Create autotuning.jl

autotuning.jl

@@ -0,0 +1,1831 @@
+### 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
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+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
+git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
+uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
+version = "1.7.10+4"
+
+[[deps.Xorg_libXinerama_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
+git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
+uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
+version = "1.1.4+4"
+
+[[deps.Xorg_libXrandr_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
+git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
+uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
+version = "1.5.2+4"
+
+[[deps.Xorg_libXrender_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
+git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
+uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
+version = "0.9.10+4"
+
+[[deps.Xorg_libpthread_stubs_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
+uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
+version = "0.1.0+3"
+
+[[deps.Xorg_libxcb_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
+git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
+uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
+version = "1.13.0+3"
+
+[[deps.Xorg_libxkbfile_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
+git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
+uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
+version = "1.1.0+4"
+
+[[deps.Xorg_xcb_util_image_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
+uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
+version = "0.4.0+1"
+
+[[deps.Xorg_xcb_util_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
+git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
+uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
+version = "0.4.0+1"
+
+[[deps.Xorg_xcb_util_keysyms_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
+uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
+version = "0.4.0+1"
+
+[[deps.Xorg_xcb_util_renderutil_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
+uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
+version = "0.3.9+1"
+
+[[deps.Xorg_xcb_util_wm_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
+uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
+version = "0.4.1+1"
+
+[[deps.Xorg_xkbcomp_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
+git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
+uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
+version = "1.4.2+4"
+
+[[deps.Xorg_xkeyboard_config_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
+git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
+uuid = "33bec58e-1273-512f-9401-5d533626f822"
+version = "2.27.0+4"
+
+[[deps.Xorg_xtrans_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
+uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
+version = "1.4.0+3"
+
+[[deps.Zlib_jll]]
+deps = ["Libdl"]
+uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
+version = "1.2.12+3"
+
+[[deps.Zstd_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
+uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
+version = "1.5.2+0"
+
+[[deps.ZygoteRules]]
+deps = ["MacroTools"]
+git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
+uuid = "700de1a5-db45-46bc-99cf-38207098b444"
+version = "0.2.2"
+
+[[deps.libass_jll]]
+deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
+git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
+uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
+version = "0.15.1+0"
+
+[[deps.libblastrampoline_jll]]
+deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
+uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
+version = "5.1.0+0"
+
+[[deps.libfdk_aac_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
+uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
+version = "2.0.2+0"
+
+[[deps.libpng_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
+git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
+uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
+version = "1.6.38+0"
+
+[[deps.libvorbis_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
+git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
+uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
+version = "1.3.7+1"
+
+[[deps.nghttp2_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
+version = "1.41.0+1"
+
+[[deps.p7zip_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
+version = "17.4.0+0"
+
+[[deps.x264_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
+uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
+version = "2021.5.5+0"
+
+[[deps.x265_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
+uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
+version = "3.5.0+0"
+
+[[deps.xkbcommon_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
+git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
+uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
+version = "0.9.1+5"
+"""
+
+# ╔═╡ Cell order:
+# ╟─6147e0a7-cd0d-4170-b1f8-af6263cac3b8
+# ╠═74229db1-870c-45aa-8d75-1c85a390c3b4
+# ╟─a9f8a84d-b006-497c-b345-b56886b0fad0
+# ╠═c2663176-98b7-407b-84d1-7a133996e3a7
+# ╠═e5783b41-f38e-4318-8c85-8ac9c2785a83
+# ╟─ea7c6a3e-bda7-4bdf-9562-51bec9c12693
+# ╠═548da4c7-3696-457d-9a1a-1d2be7dc07f4
+# ╠═6d53dc20-20d8-4f8d-8e34-f2771cf57c27
+# ╠═869e841e-3ee0-4266-932f-1259b325a0f7
+# ╟─0ecff517-8611-4aba-8902-0ee442452e18
+# ╠═a2a20f1d-ed49-453a-a4cc-8cc0b2da6de2
+# ╟─d42dbb54-a4a9-4318-b7ae-39011a2f703b
+# ╠═fd56b213-f975-4280-8d48-7586f65df392
+# ╠═6b954e81-3582-44e5-adcf-1289d1db3234
+# ╟─0aec68aa-7126-410b-b8f3-8b1d5bfecdf0
+# ╠═24f710ff-a5ba-4790-96ac-5c15cedbc237
+# ╠═a40a824f-e2d6-4487-924f-cf6256386a56
+# ╟─00000000-0000-0000-0000-000000000001
+# ╟─00000000-0000-0000-0000-000000000002