{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Quadratically optimal control on a fixed and finite time horizon for a linear discrete-time system - direct formulations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we consider a standard discrete-time LTI system modelled by state equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\mathbf x_{k+1} = \\mathbf A \\mathbf x_k + \\mathbf B \\mathbf u_k,\\qquad \\mathbf x_0 = \\mathrm{given},$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $\\mathbf x\\in\\mathbb{R}^n$, $\\mathbf u\\in\\mathbb{R}^m$, $\\mathbf A\\in\\mathbb{R}^{n\\times n}$ and $\\mathbf B\\in\\mathbb{R}^{n\\times m}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We want to minimize the popular quadratic criterion "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " \n",
    "$$\\underset{\\mathbf u_0,\\ldots, \\mathbf u_{N-1}, (\\mathbf x_{0}),\\ldots, \\mathbf x_N}{\\mathrm{minimize}} \\quad \\frac{1}{2} \\mathbf x_N^T \\mathbf S \\mathbf x_N + \\frac{1}{2} \\sum_{k=0}^{N-1} \\left(\\mathbf x_k^T \\mathbf Q \\mathbf x_k + \\mathbf u_k^T \\mathbf R \\mathbf u_k \\right)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "for a given initial value vector $\\mathbf x_0\\in\\mathbb{R}^n$ and weighting matrices $\\mathbf S\\geq 0$, $\\mathbf Q\\geq 0$ and $\\mathbf R>0$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below we reformulate the optimal control problem as a numerical optimization problem. We offer two formulations: simultaneous and sequential."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simultaneous (sparse) formulation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The adjective comes from the fact that we solve the optimization over the states $x$ and controls $u$ simultaneously. It is also refered to as sparse because the corresponding matrices are sparse, which can be exploited by numerical solvers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Building the linear discrete-time dynamical system to be controlled"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Discrete-time model of a double integrator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Float64,2}:\n",
       " 1.0  1.0\n",
       " 0.0  1.0"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = [1.0 1.0;0.0 1.0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       " 0.0\n",
       " 1.0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = [0.0, 1.0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Number of states"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Number of inputs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An initial state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Int64,1}:\n",
       " 1\n",
       " 3"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = [1, 3] "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Setting the discrete-time LQ optimal control problem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we set the time horizon on which the control problem is solved. You are invited to experiment with this value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = 10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Within the quadratic cost function we restrict ourselves to diagonal matrices $S$, $Q$ and $R$. Practically speaking, this is not a serious restriction because we are not able to explot the fullness of those matrices anyway. The diagonals of the three matrices are"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Int64,1}:\n",
       " 1\n",
       " 2"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s = [1, 2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Int64,1}:\n",
       " 1\n",
       " 2"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = [1, 2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-element Array{Int64,1}:\n",
       " 1"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r = [1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Feel free to experiment with these numbers (and rerun the code)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We assemble the matrices from the diagonals here."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Int64,2}:\n",
       " 1  0\n",
       " 0  2"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S = diagm(0=>s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Int64,2}:\n",
       " 1  0\n",
       " 0  2"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q = diagm(0=>q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×1 Array{Int64,2}:\n",
       " 1"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = diagm(0=>r)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Formulating the quadratic optimization problem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will use some functionality for building block and sparse matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "using BlockArrays, SparseArrays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "Qbar = BlockArray(spzeros(N*n,N*n),repeat([n],N),repeat([n],N));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i=1:(N-1)\n",
    "    Qbar[Block(i,i)] = Q\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Int64,2}:\n",
       " 1  0\n",
       " 0  2"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Qbar[Block(N,N)] = S"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "Rbar = BlockArray(spzeros(N*m,N*m),repeat([m],N),repeat([m],N));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i=1:N\n",
