MAn_Case-1_Liquid-level-in-tanks.ipynb

{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Model Analysis\n",
    "## Case 1: Liquid Level in Tanks\n",
    "### Author: Bernt Lie, University of South-Eastern Norway\n",
    "### Date: March 22, 2023\n",
    "### Installing packages"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using ModelingToolkit\n",
    "using ModelingToolkit: inputs, outputs\n",
    "#using ModelingToolkitStandardLibrary\n",
    "using DifferentialEquations\n",
    "using Plots, LaTeXStrings\n",
    "using ControlSystems, ControlSystemsMTK\n",
    "using LinearAlgebra\n",
    "#using Latexify"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Constants"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Specifying line properties as constants makes it possible to globally \n",
    "# change these values and keep consistency\n",
    "LW1 = 2.5\n",
    "LW2 = 2\n",
    "LC1 = :red\n",
    "LC2 = :blue\n",
    "LC3 = :green\n",
    "LC4 = :orange\n",
    "LS1 = :solid\n",
    "LS2 = :dash\n",
    "LS3 = :dashdot\n",
    "LS4 = :dot\n",
    "LA1 = 1\n",
    "LA2 = 0.3\n",
    "\n",
    "figpath = \"C:/Users/Bernt/OneDrive/Documents/booksBLSOL/ModDynSyst/figs/\"\n",
    "figtype = \".svg\"\n",
    "SAVEFIG = true\n",
    ";"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Utility functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "function get_symbols(eqs)\n",
    "    symbs = [reduce(vcat,[Symbolics.get_variables.(eq.lhs) for eq in eqs]); \n",
    "        reduce(vcat,[Symbolics.get_variables.(eq.rhs) for eq in eqs])]\n",
    "    idxs_params = isequal.(expand_derivatives.(Differential(t).(symbs)),0)\n",
    "    params = unique(symbs[idxs_params])\n",
    "    vars = unique(symbs[isequal.(idxs_params,0)])\n",
    "    return vars, params\n",
    "end\n",
    ";"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "function linearize_symbolic_dae01(eqs, u_mod, y_mod; x_opt = [])\n",
    "    #\n",
    "    # This function assumes that `eqs` represents an index 1 DAE \n",
    "    #\n",
    "    # eqs: Symbolics partial model equations, i.e., with inputs undefined\n",
    "    # u_mod: vector of variables which are inputs in `eqs`\n",
    "    # y_mod: vector of variables in `eqs` which are outputs (inputs not allowed)\n",
    "    # x_opt: optional state vector ξ if other than differential variables x\n",
    "    #\n",
    "    # Unknown dependent variables v \n",
    "    vars,_ = get_symbols(eqs)\n",
    "    n_v = length(vars)\n",
    "    #\n",
    "    # Equations eqs split up into lhs and rhs\n",
    "    eqs_lhs = [eq.lhs for eq in eqs]\n",
    "    eqs_rhs = [eq.rhs for eq in eqs]\n",
    "    n_eqs = length(eqs)\n",
    "    # -- getting vec_field to be of correct size and type\n",
    "    vec_field = eqs_rhs\n",
    "    #\n",
    "    # Determining which equations are differential, and which are algebraic\n",
    "    # -- vector of derivatives; used for finding differential variables\n",
    "    dvdt_mod = Differential(t).(vars)\n",
    "    # -- assuming that only eqs_lhs has time derivatives of variables\n",
    "    idxs_eq_d = filter(!isnothing,[findfirst(isequal.(eq,dvdt_mod)) for eq in eqs_lhs])\n",
    "    idxs_eq_a = setdiff(1:n_eqs,idxs_eq_d)\n",
    "    #\n",
    "    # Finishing up vec_field\n",
    "    vec_field[idxs_eq_d] .= eqs_rhs[idxs_eq_d]\n",
    "    vec_field[idxs_eq_a] .= eqs_rhs[idxs_eq_a] .- eqs_lhs[idxs_eq_a]\n",
    "    #\n",
    "    # Finding Jacobian\n",
    "    J = Symbolics.jacobian(vec_field,vars)\n",
    "    #\n",
    "    # Finding vars indices of inputs, differential variables, algebraic variables, outputs\n",
    "    idxs_u = [findmax(isequal.(u,vars))[2] for u in u_mod]\n",
    "    # -- differential and algebraic variables\n",
    "    idxs_x = filter(!isnothing,[findfirst(isequal.(dvdt_mod,eq)) for eq in eqs_lhs])\n",
    "    idxs_z = setdiff(1:n_v, idxs_u, idxs_x)\n",
    "    # -- checking that inputs are not differential variables\n",
    "    if !isdisjoint(idxs_x,idxs_u)\n",
    "        println(\"Inputs can not be differential variables.\")\n",
    "        println(\"Inputs: $(u_mod)\")\n",
    "        println(\"Differential variables: $(v_mod[idxs_x])\")\n",
    "    end\n",
    "    # -- outputs\n",
    "    idxs_y = [findmax(isequal.(y,vars))[2] for y in y_mod]\n",
    "    # -- checking that outputs are not inputs\n",
    "    if !isdisjoint(idxs_y,idxs_u)\n",
    "        println(\"Outputs can not be inputs.\")\n",
    "        println(\"Outputs: $(y_mod)\")\n",
    "        println(\"Inputs: $(u_mod)\")\n",
    "    end\n",
    "    #\n",
    "    # Deconstructing Jacobian\n",
    "    J_d_x = J[idxs_eq_d,idxs_x]\n",
    "    J_d_z = J[idxs_eq_d,idxs_z]\n",
    "    J_d_u = J[idxs_eq_d,idxs_u]\n",
    "    J_a_x = J[idxs_eq_a,idxs_x]\n",
    "    J_a_z = J[idxs_eq_a,idxs_z]\n",
    "    J_a_u = J[idxs_eq_a,idxs_u]\n",
    "    #\n",
    "    # Jacobian matrix for outputs; contains ones and zeros\n",
    "    n_y = length(idxs_y)\n",
