oc = time_grid(ocp_sol) - x_oc = state(ocp_sol).(t_oc) - u_oc = control(ocp_sol).(t_oc) - v_oc = variable(ocp_sol) - o_oc = objective(ocp_sol) - i_oc = iterations(ocp_sol) - - # get relevant data from JuMP model - t_jp = time_grid(problem, nlp_jp) - x_jp = state(problem, nlp_jp).(t_jp) - u_jp = control(problem, nlp_jp).(t_jp) - o_jp = objective(problem, nlp_jp) - v_jp = variable(problem, nlp_jp) - i_jp = iterations(problem, nlp_jp) - - x_vars = metadata[problem][:state_name] - u_vars = metadata[problem][:control_name] - v_vars = metadata[problem][:variable_name] - - println("┌─ ", string(problem)) - println("│") - - # number of iterations - println("├─ Number of iterations") - println("│") - println("│ OptimalControl : ", i_oc) - println("│ JuMP : ", i_jp) - println("│") - - # state - for i in eachindex(x_vars) - xi_oc = [x_oc[k][i] for k in eachindex(t_oc)] - xi_jp = [x_jp[k][i] for k in eachindex(t_jp)] - L2_oc = L2_norm(t_oc, xi_oc) - L2_jp = L2_norm(t_oc, xi_jp) - L2_ae = L2_norm(t_oc, xi_oc-xi_jp) - L2_re = L2_ae/(0.5*(L2_oc + L2_jp)) - - println("├─ State \$(x_vars[i]) (L2 norm)") - println("│") - #println("│ OptimalControl : ", L2_oc) - #println("│ JuMP : ", L2_jp) - println("│ Absolute error : ", L2_ae) - println("│ Relative error : ", L2_re) - println("│") + ```@example main + function L2_norm(T, X) + # T and X are supposed to be one dimensional + s = 0.0 + for i in 1:(length(T) - 1) + s += 0.5 * (X[i]^2 + X[i + 1]^2) * (T[i + 1]-T[i]) + end + return √(s) end - - # control - for i in eachindex(u_vars) - ui_oc = [u_oc[k][i] for k in eachindex(t_oc)] - ui_jp = [u_jp[k][i] for k in eachindex(t_jp)] - L2_oc = L2_norm(t_oc, ui_oc) - L2_jp = L2_norm(t_oc, ui_jp) - L2_ae = L2_norm(t_oc, ui_oc-ui_jp) - L2_re = L2_ae/(0.5*(L2_oc + L2_jp)) - - println("├─ Control \$(u_vars[i]) (L2 norm)") - println("│") - #println("│ OptimalControl : ", L2_oc) - #println("│ JuMP : ", L2_jp) - println("│ Absolute error : ", L2_ae) - println("│ Relative error : ", L2_re) + + function print_numerical_comparisons(problem, docp, nlp_oc_sol, nlp_jp) + + # get relevant data from OptimalControl model + ocp_sol = build_ocp_solution(docp, nlp_oc_sol) + t_oc = time_grid(ocp_sol) + x_oc = state(ocp_sol).(t_oc) + u_oc = control(ocp_sol).(t_oc) + v_oc = variable(ocp_sol) + o_oc = objective(ocp_sol) + i_oc = iterations(ocp_sol) + + # get relevant data from JuMP model + t_jp = time_grid(problem, nlp_jp) + x_jp = state(problem, nlp_jp).(t_jp) + u_jp = control(problem, nlp_jp).(t_jp) + o_jp = objective(problem, nlp_jp) + v_jp = variable(problem, nlp_jp) + i_jp = iterations(problem, nlp_jp) + + x_vars = metadata[problem][:state_name] + u_vars = metadata[problem][:control_name] + v_vars = metadata[problem][:variable_name] + + println("┌─ ", string(problem)) println("│") - end - - # variable - if !isnothing(v_vars) - for i in eachindex(v_vars) - vi_oc = v_oc[i] - vi_jp = v_jp[i] - vi_ae = abs(vi_oc-vi_jp) - vi_re = vi_ae/(0.5*(abs(vi_oc) + abs(vi_jp))) - - println("├─ Variable \$(v_vars[i])") - println("│") - #println("│ OptimalControl : ", vi_oc) - #println("│ JuMP : ", vi_jp) - println("│ Absolute error : ", vi_ae) - println("│ Relative error : ", vi_re) - println("│") + println("├─ Number of Iterations") + @printf("│ OptimalControl : %d JuMP : %d\\n", i_oc, i_jp) + + # States + println("├─ States (L2 Norms)") + for i in eachindex(x_vars) + xi_oc = [x_oc[k][i] for k in eachindex(t_oc)] + xi_jp = [x_jp[k][i] for k in eachindex(t_jp)] + L2_ae = L2_norm(t_oc, xi_oc - xi_jp) + L2_re = L2_ae / (0.5 * (L2_norm(t_oc, xi_oc) + L2_norm(t_oc, xi_jp))) + @printf("│ %-6s Abs: %.3e Rel: %.3e\\n", x_vars[i], L2_ae, L2_re) end - end - # objective - o_ae = abs(o_oc-o_jp) - o_re = o_ae/(0.5*(abs(o_oc) + abs(o_jp))) + # Controls + println("├─ Controls (L2 Norms)") + for i in eachindex(u_vars) + ui_oc = [u_oc[k][i] for k in eachindex(t_oc)] + ui_jp = [u_jp[k][i] for k in eachindex(t_jp)] + L2_ae = L2_norm(t_oc, ui_oc - ui_jp) + L2_re = L2_ae / (0.5 * (L2_norm(t_oc, ui_oc) + L2_norm(t_oc, ui_jp))) + @printf("│ %-6s Abs: %.3e Rel: %.3e\\n", u_vars[i], L2_ae, L2_re) + end - println("├─ objective") - println("│") - #println("│ OptimalControl : ", o_oc) - #println("│ JuMP : ", o_jp) - println("│ Absolute error : ", o_ae) - println("│ Relative error : ", o_re) - println("│") - println("└─") + # Variables + if !isnothing(v_vars) + println("├─ Variables") + for i in eachindex(v_vars) + vi_oc = v_oc[i] + vi_jp = v_jp[i] + vi_ae = abs(vi_oc - vi_jp) + vi_re = vi_ae / (0.5 * (abs(vi_oc) + abs(vi_jp))) + @printf("│ %-6s Abs: %.3e Rel: %.3e\\n", v_vars[i], vi_ae, vi_re) + end + end - return nothing - end - nothing # hide - ``` + # Objective + o_ae = abs(o_oc - o_jp) + o_re = o_ae / (0.5 * (abs(o_oc) + abs(o_jp))) + println("├─ Objective") + @printf("│ Abs: %.3e Rel: %.3e\\n", o_ae, o_re) + println("└─") + return nothing + end + nothing # hide + ``` - ```@raw html -
-
- ``` + ```@raw html +
+ ``` ```@example main - numerical_comparison(:$PROBLEM, docp, nlp_oc_sol, nlp_jp) + print_numerical_comparisons(:$PROBLEM, docp, nlp_oc_sol, nlp_jp) ``` - ## Plot the solutions + ## Plotting the solutions - Visualise states, costates, and controls from the OptimalControl and JuMP solutions: + In this section, we visualise the trajectories of the states, costates, and controls obtained from both the OptimalControl and JuMP solutions. The plots provide an intuitive way to compare the two approaches and to observe how the constraints and the optimal control influence the system dynamics. + + For each variable, the OptimalControl solution is shown in solid lines, while the JuMP solution is overlaid using dashed lines. Since both models represent the same mathematical problem, their trajectories should closely coincide, highlighting the consistency between the two formulations. ```@example main # build an ocp solution to use the plot from OptimalControl package @@ -367,16 +364,12 @@ function generate_documentation( # from OptimalControl solution plt = plot( ocp_sol; - state_style=(color=1,), - costate_style=(color=1, legend=:none), - control_style=(color=1, legend=:none), - path_style=(color=1, legend=:none), - dual_style=(color=1, legend=:none), + color=1, size=(816, 240*(n+m)), label="OptimalControl", - leftmargin=20mm, + leftmargin=$LEFT_MARGIN, ) - for i in 2:n + for i in 2:length(plt) plot!