Notebook for the control class on Linear Quadratic Regulator

LQR.ipynb

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    "<img src=\"https://assets.ensam.eu/logo/fr/logo-trans-322x84.png\" style=\"width:256px\" >"
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    "# **Linear Quadratic Regulator**\n",
    "\n",
    "Let us consider a discrete-time linear time invariant dynamical system\n",
    "\n",
    "$$\n",
    "    \\mathbf{x}_{t+1} = \\mathbf{Ax}_t + \\mathbf{Bu}_t\n",
    "$$\n",
    "\n",
    "where $\\mathbf{x} \\in \\mathbb{R}^n$ is the state-vector of the system, and $\\mathbf{u} \\in \\mathbb{R}^p$ its inputs.\n",
    "The matrix $\\mathbf{A} \\in \\mathbb{R}^{n \\times n}$ describes the natural dynamics of the system while $\\mathbf{B} \\in \\mathbb{R}^{n \\times p}$ describes how our set of actuators can influence the dynamics.\n",
    "Our objective in this notebook will be to find a control law $\\left\\{ \\mathbf{u}_t \\right\\}_{t=0, T-1}$ solution to the following optimization problem\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "    \\mathrm{minimize} & \\quad \\dfrac12 \\sum_{t=0}^{T-1} \\left( \\mathbf{x}^*_t \\mathbf{Q} \\mathbf{x}_t + \\mathbf{u}_t^* \\mathbf{R} \\mathbf{u}_t \\right) + \\dfrac12 \\mathbf{x}^*_T \\mathbf{Q} \\mathbf{x}_T \\\\\n",
    "    \\mathrm{subject~to} & \\quad \\mathbf{x}_{t+1} = \\mathbf{Ax}_t + \\mathbf{Bu}_t \\quad \\forall t \\in \\left[ 0, T-1 \\right] \\\\\n",
    "                        & \\quad \\mathbf{x}_0 = \\mathbf{x}(0),\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "where $\\mathbf{Q} \\in \\mathbb{R}^{n \\times n}$ is a symmetric positive semi-definite matrix and $\\mathbf{R} \\in \\mathbb{R}^{p \\times p}$ is symmetric positive definite.\n",
    "We will solve this problem using three different approaches, namely:\n",
    "1. Gradient-based optimization,\n",
    "2. Dynamic Programming,\n",
    "3. Ricatti-based feedback control for $T \\to \\infty$.\n",
    "\n",
    "The system will consider is the inverted pendulum on a cart, also known as **cartpole**.\n",
    "It is a classic problem in control theory.\n",
    "A depiction of this sytem is shown below\n",
    "\n",
    "<img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/0/00/Cart-pendulum.svg/300px-Cart-pendulum.svg.png\" style=\"margin:auto\"/>\n",
    "\n",
    "</br>\n",
    "\n",
    "Starting from Newton's principles, the equations of motion linearized in the vicinity of the upright equilibrium position are given in state-space form by\n",
    "\n",
    "$$\n",
    "\\dfrac{\\mathrm{d}}{\\mathrm{d}t}\n",
    "\\begin{bmatrix}\n",
    " x \\\\ \\dot{x} \\\\ \\vartheta \\\\ \\dot{\\vartheta}\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    " 0 & 1 & 0 & 0 \\\\\n",
    " 0 & -\\dfrac{\\delta}{M} & \\dfrac{mg}{M} & 0 \\\\\n",
    " 0 & 0 & 0 & 1 \\\\\n",
    " 0 & -\\dfrac{\\delta}{ML} & -\\dfrac{(m+M)g}{ML} & 0 \\\\\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    " x \\\\ \\dot{x} \\\\ \\vartheta \\\\ \\dot{\\vartheta}\n",
    "\\end{bmatrix}\n",
    "+\n",
    "\\begin{bmatrix}\n",
    "0 \\\\ \\dfrac{1}{M} \\\\ 0 \\\\ \\dfrac{1}{ML}\n",
    "\\end{bmatrix}\n",
    "u(t)\n",
    "$$\n",
    "\n",
    "where $x \\in \\mathbb{R}$ denotes the cart's horizontal position and $\\dot{x}$ its velocity.\n",
    "Likewise, $\\vartheta$ denotes the angular position of the pendulum (with $\\vartheta = 0$ being the upright position) and $\\dot{\\vartheta}$ its angular velocity.\n",
    "Additionally, $\\delta$ denotes the friction coefficient between the cart and its rails, $M$ is the mass of the cart, $m$ that of the pendulum, $L$ its length and $g$ is the gravity.\n",
    "Our objective thus is to find the sequence of lateral forces $\\mathbf{F}_t$ (hereafter denoted $\\mathbf{u}_t$ with the standard control theory notation) such that the pendulum stays in the upward position.\n",
    "\n",
    "In what follows, we will use `scipy` built-in functions to automatically discretize these equations and obtain a discrete-time linear time invariant dynamical system.\n",
    "The cell below imports all of the packages and/or functions you will need.\n",
    "It also sets up the state-space model will use throughout this notebook."
