# 16 Pre-Class Assignment: Linear Dynamical Systems¶

## Readings for this topic (Recommended in bold)¶

## Goals for today’s pre-class assignment¶

## 1. Linear Dynamical Systems¶

A linear dynamical system is a simple model of how a system changes with time. These systems can be represented by the following “dynamics” or “update equation”:

Where \(t\) is an integer representing th progress of time and \(A_t\) are an \(n \times n\) matrix called the dynamics matrices. Often the above matrix does not change with \(t\). In this case the system is called “time-invariant”.

We have seen a few “time-invarient” examples in class.

✅ **DO THIS:** Review ** Chapter 9 in the Boyd and Vandenberghe** text and become familiar with the contents and the basic terminology.

## 2. Markov Models¶

This is not the first time we have used Dynamical Linear Systems.

✅ **DO THIS:** Review markov models in 10–Eigenproblems_pre-class-assignment.ipynb. See how this is a special type of linear dynamical systems that work with state probabilities.

### Example¶

The dynamics of infection and the spread of an epidemic can be modeled as a linear dynamical system.

We count the fraction of the population in the following four groups:

Susceptible: the individuals can be infected next day

Infected: the infected individuals

Recovered (and immune): recovered individuals from the disease and will not be infected again

Decreased: the individuals died from the disease

We denote the fractions of these four groups in \(x(t)\). For example \(x(t)=(0.8,0.1,0.05,0.05)\) means that at day \(t\), 80% of the population are susceptible, 10% are infected, 5% are recovered and immuned, and 5% died.

We choose a simple model here. After each day,

5% of the susceptible individuals will get infected

3% of infected inviduals will die

10% of infected inviduals will recover and immuned to the disease

4% of infected inviduals will recover but not immuned to the disease

83% of the infected inviduals will remain

✅ **Do this:** Write the dynamics matrix for the above markov linear dynamical system. Come to class ready to discuss the matrix. (hint the columns of the matrix should add to 1).

```
# Put your matrix here
```

✅ **Do this:** Review how we solved for the long term steady state of the markov system. See if you can find these probabilities for your dyamics matrix.

```
# Put your matrix here
```

## 3. Ordinary Differential Equations¶

Ordinary Differential Equations (ODEs) are yet another for of linear dynamical systems and are a scientific model used in a wide range of problems of the basic form:

### $\(\dot{x} = A x\)$¶

These are equations such that the is the instantaneous rate of change in \(x\) (i.e. \(\dot{x}\) is the derivative of \(x\)) is dependent on \(x\). Many systems can be modeled with these types of equations.

Here is a quick video that introduces the concepts of Differential Equations. The following is a good review of general ODEs.

```
from IPython.display import YouTubeVideo
YouTubeVideo("8QeCQn7uxnE",width=640,height=360, cc_load_policy=True)
```

Now consider an ODE as a system of linear equations:

Based on the current \(x\) vector at time \(t\) and the matrix \(A\), we can calculate the derivative at \(\dot{x}\) at time \(t\). Once we know the derivative, we can increment the time to by some small amount \(dt\) and calculate a new value of \(x\) as follows:

Then we can do the exact sequence of calculations again for \(t+2\). The following function has the transition matrix (\(A\)), the starting state vector (\(x_0\)) and a number of time steps (\(N\)) and uses the above equations to calculate each state and return all of the \(x\) statues:

The following code generates a trajectory of points starting from `x_0`

, applying the matrix \(A\) to get \(x_1\) and then applying \(A\) again to see how the system progresses from the start state.

```
%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
sym.init_printing()
```

```
---------------------------------------------------------------------------
ModuleNotFoundError Traceback (most recent call last)
<ipython-input-4-97c04e533e83> in <module>
----> 1 get_ipython().run_line_magic('matplotlib', 'inline')
2 import matplotlib.pylab as plt
3 import numpy as np
4 import sympy as sym
5 sym.init_printing()
~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/interactiveshell.py in run_line_magic(self, magic_name, line, _stack_depth)
2342 kwargs['local_ns'] = self.get_local_scope(stack_depth)
2343 with self.builtin_trap:
-> 2344 result = fn(*args, **kwargs)
2345 return result
2346
~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/decorator.py in fun(*args, **kw)
230 if not kwsyntax:
231 args, kw = fix(args, kw, sig)
--> 232 return caller(func, *(extras + args), **kw)
233 fun.__name__ = func.__name__
234 fun.__doc__ = func.__doc__
~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/magic.py in <lambda>(f, *a, **k)
185 # but it's overkill for just that one bit of state.
186 def magic_deco(arg):
--> 187 call = lambda f, *a, **k: f(*a, **k)
188
189 if callable(arg):
~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/magics/pylab.py in matplotlib(self, line)
97 print("Available matplotlib backends: %s" % backends_list)
98 else:
---> 99 gui, backend = self.shell.enable_matplotlib(args.gui.lower() if isinstance(args.gui, str) else args.gui)
100 self._show_matplotlib_backend(args.gui, backend)
101
~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/interactiveshell.py in enable_matplotlib(self, gui)
3511 """
3512 from IPython.core import pylabtools as pt
-> 3513 gui, backend = pt.find_gui_and_backend(gui, self.pylab_gui_select)
3514
3515 if gui != 'inline':
~/REPOS/MTH314_Textbook/MakeTextbook/envs/lib/python3.9/site-packages/IPython/core/pylabtools.py in find_gui_and_backend(gui, gui_select)
278 """
279
--> 280 import matplotlib
281
282 if gui and gui != 'auto':
ModuleNotFoundError: No module named 'matplotlib'
```

```
def traj(A, x, n):
dt = 0.01
x_all = np.matrix(np.zeros((len(x),n))) # Store all points on the trajectory
for i in range(n):
x_dot = A*x # First we transform x into the derrivative
x = x + x_dot*dt # Then we estimate x based on the previous value and a small increment of time.
x_all[:,i] = x[:,0]
return x_all
```

For example the following code uses the matrix \(A= \begin{bmatrix}1 & 1 \\ 1 & -2\end{bmatrix}\) and the starting point (0,0) over 50 timesteps to get a graph:

```
A = np.matrix([[1,1],[1,-2]])
x0 = np.matrix([[1],[1]])
x_all = traj(A, x0, 50)
plt.scatter(np.asarray(x_all[0,:]),np.asarray(x_all[1,:]))
plt.scatter(list(x0[0,:]),list(x0[1,:])) #Plot the start point as a refernce
```

✅ **Do this:** Let
$\(A= \begin{bmatrix}2 & 3 \\ 4 & -2\end{bmatrix}\)$

Write a loop over the points \((1.5,1)\), \((-1.5,-1)\), \((-1,2)\) and plot the results of the `traj`

function:

```
A = np.matrix([[2,3],[4,-2]])
x0 = np.matrix([[1.5, -1.5, -1, 1, 2],[1, -1, 2, -2, -2]])
```

```
# Put your code here
```

✅ **Do this:** Let
$\(A= \begin{bmatrix}6 & -1 \\ 1 & 4\end{bmatrix}\)$

Write a loop over the points \((1.5,1)\), \((-1.5,-1)\), \((-1,2)\), \((1,-2)\) and \((2,-2)\) and plot the results of the `traj`

function:

```
# Put your code here
```

✅ **Do this:** Let
$\(A= \begin{bmatrix}5 & 2 \\ -4 & 1\end{bmatrix}\)$

Write a loop over the points \((1.5,1)\), \((-1.5,-1)\), \((-1,2)\), \((1,-2)\) and \((2,-2)\) and plot the results of the `traj`

function:

```
# Put your code here
```

## 4. Assignment wrap up¶

✅ **Assignment-Specific QUESTION:** Where you able to get the ODE code working in the above example. If not, where did you get stuck?

Put your answer to the above question here

✅ **QUESTION:** Summarize what you did in this assignment.

Put your answer to the above question here

✅ **QUESTION:** What questions do you have, if any, about any of the topics discussed in this assignment after working through the jupyter notebook?

Put your answer to the above question here

✅ **QUESTION:** How well do you feel this assignment helped you to achieve a better understanding of the above mentioned topic(s)?

Put your answer to the above question here

✅ **QUESTION:** What was the **most** challenging part of this assignment for you?

Put your answer to the above question here

✅ **QUESTION:** What was the **least** challenging part of this assignment for you?

Put your answer to the above question here

✅ **QUESTION:** What kind of additional questions or support, if any, do you feel you need to have a better understanding of the content in this assignment?

Put your answer to the above question here

✅ **QUESTION:** Do you have any further questions or comments about this material, or anything else that’s going on in class?

Put your answer to the above question here

✅ **QUESTION:** Approximately how long did this pre-class assignment take?

Put your answer to the above question here

Written by Dr. Dirk Colbry, Michigan State University

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.