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\ No newline at end of file From 6185b0167ef11099b101a0935882d3011ef8c0a9 Mon Sep 17 00:00:00 2001 From: docldai-22 Date: Thu, 4 Sep 2025 08:36:36 -0400 Subject: [PATCH 5/5] Add files via upload --- class03/notebook.jl | 3436 +++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 3436 insertions(+) create mode 100644 class03/notebook.jl diff --git a/class03/notebook.jl b/class03/notebook.jl new file mode 100644 index 0000000..2720b84 --- /dev/null +++ b/class03/notebook.jl @@ -0,0 +1,3436 @@ +### A Pluto.jl notebook ### +# v0.20.17 + +using Markdown +using InteractiveUtils + +# ╔═╡ d12bda70-894b-11f0-2944-ddded1b41f2c +begin + using LinearAlgebra # I(n), Diagonal + using ControlSystems # ss, care, c2d, bode, etc. + using OrdinaryDiffEq # CT simulations + using Plots # plotting + using OSQP, SparseArrays # QP/MPC + using Random # for LQG noise simulation +end + +# ╔═╡ 6e282c8b-54e8-4e0a-90a7-25967b8d14f7 +# Topic 2 case1: Single Shooting for double integrator (minimum energy, fixed terminal state) +# Dynamics: ẋ1 = x2, ẋ2 = u +# Cost: J = 1/2 ∫ u(t)^2 dt (PMP ⇒ u(t) = c1*t - c2) +# Boundary: x(0) = [x10, x20], x(T) = [0, 0] +# Unknowns (shooting params): c = [c1, c2] + +let + using DifferentialEquations, NLsolve + + # ----- Problem setup ----- + T = 2.0 + x10 = 1.0 + x20 = 0.0 + x0 = [x10, x20] # initial state + xT = [0.0, 0.0] # desired terminal state + + # ----- Forward rollout given c = [c1, c2] ----- + # Control u(t) = c1*t - c2 (from PMP optimality) + function rollout(c; save_traj::Bool=false) + c1, c2 = c + u(t) = c1*t - c2 + f!(dx, x, p, t) = begin + dx[1] = x[2] + dx[2] = u(t) + end + prob = ODEProblem(f!, x0, (0.0, T)) + if save_traj + sol = solve(prob; reltol=1e-10, abstol=1e-10, saveat=range(0, T; length=401)) + return sol, [u(t) for t in sol.t] + else + sol = solve(prob; reltol=1e-10, abstol=1e-10, save_everystep=false) + return sol(T) # terminal state only + end + end + + # ----- Residual for NLsolve: F(c) = x(T; c) - xT ----- + function residual!(F, c) + xT_pred = rollout(c) + F[1] = xT_pred[1] - xT[1] + F[2] = xT_pred[2] - xT[2] + end + + # Initial guess for c = [c1, c2] (you can tweak if needed) + c0 = [0.0, 0.0] + + # Solve F(c) = 0 by Newton/Broyden (NLsolve picks a method automatically) + sol = nlsolve(residual!, c0; xtol=1e-12, ftol=1e-12) + @assert sol.f_converged "Shooting did not converge. Try a different c0." + ĉ = sol.zero + println("Optimal shooting params (c1, c2) = ", ĉ) + + # ----- Reconstruct full optimal trajectories ----- + soltraj, utraj = rollout(ĉ; save_traj=true) + ts = soltraj.t + xs = hcat(soltraj.u...)' # size (N, 2) + x1s = xs[:,1]; x2s = xs[:,2] + + # ----- Checks ----- + println("Boundary check: x(0) = ", xs[1,:], " target = ", x0) + println("Boundary check: x(T) = ", xs[end,:], " target = ", xT) + # Residual norm at solution (should be ~0) + Ffin = zeros(2); residual!