    "    Rbar[Block(i,i)] = R\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "Qtilde = blockdiag(sparse(Qbar),sparse(Rbar));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "Bbar = BlockArray(spzeros(N*n,N*m),repeat([n],N),repeat([m],N));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i=1:N\n",
    "    Bbar[Block(i,i)] = sparse(B)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "Abar = BlockArray(sparse(-1.0*I,n*N,n*N),repeat([n],N),repeat([n],N));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i=2:N\n",
    "    Abar[Block(i,(i-1))] = sparse(A)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that here `Abar` is actually `Abar-I` from the lecture notes. I combined them into a single variable. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "Atilde = sparse([Abar Bbar]); "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "A0bar = spzeros(n*N,n); A0bar[1:n,1:n] = A;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "btilde = A0bar*sparse(x0);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solving the optimization problem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are about to solve the problem \n",
    "\\begin{align}\n",
    " \\underset{\\tilde{\\mathbf{x}}\\in\\mathbb{R}^{2N}}{\\text{minimize}} &\\qquad\\qquad \\frac{1}{2}\\tilde{\\mathbf{x}}^T \\tilde{\\mathbf{Q}} \\tilde{\\mathbf{x}}\\\\\n",
    " \\text {subject to} &\\qquad\\qquad \\tilde{\\mathbf{A}} \\tilde{\\mathbf{x}} + \\tilde{\\mathbf{b}} = \\mathbf 0.\n",
    " \\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we can either proceed on our own and formulate the KKT linear system and solve it using a specialized solver for symmetric but possibly indefinite systems (relying on LDL factorization), or we can resort to calling some QP solver. We opt for the latter since we would be forced to do it as soon as we want to add some additional inequality-type constraints. We choose [Convex.jl](https://github.com/JuliaOpt/Convex.jl) package."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Convex, SCS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Variable\n",
       "size: (30, 1)\n",
       "sign: real\n",
       "vexity: affine\n",
       "id: 151…642"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xtilde = Variable((n+m)*N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "minimize\n",
       "└─ * (convex; positive)\n",
       "   ├─ 0.5\n",
       "   └─ * (convex; positive)\n",
       "      ├─ 1\n",
       "      └─ qol_elem (convex; positive)\n",
       "         ├─ …\n",
       "         └─ …\n",
       "subject to\n",
       "└─ == constraint (affine)\n",
       "   ├─ + (affine; real)\n",
       "   │  ├─ * (affine; real)\n",
       "   │  │  ├─ …\n",
       "   │  │  └─ …\n",
       "   │  └─ 20-element SparseVector{Float64,Int64}\n",
       "   └─ 0\n",
       "\n",
       "status: `solve!` not called yet"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "problem = minimize(1/2*quadform(xtilde,Qtilde),[Atilde*xtilde + btilde == 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "solve!(problem, () -> SCS.Optimizer(verbose=false))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "OPTIMAL::TerminationStatusCode = 1"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "problem.status"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×10 Array{Float64,2}:\n",
       "  4.0       2.90983   1.63933    0.827815  …   0.0161349    0.00855227\n",
       " -1.09017  -1.2705   -0.811515  -0.434556     -0.00758277  -0.00252759"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xopt = reshape(xtilde.value[1:(n*N)],(2,:))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×10 Array{Float64,2}:\n",
       " -4.09017  -0.180336  0.458988  0.376959  …  0.0254833  0.0116684  0.00505516"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "uopt = reshape(xtilde.value[(n*N+1):end],(m,:))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting the responses"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Plots.PyPlotBackend()"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pyplot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(0:N,hcat(x0,xopt)',marker=:diamond,label=[\"x1\" \"x2\"],linetype=:steppost)\n",
    "xlabel!(\"k\")\n",
    "ylabel!(\"x\") # beware the conflict in notation: x_1 vs x_k. I could have used x(k) for dependence on the discrete time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(0:(N-1),uopt',marker=:diamond,label=\"u\",linetype=:steppost)\n",
    "xlabel!(\"k\")\n",
    "ylabel!(\"u\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sequential (condensed) formulation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the alternative formulation of the optimization problem, we eliminate the states altogether by expressing them as functions of the inputs. The resulting optimization is then conducted only over the controls $u$ and (and the initial state $x_0$ should it be left free). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{equation}\n",
    " \\tilde J(\\mathbf u,\\mathbf x_0) = \\frac{1}{2}\\mathbf u^T\\mathbf H\\mathbf u +  \\mathbf x_0^T\\mathbf F^T \\mathbf u.\n",
