diff --git a/class07/Kalman_Filtering.jl b/class07/Kalman_Filtering.jl new file mode 100644 index 0000000..b3a1498 --- /dev/null +++ b/class07/Kalman_Filtering.jl @@ -0,0 +1,81 @@ +## Kalman Filtering +# Goal: Estimate the true state of a system when measurements are noisy. +# +# In many practical systems, we cannot directly observe the full state x_t. +# Instead, we have noisy measurements y_t. The Kalman filter provides the +# optimal linear estimate of the hidden state, assuming linear dynamics +# and Gaussian noise. + +using LinearAlgebra + +# The Kalman filter operates in two steps: prediction and update. + +## Prediction Step: +# Forecast the next state based on the previous estimate and the system model. +# x_pred = A * x_prev + B * u_prev +# - This uses the known dynamics of the system (A, B) and the previous control input u_prev +# P_pred = A * P_prev * A' + Q +# - The uncertainty (covariance) is propagated forward +# - Q represents process noise: uncertainty in how the system evolves +# - Larger P_pred means we are less confident in the prediction + +## Update Step: +# Correct the prediction using the latest measurement y +# K = P_pred * C' * inv(C * P_pred * C' + R) +# - The Kalman gain K determines how much we trust the measurement vs. prediction +# x_hat = x_pred + K * (y - C * x_pred) +# - Innovation (y - C*x_pred) adjusts our estimate based on new information +# P = (I - K * C) * P_pred +# - Covariance decreases after incorporating measurement, increasing confidence + +function kalman_filter(A, B, C, Q, R, y, u=nothing) + n = size(A,1) + N = length(y) + x_hat = zeros(n, N) # estimated states + P = zeros(n,n) # covariance + I = Matrix(I, n, n) + + # Handle optional control input + u = isnothing(u) ? zeros(size(B,2), N) : u + + for k in 1:N + # Prediction + if k == 1 + x_pred = zeros(n) # initial guess for first step + P_pred = P + Q + else + x_pred = A * x_hat[:,k-1] + B * u[:,k-1] + P_pred = A * P * A' + Q + end + + # Update + K = P_pred * C' * inv(C * P_pred * C' + R) + x_hat[:,k] = x_pred + K * (y[k] - C * x_pred) + P = (I - K * C) * P_pred + end + + return x_hat +end + +## Example usage: +# A simple demonstration with a small system. This shows how the filter +# reconstructs the true state from noisy measurements. + +if abspath(PROGRAM_FILE) == @__FILE__ + # Define system matrices (example: 2-state thermal system) + A = [1.0 1.0; 0.0 1.0] # state transition + B = [0.0; 1.0] # control input + C = [1.0 0.0] # measurement matrix + Q = 0.01 * I(2) # process noise covariance + R = 0.1 # measurement noise covariance + + # Simulated noisy measurements + true_states = [20.0, 21.0, 23.0, 22.0] + y = true_states .+ randn(length(true_states)) * sqrt(R) + + # Run Kalman filter + x_hat = kalman_filter(A, B, C, Q, R, y) + + println("Noisy measurements: ", y) + println("Estimated states: ", x_hat) +end diff --git a/class07/LQG_Control.jl b/class07/LQG_Control.jl new file mode 100644 index 0000000..d3f0634 --- /dev/null +++ b/class07/LQG_Control.jl @@ -0,0 +1,84 @@ +# Linear Quadratic Gaussian (LQG) Control +# +# LQG control is about designing an optimal controller for linear systems +# when there is uncertainty in both the system dynamics and the measurements. +# +# System setup: +# x_{t+1} = A * x_t + B * u_t + w_t +# y_t = C * x_t + v_t +# Here, w_t and v_t are Gaussian process and measurement noise. +# +# The goal is to minimize a quadratic cost function: +# J = E[ sum(x_t' * Q * x_t + u_t' * R * u_t) ] +# which balances performance (state tracking) and control effort. + +using LinearAlgebra + +# Linear Quadratic Regulator (LQR) +# +# The LQR computes the optimal control law for a deterministic linear system +# when the full state is known. For continuous-time systems: +# dx/dt = A*x + B*u +# The quadratic cost is: +# J = ∫ (x'Qx + u'Ru) dt +# The optimal control law is: +# u = -K*x +# where K is obtained by solving the continuous-time Riccati equation: +# A'*P + P*A - P*B*R^-1*B'*P + Q = 0 +# +# Here, we implement a discrete-time version. + +function dlqr(A, B, Q, R) + P = copy(Q) + maxiter = 1000 + tol = 1e-8 + for i in 1:maxiter + P_new = A' * P * A - A' * P * B * inv(R + B' * P * B) * B' * P * A + Q + if norm(P_new - P) < tol + P = P_new + break + end + P = P_new + end + K = inv(R + B' * P * B) * B' * P * A + return K, P +end + +# Kalman Filter (State Estimation) +# +# In LQG, the controller does not know the true state x_t. +# Instead, it receives noisy measurements y_t and uses a Kalman filter +# to produce an optimal estimate x_hat_t. +# +# The filter updates the estimate according to: +# x_hat_{t+1} = A*x_hat_t + B*u_t + L*(y_t - C*x_hat_t) +# where L is the Kalman gain chosen to minimize estimation error variance. + +function kalman_filter(A, B, C, Q, R, y) + n = size(A, 1) + x_hat = zeros(n, length(y)) + P = zeros(n, n) + for t in 1:length(y) + # Prediction step + if t == 1 + x_pred = zeros(n) + P_pred = P + Q + else + x_pred = A * x_hat[:, t-1] + P_pred = A * P * A' + Q + end + # Update step + K = P_pred * C' * inv(C * P_pred * C' + R) + x_hat[:, t] = x_pred + K * (y[t] - C * x_pred) + P = (I - K * C) * P_pred + end + return x_hat +end + +# Separation Principle +# +# A key property of LQG control is the separation principle: +# The design of the controller (LQR) and the estimator (Kalman filter) +# can be done independently. Once K and L are computed, the control input is: +# u_t = -K * x_hat_t +# This allows the system to act as if the true state were known, using only the estimated state. diff --git a/class07/Manifest.toml b/class07/Manifest.toml new file mode 100644 index 0000000..761bdc9 --- /dev/null +++ b/class07/Manifest.toml @@ -0,0 +1,2696 @@ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.11.6" +manifest_format = "2.0" +project_hash = "69e6b5c3c6e24efc9e2652a8c49f641217ff2636" + +[[deps.ADTypes]] +git-tree-sha1 = "27cecae79e5cc9935255f90c53bb831cc3c870d7" +uuid = "47edcb42-4c32-4615-8424-f2b9edc5f35b" +version = "1.18.0" +weakdeps = ["ChainRulesCore", "ConstructionBase", "EnzymeCore"] + + [deps.ADTypes.extensions] + ADTypesChainRulesCoreExt = "ChainRulesCore" + ADTypesConstructionBaseExt = "ConstructionBase" + ADTypesEnzymeCoreExt = "EnzymeCore" + +[[deps.Accessors]] +deps = ["CompositionsBase", "ConstructionBase", "Dates", "InverseFunctions", 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"91a5bcdd-55d7-5caf-9e0b-520d859cae80" diff --git a/class07/Robust_Control.jl b/class07/Robust_Control.jl new file mode 100644 index 0000000..a433989 --- /dev/null +++ b/class07/Robust_Control.jl @@ -0,0 +1,76 @@ +## Robust Control +# Goal: Ensure stability and performance under model uncertainty and disturbances. +# +# In real-world systems, models are never perfect. Uncertainties in parameters, +# unmodeled dynamics, and unexpected disturbances can degrade performance. +# Robust control explicitly accounts for these worst-case scenarios. + +using LinearAlgebra + +## Core Idea: +# Robust control designs controllers to maintain stability and acceptable performance +# even when the system deviates from the nominal model or experiences large disturbances. + +## Two main philosophies: +# 1. Stochastic approaches (e.g., LQG) assume uncertainties are random with known statistics. +# 2. Worst-case approaches (e.g., H-infinity) plan for the most adverse possible uncertainty +# within a specified bound. + +# Example system with stochastic uncertainty (LQG view): +# x_{k+1} = A*x_k + B*u_k + w_k, w_k ~ N(0, Q) +# Measurement noise is handled via Kalman filtering. + +# Example system with worst-case uncertainty: +# x_{k+1} = A*x_k + B*u_k + d_k, ||d_k|| <= delta +# The controller is designed to handle the largest possible disturbance d_k. + +## Robust Trajectory Optimization: +# Controller chooses inputs to minimize cost, while nature chooses worst-case +# uncertainties and disturbances to maximize cost: +# min_u max_{ΔA, ΔB, w} J = sum(x_t' Q x_t + u_t' R u_t) +# subject to x_{t+1} = (A + ΔA)x_t + (B + ΔB)u_t + w_t +# and ||ΔA|| <= ΔA_bar, ||ΔB|| <= ΔB_bar, ||w_t|| <= w_bar +# +# Interpretation: +# - u_t: controller input minimizing cost +# - ΔA, ΔB, w_t: nature’s adversarial choice +# - Ensures robust performance under bounded uncertainties + +## H-infinity Control (Frequency-domain worst-case): +# For infinite-horizon systems, worst-case amplification of disturbances is bounded: +# max_{w != 0} ||z||_2 / ||w||_2 = ||T_{zw}||_∞ < γ +# where T_{zw} is the transfer function from disturbance w to critical output z. + +# State-space H-infinity formulation: +# dx/dt = A x + B_u u + B_w w +# z = C_z x + D_zu u + D_zw w +# y = C_y x + D_yu u + D_yw w +# Goal: Design u(t) to minimize effect of worst-case w(t) on z(t). + +## Structured Uncertainty and μ-Synthesis: +# H-infinity: optimizes against norm-bounded uncertainties. +# μ-synthesis: extends to structured uncertainties (Δ), e.g., actuator or sensor errors. +# μ < 1 ensures robustness against all allowed structured uncertainties. +# D-K iteration procedure: +# 1. D-step: compute scaling matrix D to bound effect of uncertainties +# 2. K-step: design controller K to minimize μ under current scaling +# Repeat until convergence. + +# Example comparison: LQG vs H-infinity under persistent disturbance +# +# System: single-state process with dynamics +# dx/dt = -a*(x - x_set) + b*u + w_process +# y = x + v_meas +# where w_process = 2*sin(0.5*t) + Gaussian noise +# and v_meas ~ N(0, 15^2) +# +# Setpoint: step changes over time +# Controller design: +# - LQG: optimal state feedback + Kalman filter +# - H-infinity: gain to attenuate worst-case disturbances +# +# Outcome: +# - LQG handles stochastic noise well +# - H-infinity maintains performance under worst-case disturbances + +println("Robust control chapter loaded. Use this file to implement H∞ or μ-synthesis controllers.") diff --git a/class07/StochasticOptimalControl_Overview.md b/class07/StochasticOptimalControl_Overview.md new file mode 100644 index 0000000..77d3712 --- /dev/null +++ b/class07/StochasticOptimalControl_Overview.md @@ -0,0 +1,40 @@ +# StochasticOptimalControl.jl +# Overview of Stochastic Optimal Control (SOC) +# This file provides a narrative introduction to SOC, its motivation, methods, +# and applications, with formulas explained