Updates and fixes

notebooks/nb_10_introduction.ipynb

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@@ -746,7 +759,12 @@
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     "# Optimal Control\n",
     "\n",
-    "Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. The fundamental idea in optimal control is to formulate the goal of control as the long-term optimization of a scalar cost function as opposed to formulating the objective as direct constraints on the system's behaviour (e.g. overshoot) as done in classical control.\n",
+    "Optimal control theory is a branch of control theory that emerged an independent field emerged in the 1950s. It deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. The fundamental idea in optimal control is to formulate the goal of control as the long-term optimization of a scalar cost function as opposed to formulating the objective as direct constraints on the system's behaviour (e.g. overshoot) as done in classical control.\n",
+    "\n",
+    "Optimal control is one of the most useful systematic methods for controller design. It has several advantages:\n",
+    "- It gives a systematic approach to the solution of control problems.\n",
+    "- There are normally many possible solutions to a control problem. Some are good, others are poor.\n",
+    "  Optimal control reduces this redundancy by selecting a controller that isoptimalbest according to some cost function.\n",
     "\n",
     "The optimal control problem is to find a control $u^* \\in \\mathbf{U}$ which causes the system $\\dot{x}(t) = f(x(t), u(t))$ to follow a trajectory $x^* \\in \\mathbf{X}$ that minimizes the cost (performance measure)."
    ]
@@ -802,11 +820,9 @@
     "- Infinite Horizon:\n",
     "\n",
     "  $$\n",
-    "  J(x_0, u) = \\sum \\limits_{k = 0}^{\\infty} \\gamma^k c_k(x_k, u_k)\n",
+    "  J(x_0, u) = \\sum \\limits_{k = 0}^{\\infty} c_k(x_k, u_k)\n",
     "  $$\n",
     "\n",
-    "  Where $0 < \\gamma \\le 1$ is a discount factor.\n",
-    "\n",
     "- Stochastic finite horizon:\n",
     "\n",
     "  $$\n",
@@ -855,10 +871,10 @@
     "Optimal control problems solving methods can be classified in three main families:\n",
     "Dynamic Programming (DP), Indirect Methods based on calculus of variation and Direct Methods.\n",
     "\n",
-    "- DP is helpful where the number of states is limited and the dynamics are known. It divides an optimal control issue into smaller subproblems and recursively solves each.\n",
+    "- DP is helpful where the number of states is limited and the dynamics are known. It divides an optimal control problem into smaller subproblems and recursively solves each one of them.\n",
     "\n",
-    "- Indirect methods rely on Pontryagin’s Minimum Principle (PMP) to derive the necessary conditions\n",
-    "  for optimality. This method uses the Hamiltonian of the system to reduce the global optimal control problem\n",
+    "- Indirect methods rely on Pontryagin’s Minimum Principle (PMP) to derive the necessary conditions for optimality.\n",
+    "  This method uses the Hamiltonian of the system to reduce the global optimal control problem\n",
     "  to the solution of a system of $2N$ equations given in the form of a two-point boundary value problem (BVP).  \n",
     "  Problems involving continuous states and control inputs benefit most from it.\n",
     "\n",