   ]
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    "# --> Standard Python packages.\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# --> Convex optimization in Python.\n",
    "import cvxpy as cp\n",
    "\n",
    "# --> SciPy built-in minimization function.\n",
    "from scipy.optimize import minimize\n",
    "\n",
    "# --> LTI system related utilities.\n",
    "from scipy.signal import lti, dlti, dlsim"
   ]
  },
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    "# --> State-space model.\n",
    "def state_space_model(params, dt=0.01):\n",
    "    \"\"\"\n",
    "    This function sets up the state-space model for CartPole.\n",
    "\n",
    "    INPUT\n",
    "    -----\n",
    "\n",
    "    params :    python list\n",
    "                Physical parameters of the system. \n",
    "\n",
    "    dt     :    float\n",
    "                Sampling period for the discretization of the dynamics.\n",
    "\n",
    "    RETURN\n",
    "    ------\n",
    "\n",
    "    dynsys :    scipy.signal.dlti\n",
    "                State-space model encapsulated in the scipy-specific object.\n",
    "    \"\"\"\n",
    "\n",
    "    # --> Unpack parameters.\n",
    "    m, M, L, g, δ = params\n",
    "\n",
    "    # --> Dynamics matrix.\n",
    "    A = np.array([\n",
    "        [0.0, 1.0, 0.0, 0.0],                # Equation for the velocity dx/dt = ẋ\n",
    "        [0, -δ/M, m*g/M, 0.0],               # Equation for the cart's acceleration.\n",
    "        [0.0, 0.0, 0.0, 1.0],                # Equation for the pendulum angular velocity dθ/dt = θ̇\n",
    "        [0.0, -δ/(M*L), -(m+M)*g/(M*L), 0.0] # Equation for the pendulum angular acceleration.\n",
    "    ])\n",
    "\n",
    "    # --> Input-to-state matrix.\n",
    "    B = np.array([\n",
    "        [0.0],\n",
    "        [1.0/M],\n",
    "        [0.0],\n",
    "        [1.0/(M*L)]\n",
    "    ])\n",
    "\n",
    "    # --> Measurement operator.\n",
    "    C = np.eye(4)\n",
    "\n",
    "    # --> Feedthrough matrix.\n",
    "    D = np.zeros((4, 1))\n",
    "\n",
    "    # --> Continuous-time state space model.\n",
    "    ssm = lti(A, B, C, D)\n",
    "\n",
    "    # --> Transform to a discrete-time system.\n",
    "    ssm = ssm.to_discrete(dt, method=\"bilinear\")\n",
    "\n",
    "    return ssm"
   ]
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    "# --> Parameters of the system.\n",
    "params = m, M, L, g, δ = 1, 5, 2, -10, 1\n",
    "\n",
    "# --> Create the discrete-time state-space model.\n",
    "ssm = state_space_model(params)"
   ]
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    "We also define the $\\mathbf{Q}$ and $\\mathbf{R}$ matrices in the cell below.\n",
    "Different versions of these matrices are provided.\n",
    "You can play with these to get a better intuition of their physical meaning."
   ]
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    "# --> Choose between case = 0, 1, 2.\n",
    "case = 0\n",
    "\n",
    "if case == 0:\n",
    "    #################################################################\n",
    "    #####     STATE COST MORE IMPORTANT THAN ACTUATION COST     #####\n",
    "    #################################################################\n",
    "    \n",
    "    # --> SPD matrix for the state cost.\n",
    "    Q = np.eye(4)\n",
    "    \n",
    "    # --> SPD matrix for the actuation cost.\n",
    "    R = 1e-3 * np.eye(1)\n",
    "\n",
    "elif case == 1:\n",
    "    #################################################################\n",
    "    #####     ACTUATION COST MORE IMPORTANT THAN STATE COST     #####\n",
    "    #################################################################\n",
    "    \n",
    "    # --> State-cost.\n",
    "    Q = 1e-3 * np.eye(4)\n",
    "    \n",
    "    # --> Actuation cost.\n",
    "    R = np.eye(1)\n",
    "\n",
    "elif case == 2:\n",
    "    ###################################################################\n",
    "    #####     STATE COST AND ACTUATION COST EQUALLY IMPORTANT     #####\n",
    "    ###################################################################\n",
    "\n",
    "    # --> State-cost.\n",
    "    Q = np.eye(4)\n",
    "\n",
    "    # --> Actuation cost.\n",
    "    R = np.eye(1)"
   ]
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    "## **Finite-horizon optimal control**\n",
    "\n",
    "As a starting point, let us consider the finite-horizon problem (i.e. $T < \\infty$), i.e. the same as before:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "    \\mathrm{minimize} & \\quad \\dfrac12 \\sum_{t=0}^{T-1} \\left( \\mathbf{x}^*_t \\mathbf{Q} \\mathbf{x}_t + \\mathbf{u}_t^* \\mathbf{R} \\mathbf{u}_t \\right) + \\dfrac12 \\mathbf{x}^*_T \\mathbf{Q} \\mathbf{x}_T \\\\\n",
    "    \\mathrm{subject~to} & \\quad \\mathbf{x}_{t+1} = \\mathbf{Ax}_t + \\mathbf{Bu}_t \\quad \\forall t \\in \\left[ 0, T-1 \\right] \\\\\n",
    "                        & \\quad \\mathbf{x}_0 = \\mathbf{x}(0).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "We will use different approaches to solve this problem in order to illustrate their pros and cons.\n",
    "These three approaches are\n",
    "1. Gradient-based optimization using `scipy.optimize.minimize`\n",
    "2. Convex optimization with `cvxpy`\n",
    "3. Dynamic Programming\n",
    "\n",
    "For the time being, all the different solvers are already implemented for you.\n",
    "Study them carefully as you will have to implement similar solvers eventually.\n",
    "The cell below sets up some of the parameters for the computation."