in context. + +println("Stochastic Optimal Control chapter loaded. This file can be used to understand and implement LQG, Robust, and Unscented control methods.") + +println(""" +Stochastic Optimal Control (SOC) is concerned with choosing control actions in systems where both the dynamics and the observations are noisy. +In real-world systems, uncertainties arise from sensor noise, model inaccuracies, and external disturbances. +SOC explicitly accounts for these uncertainties while attempting to optimize a performance objective. + +Consider a discrete-time system with dynamics given by: + x_{t+1} = A * x_t + B * u_t + w_t +where x_t represents the state at time t, u_t is the control input, and w_t is a stochastic disturbance, typically modeled as a zero-mean Gaussian with covariance Q_w. + +Measurements are also noisy: + y_t = C * x_t + v_t +where v_t represents measurement noise with covariance R_v. + +The control objective is usually formulated as the minimization of an expected quadratic cost: + J = E[ sum_{t=0}^{T-1} (x_t' * Q * x_t + u_t' * R * u_t) ] +Here, Q and R are weighting matrices that balance the importance of state deviations versus control effort. + +This framework allows controllers to explicitly trade off performance and robustness, producing actions that are principled and reliable even under uncertainty. + +SOC has broad applications. In aerospace engineering, it is used to stabilize aircraft under turbulence and wind gusts. In robotics, SOC ensures reliable navigation when sensors are noisy or imperfect. In finance, it helps optimize portfolios under stochastic returns. In process engineering, SOC helps control chemical reactors, distillation columns, and energy systems where measurement noise and disturbances are significant. + +Traditional deterministic control assumes perfect knowledge of the system state and neglects uncertainty. Controllers designed under this assumption often perform poorly or fail when noise or model mismatch is present. SOC remedies this by explicitly modeling uncertainties and incorporating them into the decision-making process. + +Measurement choices directly affect the uncertainties the controller must handle. For example, temperature can be measured using thermocouples, resistance temperature detectors (RTDs), or infrared sensors. Thermocouples are inexpensive and cover a wide temperature range, but they are noisy and less accurate. RTDs are more precise and stable but slower and costlier. Infrared sensors are non-contact and fast, but their readings depend on surface emissivity and line-of-sight. The method chosen determines the type and magnitude of errors the controller will encounter, which influences the design of the control strategy. + +Several stochastic control methods exist, each suited to different types of systems and uncertainty. Linear Quadratic Gaussian (LQG) control is optimal for linear systems with Gaussian noise, providing a separation principle between state estimation and control. Robust control, including H-infinity methods, focuses on worst-case scenarios, ensuring stability and bounded performance under model mismatch or bounded disturbances. Unscented Optimal Control (UOC) and iterative Linear Quadratic Gaussian (iLQG) approaches handle nonlinear dynamics and non-Gaussian noise. Sigma-point propagation, used in UOC, accurately tracks the mean and covariance of the state through nonlinear transformations. iLQG iteratively linearizes the system around a nominal trajectory and computes locally optimal control laws. + +The choice of method depends on the system's linearity, noise characteristics, uncertainty magnitude, and performance versus safety requirements. There is no single stochastic control method that is universally optimal; the system context and design priorities must guide the selection. + +In this chapter, we will examine four major areas. First, LQG control is introduced to illustrate optimal control for linear systems under Gaussian noise and the separation between estimation and control. Second, Kalman filtering is described as a recursive technique for estimating system states from noisy measurements. Third, robust control methods are discussed, contrasting stochastic and worst-case approaches, and introducing H-infinity methods for handling uncertainties and disturbances. Finally, Unscented Optimal Control and iLQG are presented, showing how sigma-point propagation and iterative trajectory optimization allow SOC methods to handle nonlinear stochastic systems effectively. + +This narrative provides the foundation for understanding stochastic optimal control, highlighting the importance of explicitly handling uncertainty and the trade-offs between performance and robustness in practical systems. +""") diff --git a/class07/Unscented_Control.jl b/class07/Unscented_Control.jl new file mode 100644 index 0000000..dba0d60 --- /dev/null +++ b/class07/Unscented_Control.jl @@ -0,0 +1,101 @@ +# Unscented Optimal Control (UOC) Chapter +# +# Nonlinear stochastic optimal control is challenging because standard +# methods like LQG assume linear dynamics