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html"""main {
    max-width: 1050px;
    align-self: flex-start;
    margin-left: 70px;
}

pluto-helpbox {
	width: clamp(300px,calc(100vw - 781px),600px)
}
"""

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using FastDifferentiation

# ==============================================================================
## Try simple things
# ==============================================================================

x = make_variables(:x, 2)
mu = make_variables(:mu, 2)
ex = sum(abs2, x .* mu)
hs = sparse_hessian(ex, x)

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# ref: https://www.control.lth.se/fileadmin/control/Education/EngineeringProgram/FRTN05/2019/lec06eight.pdf
# ref: https://www.kth.se/social/upload/4fd0a7fdf276547736000003/lec08.pdf
using ControlSystemsBase
using ControlSystems
using ControlSystemIdentification

s = tf("s")
G = 4/(s*(s+1)^2) # A test system

nl = ControlSystemsBase.saturation(1)

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Trimming

givne a system $\dot x = f(x, u)$ where $x$ are state variables and $u$ control inputs, we want to find a trim point $x_t, u_t$ that satisfies a set of trim conditions. In the absence of any user-specified conditions, this reduces to finding a stationary point $$0 = f(x,u)$$ i.e., $\dot x = 0$ is typically an implicit trim condition.

The user-specified trim conditions may take many forms, for example

• $x_i \approx 5$ should be close to 5 (called a soft constraint)

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using PrettyTables
using ModelingToolkit, Symbolics
using ModelingToolkit: has_defaults, defaults, getbounds, AbstractODESystem, n_extra_equations, get_iv
using Symbolics: unwrap, VariableSource

function describe(io::IO, sys::AbstractODESystem; backend = Val(:text), alignment = :l)

    header = ["Variable name", "Variable type", "Default value", "Bounds", "Description"]
    x = sort(states(sys), by=string)
    p = sort(parameters(sys), by=string)

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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

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using Symbolics

"""
    lie(f, h, x)

The Lie derivative of `h` w.r.t. `x` in the direction of `f`.
"""
lie(f, h, x) = Symbolics.gradient(h, x)'f

"""

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using StaticArrays
using Statistics, LinearAlgebra
using ModelingToolkit, Symbolics

"""
    rk4(f, l, Ts)

Discretize dynamics `f` and loss function `l`using RK4 with sample time `Ts`. 
The returned function is on the form `(xₖ,uₖ,t)-> (xₖ₊₁, loss)`.
Both `f` and `l` take the arguments `(x, u, t)`.

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using StaticArrays
using Statistics, LinearAlgebra
using ModelingToolkit, Symbolics

"""
    rk4(f, l, Ts)

Discretize dynamics `f` and loss function `l`using RK4 with sample time `Ts`. 
The returned function is on the form `(xₖ,uₖ,t)-> (xₖ₊₁, loss)`.
Both `f` and `l` take the arguments `(x, u, t)`.

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using ModelingToolkit
using ModelingToolkit: renamespace
using LinearAlgebra
using ModelingToolkit: isinput, isoutput, is_bound, isdifferential
using Symbolics: jacobian, substitute

"""
    linearize(sys::ODESystem; numeric=false, x0=nothing)

Linaerize `sys` around operating point `x0`. `x0` can be `nothing` for a symbolic linearization, or a `Dict` that maps states and parameters to values.