   ]
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    "# --> Time-horizon.\n",
    "Th = 5.0\n",
    "\n",
    "# --> Number of time-steps.\n",
    "nt = int(Th / ssm.dt) + 1\n",
    "\n",
    "# --> Initial condition.\n",
    "x0 = np.array([\n",
    "    0.0,     # Initial position of the cart.\n",
    "    0.0,     # Initial velocity of the cart.\n",
    "    np.pi/4, # Initial angular position (w.r.t to the upward equilbrium)\n",
    "    0.0      # Initial angular velocity.\n",
    "])"
   ]
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    "### **Gradient-based optimization**\n",
    "\n",
    "As a starting point, let us treat this problem as a generic optimization problem and use gradient-based algorithms to solve it.\n",
    "For that purpose, we will need a `python` function returning:\n",
    "1. `cost`: the cost associated with the current estimate of the control law\n",
    "2. `grad`: the corresponding gradient. Note that we first need to actually express this gradient.\n",
    "\n",
    "The LQR optimization problem is an equality constrained quadratic program.\n",
    "We can transform it into an unconstrained problem by using *Lagrange multipliers*.\n",
    "This leads to\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\mathrm{minimize} \\quad & \\dfrac12 \\sum_{t=0}^{T-1} \\left( \\mathbf{x}_t^* \\mathbf{Q} \\mathbf{x}_t + \\mathbf{u}_t^* \\mathbf{R} \\mathbf{u}_t \\right) & \\text{(Cost to go)} \\\\\n",
    "                        & + \\dfrac12 \\mathbf{x}_T^* \\mathbf{Q} \\mathbf{x}_T & \\text{(Final cost)} \\\\\n",
    "                        & + \\sum_{t=0}^{T-1} \\mathbf{p}_{t+1}^* \\left( \\mathbf{A} \\mathbf{x}_t + \\mathbf{B} \\mathbf{u}_t - \\mathbf{x}_{t+1} \\right) & \\text{(Dynamical constraints)} \\\\\n",
    "                        & + \\mathbf{p}^*_0 \\left( \\mathbf{x}(0) - \\mathbf{x}_0 \\right) & \\text{(Initial condition)}.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "The new variables $\\mathbf{p}_i \\in \\mathbb{R}^n$ are the Lagrange multipliers used to enforce the dynamics of the system in the objective function.\n",
    "Let $\\mathcal{L}$ be this augmented objective function.\n",
    "The first order optimality conditions with respect to the actuation signal $\\left\\{ \\mathbf{u}_t \\right\\}_{t=0}^{T-1}$ read\n",
    "\n",
    "$$\n",
    "\\left.\n",
    "\\begin{aligned}\n",
    "\\nabla_{\\mathbf{u}_0} \\mathcal{L} & \\equiv \\mathbf{R} \\mathbf{u}_0 + \\mathbf{B}^* \\mathbf{p}_1 = 0\\\\\n",
    "\\nabla_{\\mathbf{u}_1} \\mathcal{L} & \\equiv \\mathbf{R} \\mathbf{u}_1 + \\mathbf{B}^* \\mathbf{p}_2 = 0\\\\\n",
    "\\nabla_{\\mathbf{u}_2} \\mathcal{L} & \\equiv \\mathbf{R} \\mathbf{u}_2 + \\mathbf{B}^* \\mathbf{p}_3 = 0\\\\\n",
    "\\vdots & \\\\\n",
    "\\nabla_{\\mathbf{u}_{T-1}} \\mathcal{L} & \\equiv \\mathbf{R} \\mathbf{u}_{T-1} + \\mathbf{B}^* \\mathbf{p}_T = 0.\n",
    "\\end{aligned}\n",
    "\\right\\} \\quad \\text{(Optimality w.r.t. the actuation sequence)}\n",
    "$$\n",
    "\n",
    "Likewise, the first order optimality conditions with respect to $\\left\\{ \\mathbf{x}_t \\right\\}_{t=0, T}$ read\n",
    "\n",
    "$$\n",
    "\\left.\n",
    "\\begin{aligned}\n",