(Ffin, ĉ) + println("Terminal residual ||F|| = ", norm(Ffin)) + + # ----- (Optional) plots ----- + if @isdefined(Plots) + p1 = plot(ts, x1s, lw=2, label="x1(t)", xlabel="t", title="State") + plot!(ts, x2s, lw=2, label="x2(t)") + + p2 = plot(ts, utraj, lw=2, label="u(t)", xlabel="t", title="Control") + + plot(p1, p2, layout=(1,2), size=(900,400)) + + end +end + + +# ╔═╡ 59f5b6c7-d211-42c0-b662-377e7ca6a3a5 +# Topic 2 case2: Multiple Shooting for Double Integrator +# Dynamics: ẋ1 = x2, ẋ2 = u +# Cost: J = 1/2 ∫ u^2 dt +# Boundary: x(0) = [1,0], x(T) = [0,0] + +let + using JuMP, Ipopt + + # ----- problem setup ----- + T = 2.0 + M = 20 # number of shooting intervals + Δt = T/M + x0 = [1.0, 0.0] + xT = [0.0, 0.0] + umax = 5.0 # control bound + + # ----- build NLP ----- + model = Model(Ipopt.Optimizer) + set_silent(model) + + # decision variables + @variables(model, begin + x[1:2, 0:M] # state at each node + -umax <= u[0:M-1] <= umax # piecewise constant control + end) + + # initial condition + @constraint(model, x[1,0] == x0[1]) + @constraint(model, x[2,0] == x0[2]) + + # dynamics continuity (Euler) + for k in 0:M-1 + @constraint(model, x[1,k+1] == x[1,k] + Δt * x[2,k]) + @constraint(model, x[2,k+1] == x[2,k] + Δt * u[k]) + end + + # terminal condition + @constraint(model, x[1,M] == xT[1]) + @constraint(model, x[2,M] == xT[2]) + + # objective: minimize sum of u^2 + @objective(model, Min, sum(0.5 * Δt * u[k]^2 for k=0:M-1)) + + # ----- solve ----- + optimize!(model) + println("Status: ", termination_status(model)) + println("Objective value: ", objective_value(model)) + + # ----- extract solution ----- + xs = [value.(x[:,k]) for k in 0:M] + us = [value(u[k]) for k in 0:M-1] + ts = collect(0:Δt:T) + + # check boundary + println("x(0) = ", xs[1], " target = ", x0) + println("x(T) = ", xs[end], " target = ", xT) + + # ----- plot results ----- + x1s = [xi[1] for xi in xs] + x2s = [xi[2] for xi in xs] + p1 = plot(ts, x1s, lw=2, label="x1(t)", xlabel="t", title="State") + plot!(ts, x2s, lw=2, label="x2(t)") + p2 = plot(ts[1:end-1], us, lw=2, label="u(t)", xlabel="t", title="Control") + plot(p1, p2, layout=(1,2), size=(900,400)) +end + + +# ╔═╡ c925a661-0a41-4fa2-a772-e7217e44a818 +# Topic 1 case: analytic example with state+control running cost + +let + + # ----- Problem parameters ----- + T = 2.5 # horizon + x0 = 1.5 # initial state + q = 1.0 # state weight (>0) + r = 0.2 # control weight (>0) + + # ----- Analytic PMP solution ----- + α = sqrt(q/r) + + # x(t) = A cosh(α t) + B sinh(α t) + A = x0 + B = -x0 * coth(α*T) # from x(T)=0 + + x_star(t) = A*cosh(α*t) + B*sinh(α*t) + xdot_star(t) = α*(A*sinh(α*t) + B*cosh(α*t)) + + u_star(t) = xdot_star(t) # because ẋ = u + λ_star(t) = -r * xdot_star(t) # because u = -λ/r + + # ----- Verify PMP conditions on a grid ----- + ts = range(0, T; length=401) + + # residual trackers (max norms over the grid) + max_res_state = 0.0 # |ẋ - ∂H/∂λ| + max_res_costate = 0.0 # |λ̇ + ∂H/∂x| + max_res_opt = 0.0 # |∂H/∂u| + + for i in 2:length(ts) + t_prev, t = ts[i-1], ts[i] + dt = t - t_prev + + # Finite differences for derivatives (for numerical verification) + xdot_fd = (x_star(t) - x_star(t_prev)) / dt + lambdadot_fd = (λ_star(t) - λ_star(t_prev)) / dt + + # Hamiltonian partials + dH_dλ = u_star(t) # ∂H/∂λ = u + dH_dx = q * x_star(t) # ∂H/∂x = q x + dH_du = r * u_star(t) + λ_star(t) # optimality residual + + max_res_state = max(max_res_state, abs(xdot_fd - dH_dλ)) + max_res_costate = max(max_res_costate, abs(lambdadot_fd + dH_dx)) + max_res_opt = max(max_res_opt, abs(dH_du)) + end + + println("PMP residuals over grid (should be near zero):") + println(" state eqn (ẋ - ∂H/∂λ): ", max_res_state) + println(" costate eqn (λ̇ + ∂H/∂x): ", max_res_costate) + println(" optimality (∂H/∂u): ", max_res_opt) + + # Boundary checks + println("Boundary check: x(0) = ", x_star(0.0), " (target ", x0, ")") + println("Boundary check: x(T) = ", x_star(T), " (target 0.0)") + + # ----- Optional plots ----- + if @isdefined(Plots) + p1 = plot(ts, [x_star.(ts), u_star.(ts), λ_star.(ts)], + label = ["x(t)" "u(t)" "λ(t)"], xlabel="t", lw=2) + display(p1) + end + + # Report a few values + for t in (0.0, T/2, T) + println("t=$(round(t,digits=2)) x=$(x_star(t)) u=$(u_star(t)) λ=$(λ_star(t))") + end +end + + +# ╔═╡ 8b8d8f05-4de5-48bd-8661-787b86014a87 +# Topic 2 case 3: Multiple Shooting for the nonlinear pendulum (Euler integration) +# θ̇ = ω, ω̇ = u - sin(θ) +# Cost: +# J = Σ 1/2 Δt (qθ θ_k^2 + qω ω_k^2 + r u_k^2) + 1/2 (Qfθ θ_M^2 + Qfω ω_M^2) + +let + + # ----- Horizon and grid ----- + T = 2.5 # horizon + M = 60 # number of shooting intervals (nodes = M+1) + Δt = T / M + ts = collect(0:Δt:T) + + # ----- Initial and (soft) terminal target ----- + θ0, ω0 = pi, 0.0 # start downward, zero speed + # You can keep terminal as soft via Qf*, or add hard constraints x_M = 0. + + # ----- Weights ----- + qθ, qω, r = 1.0, 0.1, 0.05 + Qfθ, Qfω = 60.0, 12.0 # terminal weights (bigger → closer to upright rest) + + # ----- Control bound (optional but useful) ----- + umax = 2.0 + + # ----- Build NLP ----- + model = Model(Ipopt.Optimizer) + set_silent(model) # comment this to see Ipopt logs + + # Decision variables: + # x[:,k] = [θ_k; ω_k] at node k, u[k] is piecewise constant over [t_k, t_{k+1}) + @variables(model, begin + x[1:2, 0:M] + -umax <= u[0:M-1] <= umax + end) + + # Initial condition + @constraint(model, x[1,0] == θ0) + @constraint(model, x[2,0] == ω0) + + # Multiple-shooting continuity (Euler) + # θ_{k+1} = θ_k + Δt * ω_k + # ω_{k+1} = ω_k + Δt * (u_k - sin(θ_k)) + for k in 0:M-1 + @NLconstraint(model, x[1,k+1] == x[1,k] + Δt * x[2,k]) + @NLconstraint(model, x[2,k+1] == x[2,k] + Δt * (u[k] - sin(x[1,k]))) + end + + # (Optional) Hard terminal constraints (uncomment to force exact upright rest): + # @constraint(model, x[1,M] == 0.0) + # @constraint(model, x[2,M] == 0.0) + + # Objective: running cost + terminal cost + @NLobjective(model, Min, + sum(0.5 * Δt * (qθ * x[1,k]^2 + qω * x[2,k]^2 + r * u[k]^2) for k=0:M-1) + + 0.5 * (Qfθ * x[1,M]^2 + Qfω * x[2,M]^2) + ) + + # ----- Solve ----- + optimize!(model) + println("Status: ", termination_status(model)) + println("Objective value: ", objective_value(model)) + + # ----- Extract solution ----- + θ = [value(x[1,k]) for k in 0:M] + ω = [value(x[2,k]) for k in 0:M] + ustar = [value(u[k]) for k in 0:M-1] + + println("θ(0) = ", θ[1], " θ(T) = ", θ[end]) + println("ω(0) = ", ω[1], " ω(T) = ", ω[end]) + println("|u|_∞ = ", maximum(abs.(ustar))) + + # ----- Plot ----- + if @isdefined(Plots) + p1 = plot(ts, θ, lw=2, label="θ(t)", xlabel="t", title="Pendulum State") + plot!(ts, ω, lw=2, label="ω(t)") + p2 = plot(ts[1:end-1], ustar, lw=2, label="u(t)", xlabel="t", title="Control") + plot(p1, p2, layout=(1,2), size=(900,400)) + end +end + + +# ╔═╡ 78835928-e3c8-4817-97ac-d0ac1ba50c63 +begin + # 4-mass chain: x = [q1,q2,q3,q4, q̇1,q̇2,q̇3,q̇4] + # u acts on mass 1. Outputs (for visualization): q1, q4. + function chain4_msmd(; m = ones(4), + k = [800.0, 600.0, 600.0, 900.0, 700.0], # k_wallL, k12, k23, k34, k_wallR + c = [12.0, 10.0, 10.0, 15.0, 12.0], # c_wallL, c12, c23, c34, c_wallR + u_on = 1) + + n = 4 + M = Diagonal(m) + Kq = zeros(n,n) + Cq = zeros(n,n) + + # wall left + Kq[1,1] += k[1]; Cq[1,1] += c[1] + # internal springs/dampers + Kq[1,1] += k[2]; Kq[1,2] -= k[2]; Kq[2,1] -= k[2]; Kq[2,2] += k[2] + Cq[1,1] += c[2]; Cq[1,2] -= c[2]; Cq[2,1] -= c[2]; Cq[2,2] += c[2] + + Kq[2,2] += k[3]; Kq[2,3] -= k[3]; Kq[3,2] -= k[3]; Kq[3,3] += k[3] + Cq[2,2] += c[3]; Cq[2,3] -= c[3]; Cq[3,2] -= c[3]; Cq[3,3] += c[3] + + Kq[3,3] += k[4]; Kq[3,4] -= k[4]; Kq[4,3] -= k[4]; Kq[4,4] += k[4] + Cq[3,3] += c[4]; Cq[3,4] -= c[4]; Cq[4,3] -= c[4]; Cq[4,4] += c[4] + + # wall right + Kq[4,4] += k[5]; Cq[4,4] += c[5] + + Z = zeros(n,n) + Eye = I(n) + A = [ Z Eye; + -M\Kq -M\Cq ] + + Bq = zeros(n,1); Bq[u_on] = 1.0 + B = [ zeros(n,1); + M\Bq ] + + Cy = zeros(2,2n); Cy[1,1] = 1.0; Cy[2,4] = 1.0 # q1, q4 + D = zeros(2,1) + + return ss(A,B,Cy,D) # continuous-time, SISO + end + + # Build the plant globally so all later cells can use it + sys = chain4_msmd() + A,B,Cy,D = sys.A, sys.B, sys.C, sys.D + @show size(A), size(B), size(Cy) + +end + +# ╔═╡ d849f412-8b3c-4ca9-a9dd-80b70cae8dbc +begin + # ---- Cell 3 (fixed) ---- + # Build G,H for z = [ Wq*S*q ; Wv*v ], H=0 → N = G' H = 0 + function build_G_weights(; Wq = Diagonal([4.0, 4.0, 4.0, 8.0]), + Wv = 0.2I(4)) + S = [ 1 -1 0 0; + 0 1 -1 0; + 0 0 1 -1; + 0 0 0 1 ] + Gq = Wq * S + Gv = Wv + G = [ Gq zeros(4,4); + zeros(4,4) Gv ] + H = zeros(size(G,1), 1) # no direct u in z + return G, H + end + + G, H = build_G_weights() + Q = G' * G + R = Matrix(0.5I(1)) + N = zeros(size(A,1), size(B,2)) + + + P = care(A, B, Q, R, N)[1] + K = lqr(A, B, Q, R, N) + + @show size(A), size(B), size(K) + + Acl = A - B * K + @show real.