Lecture 12: Scripts

scripts/lecture_12/Project.toml

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+[deps]
+DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa"
+LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
+
+[compat]
+DifferentialEquations = "= 6.16.0"
+Plots = "= 1.10.3"
+julia = "1.5"

scripts/lecture_12/script.jl

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+using DifferentialEquations
+using LinearAlgebra
+using Plots
+
+struct Wave
+    f
+    g
+    c
+end
+
+# Exercise
+
+
+
+# Visualization
+
+function plot_wave(y, file_name; fps = 60, kwargs...)
+    anim = @animate for (i, y_row) in enumerate(eachrow(y))
+        plot(
+            y_row;
+            title = "t = $(i-1)Δt",
+            xlabel = "x",
+            ylabel = "y(t, x)",
+            legend = false,
+            linewidth = 2,
+            kwargs...
+        )
+    end
+    gif(anim, file_name; fps, show_msg = false)
+
+    return nothing
+end
+
+# Exercise
+
+
+
+
+
+
+
+
+
+
+
+
+
+# DiferentialEquations
+
+f(u,p,t) = 0.98*u
+
+u0 = 1.0
+tspan = (0.0, 1.0)
+
+prob = ODEProblem(f, u0, tspan)
+
+sol = solve(prob)
+
+plot(sol; label="")
+
+sol(0.8)
+
+# Exercise
+
+
+
+# Lorenz system
+
+function lorenz(u, p, t)
+    σ, ρ, β = p
+    x_t = σ*(u[2]-u[1])
+    y_t = u[1]*(ρ-u[3]) - u[2]
+    z_t = u[1]*u[2] - β*u[3]
+    return [x_t; y_t; z_t]
+end
+
+u0 = [1.0; 0.0; 0.0]
+p = [10; 28; 8/3] 
+
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz, u0, tspan, p)
+
+sol = solve(prob)
+
+plot(sol)
+
+plt1 = plot(sol, vars=(1,2,3), label="")
+
+plot(sol, vars=(1,2,3), denseplot=false; label="")
+
+traj = hcat(sol.u...)
+plot(traj[1,:], traj[2,:], traj[3,:]; label="")
+
+# Exercise
+
+
+
+# Solution comparison
+
+hcat(sol(tspan[2]), sol0(tspan[2]))
+
+
+
+
+
+
+
+
+
+
+
+# Bisection setup
+
+function bisection(f, a, b; tol=1e-6)
+    fa = f(a)
+    fb = f(b)
+    fa == 0 && return a
+    fb == 0 && return b
+    fa*fb > 0 && error("Wrong initial values for bisection")
+    while b-a > tol
+        c = (a+b)/2
+        fc = f(c)
+        fc == 0 && return c
+        if fa*fc > 0
+            a = c
+            fa = fc
+        else
+            b = c
+            fb = fc
+        end
+    end
+    return (a+b)/2
+end
+
+# PMSM structure
+
+struct PMSM{T<:Real}
+    ρ::T
+    ω::T
+    A::Matrix{T}
+    invA::Matrix{T}
+
+    function PMSM(ρ, ω)
+        A = -ρ*[1 0; 0 1] -ω*[0 -1; 1 0]
+        return new{eltype(A)}(ρ, ω, A, inv(A))
+    end
+end
+
+function expA(p::PMSM, t)
+    ρ, ω = p.ρ, p.ω
+    return exp(-ρ*t)*[cos(ω*t) sin(ω*t); -sin(ω*t) cos(ω*t)]
+end
+
+ρ = 0.1
+ω = 2
+x0 = [0;-0.5]
+q = [1;0]
+
+ps = PMSM(ρ, ω)
+
+# Exercise
+
+
+
+# Exercise
+
+
+
+# Plotting trajectory
+
+ts = 0:0.0001:10
+
+xs1 = trajectory_fin_diff(ps, x0, ts, q)
+xs2 = trajectory_exact(ps, x0, ts, q)
+
+plot(xs1[1,:], xs1[2,:], label="Finite differences")
+plot!(xs2[1,:], xs2[2,:], label="True value")
+
+# Exercise
+
+
+
+# Plotting trajectory
+
+function trajectory_control(p::PMSM, x0, ts, q, U_max, p0)
+    xs = zeros(length(x0), length(ts))
+
+    for (i, t) in enumerate(ts)
+        eAt  = expA(p, t)
+        emAt = expA(p, -t)
+        xs[:, i] = eAt*(x0 + p.invA * (I - emAt)*q + U_max/ρ*(exp(ρ*t) - 1)*p0)
+    end
+    return xs
+end
+
+p0 = ps.ρ/(U_max*(exp(ps.ρ*τ)-1))*(expA(ps, -τ)*x_t - x0 - ps.invA*(I-expA(ps, -τ))*q)
+p0 /= norm(p0)
+
+ts = range(0, τ; length=100)
+
+traj = trajectory_control(ps, x0, ts, q, U_max, p0)
+
+plot(traj[1,:], traj[2,:], label="Optimal trajectory")
+scatter!([x0[1]], [x0[2]], label="Starting point")
+scatter!([x_t[1]], [x_t[2]], label="Target point")
+
+# BONUS
+
+ts = 0:0.01:10
+
+plt = plot()
+for α = 0:π/4:2*π
+    trj = trajectory_control(ps, x0, ts, q, U_max, [sin(α); cos(α)])
+    plot!(plt, trj[1,:], trj[2,:], label="")
+end
+display(plt)
+
+
+
+

scripts/lecture_12/script_init.jl

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+using Pkg
+Pkg.activate(".")
+Pkg.instantiate()
+
+using DifferentialEquations
+using LinearAlgebra
+using Plots
+
+f(u,p,t) = 0.98*u
+
+u0 = 1.0
+tspan = (0.0, 1.0)
+
+prob = ODEProblem(f, u0, tspan)
+
+sol = solve(prob)
+
+plot(sol; label="")
+
+sol(0.8)
+
+function lorenz(u, p, t)
+    σ, ρ, β = p
+    x_t = σ*(u[2]-u[1])
+    y_t = u[1]*(ρ-u[3]) - u[2]
+    z_t = u[1]*u[2] - β*u[3]
+    return [x_t; y_t; z_t]
+end
+
+u0 = [1.0; 0.0; 0.0]
+p = [10; 28; 8/3] 
+
+tspan = (0.0, 100.0)
+prob = ODEProblem(lorenz, u0, tspan, p)