(plt[i]; legend=:none) end @@ -447,3 +440,11 @@ function generate_documentation_problems(; return problems_pages end + +function get_left_margin(problem) + return if problem == :beam + "5mm" + else + "20mm" + end +end \ No newline at end of file diff --git a/docs/src/assets/Manifest.toml b/docs/src/assets/Manifest.toml index 8cd9666c..80649039 100644 --- a/docs/src/assets/Manifest.toml +++ b/docs/src/assets/Manifest.toml @@ -2,7 +2,7 @@ julia_version = "1.11.6" manifest_format = "2.0" -project_hash = "e4420c03c5c208c963d869daa22ca1e9ba918652" +project_hash = "a19476c10cc83ea3d3a7dbb184d905b818a8ca62" [[deps.ADNLPModels]] deps = ["ADTypes", "ForwardDiff", "LinearAlgebra", "NLPModels", "Requires", "ReverseDiff", "SparseArrays", "SparseConnectivityTracer", "SparseMatrixColorings"] diff --git a/docs/src/assets/Project.toml b/docs/src/assets/Project.toml index 95e11132..622a5962 100644 --- a/docs/src/assets/Project.toml +++ b/docs/src/assets/Project.toml @@ -12,6 +12,7 @@ NLPModelsIpopt = "f4238b75-b362-5c4c-b852-0801c9a21d71" OptimalControl = "5f98b655-cc9a-415a-b60e-744165666948" OptimalControlProblems = "59046045-fb9c-4c23-964f-ff0a25704f96" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +Printf = "de0858da-6303-5e67-8744-51eddeeeb8d7" TOML = "fa267f1f-6049-4f14-aa54-33bafae1ed76" [compat] @@ -27,4 +28,5 @@ NLPModels = "0.21" NLPModelsIpopt = "0.10" OptimalControl = "1" Plots = "1" +Printf = "1" TOML = "1" diff --git a/docs/src/problems-introduction.md b/docs/src/problems-introduction.md index 81edcd65..548a8817 100644 --- a/docs/src/problems-introduction.md +++ b/docs/src/problems-introduction.md @@ -207,7 +207,6 @@ We detail below the characteristics of the optimal control problems and their as ```@raw html -
``` For each optimal control problem, we provide the dimensions of the state, control, and variable. We also specify the type of objective function (Mayer, Lagrange, or Bolza), whether the final time is free or fixed, and whether there are constraints on the state (`x`), control (`u`), variable (`v`), path (`c`—from *chemin* in French, since `p` is reserved for the costate), or boundary (`b`). diff --git a/ext/Descriptions/beam.md b/ext/Descriptions/beam.md index fa0afde4..f9667af2 100644 --- a/ext/Descriptions/beam.md +++ b/ext/Descriptions/beam.md @@ -1,9 +1,11 @@ -The *beam problem* is a classical benchmark in constrained optimal control. It originates from the Euler–Bernoulli beam model (see Bryson et al. 1963). The system describes the deflection of a clamped flexible beam under an external force. Both the state trajectory $x(\cdot)$ and the control $u(\cdot)$ are decision variables. The aim is to minimise the control effort subject to boundary conditions and a state inequality constraint on the displacement. +The *(clamped) beam problem* is a classical benchmark in constrained optimal control. It originates from the Euler–Bernoulli beam model (see Bryson et al. 1963). The system describes the deflection of a clamped flexible beam under an external force. Both the state trajectory $x(\cdot)$ and the control $u(\cdot)$ are decision variables. The aim is to minimise the control effort subject to boundary conditions and a state inequality constraint on the displacement. + +### Mathematical