    "    J_y = zeros(n_y,n_v)\n",
    "    for i in 1:n_y\n",
    "        J_y[i,:] = isequal.(y_mod[i],vars)\n",
    "    end\n",
    "    J_y_x = J_y[:,idxs_x]\n",
    "    J_y_z = J_y[:,idxs_z]\n",
    "    J_y_u = J_y[:,idxs_u]\n",
    "    #\n",
    "    # Finding LTI Matrices assuming differential variables are states\n",
    "    # -- A and B matrices\n",
    "    AA = simplify.(J_d_x - J_d_z*(J_a_z\\J_a_x))\n",
    "    BB = simplify.(J_d_u - J_d_z*(J_a_z\\J_a_u))\n",
    "    # -- C and D matrices\n",
    "    CC = simplify.(J_y_x - J_y_z*(J_a_z\\J_a_x))\n",
    "    DD = simplify.(J_y_u - J_y_z*(J_a_z\\J_a_u))\n",
    "    #\n",
    "    #-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#\n",
    "    # Checking whether optional states are requested\n",
    "    #-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#-#\n",
    "    n_x = length(idxs_x)\n",
    "    n_ξ = length(x_opt)\n",
    "    #\n",
    "    if n_ξ == 0\n",
    "        return AA, BB, CC, DD, vars[idxs_x]\n",
    "    elseif n_ξ != n_x\n",
    "        println(\"Incorrect size of optional state\")\n",
    "        return AA, BB, CC, DD, vars[idxs_x]\n",
    "    else\n",
    "        # Finding transformation matrix TT from original to new state\n",
    "        # -- indices for new state\n",
    "        idxs_ξ = [findmax(isequal.(x,vars))[2] for x in x_opt]\n",
    "        # Jacobian matrix for new state; contains ones and zeros\n",
    "        J_ξ = zeros(n_ξ,n_v)\n",
    "        for i in 1:n_ξ\n",
    "            J_ξ[i,:] = isequal.(x_opt[i],vars)\n",
    "        end\n",
    "        J_ξ_x = J_ξ[:,idxs_x]\n",
    "        J_ξ_z = J_ξ[:,idxs_z]\n",
    "        J_ξ_u = J_ξ[:,idxs_u]\n",
    "        #\n",
    "        TT = simplify.(J_ξ_x - J_ξ_z*(J_a_z\\J_a_x))\n",
    "        TT_u = simplify.(J_ξ_u - J_ξ_z*(J_a_z\\J_a_u))\n",
    "        # -- not allowed to have inputs in ξ, i.e., must require TT_u ≡ 0\n",
    "        if sum(isequal.(isequal.(TT_u,0),0)) != 0\n",
    "            println(\"Optional states lead to input derivatives\")\n",
    "            return AA, BB, CC, DD, vars[idxs_x]\n",
    "        end\n",
    "        AA_ = simplify.(TT*AA*inv(TT))\n",
    "        BB_ = simplify.(TT*BB)\n",
    "        CC_ = simplify.(CC*inv(TT))\n",
    "        DD_ = DD\n",
    "        #\n",
    "        return AA_, BB_, CC_, DD_, x_opt\n",
    "    end\n",
    "end\n",
    ";"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Constant area tank with influent\n",
    "When the model has an influent, it will not become empty, so there is little chance of the level $h$ to become negative by numerical error. Because of this, we can now use the simple expression for effluent.\n",
    "\n",
    "### Input functions"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will use two input functions: one with constant value of 2, `ṁ_const(t)`, and one which starts at value 2 and then jumps down to value 1.5 at time = 1, `ṁ_step(t)`. The actual influent function is denoted `ṁ(t)`, and we set `ṁ(t)` equal to one of `ṁ_const(t)` and `ṁ_step(t)`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "ṁ_const(t) = 2\n",
    "ṁ_step(t) = t < 1 ? 2 : 1.5    # meaning: if t < 1, then 2, else 1.5\n",
    ";"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Constant influent\n",
    "We create a function `ṁ(t)`which equals `ṁ_const(t)`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "ṁ(t) = ṁ_const(t);"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because function `ṁ(t)` is unknown to `ModelingToolkit.jl`, it must be *registered* (standard built-in Julia functions are by default registered) before we can use it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "@register_symbolic ṁ(t)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Model"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now create the model with influent `ṁ_i` which we will set equal to function `ṁ`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{align}\n",
       "\\frac{\\mathrm{d} m\\left( t \\right)}{\\mathrm{d}t} =&  - \\textnormal{\\.{m}}_{e}\\left( t \\right) + \\textnormal{\\.{m}}_{i}\\left( t \\right) \\\\\n",
       "m\\left( t \\right) =& \\rho V\\left( t \\right) \\\\\n",
       "V\\left( t \\right) =& A h\\left( t \\right) \\\\\n",
       "\\textnormal{\\.{m}}_{e}\\left( t \\right) =& K \\sqrt{\\frac{h\\left( t \\right)}{h_{\\varsigma}}} \\\\\n",
       "\\textnormal{\\.{m}}_{i}\\left( t \\right) =& \\textnormal{\\.{m}}\\left( t \\right)\n",
       "\\end{align}\n",
       " $$"
      ],
      "text/plain": [
       "\u001b[0m\u001b[1mModel tank_i with 5 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
       "\u001b[0m\u001b[1mStates (5):\u001b[22m\n",
       "  m(t) [defaults to 1.5A*ρ]\n",
       "  ṁ_e(t)\n",
       "  ṁ_i(t)\n",
       "  V(t)\n",
       "⋮\n",
       "\u001b[0m\u001b[1mParameters (4):\u001b[22m\n",
       "  ρ [defaults to 1]\n",
       "  A [defaults to 5]\n",
       "  h_ς [defaults to 3]\n",
