vfdata[i, 2:] = sys._rhs(0, x, 0)

A collection of Jupyter notebooks are available on [Google Colab](),

where they can be executed through a web browser:

"In this lecture we describe tools in the Python Control Systems Toolbox ([python-control](https://python-control.org]) that can be used to analyze linear systems, including some of the options available to present the information in different ways.\n"

"Another way to get these same dynamics is to define and input/output system:"

"The `initial_response` function can be used to compute the response of the system with no input, but starting from a given initial condition. This function returns a response object, we can be used for plotting."

" u = np.atleast_1d(u) # Fix python-control bug\n",

"We see that the controller no longer stabilizes the equilibrium point (shown with the dashed lines). In particular, the steady state value of the Lynx population does to almost twice the original value.\n",

" predprey_pi, T, xe[1], [25, 25], params={'r': 1.65})\n",

"y = \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix} x.\n",

"y = \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix} x.\n",

"y = \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix} x.\n",

"# Part 2: PVTOL Linear Quadratic Regulator Example\n",

"Note that this is slight different than the first equation: here we are using $x_\\text{d}$ instead of $x_\\text{e}$ and $u_\\text{d}$ instead of $u_\\text{e}$. This is because we want our controller to track a desired trajectory $(x_\\text{d}(t), u_\\text{d}(t))$ rather than just stabilize the equilibrium point $(x_\\text{e}, u_\\text{e})$."

"This command will usually generating a warning saying that python control \"cannot verify system output is system state\". This happens because we specified an output function `pvtol_output` when we created the system model, and python-control does not have a way of checking that the output function runs the entire state (which is needed if we are going to do full-state feedback).\n",

"Next we generate a step response and plot the results. Because our closed loop system takes as inputs $x_\\text{d}$ and $u_\\text{d}$, we need to set those variable to values that would corresponding to our step input. In this case, we are taking a step in the $x$ coordinate, so we set $x_\\text{d}$ to be $1$ in that coordinate starting at $t = 0$ and continuing for some sufficiently long period of time ($15$ seconds):"

"In this lecture, we will show how to construct an observer for a system in the presence of noise and distrubances.\n",

"A Kalman filter allows us to estimate the optimal state give measurements of the inputs and outputs, as well as knowldge of the covariance of the signals."

"print(\"ct.interconnect(syslist, connections=None, inplist=None, outlist=None, params=None, check_unused=True, add_unused=False, ignore_inputs=None, ignore_outputs=None, warn_duplicate=None, debug=False, **kwargs)\\n\\t\")\n",

"print(ct.interconnect.__doc__)"

"Another standard set of analysis tools is to identify the gain, phase, and stability margins for the sytem:\n",

"The first two of the items can be computed either by looking at the frequeny response or by using the `margin` command.\n",

"# Find the equiblirum point corresponding to the origin\n",

"# Define the time horizon for the maneuver\n",

"# Re-define the time horizon for the maneuver\n",

"We see that with the addition of disturbances and noise, we sometimes violate the constraint 'nicolas' (if you plot doesn't show an intersection with the bottom dashed curve, try regenerating the noise and running the simulation again). This can be fixed by establishing a more conservative constraint (see commented out constraint in code block above)."

"* [Python Control Systems Library (python-control) documentation](https://python-control.readthedocs.io/en/0.9.3.post2/)"