formulation The problem can be written as ```math -\min_{x,\,u} J(x,u) = \int_0^1 u(t)^2 \, dt +\min_{x,\,u} J(x,u) = \int_0^1 u^2(t) \, \mathrm{d}t ``` subject to the dynamics @@ -18,9 +20,8 @@ x_1(0) = 0, \quad x_1(1) = 0, \qquad x_2(0) = 1, \quad x_2(1) = -1, ``` -and the constraints +and the state constraint ```math --10 \le u(t) \le 5, \qquad 0 \le x_1(t) \le a, ``` @@ -28,11 +29,37 @@ where $a$ denotes the maximal admissible deflection (e.g. $a=0.1$ in the current ### Qualitative behaviour -- If $a \geq 1/4$, the inequality constraint is inactive and the solution is $x(t) = t(1-t)$. -- If $a \in [1/6, 1/4]$, there is a single touch point at $t=1/2$. -- If $a < 1/6$, a boundary arc appears, with the constraint active on an interval. +This problem features a second-order state constraint. In particular, the control can be recovered by differentiating the constraint twice: + +```math +\dot{x}_1(t) = x_2(t), \quad \ddot{x}_1(t) = u(t) +``` + +Along a **boundary arc**, where $x_1$ remains constant, both the first and second derivatives of $x_1$ vanish. As a result, the control along such an arc is zero: + +```math +u(t) = 0 +``` + +The qualitative behaviour of the optimal trajectory depends on the value of the parameter $a$: + +- **Case 1: $a \ge 1/4$**. In this regime, the inequality constraint is **inactive**, meaning it does not influence the trajectory. The optimal solution is simply + +```math +x(t) = t(1-t) +``` + +which is the unconstrained trajectory. + +- **Case 2: $1/6 \le a \le 1/4$**. Here, the trajectory **touches the boundary exactly once**, at $t = 1/2$. There is no interval where the constraint is active, but the single touch reflects the influence of the state inequality. + +- **Case 3: $a < 1/6$**. For smaller values of $a$, a **boundary arc appears**, meaning the state constraint becomes active over a finite interval. Along this interval, the control is zero: + +```math +u(t) = 0. +``` -These features illustrate the role of state inequality constraints in shaping optimal trajectories and controls. +These three cases clearly illustrate how **state inequality constraints** shape both the form of the optimal trajectories and the corresponding control actions. In particular, they show that second-order state constraints can either have no effect, a single-touch effect, or influence a whole interval of the trajectory. ### Characteristics @@ -44,4 +71,4 @@ These features illustrate the role of state inequality constraints in shaping op ### References - Bryson, A.E., Denham, W.F., & Dreyfus, S.E. (1963). *Optimal programming problems with inequality constraints I: necessary conditions for extremal solutions*. AIAA Journal. -- BOCOP examples: Clamped Beam problem. [Examples-BOCOP.pdf](https://project.inria.fr/bocop/files/2017/05/Examples-BOCOP.pdf) +- BOCOP examples: Clamped Beam problem. [Examples-BOCOP.pdf](https://project.inria.fr/bocop/files/2017/05/Examples-BOCOP.pdf). diff --git a/ext/Descriptions/bioreactor.md b/ext/Descriptions/bioreactor.md index 92290e35..c0579bdb 100644 --- a/ext/Descriptions/bioreactor.md +++ b/ext/Descriptions/bioreactor.md @@ -4,7 +4,7 @@ The control variable $u(t)$ represents the