       "  K [defaults to 5]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@parameters ρ=1 A=5 K=5 h_ς=3\n",
    "@variables t m(t)=1.5*ρ*A ṁ_i(t) ṁ_e(t) V(t) h(t)\n",
    "Dt = Differential(t)\n",
    "\n",
    "eqs_p = [Dt(m) ~ ṁ_i-ṁ_e,\n",
    "        m ~ ρ*V,\n",
    "        V ~ A*h,\n",
    "        ṁ_e ~ K*sqrt(h/h_ς)]\n",
    "\n",
    "@named tank_p = ODESystem(eqs_p)    # Partial tank model, i.e., without inputs\n",
    "\n",
    "eqs_i = [ṁ_i ~ ṁ(t)]\n",
    "@named tank_i = ODESystem(eqs_i)\n",
    "\n",
    "tank = extend(tank_i, tank_p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{align}\n",
       "\\frac{\\mathrm{d} m\\left( t \\right)}{\\mathrm{d}t} =&  - \\textnormal{\\.{m}}_{e}\\left( t \\right) + \\textnormal{\\.{m}}_{i}\\left( t \\right)\n",
       "\\end{align}\n",
       " $$"
      ],
      "text/plain": [
       "\u001b[0m\u001b[1mModel tank_i with 1 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
       "\u001b[0m\u001b[1mStates (1):\u001b[22m\n",
       "  m(t) [defaults to 1.5A*ρ]\n",
       "\u001b[0m\u001b[1mParameters (4):\u001b[22m\n",
       "  ρ [defaults to 1]\n",
       "  A [defaults to 5]\n",
       "  h_ς [defaults to 3]\n",
       "  K [defaults to 5]\n",
       "\u001b[35mIncidence matrix:\u001b[39m1×2 SparseArrays.SparseMatrixCSC{Num, Int64} with 2 stored entries:\n",
       " ×  ×"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tank_simp = structural_simplify(tank)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{align}\n",
       "\\frac{\\mathrm{d} m\\left( t \\right)}{\\mathrm{d}t} =&  - K \\sqrt{\\frac{m\\left( t \\right)}{A h_{\\varsigma} \\rho}} + \\textnormal{\\.{m}}\\left( t \\right)\n",
       "\\end{align}\n",
       " $$"
      ],
      "text/plain": [
       "1-element Vector{Equation}:\n",
       " Differential(t)(m(t)) ~ ṁ(t) - K*sqrt(m(t) / (A*h_ς*ρ))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "full_equations(tank_simp)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Running simulation to find (numeric) steady state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "tspan = (0.0,1e3)\n",
    "\n",
    "prob = ODEProblem(tank_simp, [], tspan)\n",
    "sol = solve(prob)\n",
    "name_s = states(tank)\n",
    "s_ss = sol(tspan[2]; idxs = name_s)\n",
    "m_ss = sol(tspan[2]; idxs = m)\n",
    "h_ss = sol(tspan[2]; idxs = h)\n",
    ";"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Symbolic linearization of model\n",
    "#### Default state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{align}\n",
       "\\frac{\\mathrm{d} m\\left( t \\right)}{\\mathrm{d}t} =&  - \\textnormal{\\.{m}}_{e}\\left( t \\right) + \\textnormal{\\.{m}}_{i}\\left( t \\right) \\\\\n",
       "m\\left( t \\right) =& \\rho V\\left( t \\right) \\\\\n",
       "V\\left( t \\right) =& A h\\left( t \\right) \\\\\n",
       "\\textnormal{\\.{m}}_{e}\\left( t \\right) =& K \\sqrt{\\frac{h\\left( t \\right)}{h_{\\varsigma}}}\n",
       "\\end{align}\n",
       " $$"
      ],
      "text/plain": [
       "\u001b[0m\u001b[1mModel tank_p with 4 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
       "\u001b[0m\u001b[1mStates (5):\u001b[22m\n",
       "  m(t) [defaults to 1.5A*ρ]\n",
       "  ṁ_e(t)\n",
       "  ṁ_i(t)\n",
       "  V(t)\n",
       "⋮\n",
       "\u001b[0m\u001b[1mParameters (4):\u001b[22m\n",
       "  ρ [defaults to 1]\n",
       "  A [defaults to 5]\n",
       "  h_ς [defaults to 3]\n",
       "  K [defaults to 5]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tank_p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Vector{SymbolicUtils.BasicSymbolic{Real}}:\n",
       " m(t)\n",
       " ṁ_e(t)\n",
       " ṁ_i(t)\n",
       " V(t)\n",
       " h(t)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "states(tank_p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-element Vector{Any}:\n",
       " m(t)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "u_tank = [ṁ_i]\n",
    "y_tank = [h]\n",
    "x_tank = [h]\n",
    "\n",
    "As, Bs, Cs, Ds, x_name = linearize_symbolic_dae01(eqs_p, u_tank, y_tank)\n",
    "x_name"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{c}\n",
       "\\frac{ - K}{2 A h_{\\varsigma} \\rho \\sqrt{\\frac{h\\left( t \\right)}{h_{\\varsigma}}}} \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "1×1 Matrix{Num}:\n",
       " (-K) / (2A*h_ς*ρ*sqrt(h(t) / h_ς))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "As"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{c}\n",
       "1.0 \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "1×1 Matrix{Num}:\n",
       " 1.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Bs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{c}\n",
       "\\frac{1}{A \\rho} \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "1×1 Matrix{Num}:\n",
       " 1 / (A*ρ)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Cs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{c}\n",
       "0.0 \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "1×1 Matrix{Num}:\n",