(eigvals(Acl)) + + # CT 仿真 + x0 = [0.0, 0.0, 0.0, 0.05, 0.0, 0.0, 0.0, 0.0] + + function cl_lqr!(dx, x, p, t) + u = (K * x)[1] + dx[:] = A * x - B * u + end + + Tsim = 8.0 + sol = solve(ODEProblem(cl_lqr!, x0, (0.0,Tsim)), Tsit5(); abstol=1e-9, reltol=1e-9) + + t = sol.t + X = reduce(hcat, sol.u) + uL = [(K * X[:,i])[1] for i in eachindex(t)] .* (-1) + + plt1 = plot(t, X[4,:], lw=2, xlabel="t [s]", ylabel="q4 [m]", title="CT LQR: end mass deflection") + plt2 = plot(t, uL, lw=2, xlabel="t [s]", ylabel="u [N]", title="CT LQR: control") + plot(plt1, plt2, layout=(2,1), size=(800,600)) + +end + +# ╔═╡ 33e04661-b759-40f6-815a-5b0c60a708be +begin + # —— Discretize —— + Ts = 0.02 + sysd = c2d(sys, Ts) + Ad, Bd, Cd, Dd = sysd.A, sysd.B, sysd.C, sysd.D + n, m = size(Ad,1), size(Bd,2) # n=8, m=1 + + + Qd = Matrix(Q) # 8×8 + Rd = Matrix(R) # 1×1 + Nd = zeros(n, m) # 8×1 + Qf = 50.0I(n) + + + # X = Acal*x0 + Bcal*U + function stack_dyn(Ad, Bd, Np) + n, m = size(Ad,1), size(Bd,2) + Acal = [Ad^k for k in 1:Np] |> x -> reduce(vcat, x) + Bcal = spzeros(n*Np, m*Np) + for i in 1:Np, j in 1:i + Bcal[(n*(i-1)+1):(n*i), (m*(j-1)+1):(m*j)] = Ad^(i-j) * Bd + end + return Acal, Bcal + end + + + function make_Qblk(Qd, Qf, Np) + n = size(Qd,1) + Qblk = spzeros(n*Np, n*Np) + for k in 1:Np + Qblk[(n*(k-1)+1):(n*k), (n*(k-1)+1):(n*k)] = sparse(k < Np ? Qd : Qf) + end + return Qblk + end + + function make_Rblk(Rd, Np) + m = size(Rd,1) + Rblk = spzeros(m*Np, m*Np) + for k in 1:Np + Rblk[(m*(k-1)+1):(m*k), (m*(k-1)+1):(m*k)] = sparse(Rd) + end + return Rblk + end + + function make_Nblk(Nd, Np) + n, m = size(Nd) + Nblk = spzeros(n*Np, m*Np) + for k in 1:(Np-1) + Nblk[(n*(k-1)+1):(n*k), (m*(k-1)+1):(m*k)] = sparse(Nd) + end + return Nblk + end + + function condensed_qp_mats(Ad,Bd,Qd,Rd,Nd,Qf,Np) + Acal, Bcal = stack_dyn(Ad,Bd,Np) + Qblk = make_Qblk(Qd,Qf,Np) # 640×640 + Rblk = make_Rblk(Rd,Np) # 80×80 + Nblk = make_Nblk(Nd,Np) # 640×80 + + # H 和 f(x0) + H = Bcal' * Qblk * Bcal + (Bcal' * Nblk + Nblk' * Bcal) + Rblk + f_of = x0 -> (Bcal' * Qblk * Acal + Nblk' * Acal) * x0 + return H, f_of, Acal, Bcal, Qblk, Rblk, Nblk + end + + Np = 80 + Hqp, f_of, Acal, Bcal, Qblk, Rblk, Nblk = condensed_qp_mats(Ad,Bd,Qd,Rd,Nd,Qf,Np) + + # (640,80), (640,640), (80,80), (640,80)) + @show size(Acal), size(Qblk), size(Rblk), size(Nblk) + # Hessian (m*Np, m*Np) = (80,80) + @show size(Hqp) + +end + +# ╔═╡ 04291184-92d8-4c52-b4e7-e2d487e5d87b +begin + # ==== Cell 5: Condensed constraints (|u| and |q4|) ==== + + # |u_k| ≤ umax;|q4_k| ≤ qmax + umax = 