input flow rate between the two units The dynamics include algal growth driven by light, substrate consumption, and biomass evolution. The aim is to maximise methane production over a fixed time horizon under biological and operational constraints. -### Problem formulation +### Mathematical formulation We minimise diff --git a/ext/JuMPModels/ducted_fan.jl b/ext/JuMPModels/ducted_fan.jl index 9fbc16c1..3208486d 100644 --- a/ext/JuMPModels/ducted_fan.jl +++ b/ext/JuMPModels/ducted_fan.jl @@ -51,7 +51,7 @@ function OptimalControlProblems.ducted_fan( @variable(model, vα[0:N], start = 0.1) @variable(model, -5 <= u₁[0:N] <= 5, start = 0.1) # [N] @variable(model, 0 <= u₂[0:N] <= 17, start = 1) # [N] - @variable(model, 0.1 <= tf, start = 1) + @variable(model, 0.1 <= tf, start = 1.5) # Boundary constraints @constraints( diff --git a/ext/OptimalControlModels/ducted_fan.jl b/ext/OptimalControlModels/ducted_fan.jl index 81bebc22..ba077e2e 100644 --- a/ext/OptimalControlModels/ducted_fan.jl +++ b/ext/OptimalControlModels/ducted_fan.jl @@ -95,7 +95,7 @@ function OptimalControlProblems.ducted_fan( # initial guess xinit = [0.1, 0.1, -0.1, 0.1, 0.1, 0.1] # [x₁, v₁, x₂, v₂, α, vα] uinit = [0.1, 1] # [u₁, u₂] - varinit = [1] # [tf] + varinit = [1.5] # [tf] init = (state=xinit, control=uinit, variable=varinit) # discretise the optimal control problem diff --git a/test/Project.toml b/test/Project.toml index 89cf12fd..2e076448 100644 --- a/test/Project.toml +++ b/test/Project.toml @@ -5,6 +5,7 @@ CTDirect = "790bbbee-bee9-49ee-8912-a9de031322d5" Interpolations = "a98d9a8b-a2ab-59e6-89dd-64a1c18fca59" Ipopt = "b6b21f68-93f8-5de0-b562-5493be1d77c9" JuMP = "4076af6c-e467-56ae-b986-b466b2749572" +NLPModels = "a4795742-8479-5a88-8948-cc11e1c8c1a6" NLPModelsIpopt = "f4238b75-b362-5c4c-b852-0801c9a21d71" OptimalControl = "5f98b655-cc9a-415a-b60e-744165666948" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" @@ -18,6 +19,7 @@ CTDirect = "0.16" Interpolations = "0.15, 0.16" Ipopt = "1" JuMP = "1" +NLPModels = "0.21" NLPModelsIpopt = "0.10" OptimalControl = "1" Plots = "1" diff --git a/test/extras/beam.jl b/test/extras/beam.jl new file mode 100644 index 00000000..d2a80da0 --- /dev/null +++ b/test/extras/beam.jl @@ -0,0 +1,28 @@ +using Pkg +Pkg.activate("test") +using OptimalControl +using NLPModelsIpopt +using Plots + +function Beam(a) + + tf = 1 + + ocp = @def begin + t ∈ [0, tf], time + x ∈ R², state + u ∈ R, control + x(0) == [0, 1] + x(tf) == [0, -1] + ẋ(t) == [x₂(t), u(t)] + 0 ≤ x₁(t) ≤ a + ∫(u(t)^2) → min + end + + return ocp + +end + +plot(solve(Beam(1/2))) # state constraint inactive +plot(solve(Beam(1/5))) # state constraint active: touch point +plot(solve(Beam(1/8))) # state constraint active: arc \ No newline at end of file diff --git a/test/figures/init/beam.pdf b/test/figures/init/beam.pdf index 4631f547ed057b5f0a8aaf324f273ef98790baaa..2280a2a2bce60ede941701e46d0362c76db5e789 100644 GIT binary patch literal 228976 zcmeFa-L7Uya-MfyS5YSdBm%D4-_QOjgAgE-CN0T=XpI3IAb2q1G>1cU&saUp5p|UY zbPM){_<7tu7lu6iq1W=2FtM!p%DYkmLXFaGq;{)fw_`-gw=|NcM! z_kaIz`S|ql*Z=wNKK$_?fB4CteEr4GzWMdnAOGc_|NEc*tAfA$^{@W)XWxAB@n8Pw 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