       " 0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Ds"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Specified state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{c}\n",
       "h\\left( t \\right) \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "1-element Vector{Num}:\n",
       " h(t)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A1s, B1s, C1s, D1s, x_name = linearize_symbolic_dae01(eqs_p, u_tank, y_tank, x_opt=x_tank)\n",
    "x_name"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{c}\n",
       "\\frac{ - K}{2 A h_{\\varsigma} \\rho \\sqrt{\\frac{h\\left( t \\right)}{h_{\\varsigma}}}} \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "1×1 Matrix{Num}:\n",
       " (-K) / (2A*h_ς*ρ*sqrt(h(t) / h_ς))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A1s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{c}\n",
       "\\frac{1}{A \\rho} \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "1×1 Matrix{Num}:\n",
       " 1 / (A*ρ)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "B1s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{c}\n",
       "1 \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "1×1 Matrix{Num}:\n",
       " 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "C1s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{c}\n",
       "0.0 \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "1×1 Matrix{Num}:\n",
       " 0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "D1s"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Numeric evaluation of symbolically linearized model\n",
    "#### Default state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 5 entries:\n",
       "  h_ς  => 3\n",
       "  K    => 5\n",
       "  m(t) => 1.5A*ρ\n",
       "  A    => 5\n",
       "  ρ    => 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "default_vals = ModelingToolkit.defaults(tank_p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Vector{SymbolicUtils.BasicSymbolic{Real}}:\n",
       " m(t)\n",
       " ṁ_e(t)\n",
       " ṁ_i(t)\n",
       " V(t)\n",
       " h(t)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "name_s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×1 Matrix{Float64}:\n",
       " -0.4166666666386296"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Asn = substitute(As, Dict(name_s .=> s_ss)) |> x -> substitute(x, default_vals) |> x -> Symbolics.unwrap.(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×1 Matrix{Float64}:\n",
       " 1.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Bsn = substitute(Bs, Dict(name_s .=> s_ss)) |> x -> substitute(x, default_vals) |> x -> Symbolics.unwrap.(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×1 Matrix{Float64}:\n",
       " 0.2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Csn = substitute(Cs, default_vals) |> x -> Symbolics.unwrap.(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×1 Matrix{Float64}:\n",
       " 0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Dsn = substitute(Ds, default_vals) |> x -> Symbolics.unwrap.(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Specified state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×1 Matrix{Float64}:\n",
       " -0.4166666666386296"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A1sn = substitute(A1s, Dict(name_s .=> s_ss)) |> x -> substitute(x, default_vals) |> x -> Symbolics.unwrap.(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×1 Matrix{Float64}:\n",
       " 0.2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "B1sn = substitute(B1s, Dict(name_s .=> s_ss)) |> x -> substitute(x, default_vals) |> x -> Symbolics.unwrap.(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×1 Matrix{Int64}:\n",
       " 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "C1sn = substitute(C1s, Dict(name_s .=> s_ss)) |> x -> substitute(x, default_vals) |> x -> Symbolics.unwrap.(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×1 Matrix{Float64}:\n",
       " 0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "D1sn = substitute(D1s, Dict(name_s .=> s_ss)) |> x -> substitute(x, default_vals) |> x -> Symbolics.unwrap.(x)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transfer Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}\n",
       "   0.19999999999999996\n",
       "-------------------------\n",
       "1.0s + 0.4166666666386296\n",
       "\n",
       "Continuous-time transfer function model"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sys_sn = ss(Asn,Bsn,Csn,Dsn) |> tf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}\n",
       "   0.19999999999999996\n",
       "-------------------------\n",
       "1.0s + 0.4166666666386296\n",
       "\n",
