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autotuning.jl
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autotuning.jl
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### A Pluto.jl notebook ###
# v0.19.8
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)
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
end
# ╔═╡ 74229db1-870c-45aa-8d75-1c85a390c3b4
begin
using ControlSystems, OrdinaryDiffEq, Plots, Pkg, LinearAlgebra, Statistics, PlutoUI
gr(show=false, size=(500,400)) # Set defaults for plotting
end
# ╔═╡ 548da4c7-3696-457d-9a1a-1d2be7dc07f4
using NLopt, ForwardDiff
# ╔═╡ 6147e0a7-cd0d-4170-b1f8-af6263cac3b8
md"""
# Auto tuning
## The processes
The processes studied in this session are third order systems with two poles in $s=-0.5$ and one pole in -2.
$$P(s) = \dfrac{1}{(2s+1)^2(0.5s+1)}$$
with or without input nonlinearity.
## Installation
- First of all, you need Julia installed, see https://julialang.org/downloads/
- You also need [IJulia](https://github.com/JuliaLang/IJulia.jl#installation), which in turn requires Jupyter.
- Once Julia is up and running, the following julia code only has to be run once, it installs the relevant packages.
The code below must be run once every time the kernel is restarted
"""
# ╔═╡ a9f8a84d-b006-497c-b345-b56886b0fad0
md"""
We now proceed to define the process model and some parameters for our simulation
"""
# ╔═╡ c2663176-98b7-407b-84d1-7a133996e3a7
begin
P = tf(1,[2.,1])^2*tf(1,[0.5,1]) # Process model
h = 0.1 # Sample time (only used for plots)
Ps = ss(P) # State-space representation of process model
end;
# ╔═╡ e5783b41-f38e-4318-8c85-8ac9c2785a83
begin
s = Simulator(Ps)
x0 = [0.,0,0] # Initial state
Tf = 20 # Length of experiments (seconds)
t = 0:h:Tf # Time vector
tspan = (0.0,Tf)
end;
# ╔═╡ ea7c6a3e-bda7-4bdf-9562-51bec9c12693
md"""
## Auto tuning
Even with interactive tools, tuning a PID controller by hand can be difficult. A basic control course teaches us how to design a PID controller based on, e.g., pole placement. Here, we will formulate an optimization problem where we optimize a cost function $J(k_p,k_i,k_d)$ based on the step response of the closed-loop system
$$J(k_p,k_i,k_d) \quad=\quad \dfrac{1}{T_f}\int_0^{T_f} |r(t)-y(t)| dt \quad=\quad \dfrac{1}{T_f}\int_0^{T_f} |e(t)|dt$$
"""
# ╔═╡ 6d53dc20-20d8-4f8d-8e34-f2771cf57c27
begin
p = [0.1, 0.1, 0.1] # Initial guess [kp, ki, kd]
Kpid(kp,ki,kd) = pid(kp=kp, ki=ki, kd=kd)
Kpid(p) = K(p...)
function timedomain(p)
C = Kpid(p[1], p[2], p[3])
L = feedback(P*C) |> ss
step(L, t).y
end
function costfun(p)
y = timedomain(p)
mean(abs,1 .- y) # ~ Integrated absolute error IAE
end
f_cfg = ForwardDiff.GradientConfig(costfun, p)
function f(p::Vector, grad::Vector=[])
if length(grad) > 0
grad .= ForwardDiff.gradient(costfun,p,f_cfg)
end
costfun(p)
end
function runopt(p; f_tol = 1e-5, x_tol = 1e-3)
opt = Opt(:LD_MMA, 3)
opt.lower_bounds = 1e-6ones(3)
opt.xtol_rel = x_tol
opt.ftol_rel = f_tol
opt.min_objective = f
NLopt.optimize(opt, p)[2]
end
end
# ╔═╡ 869e841e-3ee0-4266-932f-1259b325a0f7
begin
@info "Starting Optimization"
@time popt = runopt(p, x_tol=1e-6)
y = timedomain(popt)
plot(t,y')
end
# ╔═╡ 0ecff517-8611-4aba-8902-0ee442452e18
md"""
If we look at the sensitivity function of the closed-loop system obtained with the optimized controller, we see that the optimization procedure produced a quite aggressive controller with poor robustness.
"""
# ╔═╡ a2a20f1d-ed49-453a-a4cc-8cc0b2da6de2
begin
C = Kpid(p...)
gangoffourplot(P,C, exp10.(LinRange(-1,3,500)), legend=false)
end
# ╔═╡ d42dbb54-a4a9-4318-b7ae-39011a2f703b
md"""
To combat this, we can add bounds on the maximum value of the sensitivity functions:
"""
# ╔═╡ fd56b213-f975-4280-8d48-7586f65df392
begin
Ω = exp10.(LinRange(-1,2,150))
p0 = [0.1,0.1,0.1]
function freqdomain(p)
C = Kpid(p[1], p[2], p[3])
PC = P*C
S = feedback(1, PC) # Sensitivity fun
T = feedback(PC) # Comp. Sensitivity fun
Sw = bodev(S,Ω,unwrap=false)[1] # Freq. domain constraints
Tw = bodev(T,Ω,unwrap=false)[1] # Freq. domain constraints
Sw,Tw
end
function constraintfun(p)
Sw,Tw = freqdomain(p)
[maximum(Sw); maximum(Tw)]
end
g_cfg = ForwardDiff.JacobianConfig(constraintfun, p)
end
# ╔═╡ 0aec68aa-7126-410b-b8f3-8b1d5bfecdf0
begin
Ms_gui = @bind Ms Slider(1:0.1:2, default=1.1, show_value=true)
Mt_gui = @bind Mt Slider(1:0.1:2, default=1.1, show_value=true)
md"``M_S`` = $Ms_gui ``M_T`` = $Mt_gui"
end
# ╔═╡ 6b954e81-3582-44e5-adcf-1289d1db3234
begin
function c(result, p::Vector, grad)
if length(grad) > 0
grad .= ForwardDiff.jacobian(constraintfun,p,g_cfg)'
end
result .= constraintfun(p) .- [Ms, Mt]
end
function runopt2(p; f_tol = 1e-5, x_tol = 1e-3, c_tol = 1e-8)
opt = Opt(:LD_SLSQP, 3)
lower_bounds!(opt, 1e-6ones(3))
xtol_rel!(opt, x_tol)
ftol_rel!(opt, f_tol)
min_objective!(opt, f)
inequality_constraint!(opt, c, c_tol*ones(2))
NLopt.optimize(opt, p)[2]
end
end
# ╔═╡ 24f710ff-a5ba-4790-96ac-5c15cedbc237
begin
popt2 = runopt2(p, x_tol=1e-6)
y2 = timedomain(popt2)
Sw,Tw = freqdomain(popt2)
plot(t,y2', layout=2, size=(800,400))
plot!(Ω, [Sw Tw] , lab=["Sw" "Tw"], subplot=2, xscale=:log10, yscale=:log10)
hline!([Ms], c = 1, l=:dash, subplot=2, lab="Ms")
hline!([Mt], c = 2, l=:dash, subplot=2, lab="Mt", ylims=(0.01,3))
end
# ╔═╡ a40a824f-e2d6-4487-924f-cf6256386a56
begin
C2 = Kpid(popt2...)
gangoffourplot(P,C2, exp10.(LinRange(-1,3,500)), legend=false)
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
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Plots = "~1.30.0"
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"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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