    "\\nabla_{\\mathbf{x}_0} \\mathcal{L} & \\equiv \\mathbf{Q} \\mathbf{x}_0 + \\mathbf{A}^* \\mathbf{p}_1 - \\mathbf{p}_0 = 0 \\\\\n",
    "\\nabla_{\\mathbf{x}_1} \\mathcal{L} & \\equiv \\mathbf{Q} \\mathbf{x}_1 + \\mathbf{A}^* \\mathbf{p}_2 - \\mathbf{p}_1 = 0 \\\\\n",
    "\\nabla_{\\mathbf{x}_2} \\mathcal{L} & \\equiv \\mathbf{Q} \\mathbf{x}_2 + \\mathbf{A}^* \\mathbf{p}_3 - \\mathbf{p}_2 = 0 \\\\\n",
    "\\vdots & \\\\\n",
    "\\nabla_{\\mathbf{x}_{T-1}} \\mathcal{L} & \\equiv \\mathbf{Q} \\mathbf{x}_{T-1} + \\mathbf{A}^* \\mathbf{p}_T - \\mathbf{p}_{T-1} = 0 \\\\\n",
    "\\nabla_{\\mathbf{x}_T} \\mathcal{L} & \\equiv \\mathbf{Q} \\mathbf{x}_T - \\mathbf{p}_T = 0.\n",
    "\\end{aligned}\n",
    "\\right\\} \\quad \\text{(Optimality w.r.t. to the state sequence)}\n",
    "$$\n",
    "\n",
    "Finally, the first order optimality conditions with respect to the set of Lagrange multipliers simply correspond to the dynamical constraints $\\mathbf{x}_{t+1} = \\mathbf{Ax}_t + \\mathbf{Bu}_t$.\n",
    "It needs to be emphasized that the optimality conditions with respect to the state sequence $\\left\\{ \\mathbf{x}_t \\right\\}_{t=0, T}$ are a bit peculiar in that they actually define a linear system integrated backward in time with the state sequence as input.\n",
    "Starting from $\\mathbf{p}_T = \\mathbf{Q} \\mathbf{x}_T$, this backward recursion is given by\n",
    "\n",
    "$$\n",
    "\\mathbf{p}_{t-1} = \\mathbf{A}^* \\mathbf{p}_t + \\mathbf{Q} \\mathbf{x}_{t-1} \\quad \\forall t \\in \\left[ 1, T \\right],\n",
    "$$\n",
    "\n",
    "In the control theory community, the Lagrange multiplier $\\mathbf{p} \\in \\mathbb{R}^n$ is often denoted as the *co-state* or *adjoint state vector*.\n",
    "From these different conditions, we can devise the following procedure to evaluate the gradient of the cost function with respect to the input sequence $\\left\\{ \\mathbf{u}_t \\right\\}_{t=0, T-1}$ we wish to optimize.\n",
    "This procedure requires three steps:\n",
    "1. Given the current estimate $\\left\\{ \\mathbf{u}_t \\right\\}_{t=0, T-1}$ of the optimal actuation signal, integrate the system $(\\mathbf{x}_{t+1} = \\mathbf{Ax}_t + \\mathbf{B} \\mathbf{u}_t)$ forward in time with initial condition $\\mathbf{x}_0 = \\mathbf{x}(0)$ to obtain the state sequence $\\left\\{ \\mathbf{x}_t \\right\\}_{t=0, T}$.\n",
    "2. Given the state sequence $\\left\\{ \\mathbf{x}_t \\right\\}_{t=0, T}$ just computed, integrate the adjoint system $(\\mathbf{p}_{t-1} = \\mathbf{A}^* \\mathbf{p}_t + \\mathbf{Q}\\mathbf{x}_{t-1})$ backward in time starting from the final condition $\\mathbf{p}_T = \\mathbf{Q} \\mathbf{x}_T$.\n",
    "3. Given this adjoint state sequence $\\left\\{ \\mathbf{p}_t \\right\\}_{t=1, T}$, evaluate each component of the gradient of the cost function with respect to the actuation signal as $\\nabla_{\\mathbf{u}_t} \\mathcal{L} = \\mathbf{R} \\mathbf{u}_t + \\mathbf{B}^* \\mathbf{p}_{t+1}$.\n",
    "\n",
    "This procedure is implemented in the function below.\n",
    "Once this gradient has been evaluated, you can pass it on to any gradient-based optimizer you like (e.g. [BFGS](https://en.wikipedia.org/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm), [Conjugate Gradient](https://en.wikipedia.org/wiki/Conjugate_gradient_method), etc)."