       "Continuous-time transfer function model"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sys1_sn = ss(A1sn,B1sn,C1sn,D1sn) |> tf"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Time constant from symbolic expression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\frac{2 A h_{\\varsigma} \\rho \\sqrt{\\frac{h\\left( t \\right)}{h_{\\varsigma}}}}{K}\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "(2A*h_ς*ρ*sqrt(h(t) / h_ς)) / K"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "τ = -1/A1s[]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.4000000001614934"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "substitute(τ, Dict(name_s .=> s_ss)) |> x -> substitute(x, default_vals) |> x -> Symbolics.unwrap.(x)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linearizing the model using MTK"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "StateSpace{Continuous, Float64}\n",
       "A = \n",
       " -0.4166666666386296\n",
       "B = \n",
       " 1.0\n",
       "C = \n",
       " 0.2\n",
       "D = \n",
       " -0.0\n",
       "\n",
       "Continuous-time state-space model"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mats, tank_ = linearize(tank_p, [ṁ_i], [h], op=Dict(m=>m_ss, ṁ_i=>ṁ(0)))\n",
    "sys = ss(mats...)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Any}:\n",
       " m(t)\n",
       " ṁ_i(t)\n",
       " h(t)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "[states(tank_); inputs(tank_); outputs(tank_)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}\n",
       "   0.19999999999999996\n",
       "-------------------------\n",
       "1.0s + 0.4166666666386296\n",
       "\n",
       "Continuous-time transfer function model"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tf(sys)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linearization using ControlSystemsMTK"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "NamedStateSpace{Continuous, Float64}\n",
       "A = \n",
       " -0.4166666666386296\n",
       "B = \n",
       " 1.0\n",
       "C = \n",
       " 0.2\n",
       "D = \n",
       " -0.0\n",
       "\n",
       "Continuous-time state-space model\n",
       "With state  names: m(t)\n",
       "     input  names: ṁ_i(t)\n",
       "     output names: h(t)\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "named_ss(tank_p, [ṁ_i], [h]; op=Dict(m=>m_ss, ṁ_i=>ṁ(0)))"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Time constant"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.4000000001614934"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "λ = eigvals(sys.A)[]\n",
    "# Timeconstant(s)\n",
    "\n",
    "τ = 1.0/abs.(λ)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting nonlinear & linearized model with time constant"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "ṁ(t) = ṁ_step(t)\n",
    "\n",
    "tspan = (0.0,1e3)\n",
    "\n",
    "prob = ODEProblem(tank_simp, [m_ss], tspan)\n",
    "sol = solve(prob)\n",
    "m_ss_ = sol(tspan[2]; idxs = m)\n",
    "h_ss_ = sol(tspan[2]; idxs = h)\n",
    ";"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "tspan = (0.0,10)\n",
    "prob = ODEProblem(tank_simp, [m_ss], tspan)\n",
    "sol = solve(prob, QBDF())\n",
    "#\n",
    "# Non-linear solution\n",
    "plot(sol, idxs=h, lw=LW1, lc=LC1, label=L\"h_\\mathrm{nonlin}\")\n",
    ";"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Finding linear case steady level after step\n",
    "h_δ,t,x,ṁ_δ = lsim(sys, (x,t) -> [ṁ_step(t) - ṁ_step(0)], 0:0.01:1e3)\n",
    "h_δ_ss_ = h_δ[end]\n",
    "# Simulating linear case\n",
    "h_δ,t,x,ṁ_δ = lsim(sys, (x,t) -> [ṁ_step(t) - ṁ_step(0)], 0:0.01:10)\n",
    "\n",
    "# Tangent line\n",
    "h_tangent(t) = h_ss + (t-1)*h_δ_ss_/τ\n",
    ";"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Non-linear vs linear solution\n",
    "plot!(t, permutedims(h_δ).+h_ss, lw=LW1, lc=LC2, label=L\"h_\\mathrm{lin}\")\n",
    "# Initial value\n",
    "hline!([h_ss], lw=LW2, lc=LC3, ls=LS2, label=L\"h(0)\")\n",
    "# Final value: nonlinear vs. linear\n",
    "hline!([h_ss_], lw=LW2, lc=LC1, ls=LS2, la=LA2,label=L\"h_\\mathrm{nonlin}(\\infty)\")\n",
    "hline!([h_ss + h_δ_ss_], lw=LW2, lc=LC2, ls=LS2, la=LA2,label=L\"h_\\mathrm{lin}(\\infty)\")\n",
    "# Time where step takes place & step time + time constant\n",
    "vline!([1], lw=LW2, lc=LC3,ls=LS3, label=L\"t_\\mathrm{step}\")\n",
    "vline!([1+τ], lw=LW2, lc=LC4,ls=LS3, label=L\"t_\\mathrm{step}+\\tau\")\n",
    "# Tanget line for linear model at step\n",
    "plot!(h_tangent, lw=LW2, lc=:gray,ls=LS3, label=L\"h_\\mathrm{tangent}\")\n",
    "# 63% rulers for non-linear and linear case\n",
    "hline!([h_ss + (1-exp(-1))*(h_ss_-h_ss)], lw=LW1, lc=LC1,ls=LS3, la=LA2, label=L\"\\Delta h^\\mathrm{nonlin,ss}_{63\\%}\")\n",
    "hline!([h_ss + (1-exp(-1))*h_δ_ss_], lw=LW1, lc=LC2,ls=LS3, la=LA2, label=L\"\\Delta h^\\mathrm{lin,ss}_{63\\%}\")\n",
    "# Touching up with axis labels and title\n",
    "plot!(xlabel=L\"time $t$ [s]\", ylabel = \"[dm]\")\n",
    "plot!(title=L\"Constant area: Nonlinear vs. linear model; $\\tau$\")\n",
    "# Restricting ylims because of tangent line\n",
    "plot!(ylims=((h_ss+h_δ_ss_)*0.95, h_ss*1.05))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "if SAVEFIG\n",
    "    figname = \"drivenTank_Analysis\"\n",