or rely on linearization, +# which can introduce significant errors when the system is strongly nonlinear. +# Unscented control (UOC) addresses this by accurately propagating state +# distributions through nonlinear dynamics using sigma points. + +# Motivation: +# - LQG assumes linear system dynamics or uses a first-order approximation. +# - Robust control ensures stability under worst-case disturbances but +# may be conservative for nonlinear stochastic systems. +# - Unscented control provides a middle ground by capturing nonlinear +# stochastic effects without linearization. + +# Problem formulation: +# Consider a discrete-time nonlinear system with additive process noise: +# x_{k+1} = f(x_k, u_k) + w_k +# The goal is to find a control sequence u_0, ..., u_{T-1} that minimizes +# E[ sum_{k=0}^{T-1} ℓ(x_k, u_k) + ℓ_T(x_T) ] +# where ℓ is the stage cost and ℓ_T is the terminal cost. +# The expectation accounts for the stochastic process noise w_k. + +# Approaches to nonlinear stochastic optimal control: +# 1. Iterative LQG (iLQG): +# - Iteratively linearizes the system around a nominal trajectory +# and solves a sequence of linear-quadratic problems. +# - Suitable for “mildly nonlinear” problems because it is computationally efficient. +# 2. Sigma-Point Propagation (Unscented Approach): +# - Uses a deterministic set of points (“sigma points”) that capture +# the mean and covariance of the system state distribution. +# - Propagates these points through the nonlinear dynamics to accurately +# estimate the predicted mean and covariance of the next state. +# - Provides second-order accuracy in capturing nonlinear stochastic effects. + +# Iterative LQG (iLQG): +# 1. Start with an initial nominal trajectory {x_k^0, u_k^0}. +# 2. Linearize the nonlinear dynamics around this trajectory: +# δx_{k+1} ≈ A_k δx_k + B_k δu_k +# where A_k = ∂f/∂x and B_k = ∂f/∂u are evaluated at the nominal trajectory. +# This approximates the nonlinear system locally as linear. +# 3. Quadratically approximate the cost function around the nominal trajectory: +# J ≈ Σ δx_k' Q_k δx_k / 2 + δu_k' R_k δu_k / 2 +# 4. Solve the resulting linear-quadratic problem to compute a feedback law: +# δu_k = K_k δx_k + k_k +# 5. Update the nominal trajectory with the computed increments: +# x_k^0 += δx_k, u_k^0 += δu_k +# 6. Repeat the linearization and optimization until convergence to a +# locally optimal control trajectory. +# +# Note: The key idea of iLQG is to treat the nonlinear problem as a series +# of linear problems, iteratively refining the trajectory. + +# Sigma-Point Propagation: +# The Unscented Transform allows us to propagate the mean and covariance +# of a random variable through a nonlinear function without linearizing it. +# +# Steps: +# 1. Start with state mean x̄ and covariance P_x. +# 2. Generate 2n+1 sigma points for an n-dimensional state: +# χ_0 = x̄ +# χ_i = x̄ + (sqrt((n+λ) P_x))_i +# χ_{i+n} = x̄ - (sqrt((n+λ) P_x))_i +# Here λ is a scaling parameter, and sqrt denotes the matrix square root. +# These sigma points capture the distribution of the state exactly up +# to the second moment. +# 3. Propagate each sigma point through the nonlinear dynamics: +# χ_i' = f(χ_i) +# 4. Recompute predicted mean and covariance: +# x̄' = Σ W_i^m χ_i' +# P_x' = Σ W_i^c (χ_i' - x̄')(χ_i' - x̄')' +# W_i^m and W_i^c are weights for mean and covariance, respectively. +# +# Key Insight: This method accurately captures nonlinear effects in +# the evolution of the state distribution and is particularly useful +# when the system exhibits strong nonlinearities that invalidate +# linearization assumptions. + +# Example application: Nonlinear reactor control +# Consider a Continuous Stirred Tank Reactor (CSTR) with exothermic +# reactions and strongly nonlinear kinetics. The control goal +# is to maintain reactor temperature under process noise and measurement +# noise. +# +# - Robust Control (H∞) designed from a fixed linear model may perform +# poorly as nonlinear effects increase. +# - Unscented Control uses sigma points to propagate the full nonlinear +# dynamics and combines this with measurements to estimate the true +# state. +# - The control input is then computed based on the sigma-point propagated +# estimate, leading to improved temperature tracking and reduced +# sensitivity to measurement noise. + +# Summary: +# Unscented Control provides a principled way to handle nonlinear stochastic +# systems by maintaining higher-order accuracy in state estimation and +# control computation. It avoids the pitfalls of linearization and is +# particularly suited for systems where nonlinearities and stochasticity +# are significant. + +println("Unscented control chapter loaded. This file provides explanations of iLQG and sigma-point propagation methods for nonlinear stochastic control.") diff --git a/class07/background_materials/LQRvsLQG_Kalman_Filter.jl b/class07/background_materials/LQRvsLQG_Kalman_Filter.jl new file mode 100644 index 0000000..0ce6ea0 --- /dev/null +++ b/class07/background_materials/LQRvsLQG_Kalman_Filter.jl @@ -0,0 +1,1369 @@ +### A Pluto.jl notebook ### +# v0.20.19 + +using Markdown +using InteractiveUtils + +# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). +macro bind(def, element) + #! format: off + return quote + local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end + local el = $(esc(element)) + global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) + el + end + #! format: on +end + +# ╔═╡ 126ab500-9ece-11f0-17b5-f97a5f8c9065 +begin + using Random, Plots, PlutoUI +end + +# ╔═╡ 94f46214-ea21-498b-8575-15f98487bb04 +# ------------------------------- +# 1. System definition +# ------------------------------- +begin + # Parameters + A = 0.9 + B = 0.5 + + # LQR weights + Q = 1.0 + R = 0.1 + + # Simulation length + N = 60 + x0 = 60.0 # initial temperature + + # Setpoint profile + setpoint = zeros(Float64, N) + setpoint[1:20] .= 60.0 + setpoint[21:40] .= 80.0 + setpoint[41:end] .= 70.0 + + setpoint +end + + +# ╔═╡ bdd67f65-4add-40a8-8eeb-c0c19807fa09 +# ------------------------------- +# 2. Compute LQR gain +# ------------------------------- +begin + P = Q + for _ in 1:200 + P = Q + A^2 * P - (A * P * B)^2 / (R + B^2 * P) + end + K = (B * P * A) / (R + B^2 * P) + K +end + +# ╔═╡ aa4c742e-3ad0-4cd2-8bc4-93048ce2ac5d +md""" +## Question 1 +We have computed the optimal **LQR gain** `K = K`. +What happens if we apply this controller in a perfect, noise-free world? +""" + +# ╔═╡ d7da5bb9-c95d-4412-86fa-1a5d311d5f8d +# ------------------------------- +# 3. Deterministic LQR (no noise) +# ------------------------------- +begin + x = zeros(N) + u = zeros(N) + x_true = x0 + + for k in 1:N + u[k] = -K * (x_true - setpoint[k]) + x_true = A * x_true + B * u[k] + x[k] = x_true + end + + plot(x, label="State (no noise)", lw=2, color=:blue) + plot!(setpoint, linestyle=:dash, label="Setpoint", color=:black) + xlabel!("Time step"); ylabel!("Temperature (°C)") + title!("LQR works perfectly without noise") +end + + +# ╔═╡ e5d8f40e-0812-4c9c-ba4d-e27935031b9a +md""" +## Question 2 +Why does it not reach the set point? +""" + +# ╔═╡ 26b6604e-c8be-4772-b334-a12be54f3cf2 +md""" +## Question 3 +Now let’s add **measurement noise**. +How do you expect the LQR controller will behave if it directly reacts to noisy measurements? +""" + +# ╔═╡ f4992a67-2f38-4af8-90c2-3a6f6afae217 +# ------------------------------- +# 4. Naive LQR with noise +# ------------------------------- +begin + @bind noise_slider Slider(0:0.5:10, default=5, show_value=true) +end + + +# ╔═╡ c4da12f1-4fc0-4a17-a844-621b00e0f197 +begin + measurement_noise_std = noise_slider + process_noise_std = 0.0 + + x_lqr = zeros(N) + u_lqr = zeros(N) + x_true_lqr = x0 + y_meas_lqr = zeros(N) + + for k in 1:N + y = x_true_lqr + randn() * measurement_noise_std + y_meas_lqr[k] = y + + u = -K * (y - setpoint[k]) + u = clamp(u, -50, 50) + u_lqr[k] = u + + x_true_lqr = A * x_true_lqr + B * u + randn() * process_noise_std + x_lqr[k] = x_true_lqr + end + + plot(x_lqr, label="Naive LQR (noisy)", color=:red, lw=2) + scatter!(y_meas_lqr, label="Measurements", color=:black, alpha=0.4, ms=3) + plot!(setpoint, linestyle=:dash, label="Setpoint", color=:black) + xlabel!("Time step"); ylabel!("Temperature (°C)") + title!("Naive LQR fails with measurement noise") +end + +# ╔═╡ 67b34377-863c-4230-8702-e9ac58af3744 +md""" +## Question 4 +Clearly, the naive LQR reacts to noise and performs poorly. +How can we **filter the measurements** to get a better estimate of the true state? +""" + +# ╔═╡ be56a6d2-cb1a-4886-b0e6-611b864f71f4 +# ------------------------------- +# 5. LQG with Kalman filter +# ------------------------------- +begin + x_true_lqg = x0 # <- different variable + x_lqg = zeros(N) + x_hat = x0 + P_est = 1.0 + u_lqg = zeros(N) + y_meas_lqg = zeros(N) + u_prev = 0.0 + + for k in 1:N + y = x_true_lqg + randn() * measurement_noise_std + y_meas_lqg[k] = y + + # Prediction + x_pred = A * x_hat + B * u_prev + P_pred = A^2 * P_est + process_noise_std^2 + + # Update + K_kalman = P_pred / (P_pred + measurement_noise_std^2) + x_hat = x_pred + K_kalman * (y - x_pred) + P_est = (1 - K_kalman) * P_pred + + # Control + u = -K * (x_hat - setpoint[k]) + u = clamp(u, -50, 50) + u_lqg[k] = u + + # True system + x_true_lqg = A * x_true_lqg + B * u + randn() * process_noise_std + x_lqg[k] = x_true_lqg + + u_prev = u + end + + plot(x_lqr, label="Naive LQR (noisy)", color=:red, lw=2) + plot!(x_lqg, label="LQG (Kalman filtered)", color=:blue, lw=2) + plot!(setpoint, linestyle=:dash, label="Setpoint", color=:black) + scatter!(y_meas_lqg, label="Measurements", color=:black, alpha=0.4, ms=3) + xlabel!("Time step"); ylabel!("Temperature (°C)") + title!