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Merge pull request #581 from murrayrm/fix_murray_wiki_links
Fix murray wiki links + small doc typo
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doc/optimal.rst

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A common use of optimization-based control techniques is the implementation
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of model predictive control (also called receding horizon control). In
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model predict control, a finite horizon optimal control problem is solved,
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model predictive control, a finite horizon optimal control problem is solved,
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generating open-loop state and control trajectories. The resulting control
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trajectory is applied to the system for a fraction of the horizon
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length. This process is then repeated, resulting in a sampled data feedback

examples/pvtol-lqr-nested.ipynb

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"## System Description\n",
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"This example uses a simplified model for a (planar) vertical takeoff and landing aircraft (PVTOL), as shown below:\n",
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"\n",
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"![PVTOL diagram](http://www.cds.caltech.edu/~murray/wiki/images/7/7d/Pvtol-diagram.png)\n",
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"![PVTOL diagram](https://murray.cds.caltech.edu/images/murray.cds/7/7d/Pvtol-diagram.png)\n",
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"\n",
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"![PVTOL dynamics](http://www.cds.caltech.edu/~murray/wiki/images/b/b7/Pvtol-dynamics.png)\n",
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"![PVTOL dynamics](https://murray.cds.caltech.edu/images/murray.cds/b/b7/Pvtol-dynamics.png)\n",
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"\n",
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"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",
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"\n",
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"\n",
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"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",
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"\n",
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"<img src=http://www.cds.caltech.edu/~murray/wiki/images/3/3f/Pvtol-lateraltf.png>\n",
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"where\n",
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"<img src=http://www.cds.caltech.edu/~murray/wiki/images/math/3/6/4/364e56f7893637e12edb0e0ac6c45722.png> \n",
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"<img src=https://murray.cds.caltech.edu/images/murray.cds/3/3f/Pvtol-lateraltf.png>\n",
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"\n",
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"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",
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"<img src=http://www.cds.caltech.edu/~murray/wiki/images/f/f1/Pvtol-nested-1.png>\n",
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"<img src=https://murray.cds.caltech.edu/images/murray.cds/f/f1/Pvtol-nested-1.png>\n",
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"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",
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"\n",
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"The following code imports the libraries that are required and defines the dynamics:"
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.7.9"
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"version": "3.9.1"
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}
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},
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"nbformat": 4,

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