    "    savefig(figpath*figname*figtype)\n",
    "end\n",
    ";"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Varying area tank with input - linearization"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{align}\n",
       "\\frac{\\mathrm{d} h\\left( t \\right)}{\\mathrm{d}t} =& \\frac{ - \\textnormal{\\.{m}}_{e}\\left( t \\right) + \\textnormal{\\.{m}}_{i}\\left( t \\right)}{\\rho \\mathrm{Ar}\\left( t \\right)} \\\\\n",
       "\\mathrm{Ar}\\left( t \\right) =& \\left( r\\left( t \\right) \\right)^{2} \\mathrm{identity}\\left( \\pi \\right) \\\\\n",
       "r\\left( t \\right) =& \\frac{R_{\\varsigma} h\\left( t \\right)}{h_{\\varsigma}} \\\\\n",
       "\\textnormal{\\.{m}}_{e}\\left( t \\right) =& K \\sqrt{\\frac{h\\left( t \\right)}{h_{\\varsigma}}} \\\\\n",
       "\\textnormal{\\.{m}}_{i}\\left( t \\right) =& \\textnormal{\\.{m}}\\left( t \\right)\n",
       "\\end{align}\n",
       " $$"
      ],
      "text/plain": [
       "\u001b[0m\u001b[1mModel tank_i with 5 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
       "\u001b[0m\u001b[1mStates (5):\u001b[22m\n",
       "  h(t) [defaults to 1.5]\n",
       "  ṁ_e(t)\n",
       "  ṁ_i(t)\n",
       "  Ar(t)\n",
       "⋮\n",
       "\u001b[0m\u001b[1mParameters (4):\u001b[22m\n",
       "  ρ [defaults to 1]\n",
       "  R_ς [defaults to 2.1851]\n",
       "  h_ς [defaults to 3]\n",
       "  K [defaults to 5]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@parameters ρ=1 R_ς=sqrt(15/π) K=5 h_ς=3\n",
    "@variables t h(t)=1.5 ṁ_i(t) ṁ_e(t) r(t) Ar(t)\n",
    "\n",
    "ṁ(t) = ṁ_const(t)\n",
    "\n",
    "eqs_p = [Dt(h) ~ (ṁ_i-ṁ_e)/(ρ*Ar),\n",
    "        Ar ~ Num(π)*r^2,\n",
    "        r ~ R_ς/h_ς*h,\n",
    "        ṁ_e ~ K*sqrt(h/h_ς)]\n",
    "\n",
    "@named tank_p = ODESystem(eqs_p)    # Partial model\n",
    "\n",
    "eqs_i = [ṁ_i ~ ṁ(t)]\n",
    "@named tank_i = ODESystem(eqs_i)    # Inputs: connecting driving functions to input variables\n",
    "\n",
    "tank = extend(tank_i, tank_p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{align}\n",
       "\\frac{\\mathrm{d} h\\left( t \\right)}{\\mathrm{d}t} =& \\frac{ - \\textnormal{\\.{m}}_{e}\\left( t \\right) + \\textnormal{\\.{m}}_{i}\\left( t \\right)}{\\rho \\mathrm{Ar}\\left( t \\right)}\n",
       "\\end{align}\n",
       " $$"
      ],
      "text/plain": [
       "\u001b[0m\u001b[1mModel tank_i with 1 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
       "\u001b[0m\u001b[1mStates (1):\u001b[22m\n",
       "  h(t) [defaults to 1.5]\n",
       "\u001b[0m\u001b[1mParameters (4):\u001b[22m\n",
       "  ρ [defaults to 1]\n",
       "  R_ς [defaults to 2.1851]\n",
       "  h_ς [defaults to 3]\n",
       "  K [defaults to 5]\n",
       "\u001b[35mIncidence matrix:\u001b[39m1×2 SparseArrays.SparseMatrixCSC{Num, Int64} with 2 stored entries:\n",
       " ×  ×"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tank_simp = structural_simplify(tank)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-element Vector{Any}:\n",
       " h(t)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "states(tank_simp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Finding steady state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "tspan = (0.0,1e3)\n",
    "\n",
    "prob = ODEProblem(tank_simp, [], tspan)\n",
    "sol = solve(prob)\n",
    "name_s = states(tank)\n",
    "s_ss = sol(tspan[2]; idxs = name_s)\n",
    "h_ss = sol(tspan[2]; idxs = h)\n",
    ";"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Symbolic linearization of model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{align}\n",
       "\\frac{\\mathrm{d} h\\left( t \\right)}{\\mathrm{d}t} =& \\frac{ - \\textnormal{\\.{m}}_{e}\\left( t \\right) + \\textnormal{\\.{m}}_{i}\\left( t \\right)}{\\rho \\mathrm{Ar}\\left( t \\right)} \\\\\n",
       "\\mathrm{Ar}\\left( t \\right) =& \\left( r\\left( t \\right) \\right)^{2} \\mathrm{identity}\\left( \\pi \\right) \\\\\n",
       "r\\left( t \\right) =& \\frac{R_{\\varsigma} h\\left( t \\right)}{h_{\\varsigma}} \\\\\n",
       "\\textnormal{\\.{m}}_{e}\\left( t \\right) =& K \\sqrt{\\frac{h\\left( t \\right)}{h_{\\varsigma}}}\n",
       "\\end{align}\n",
       " $$"
      ],
      "text/plain": [
       "\u001b[0m\u001b[1mModel tank_p with 4 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
       "\u001b[0m\u001b[1mStates (5):\u001b[22m\n",
       "  h(t) [defaults to 1.5]\n",
       "  ṁ_e(t)\n",
       "  ṁ_i(t)\n",
       "  Ar(t)\n",
       "⋮\n",
       "\u001b[0m\u001b[1mParameters (4):\u001b[22m\n",
       "  ρ [defaults to 1]\n",
       "  R_ς [defaults to 2.1851]\n",
       "  h_ς [defaults to 3]\n",
       "  K [defaults to 5]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tank_p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Vector{SymbolicUtils.BasicSymbolic{Real}}:\n",
       " h(t)\n",
       " ṁ_e(t)\n",
       " ṁ_i(t)\n",
       " Ar(t)\n",