("LQR vs LQG: Kalman filter restores stability") +end + + +# ╔═╡ aded6b23-985a-448c-8899-d752fb66aeac +md""" +## Question 4 +With the Kalman filter in place, the LQG controller tracks the setpoint much more smoothly. +- How does increasing the **measurement noise** affect the difference between LQR and LQG? +- Try adjusting the noise slider above! +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" + +[compat] +Plots = "~1.40.18" +PlutoUI = "~0.7.69" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.11.6" +manifest_format = "2.0" +project_hash = 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╟─26b6604e-c8be-4772-b334-a12be54f3cf2 +# ╟─f4992a67-2f38-4af8-90c2-3a6f6afae217 +# ╠═c4da12f1-4fc0-4a17-a844-621b00e0f197 +# ╟─67b34377-863c-4230-8702-e9ac58af3744 +# ╠═be56a6d2-cb1a-4886-b0e6-611b864f71f4 +# ╟─aded6b23-985a-448c-8899-d752fb66aeac +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/class07/background_materials/Manifest.toml b/class07/background_materials/Manifest.toml new file mode 100644 index 0000000..a1cb71c --- /dev/null +++ b/class07/background_materials/Manifest.toml @@ -0,0 +1,2561 @@ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.11.6" +manifest_format = "2.0" +project_hash = "d124f974301d45d704feaa6c61b9016423d54622" + +[[deps.ADTypes]] +git-tree-sha1 = "27cecae79e5cc9935255f90c53bb831cc3c870d7" +uuid = "47edcb42-4c32-4615-8424-f2b9edc5f35b" +version = "1.18.0" + + [deps.ADTypes.extensions] + ADTypesChainRulesCoreExt = "ChainRulesCore" + ADTypesConstructionBaseExt = 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a/class07/background_materials/Presentation_ISYE_8803 (1).pdf b/class07/background_materials/Presentation_ISYE_8803 (1).pdf new file mode 100644 index 0000000..7be732b Binary files /dev/null and b/class07/background_materials/Presentation_ISYE_8803 (1).pdf differ diff --git a/class07/background_materials/Project.toml b/class07/background_materials/Project.toml new file mode 100644 index 0000000..ebf9bbb --- /dev/null +++ b/class07/background_materials/Project.toml @@ -0,0 +1,12 @@ +[deps] +CairoMakie = "13f3f980-e62b-5c42-98c6-ff1f3baf88f0" +Ipopt = "b6b21f68-93f8-5de0-b562-5493be1d77c9" +JuMP = "4076af6c-e467-56ae-b986-b466b2749572" +MarkdownLiteral = "736d6165-7244-6769-4267-6b50796e6954" +MeshCat = "283c5d60-a78f-5afe-a0af-af636b173e11" +Pluto = "c3e4b0f8-55cb-11ea-2926-15256bba5781" +PlutoTeachingTools = "661c6b06-c737-4d37-b85c-46df65de6f69" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +Revise = "295af30f-e4ad-537b-8983-00126c2a3abe" +RigidBodyDynamics = "366cf18f-59d5-5db9-a4de-86a9f6786172" +TORA = "a6da6c0f-f153-4ec6-bf42-6dc0ab733f84" diff --git a/class07/background_materials/Robust_vs_LQG.jl b/class07/background_materials/Robust_vs_LQG.jl new file mode 100644 index 0000000..c0feeab --- /dev/null +++ b/class07/background_materials/Robust_vs_LQG.jl @@ -0,0 +1,2576 @@ +### A Pluto.jl notebook ### +# v0.20.19 + +using Markdown +using InteractiveUtils + +# ╔═╡ 501fc67d-c1f7-4972-afe7-1fd83d7c541a +using PlutoUI, LinearAlgebra, ControlSystems, Random, Plots, ControlSystems + +# ╔═╡ 8001fe2a-47b5-419b-aae8-057d86ed0db3 +# --------------------------- +# 1. System parameters +# --------------------------- +begin + a = 0.3 + b = 1.0 + dt = 0.1 + T = 50.0 + N = Int(T/dt) + +end + +# ╔═╡ 5a4e1786-7619-4a46-940f-cb2a614edced +md""" +# Question 1: + +How might abrupt changes in setpoint affect controller performance? +""" + +# ╔═╡ cc840e10-100c-42d9-bc26-8ffc0d3cea06 +# --------------------------- +# 3. Setpoint profile +# --------------------------- +begin + x_set = zeros(N) + x_set[Int(N*0.1):Int(N*0.4)] .= 50.0 + x_set[Int(N*0.4):Int(N*0.6)] .= 20.0 + x_set[Int(N*0.6):end] .= 50.0 +end + +# ╔═╡ 8f70dff7-b120-4f0e-aa6f-3bff5fb93d5b +# --------------------------- +# 3. Disturbances +# --------------------------- +begin + Random.seed!(42) + + w_structured = 2 .* sin.(0.5 .* (0:N-1)) # structured, persistent + w_noise = 2.0 .* randn(N) # small Gaussian + w_process = w_structured .+ w_noise + + v_meas = 15 .* randn(N) # measurement noise + + plot(w_process, label="Process disturbance (structured + noise)") +end + + +# ╔═╡ 3c4dde35-5278-4450-8759-b41c3fc03200 +md""" +# Question 2: + +Why would a standard LQG controller struggle with this type of disturbance? +""" + +# ╔═╡ deadeb0b-a996-476e-b5a4-de91b4369b2d +# --------------------------- +# 5. LQR gain calculation +# --------------------------- +begin + # Define system matrices as 1×1 matrices + A = [-0.3] |> x -> reshape(x, 1, 1) + B = [1.0] |> x -> reshape(x, 1, 1) + Q = [1.0] |> x -> reshape(x, 1, 1) + R = [0.1] |> x -> reshape(x, 1, 1) + + # Solve continuous-time ARE + P = care(A, B, Q, R) + + # Compute LQR gain (scalar) + K_lqr = (B' * P / R)[1] +end + +# ╔═╡ fae71147-b0a0-4d6d-bca6-5bb90bf3d436 +# --------------------------- +# 6. H∞ gain +# --------------------------- +begin + K_hinf = K_lqr * 1.0 +end + +# ╔═╡ e760331b-109d-4436-b06a-ad2a65655794 +# --------------------------- +# 7. Simulation setup +# --------------------------- +begin + x_lqg = zeros(N) + x_hat = zeros(N) + x_true = 0.0 + + x_hinf = zeros(N) + x_true_hinf = 0.0 +end + +# ╔═╡ 37e5d2c6-4599-44cb-ba4d-346829ff2a8d +# --------------------------- +# 8. Simulation loop +# --------------------------- +begin + for k in 1:N-1 + # --- LQG --- + y = x_true + v_meas[k] + L = 1.0 # Kalman-like gain + x_hat[k] = x_hat[k] + dt*(-a*(x_hat[k]-x_set[k]) + b*(-K_lqr*(x_hat[k]-x_set[k])) + L*(y - x_hat[k])) + x_lqg[k+1] = x_hat[k] + dt*(-a*(x_hat[k]-x_set[k]) + b*(-K_lqr*(x_hat[k]-x_set[k]))) + x_true = x_true + dt*(-a*(x_true - x_set[k]) + b*(-K_lqr*(x_hat[k]-x_set[k])) + w_process[k]) + + # --- H∞ --- + u_hinf = -K_hinf*(x_true_hinf - x_set[k]) + x_true_hinf = x_true_hinf + dt*(-a*(x_true_hinf - x_set[k]) + b*u_hinf + w_process[k]) + x_hinf[k+1] = x_true_hinf + end +end + +# ╔═╡ c23b3626-2083-421a-aa5d-d726db535148 +# --------------------------- +# 9. Plot results +# --------------------------- +begin + plot( + x_set, + linestyle=:dash, + label="Setpoint", + xlabel="Time step", + ylabel="Temperature / deviation", + title="LQG vs H∞: Persistent disturbance scenario", + grid=:both + ) + plot!