   ]
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    "def cost_function(u, x0, ssm, Q, R):\n",
    "    \"\"\"\n",
    "    Procedure to evaluate the LQR cost function.\n",
    "\n",
    "    INPUT\n",
    "    -----\n",
    "\n",
    "    u : numpy array, shape (p, nt)\n",
    "        Control law to be evaluated.\n",
    "\n",
    "    x0 : numpy array, shape (n,)\n",
    "         Initial condition of the system.\n",
    "\n",
    "    ssm : scipy dlti object\n",
    "          State-space model of the system to be controlled.\n",
    "\n",
    "    Q   : numpy array, shape (n, n)\n",
    "          Symmetric positive (semi-) definite matrix associated with the state cost.\n",
    "\n",
    "    R   : numpy array, shape (p, p)\n",
    "          Symmetric positive definite matrix associated with the actuation cost.\n",
    "\n",
    "    RETURN\n",
    "    ------\n",
    "\n",
    "    cost : float\n",
    "           Cost-to-go associated with the evaluated control law.\n",
    "\n",
    "    \n",
    "    \"\"\"\n",
    "    \n",
    "    # --> Simulate the system.\n",
    "    _, _, x = dlsim(ssm, u, x0=x0)\n",
    "\n",
    "    #################################################\n",
    "    #####     GRADIENT OF THE COST FUNCTION     #####\n",
    "    #################################################\n",
    "\n",
    "    # --> Definition of the adjoint system.\n",
    "    A, B = ssm.A, ssm.B\n",
    "    adjoint = dlti(A.T, Q, np.eye(4), np.zeros((4, 4)))\n",
    "\n",
    "    # --> Initialize data for the adjoint simulation.\n",
    "    p0 = Q @ x[-1]                                # Final condition for the adjoint.\n",
    "    v  = np.zeros_like(x) ; v[:-1] = x[:-1][::-1] # Adjoint input\n",
    "\n",
    "    # --> Run the adjoint simulation.\n",
    "    _, _, p = dlsim(adjoint, v, x0 = p0)\n",
    "\n",
    "    # --> Reverse the time evolution.\n",
    "    p = p[::-1]\n",
    "\n",
    "    # --> Compute the gradient.\n",
    "    grad = np.zeros_like(u)\n",
    "    grad[:-1] = u[:-1] @ R + p[1:] @ B # Note that scipy's u and p are already transposed.\n",
    "                                       # This is slightly different from the mathematical\n",
    "                                       # notation used during the class.\n",
    "\n",
    "    #################################\n",
    "    #####     COST FUNCTION     #####\n",
    "    #################################\n",
    "    \n",
    "    # --> Evaluate the cost function.\n",
    "    cost = 0.5*sum(x[i].T @ Q @ x[i] + u[i].T @ R @ u[i] for i in range(len(u)))\n",
    "    \n",
    "    return cost, grad"
   ]
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   "cell_type": "markdown",
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   "source": [
    "In order to have this function work with `scipy.optimize.minimize`, we need to transform its inputs and output a bit.\n",
    "The cell below illustrate how to do so."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "ced39799-3983-4503-a537-958a5ff00c7a",
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   "source": [
    "def scipy_objective_function(u_flatten, x0, ssm, Q, R):\n",
    "    \"\"\"\n",
    "    Utility function to wrap our definition of the cost function and its gradient\n",
    "    in a scipy.optimize.minimize compatible format.\n",
    "    \"\"\"\n",
    "\n",
    "    # --> Get the number of inputs to the system.\n",
    "    p = ssm.B.shape[1]\n",
    "\n",
    "    # --> Reshape the flatten control law.\n",
    "    u = u_flatten.reshape(-1, p)\n",
    "\n",
    "    # --> Evaluate the LQR cost function and its gradient.\n",
    "    cost, grad = cost_function(u, x0, ssm, Q, R)\n",
    "\n",
    "    # --> Flatten the gradient.\n",
    "    grad = grad.flatten()\n",
    "    \n",
    "    return cost, grad"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f342abbe-e6e3-471b-9ef4-582ed1dd2053",
   "metadata": {},
   "source": [
    "Let us create a wrapper around this function to simplify our life."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "6bf04263-7ed1-42a6-8b85-b7050413cf84",
   "metadata": {},
   "outputs": [],
   "source": [
    "def scipy_LQR(ssm, x0, Q, R, nt, u0=None):\n",
    "    \"\"\"\n",
    "    Implementation of the finite-horizon LQR optimization problem using scipy.\n",
    "\n",
    "    INPUT\n",
    "    -----\n",
    "\n",
    "    ssm : dlti object\n",
    "          State-space model considered.\n",
    "\n",
    "    x0  : numpy array, shape (n,)\n",
    "          Initial condition of the system.\n",
    "\n",
    "    Q   : numpy array, shape (n, n)\n",
    "          Symmetric positive (semi-) definite matrix for the state cost.\n",
    "\n",
    "    R   : numpy array, shape (p, p)\n",
    "          Symmetric positive definite matrix for the actuation cost.\n",
    "\n",
    "    nt  : int\n",
    "          Optimization horizon.\n",
    "\n",
    "    RETURN\n",
    "    ------\n",
    "\n",
    "    u    : numpy array, shape (nt, )\n",
    "           Optimized control law.\n",
    "\n",
    "    cost : float\n",
    "           Value of the cost function after optimization.\n",