15.0 + qmax = 0.06 + q4row = zeros(1, n); q4row[4] = 1.0 + + """ + condensed_constraints(Bcal, Acal, n, m, Np; umax, qrow, qmax) + + """ + function condensed_constraints(Bcal, Acal, n, m, Np; umax=15.0, qrow=zeros(1,n), qmax=0.06) + # -umax ≤ u ≤ umax → [ I; -I ] U ≤ [umax; umax] + GUu = [ I(m*Np); -I(m*Np) ] # (2mNp) × (mNp) + wu = vcat( fill(umax, m*Np), fill(umax, m*Np) ) + Su = zeros(size(GUu,1), n) # (2mNp) × n + + # |qrow * x_k| ≤ qmax (k = 1..Np) + rows = [ (n*(k-1)+1):(n*k) for k in 1:Np ] + G_list = Matrix{Float64}[]; w_list = Vector{Float64}[]; S_list = Matrix{Float64}[] + for k in 1:Np + Ak = Acal[rows[k], :] + Bk = Bcal[rows[k], :] + # +qmax + push!(G_list, qrow * Bk) + push!(w_list, [qmax]) + push!(S_list, qrow * Ak) + # -qmax + push!(G_list, -qrow * Bk) + push!(w_list, [qmax]) + push!(S_list, -qrow * Ak) + end + Gs = reduce(vcat, G_list) # (2Np) × (mNp) + ws = reduce(vcat, w_list) # (2Np) + Ss = reduce(vcat, S_list) # (2Np) × n + + # + G = vcat(GUu, Gs) # ((2mNp+2Np) × (mNp)) + w = vcat(wu, ws) # ((2mNp+2Np)) + S = vcat(Su, Ss) # ((2mNp+2Np) × n) + return G, w, S + end + + Gcon, wcon, Scon = condensed_constraints(Bcal, Acal, n, m, Np; umax=umax, qrow=q4row, qmax=qmax) + + # (2mNp+2Np, mNp)、(2mNp+2Np)、(2mNp+2Np, n)) + @show size(Gcon), size(wcon), size(Scon); + +end + +# ╔═╡ cf285f59-d4b2-4935-959b-c8c9c2669570 +function mpc_sim_condensed(Ad,Bd; H, f_of, G, w, S, Ts, Nsteps, x0) + n = size(Ad,1) + xk = copy(x0) + + tlog = Vector{Float64}(undef, Nsteps) + Ulog = Vector{Float64}(undef, Nsteps) + q4log = Vector{Float64}(undef, Nsteps) + Xlog = Matrix{Float64}(undef, n, Nsteps) + + for k in 1:Nsteps + model = Model(OSQP.Optimizer); set_silent(model) + @variable(model, U[1:size(H,1)]) # U ∈ ℝ^{m*Np} + + + Hpsd = sparse(0.5*(H + H') + 1e-9I(size(H,1))) + f = f_of(xk) # Vector{Float64} + + + @objective(model, Min, 0.5 * dot(U, Hpsd * U) + dot(f, U)) + + @constraint(model, G * U .<= w .+ S * xk) + + optimize!(model) + Ustar = value.(U) + u0 = Ustar[1] # receding horizon + xk = Ad * xk + Bd * u0 + + tlog[k] = (k-1)*Ts + Ulog[k] = u0 + q4log[k] = xk[4] + Xlog[:,k] = xk + end + return (; tlog, Ulog, q4log, Xlog) +end + +# ╔═╡ e84baea8-bd88-4e09-ac0d-83e26b9fa805 +begin + # ---- Run MPC and plot results ---- + Nsteps = Int(round(8.0 / Ts)) # simulate 8 seconds like CT case + mpc = mpc_sim_condensed(Ad, Bd; + H=Hqp, f_of=f_of, G=Gcon, w=wcon, S=Scon, + Ts=Ts, Nsteps=Nsteps, x0=x0) + + # State (tip deflection q4) with constraint lines + p1 = plot(mpc.tlog, mpc.q4log, lw=2, label="q4 (MPC)", + xlabel="t [s]", ylabel="q4 [m]", + title="MPC (QP, condensed): tip deflection") + hline!([ qmax], ls=:dash, label="qmax") + hline!([-qmax], ls=:dash, label="-qmax") + + # Control with input bounds + p2 = plot(mpc.tlog, mpc.Ulog, lw=2, label="u(t)", + xlabel="t [s]", 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