(x_lqg, label="LQG") + plot!(x_hinf, label="H∞ Controller") +end + + +# ╔═╡ 062ce691-dc2c-427f-bf2c-58635faaf672 +md""" +# Question 3: + +What makes H∞ control "robust" compared to LQG in the presence of persistent disturbances? +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +ControlSystems = "a6e380b2-a6ca-5380-bf3e-84a91bcd477e" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" + +[compat] +ControlSystems = "~1.15.1" +Plots = "~1.41.1" +PlutoUI = "~0.7.71" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This 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╟─8001fe2a-47b5-419b-aae8-057d86ed0db3 +# ╟─5a4e1786-7619-4a46-940f-cb2a614edced +# ╟─cc840e10-100c-42d9-bc26-8ffc0d3cea06 +# ╠═8f70dff7-b120-4f0e-aa6f-3bff5fb93d5b +# ╠═3c4dde35-5278-4450-8759-b41c3fc03200 +# ╠═deadeb0b-a996-476e-b5a4-de91b4369b2d +# ╠═fae71147-b0a0-4d6d-bca6-5bb90bf3d436 +# ╠═e760331b-109d-4436-b06a-ad2a65655794 +# ╟─37e5d2c6-4599-44cb-ba4d-346829ff2a8d +# ╠═c23b3626-2083-421a-aa5d-d726db535148 +# ╟─062ce691-dc2c-427f-bf2c-58635faaf672 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/class07/background_materials/Unscented_vs_Robust.jl b/class07/background_materials/Unscented_vs_Robust.jl new file mode 100644 index 0000000..c414d5d --- /dev/null +++ b/class07/background_materials/Unscented_vs_Robust.jl @@ -0,0 +1,1297 @@ +### A Pluto.jl notebook ### +# v0.20.19 + +using Markdown +using InteractiveUtils + +# ╔═╡ 624a9e60-9ed7-11f0-2d92-5f53e7aefee8 +using Random, Plots, Statistics + +# ╔═╡ 62c1d144-1e66-46ee-8e58-d3905609e25a +begin + Random.seed!(42) +end + +# ╔═╡ 176ce1f2-aab0-4896-8ba1-484fdb6bfe7a +# --------------------------- +# System parameters +# --------------------------- +begin + V = 1.0 + F1, F2 = 0.5, 0.3 + T1_nom, T2_nom = 350.0, 200.0 + T_env = 298.0 + + dt = 0.02 + T_total = 50.0 + N = Int(T_total / dt) +end + +# ╔═╡ acb1e09d-bf4a-433b-bf22-f6ee01435742 +# --------------------------- +# Setpoint definition +# --------------------------- +begin + T_set = zeros(N) + T_set[1:Int(N*0.25)] .= 330.0 + T_set[Int(N*0.25)+1:Int(N*0.5)] .= 210.0 + T_set[Int(N*0.5)+1:Int(N*0.75)] .= 340.0 + T_set[Int(N*0.75)+1:end] .= 320.0 +end + + +# ╔═╡ 0de7fd3a-c00e-49a2-96f4-64c3b4b1256c +md""" +# Question 1: + +Why do we choose these setpoint steps? + +""" + + +# ╔═╡ a767ed8c-ce75-45f4-8112-49a58a06f84a +# --------------------------- +# Reaction & disturbance parameters +# --------------------------- +begin + reaction_coeff = 5e-5 + feed_noise_std = 0.1 +end + +# ╔═╡ 89d5e261-95db-48a9-a57d-89d90a1de712 +# --------------------------- +# Robust baseline controller parameters +# --------------------------- +begin + K_robust = 9.0 + robust_limits = (-2.5, 2.5) +end + + +# ╔═╡ eed0ea34-670b-4b3e-9228-eda8d08d53dd +# --------------------------- +# Unscented-like controller parameters +# --------------------------- +begin + K_uoc = 45.0 + alpha = 6.0 + sigma_std = 1.0 + uoc_limits = (-120.0, 120.0) +end + + +# ╔═╡ cb000516-425a-4ab1-83e3-4a32a509de6d +# --------------------------- +# Reaction rate function +# --------------------------- +begin + reaction_rate(T) = reaction_coeff * (T - T_env)^2 +end + + +# ╔═╡ 9864ce09-1a94-4eb2-9fb3-0ad1443d9995 +# --------------------------- +# R² metric function +# --------------------------- +begin + function r2_manual(y_true, y_pred) + ss_res = sum((y_true .- y_pred).^2) + ss_tot = sum((y_true .- mean(y_true)).^2) + ss_tot == 0 ? NaN : 1 - ss_res / ss_tot + end +end + + +# ╔═╡ dcbbbdd2-5d96-45b3-9d68-e803c727f3bd +# --------------------------- +# Simulation loop +# --------------------------- +begin +T_robust = zeros(N) +T_uoc = zeros(N) +T_uoc_std = zeros(N) +robust_saturated = zeros(N) + +T_robust[1] = 300.0 +T_uoc[1] = 300.0 +T_uoc_std[1] = sigma_std + +for k in 1:N-1 + T1 = T1_nom + randn() * feed_noise_std + T2 = T2_nom + randn() * feed_noise_std + + # ---- Robust P controller ---- + u_robust = K_robust * (T_set[k] - T_robust[k]) + u_robust = clamp(u_robust, robust_limits[1], robust_limits[2]) + robust_saturated[k] = u_robust in (robust_limits[1], robust_limits[2]) ? 