diff --git a/doc/optimal.rst b/doc/optimal.rst
index 38bfca0db..9538c28c2 100644
--- a/doc/optimal.rst
+++ b/doc/optimal.rst
@@ -66,7 +66,7 @@ intended to hold at all instants in time along the trajectory.
A common use of optimization-based control techniques is the implementation
of model predictive control (also called receding horizon control). In
-model predict control, a finite horizon optimal control problem is solved,
+model predictive control, a finite horizon optimal control problem is solved,
generating open-loop state and control trajectories. The resulting control
trajectory is applied to the system for a fraction of the horizon
length. This process is then repeated, resulting in a sampled data feedback
diff --git a/examples/pvtol-lqr-nested.ipynb b/examples/pvtol-lqr-nested.ipynb
index ceb6424c0..59e97472a 100644
--- a/examples/pvtol-lqr-nested.ipynb
+++ b/examples/pvtol-lqr-nested.ipynb
@@ -20,9 +20,9 @@
"## System Description\n",
"This example uses a simplified model for a (planar) vertical takeoff and landing aircraft (PVTOL), as shown below:\n",
"\n",
- "\n",
+ "\n",
"\n",
- "\n",
+ "\n",
"\n",
"The position and orientation of the center of mass of the aircraft is denoted by $(x,y,\\theta)$, $m$ is the mass of the vehicle, $J$ the moment of inertia, $g$ the gravitational constant and $c$ the damping coefficient. The forces generated by the main downward thruster and the maneuvering thrusters are modeled as a pair of forces $F_1$ and $F_2$ acting at a distance $r$ below the aircraft (determined by the geometry of the thrusters).\n",
"\n",
@@ -307,11 +307,10 @@
"\n",
"To design a controller for the lateral dynamics of the vectored thrust aircraft, we make use of a \"inner/outer\" loop design methodology. We begin by representing the dynamics using the block diagram\n",
"\n",
- "
\n",
- "where\n",
- "
\n",
+ "
\n",
+ "\n",
"The controller is constructed by splitting the process dynamics and controller into two components: an inner loop consisting of the roll dynamics $P_i$ and control $C_i$ and an outer loop consisting of the lateral position dynamics $P_o$ and controller $C_o$.\n",
- "
\n",
+ "
\n",
"The closed inner loop dynamics $H_i$ control the roll angle of the aircraft using the vectored thrust while the outer loop controller $C_o$ commands the roll angle to regulate the lateral position.\n",
"\n",
"The following code imports the libraries that are required and defines the dynamics:"
@@ -547,7 +546,7 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
- "version": "3.7.9"
+ "version": "3.9.1"
}
},
"nbformat": 4,
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