    "\\end{equation} "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If no additional constraints are imposed (in particular, the bound constraints on the controls), the problem boils down to solving linear equations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\mathbf H\\mathbf u=-\\mathbf F\\mathbf x_0,$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which can be formally written as"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\mathbf u = -\\mathbf H^{-1} \\mathbf F \\mathbf x_0$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "but it goes without saying that we are certainly not going to compute the invers of the matrix $\\mathbf H$,"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Building the linear system (set of linear equations)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "Chat = BlockArray(spzeros(N*n,N*m),repeat([n],N),repeat([m],N));\n",
    "Ahat = BlockArray(spzeros(N*n,n),repeat([n],N),[n]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i=1:N\n",
    "    for j = 1:i\n",
    "        Chat[Block(i,j)] = sparse(A^(i-j)*B)\n",
    "        Ahat[Block(i,1)] = sparse(A^i)\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "H = Chat'*Qbar*Chat + Rbar;\n",
    "H = Array(H); "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "F = Chat'*Qbar*Ahat;\n",
    "F = Array(F);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solving the linear system"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The crucial computation step - solving a set of linear equations - comes now"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "uopts = -H\\(F*x0);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is worth noting that the size of the problem is $n\\times n$ and the data (the matrix) is not sparse any more."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting the responses"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we plot the optimal control sequence and notice that it is identical to the one computed in the simultaneous (sparse) framework."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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"
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(0:(N-1),uopts,marker=:diamond,label=\"u_seq\",linetype=:steppost)\n",
    "xlabel!(\"k\")\n",
    "ylabel!(\"u\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×10 reshape(::PseudoBlockArray{…,::Array{Float64,1}}, 2, 10) with eltype Float64:\n",
       "  4.0       2.90983   1.63933    0.827815  …   0.016135     0.00855228\n",
       " -1.09017  -1.2705   -0.811515  -0.434556     -0.00758275  -0.00252758"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xopts = Chat*uopts + Ahat*x0;\n",
    "xopts = reshape(xopts,(2,:))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(0:N,hcat(x0,xopts)',marker=:diamond,label=[\"x1\" \"x2\"],linetype=:steppost)\n",
    "xlabel!(\"k\")\n",
    "ylabel!(\"x\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Imposing constraints on controls"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If additional constraints are imposed, namely, the bound constraints on $\\mathbf u$, the problem becomes "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Variable\n",
       "size: (10, 1)\n",
       "sign: real\n",
       "vexity: affine\n",
       "id: 731…720"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = Variable(N*m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "umax = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "minimize\n",
       "└─ + (convex; real)\n",
       "   ├─ * (convex; positive)\n",
       "   │  ├─ 0.5\n",
       "   │  └─ * (convex; positive)\n",
       "   │     ├─ …\n",
       "   │     └─ …\n",
       "   └─ sum (affine; real)\n",
       "      └─ .* (affine; real)\n",
       "         ├─ …\n",
       "         └─ …\n",
       "subject to\n",
       "└─ <= constraint (convex)\n",
       "   ├─ abs (convex; positive)\n",
       "   │  └─ 10-element real variable (id: 731…720)\n",
       "   └─ 1\n",
       "\n",
       "status: `solve!` not called yet"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "problem = minimize(1/2*quadform(u,H) + dot(F*x0,u),[abs(u) <= umax])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "solve!(problem, () -> SCS.Optimizer(verbose=false))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "OPTIMAL::TerminationStatusCode = 1"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "problem.status"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(0:(N-1),u.value,marker=:diamond,label=\"u_seq_constr\",linetype=:steppost)\n",
    "xlabel!(\"k\")\n",
    "ylabel!(\"u\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×10 reshape(::PseudoBlockArray{…,::Array{Float64,2}}, 2, 10) with eltype Float64:\n",
       " 4.0  6.0  7.0          7.0   6.0  …   1.10341    0.55488    0.318435\n",
       " 2.0  1.0  6.63161e-8  -1.0  -2.0     -0.548532  -0.236446  -0.0788149"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xopts = Chat*u.value + Ahat*x0;\n",
    "xopts = reshape(xopts,(2,:))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(0:N,hcat(x0,xopts)',marker=:diamond,label=[\"x1\" \"x2\"],linetype=:steppost)\n",
    "xlabel!(\"k\")\n",
    "ylabel!(\"x\")"
   ]
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   "source": [
    "## References\n",
    "\n",
    "[1] ..."
   ]
  }
 ],
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