       " r(t)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "states(tank_p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-element Vector{Any}:\n",
       " h(t)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "u_tank = [ṁ_i]    # Indices for inputs\n",
    "y_tank = [h]    # Indices for outputs\n",
    "\n",
    "As, Bs, Cs, Ds, x_name = linearize_symbolic_dae01(eqs_p, u_tank, y_tank)\n",
    "x_name"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{c}\n",
       "\\frac{ - K \\mathrm{Ar}\\left( t \\right) + 12.566370614359172 R_{\\varsigma} r\\left( t \\right) \\textnormal{\\.{m}}_{e}\\left( t \\right) \\sqrt{\\frac{h\\left( t \\right)}{h_{\\varsigma}}} - 12.566370614359172 R_{\\varsigma} r\\left( t \\right) \\textnormal{\\.{m}}_{i}\\left( t \\right) \\sqrt{\\frac{h\\left( t \\right)}{h_{\\varsigma}}}}{2 \\left( \\mathrm{Ar}\\left( t \\right) \\right)^{2} h_{\\varsigma} \\rho \\sqrt{\\frac{h\\left( t \\right)}{h_{\\varsigma}}}} \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "1×1 Matrix{Num}:\n",
       " (12.566370614359172R_ς*r(t)*ṁ_e(t)*sqrt(h(t) / h_ς) - K*Ar(t) - 12.566370614359172R_ς*r(t)*ṁ_i(t)*sqrt(h(t) / h_ς)) / (2h_ς*ρ*(Ar(t)^2)*sqrt(h(t) / h_ς))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "As"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{c}\n",
       "\\frac{1}{\\rho \\mathrm{Ar}\\left( t \\right)} \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "1×1 Matrix{Num}:\n",
       " 1 / (ρ*Ar(t))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Bs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{c}\n",
       "1.0 \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "1×1 Matrix{Num}:\n",
       " 1.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Cs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{c}\n",
       "0.0 \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "1×1 Matrix{Num}:\n",
       " 0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Ds"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Numeric evaluation of symbolically linearized model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 5 entries:\n",
       "  R_ς  => 2.1851\n",
       "  h_ς  => 3\n",
       "  K    => 5\n",
       "  h(t) => 1.5\n",
       "  ρ    => 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "default_vals = ModelingToolkit.defaults(tank_p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×1 Matrix{Float64}:\n",
       " -5.425347222222794"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Asn = substitute(As, Dict(name_s .=> s_ss)) |> x -> substitute(x, default_vals) |> x -> Symbolics.unwrap.(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×1 Matrix{Float64}:\n",
       " 2.6041666666667878"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Bsn = substitute(Bs, Dict(name_s .=> s_ss)) |> x -> substitute(x, default_vals) |> x -> Symbolics.unwrap.(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×1 Matrix{Float64}:\n",
       " 1.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Csn = substitute(Cs, Dict(name_s .=> s_ss)) |> x -> substitute(x, default_vals) |> x -> Symbolics.unwrap.(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×1 Matrix{Float64}:\n",
       " 0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Dsn = substitute(Ds, Dict(name_s .=> s_ss)) |> x -> substitute(x, default_vals) |> x -> Symbolics.unwrap.(x)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transfer Function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}\n",
       "   2.6041666666667878\n",
       "------------------------\n",
       "1.0s + 5.425347222222794\n",
       "\n",
       "Continuous-time transfer function model"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sys_sn = ss(Asn,Bsn,Csn,Dsn) |> tf"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Time constant from symbolic expression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\frac{ - 2 \\left( \\mathrm{Ar}\\left( t \\right) \\right)^{2} h_{\\varsigma} \\rho \\sqrt{\\frac{h\\left( t \\right)}{h_{\\varsigma}}}}{ - K \\mathrm{Ar}\\left( t \\right) + 12.566370614359172 R_{\\varsigma} r\\left( t \\right) \\textnormal{\\.{m}}_{e}\\left( t \\right) \\sqrt{\\frac{h\\left( t \\right)}{h_{\\varsigma}}} - 12.566370614359172 R_{\\varsigma} r\\left( t \\right) \\textnormal{\\.{m}}_{i}\\left( t \\right) \\sqrt{\\frac{h\\left( t \\right)}{h_{\\varsigma}}}}\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "(-2h_ς*ρ*(Ar(t)^2)*sqrt(h(t) / h_ς)) / (12.566370614359172R_ς*r(t)*ṁ_e(t)*sqrt(h(t) / h_ς) - K*Ar(t) - 12.566370614359172R_ς*r(t)*ṁ_i(t)*sqrt(h(t) / h_ς))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "τ = -1/As[]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.18431999999998058"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "substitute(τ, Dict(name_s .=> s_ss)) |> x -> substitute(x, default_vals) |> x->Symbolics.unwrap.(x)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linearization using MTK"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "StateSpace{Continuous, Float64}\n",
       "A = \n",
       " -5.425347222222793\n",