    "    \"\"\"\n",
    "    \n",
    "    # --> Initial guess for the control law.\n",
    "    if u0 == None:\n",
    "        u = np.random.randn(nt, ssm.B.shape[1])\n",
    "    else:\n",
    "        u = u0\n",
    "    \n",
    "    # --> Run the optimizer.\n",
    "    results = minimize(\n",
    "        scipy_objective_function, # Function to be minimized (+ evaluation of the gradient).\n",
    "        u.flatten(),              # Initial guess for the control law.\n",
    "        args=(x0, ssm, Q, R),     # Arguments to be passed for the function to be minimized.\n",
    "        jac=True,                 # Specify that the function also evaluates the gradient.\n",
    "    )\n",
    "    \n",
    "    # --> Reshape the optimized control law.\n",
    "    u = results.x.reshape(nt, -1)\n",
    "\n",
    "    return u, results.fun"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "86166658-62b8-4236-bcf7-42253264a1d3",
   "metadata": {},
   "source": [
    "We can now use `scipy_LQR` to find the optimal control law."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "ae925798-cc99-4c4c-bb1c-6f599576fbd9",
   "metadata": {},
   "outputs": [],
   "source": [
    "u_opt, cost = scipy_LQR(ssm, x0, Q, R, nt)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "89b32093-94c8-474f-b122-8bc22c825847",
   "metadata": {},
   "source": [
    "Let us now simulate both the uncontrolled evolution of the system and the one resulting from the optimized control law and plot the time-series of the cart's position and pendulum's angular position."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "e293f504-b1a8-4bbf-9011-f5e511e4e774",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 800x400 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# --> Simulate the uncontrolled system.\n",
    "_, _, x_uncontrolled = dlsim(ssm, 0*u_opt, x0=x0)\n",
    "\n",
    "# --> Simulate the system with the optimized control law.\n",
    "t, _, x_opt = dlsim(ssm, u_opt, x0=x0)\n",
    "\n",
    "#####################################\n",
    "#####     PLOTS THE RESULTS     #####\n",
    "#####################################\n",
    "\n",
    "# --> Create figure.\n",
    "fig, axes = plt.subplots(3, 1, sharex=True, figsize=(8, 4))\n",
    "\n",
    "# --> Control law.\n",
    "axes[0].plot(t, u_opt)\n",
    "axes[0].set(ylim=(-abs(u_opt).max(), abs(u_opt).max()), ylabel=\"u(t)\")\n",
    "\n",
    "# --> Cart's position.\n",
    "axes[1].plot(t, x_uncontrolled[:, 0], label=\"x(t) w/o control\")\n",
    "axes[1].plot(t, x_opt[:, 0], label=\"x(t) w/ control\")\n",
    "\n",
    "axes[1].legend(loc=\"lower center\", ncols=2, bbox_to_anchor=(0.5, 2.25))\n",
    "axes[1].set(ylim=(-5, 5), ylabel=\"x(t)\")\n",
    "\n",
    "# --> Pendulum's angular position.\n",
    "axes[2].plot(t, x_uncontrolled[:, 2], label=\"θ(t) w/o control\")\n",
    "axes[2].plot(t, x_opt[:, 2], label=\"θ(t) w/ control\")\n",
    "\n",
    "axes[2].set(ylim=(-np.pi/2, np.pi/2), ylabel=\"θ(t)\")\n",
    "\n",
    "# --> General setup.\n",
    "axes[2].set(xlim=(0, Th), xlabel=\"Time t\")\n",
    "\n",
    "for ax in axes:\n",
    "    ax.axhline(0, c=\"k\", ls=\"--\", lw=0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3010a4ed-3189-4eaf-a6d9-756a59234db8",
   "metadata": {},
   "source": [
    "For a non-trained eye, understanding what the controlled system is doing from these figures might not be directly obvious.\n",
    "Because we consider a simple mechanical system, we can use `matplotlib` to create an animation of this evolution.\n",
    "This is done below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "5f9dacc1-9c96-44be-abde-8777435afdd7",
   "metadata": {},
   "outputs": [],
   "source": [
    "# --> Animation utility from matplotlib.\n",
    "import matplotlib.animation as animation\n",
    "from matplotlib.patches import Rectangle\n",
    "from IPython.display import HTML\n",
    "\n",
    "def cartpole_animation(x, params, dt=0.01):\n",
    "    # --> Extract parameters.\n",
    "    m, M, L, g, δ = params\n",
    "    \n",
    "    # --> Initialize figure.\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(8, 4))\n",
    "    \n",
    "    ax.set(xlim=(-10, 10), ylim=(-0.1, 5))\n",
    "    \n",
    "    # --> Initialize data.\n",
    "    line, = ax.plot([], [], color=\"black\", ls=\"-\", lw=4)\n",
    "    time_template = \"Time = %.1fs\"\n",
    "    time_text = ax.text(0.05, 0.9, \"\", transform=ax.transAxes)\n",
    "    \n",
    "    cart_width, cart_height = 2, 1\n",
    "    cart = Rectangle(\n",
    "        (x[0, 0] - cart_width/2, 0),\n",
    "        cart_width, cart_height, fill=True,\n",
    "        color=\"lightgray\",\n",
    "        )\n",
    "    \n",
    "    ax.add_patch(cart)\n",
    "    ax.set_aspect(\"equal\")\n",
    "    ax.axis(\"off\")\n",
    "    ax.axhline(0, c=\"k\", ls=\"-\", lw=1)\n",
    "    \n",
    "    def init():\n",
    "        line.set_data([], [])\n",
    "        time_text.set_text(\"\")\n",
    "        return line, time_text, cart\n",
    "    \n",
    "    def animate(i):\n",
    "        # --> Angular position.\n",