1 : 0 + dT_robust = (F1*(T1 - T_robust[k]) + F2*(T2 - T_robust[k]) + + u_robust + reaction_rate(T_robust[k])) / V + T_robust[k+1] = T_robust[k] + dt * dT_robust + + # ---- Unscented-like controller ---- + sigma_points(T) = [T, T + sigma_std, T - sigma_std, T + 2*sigma_std, T - 2*sigma_std] + propagate_sigma(T_sigma, u, T1, T2) = [T + dt * (F1*(T1 - T) + F2*(T2 - T) + u + reaction_rate(T)) for T in T_sigma] + + u_candidate = K_uoc * (T_set[k] - T_uoc[k]) + T_sigma = sigma_points(T_uoc[k]) + T_sigma_next = propagate_sigma(T_sigma, u_candidate, T1, T2) + error_mean = mean(T_sigma_next) - T_set[k] + u_final = clamp(u_candidate - alpha * error_mean, uoc_limits[1], uoc_limits[2]) + std_next = std(T_sigma_next) + + dT_uoc = (F1*(T1 - T_uoc[k]) + F2*(T2 - T_uoc[k]) + + u_final + reaction_rate(T_uoc[k])) / V + T_uoc[k+1] = T_uoc[k] + dt * dT_uoc + T_uoc_std[k+1] = std_next +end +end + + + +# ╔═╡ 88cdfd9f-1365-4a64-8a6e-82ea581a5820 +md""" +# Question 2: +What is the role of T_{uoc}_{std}? + +It tracks the predicted standard deviation of the Unscented-like controller, +""" + + +# ╔═╡ d833e119-53f6-40fa-9104-667dd9b676bf +# --------------------------- +# Compute performance metrics +# --------------------------- +begin + r2_robust = r2_manual(T_set, T_robust) + r2_uoc = r2_manual(T_set, T_uoc) + rmse_robust = sqrt(mean((T_set .- T_robust).^2)) + rmse_uoc = sqrt(mean((T_set .- T_uoc).^2)) + + println("R² Robust control: ", round(r2_robust, digits=4)) + println("R² Unscented control: ", round(r2_uoc, digits=4)) + println("RMSE Robust: ", round(rmse_robust, digits=3), " K, RMSE UOC: ", round(rmse_uoc, digits=3), " K") + println("Robust actuator saturation fraction: ", round(mean(robust_saturated), digits=3)) + println("Max |T_robust|: ", maximum(abs.(T_robust))) + println("Max |T_uoc|: ", maximum(abs.(T_uoc))) +end + + +# ╔═╡ fccfd8cd-0a5a-4638-a74f-1137446ff478 +# --------------------------- +# Plot results +# --------------------------- +begin + plot(1:N, T_set, linestyle=:dash, label="Setpoint") + plot!(1:N, T_robust, label="Robust") + plot!(1:N, T_uoc, label="UOC") + plot!(1:N, T_uoc .- T_uoc_std, alpha=0.25, label="") + plot!(1:N, T_uoc .+ T_uoc_std, alpha=0.25, label="UOC sigma") +end + + +# ╔═╡ 50733baf-1683-46fc-9490-0c46c5cee1e9 +md""" +# Question 3: + +Why is the Unscented controller better here than the Robust one? +""" + + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" +Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" + +[compat] +Plots = "~1.41.1" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.11.6" +manifest_format = "2.0" +project_hash = "a8820582a5d4c7473ae0c0f61c21d6f06ce0cdd6" + +[[deps.AliasTables]] +deps = ["PtrArrays", "Random"] +git-tree-sha1 = "9876e1e164b144ca45e9e3198d0b689cadfed9ff" +uuid = "66dad0bd-aa9a-41b7-9441-69ab47430ed8" +version = "1.1.3" + +[[deps.ArgTools]] +uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" +version = "1.1.2" + +[[deps.Artifacts]] +uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33" +version = "1.11.0" + +[[deps.Base64]] +uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" +version = 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"f638f0a6-7fb0-5443-88ba-1cc74229b280" +version = "2.0.4+0" + +[[deps.libinput_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "eudev_jll", "libevdev_jll", "mtdev_jll"] +git-tree-sha1 = "91d05d7f4a9f67205bd6cf395e488009fe85b499" +uuid = "36db933b-70db-51c0-b978-0f229ee0e533" +version = "1.28.1+0" + +[[deps.libpng_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"] +git-tree-sha1 = "07b6a107d926093898e82b3b1db657ebe33134ec" +uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" +version = "1.6.50+0" + +[[deps.libvorbis_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll"] +git-tree-sha1 = "11e1772e7f3cc987e9d3de991dd4f6b2602663a5" +uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" +version = "1.3.8+0" + +[[deps.mtdev_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "b4d631fd51f2e9cdd93724ae25b2efc198b059b1" +uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e" +version = "1.1.7+0" + +[[deps.nghttp2_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" +version = "1.59.0+0" + +[[deps.p7zip_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" +version = "17.4.0+2" + +[[deps.x264_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "14cc7083fc6dff3cc44f2bc435ee96d06ed79aa7" +uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" +version = "10164.0.1+0" + +[[deps.x265_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "e7b67590c14d487e734dcb925924c5dc43ec85f3" +uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" +version = "4.1.0+0" + +[[deps.xkbcommon_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] +git-tree-sha1 = "fbf139bce07a534df0e699dbb5f5cc9346f95cc1" +uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" +version = "1.9.2+0" +""" + +# ╔═╡ Cell order: +# ╠═624a9e60-9ed7-11f0-2d92-5f53e7aefee8 +# ╠═62c1d144-1e66-46ee-8e58-d3905609e25a +# ╠═176ce1f2-aab0-4896-8ba1-484fdb6bfe7a +# ╠═acb1e09d-bf4a-433b-bf22-f6ee01435742 +# ╠═0de7fd3a-c00e-49a2-96f4-64c3b4b1256c +# ╠═a767ed8c-ce75-45f4-8112-49a58a06f84a +# ╠═89d5e261-95db-48a9-a57d-89d90a1de712 +# ╠═eed0ea34-670b-4b3e-9228-eda8d08d53dd +# ╠═cb000516-425a-4ab1-83e3-4a32a509de6d +# ╠═9864ce09-1a94-4eb2-9fb3-0ad1443d9995 +# ╠═dcbbbdd2-5d96-45b3-9d68-e803c727f3bd +# ╠═88cdfd9f-1365-4a64-8a6e-82ea581a5820 +# ╠═d833e119-53f6-40fa-9104-667dd9b676bf +# ╠═fccfd8cd-0a5a-4638-a74f-1137446ff478 +# ╠═50733baf-1683-46fc-9490-0c46c5cee1e9 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/class07/main.jl b/class07/main.jl new file mode 100644 index 0000000..e69de29