       "B = \n",
       " 2.6041666666667878\n",
       "C = \n",
       " 1.0\n",
       "D = \n",
       " 0.0\n",
       "\n",
       "Continuous-time state-space model"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mats, tank_ = linearize(tank_p, [ṁ_i], [h], op=Dict(h=>h_ss, ṁ_i=>ṁ(0)))\n",
    "sys = ss(mats...)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Any}:\n",
       " h(t)\n",
       " ṁ_i(t)\n",
       " h(t)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "[states(tank_); inputs(tank_); outputs(tank_)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TransferFunction{Continuous, ControlSystemsBase.SisoRational{Float64}}\n",
       "   2.6041666666667886\n",
       "------------------------\n",
       "1.0s + 5.425347222222793\n",
       "\n",
       "Continuous-time transfer function model"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tf(sys)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linearization using ControlSystemsMTK"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "NamedStateSpace{Continuous, Float64}\n",
       "A = \n",
       " -5.425347222222793\n",
       "B = \n",
       " 2.6041666666667878\n",
       "C = \n",
       " 1.0\n",
       "D = \n",
       " 0.0\n",
       "\n",
       "Continuous-time state-space model\n",
       "With state  names: h(t)\n",
       "     input  names: ṁ_i(t)\n",
       "     output names: h(t)\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "named_ss(tank_p, [ṁ_i], [h]; op=Dict(h=>h_ss, ṁ_i=>ṁ(0)))"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Time constant"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.18431999999998058"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "λ = eigvals(sys.A)[]\n",
    "# Timeconstant(s)\n",
    "\n",
    "τ = 1.0/abs.(λ)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting nonlinear & linearized model with time constant"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "ṁ(t) = ṁ_step(t)\n",
    "\n",
    "tspan = (0.0,1e3)\n",
    "\n",
    "prob = ODEProblem(tank_simp, [h_ss], tspan)\n",
    "sol = solve(prob, QBDF())\n",
    "h_ss_ = sol(tspan[2]; idxs = h)\n",
    ";"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "tspan = (0.0,10)\n",
    "prob = ODEProblem(tank_simp, [h_ss], tspan)\n",
    "sol = solve(prob, QBDF())\n",
    "#\n",
    "# Non-linear solution\n",
    "plot(sol, idxs=h, lw=LW1, lc=LC1, label=L\"h_\\mathrm{nonlin}\")\n",
    ";"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Finding linear case steady level after step\n",
    "h_δ,t,x,ṁ_δ = lsim(sys, (x,t) -> [ṁ_step(t) - ṁ_step(0)], 0:0.01:1e3)\n",
    "h_δ_ss_ = h_δ[end]\n",
    "# Simulating linear case\n",
    "h_δ,t,x,ṁ_δ = lsim(sys, (x,t) -> [ṁ_step(t) - ṁ_step(0)], 0:0.01:10)\n",
    "\n",
    "# Tangent line\n",
    "h_tangent(t) = h_ss + (t-1)*h_δ_ss_/τ\n",
    ";"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Non-linear vs linear solution\n",
    "plot!(t, permutedims(h_δ).+h_ss, lw=LW1, lc=LC2, label=L\"h_\\mathrm{lin}\")\n",
    "# Initial value\n",
    "hline!([h_ss], lw=LW2, lc=LC3, ls=LS2, label=L\"h(0)\")\n",
    "# Final value: nonlinear vs. linear\n",
    "hline!([h_ss_], lw=LW2, lc=LC1, ls=LS2, la=LA2,label=L\"h_\\mathrm{nonlin}(\\infty)\")\n",
    "hline!([h_ss + h_δ_ss_], lw=LW2, lc=LC2, ls=LS2, la=LA2,label=L\"h_\\mathrm{lin}(\\infty)\")\n",
    "# Time where step takes place & step time + time constant\n",
    "vline!([1], lw=LW2, lc=LC3,ls=LS3, label=L\"t_\\mathrm{step}\")\n",
    "vline!([1+τ], lw=LW2, lc=LC4,ls=LS3, label=L\"t_\\mathrm{step}+\\tau\")\n",
    "# Tanget line for linear model at step\n",
    "plot!(h_tangent, lw=LW2, lc=:gray,ls=LS3, label=L\"h_\\mathrm{tangent}\")\n",
    "# 63% rulers for non-linear and linear case\n",
    "hline!([h_ss + (1-exp(-1))*(h_ss_-h_ss)], lw=LW1, lc=LC1,ls=LS3, la=LA2, label=L\"\\Delta h^\\mathrm{nonlin,ss}_{63\\%}\")\n",
    "hline!([h_ss + (1-exp(-1))*h_δ_ss_], lw=LW1, lc=LC2,ls=LS3, la=LA2, label=L\"\\Delta h^\\mathrm{lin,ss}_{63\\%}\")\n",
    "# Touching up with axis labels and title\n",
    "plot!(xlabel=L\"time $t$ [s]\", ylabel = \"[dm]\")\n",
    "plot!(title=L\"Varying area: Nonlinear vs. linear model; $\\tau$\")\n",
    "# Restricting ylims because of tangent line\n",
    "plot!(ylims=((h_ss+h_δ_ss_)*0.95, h_ss*1.05), xlims=(0.5,2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [],
   "source": [
    "if SAVEFIG\n",
    "    figname = \"drivenFunnelTank_Analysis\"\n",
    "    savefig(figpath*figname*figtype)\n",
    "end\n",
    ";"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Testing `linearize_symbolic` for `ModelingToolkit` "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [],
   "source": [
    "#=\n",
    "using Pkg\n",
    "Pkg.activate(temp=true)\n",
    "Pkg.add(\"ModelingToolkit#master\")\n",
    "matrices, _ = ModelingToolkit.linearize_symbolic(tank_p, [ṁ_i], [h])\n",
    "=#"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
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