    "        θ = x[i, 2]\n",
    "        pendulum_x, pendulum_y = [x[i, 0], x[i, 0] - L*np.sin(θ)], [cart_height, L*np.cos(θ) + cart_height]\n",
    "    \n",
    "        line.set_data(pendulum_x, pendulum_y)\n",
    "        time_text.set_text(time_template % (i*dt))\n",
    "    \n",
    "        cart.set_xy((x[i, 0] - cart_width/2, 0))\n",
    "        return line, time_text, cart\n",
    "    \n",
    "    anim = animation.FuncAnimation(fig, animate, np.arange(len(x)-1), interval=25, blit=False, init_func=init)\n",
    "    plt.close()\n",
    "    return HTML(anim.to_jshtml())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "baab11f6-717c-4e8b-9db8-164a34204559",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<link rel=\"stylesheet\"\n",
       "href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/css/font-awesome.min.css\">\n",
       "<script language=\"javascript\">\n",
       "  function isInternetExplorer() {\n",
       "    ua = navigator.userAgent;\n",
       "    /* MSIE used to detect old browsers and Trident used to newer ones*/\n",
       "    return ua.indexOf(\"MSIE \") > -1 || ua.indexOf(\"Trident/\") > -1;\n",
       "  }\n",
       "\n",
       "  /* Define the Animation class */\n",
       "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
       "    this.img_id = img_id;\n",
       "    this.slider_id = slider_id;\n",
       "    this.loop_select_id = loop_select_id;\n",
       "    this.interval = interval;\n",
       "    this.current_frame = 0;\n",
       "    this.direction = 0;\n",
       "    this.timer = null;\n",
       "    this.frames = new Array(frames.length);\n",
       "\n",
       "    for (var i=0; i<frames.length; i++)\n",
       "    {\n",
       "     this.frames[i] = new Image();\n",
       "     this.frames[i].src = frames[i];\n",
       "    }\n",
       "    var slider = document.getElementById(this.slider_id);\n",
       "    slider.max = this.frames.length - 1;\n",
       "    if (isInternetExplorer()) {\n",
       "        // switch from oninput to onchange because IE <= 11 does not conform\n",
       "        // with W3C specification. It ignores oninput and onchange behaves\n",
       "        // like oninput. In contrast, Microsoft Edge behaves correctly.\n",
       "        slider.setAttribute('onchange', slider.getAttribute('oninput'));\n",
       "        slider.setAttribute('oninput', null);\n",
       "    }\n",
       "    this.set_frame(this.current_frame);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.get_loop_state = function(){\n",
       "    var button_group = document[this.loop_select_id].state;\n",
       "    for (var i = 0; i < button_group.length; i++) {\n",
       "        var button = button_group[i];\n",
       "        if (button.checked) {\n",
       "            return button.value;\n",
       "        }\n",
       "    }\n",
       "    return undefined;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.set_frame = function(frame){\n",
       "    this.current_frame = frame;\n",
       "    document.getElementById(this.img_id).src =\n",
       "            this.frames[this.current_frame].src;\n",
       "    document.getElementById(this.slider_id).value = this.current_frame;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.next_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.previous_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.first_frame = function()\n",
       "  {\n",
       "    this.set_frame(0);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.last_frame = function()\n",
       "  {\n",
       "    this.set_frame(this.frames.length - 1);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.slower = function()\n",
       "  {\n",
       "    this.interval /= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.faster = function()\n",
       "  {\n",
       "    this.interval *= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_forward = function()\n",
       "  {\n",
       "    this.current_frame += 1;\n",
       "    if(this.current_frame < this.frames.length){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.first_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.last_frame();\n",
       "        this.reverse_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.last_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_reverse = function()\n",
       "  {\n",
       "    this.current_frame -= 1;\n",
       "    if(this.current_frame >= 0){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.last_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.first_frame();\n",
       "        this.play_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.first_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.pause_animation = function()\n",
       "  {\n",
       "    this.direction = 0;\n",
       "    if (this.timer){\n",
       "      clearInterval(this.timer);\n",
       "      this.timer = null;\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.play_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = 1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_forward();\n",
       "    }, this.interval);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.reverse_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = -1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_reverse();\n",
       "    }, this.interval);\n",
       "  }\n",
       "</script>\n",
       "\n",
       "<style>\n",
       ".animation {\n",
       "    display: inline-block;\n",
       "    text-align: center;\n",
       "}\n",
       "input[type=range].anim-slider {\n",
       "    width: 374px;\n",
       "    margin-left: auto;\n",
       "    margin-right: auto;\n",
       "}\n",
       ".anim-buttons {\n",
       "    margin: 8px 0px;\n",
       "}\n",
       ".anim-buttons button {\n",
       "    padding: 0;\n",
       "    width: 36px;\n",
       "}\n",
       ".anim-state label {\n",
       "    margin-right: 8px;\n",
       "}\n",
       ".anim-state input {\n",
       "    margin: 0;\n",
       "    vertical-align: middle;\n",
       "}\n",
       "</style>\n",
       "\n",
       "<div class=\"animation\">\n",
       "  <img id=\"_anim_img755ea2af2f7146938d460829d35779bb\">\n",
       "  <div class=\"anim-controls\">\n",
       "    <input id=\"_anim_slider755ea2af2f7146938d460829d35779bb\" type=\"range\" class=\"anim-slider\"\n",
       "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
       "           oninput=\"anim755ea2af2f7146938d460829d35779bb.set_frame(parseInt(this.value));\">\n",
       "    <div class=\"anim-buttons\">\n",
       "      <button title=\"Decrease speed\" aria-label=\"Decrease speed\" onclick=\"anim755ea2af2f7146938d460829d35779bb.slower()\">\n",
       "          <i class=\"fa fa-minus\"></i></button>\n",
       "      <button title=\"First frame\" aria-label=\"First frame\" onclick=\"anim755ea2af2f7146938d460829d35779bb.first_frame()\">\n",
       "        <i class=\"fa fa-fast-backward\"></i></button>\n",
       "      <button title=\"Previous frame\" aria-label=\"Previous frame\" onclick=\"anim755ea2af2f7146938d460829d35779bb.previous_frame()\">\n",
       "          <i class=\"fa fa-step-backward\"></i></button>\n",
       "      <button title=\"Play backwards\" aria-label=\"Play backwards\" onclick=\"anim755ea2af2f7146938d460829d35779bb.reverse_animation()\">\n",
       "          <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
       "      <button title=\"Pause\" aria-label=\"Pause\" onclick=\"anim755ea2af2f7146938d460829d35779bb.pause_animation()\">\n",
       "          <i class=\"fa fa-pause\"></i></button>\n",
       "      <button title=\"Play\" aria-label=\"Play\" onclick=\"anim755ea2af2f7146938d460829d35779bb.play_animation()\">\n",
       "          <i class=\"fa fa-play\"></i></button>\n",
       "      <button title=\"Next frame\" aria-label=\"Next frame\" onclick=\"anim755ea2af2f7146938d460829d35779bb.next_frame()\">\n",
       "          <i class=\"fa fa-step-forward\"></i></button>\n",
       "      <button title=\"Last frame\" aria-label=\"Last frame\" onclick=\"anim755ea2af2f7146938d460829d35779bb.last_frame()\">\n",
       "          <i class=\"fa fa-fast-forward\"></i></button>\n",
       "      <button title=\"Increase speed\" aria-label=\"Increase speed\" onclick=\"anim755ea2af2f7146938d460829d35779bb.faster()\">\n",
       "          <i class=\"fa fa-plus\"></i></button>\n",
       "    </div>\n",
       "    <form title=\"Repetition mode\" aria-label=\"Repetition mode\" action=\"#n\" name=\"_anim_loop_select755ea2af2f7146938d460829d35779bb\"\n",
       "          class=\"anim-state\">\n",
       "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_755ea2af2f7146938d460829d35779bb\"\n",
       "             >\n",
       "      <label for=\"_anim_radio1_755ea2af2f7146938d460829d35779bb\">Once</label>\n",
       "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_755ea2af2f7146938d460829d35779bb\"\n",
       "             checked>\n",
       "      <label for=\"_anim_radio2_755ea2af2f7146938d460829d35779bb\">Loop</label>\n",
       "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_755ea2af2f7146938d460829d35779bb\"\n",
       "             >\n",
       "      <label for=\"_anim_radio3_755ea2af2f7146938d460829d35779bb\">Reflect</label>\n",
       "    </form>\n",
       "  </div>\n",
       "</div>\n",
       "\n",
       "\n",
       "<script language=\"javascript\">\n",
       "  /* Instantiate the Animation class. */\n",
       "  /* The IDs given should match those used in the template above. */\n",
       "  (function() {\n",
       "    var img_id = \"_anim_img755ea2af2f7146938d460829d35779bb\";\n",
       "    var slider_id = \"_anim_slider755ea2af2f7146938d460829d35779bb\";\n",
       "    var loop_select_id = \"_anim_loop_select755ea2af2f7146938d460829d35779bb\";\n",
       "    var frames = new Array(500);\n",
       "    \n",
       "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAGQCAYAAABWJQQ0AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90\\\n",
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       "\"\n",
       "  frames[1] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAGQCAYAAABWJQQ0AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90\\\n",
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