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Transfer Functions

Background

In control theory, a system is some mathematical relation between an input and an output.  Systems are usually represented as a rectangle connecting some input function to an output function. Figure 1  depicts a SISO (single input, single output) system taking in an input function of time $u(t)$ and returning an output function of time $y(t)$.

generalSISO
Figure 1: A SISO system

Generally, ordinary differential equations (ODEs) are the simplest way to represent a system.

  • Note: dependent variables for ODEs (usually, this is time) are generally positive real numbers

Laplace Transform

We use the Laplace transform to switch between a function of real numbers (for our purposes, lets assume this is time $t$) to a function of some complex variable $s$ (Figure 2). Note that our $s$ variable turns out to denote frequency since our original function is a function of time.

laplace
Figure 2: Laplace transfrom for our input function $u(t)$ and our output function $y(t)$ from Figure 1.

Specifically, the expression for the Laplace transfrom of a single-variable function of time $f(t)$ is \begin{equation} F(s) = \mathcal{L} \left[ f(t) \right] = \int_{0}^{\infty} f(t)e^{-st} dt \end{equation}

The following equations are simplified expressions of the Laplace transform for $f(t)$ and its corresponding first and second order ODEs:

\begin{equation*} \mathcal{L} \left[ f(t) \right]  = F(s) \end{equation}

\begin{equation*} \mathcal{L} \left[ f'(t) \right]  = sF(s) – f(0) \end{equation}

\begin{equation*} \mathcal{L} \left[ f”(t) \right] = s^2 F(s) – sf(0) – f'(0) \end{equation}

Transfer Function (SISOs)

The transfer function $H(s)$ for a dynamic SISO system relates an input $u(t)$ with an output $y(t)$ as shown in Figure 3:

transferfunctionSISO
Figure 3: Transfer function for a SISO system

Examples

Example 1: Mechanical System

Find the transfer function for a single translational damped mass-spring system (depicted in Figure 4)

mass_spring1
Figure 4: Translational mass-spring system with damping
  • Aside: we can find the equation of motion for such a system using Newton’s and D’Alembert equations
  • The ODE describing our mass-spring system turns out to be…  \begin{equation} F(t)=m \ddot{x}(t) + c\dot{x}(t) + kx(t) \end{equation}
    • Variables:
      • $F(t)=$ an external force $[\mathrm{N}]$ acting on our mass $m \, [\mathrm{kg}]$
        • Note: this is the input for our system
      • $x(t)=$ the displacement $[\mathrm{m}]$ of our object due to $F(t)$
        • Note: this is the output for our system
      • $c=$ the damping coefficient $[\mathrm{Ns/m}]$
      • $k=$ the spring constant (i.e., stiffness) $[\mathrm{N/m}]$

Assume our initial conditions are $x(0)=0$ and $\dot{x}(0)=0$

First, let’s apply the Laplace transform to each individual term in our ODE for the mass-spring system:

\begin{equation*} \mathcal{L}[\ddot{x}]  = s^2 X(s) – sx(0) – \dot{x}(0) = s^2 X(s) \end{equation}

\begin{equation*} \mathcal{L}[\dot{x}]  = sX(s) – x(0) = sX(s) \end{equation}

\begin{equation*} \mathcal{L}[x(t)]  = X(s) \end{equation}

\begin{equation*} \mathcal{L}[F(t)]  = F(s) \end{equation}

Subsituting these expressions into our ODE is equivalent to taking the Laplacian of both sides:

\begin{equation} F(s) = ms^2 X(s) + cs X(s) + k X(s) \  =  X(s) (ms^2 + cs + k)  \end{equation}

Rearragning our $F(s)$ equation as follows gives us our transfer function $H(s)$ for the sytem:

\begin{equation} H(s) = \frac{X(s)}{F(s)} = \frac{1}{ms^2 + cs + k}  \end{equation}

Thus, we have defined our mechanical system as a second order ODE and as a transfer function. Figure 5 depicts the corresponding system diagrams in terms of the ODE and in terms of the transfer function.

mechanicalsystem
Figure 5: Two equivalent system diagrams for our mass-spring system. The top system diagram represents our mass-spring system in terms of a second order ODE. The bottom system diagram represents our mass-spring system in terms of a transfer function.

Sources

How to find the transfer function of a system

  • Note: most of the content in the article originally comes from the above link
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Experiment 1

[latexpage]

Purpose:

  • Validate for myself whether all this pyramid talk is legitamite/worth looking into or nonsense

Links to similar experiments:

  • https://www.yout1ube.com/watch?v=YcpJ_y_N780
  • I will probably implement this link in my experiment
  • I read a version of this experiment that I really liked a few months ago. I just can’t find the original source. Hopefully I’ll find it, because it was well written and you could tell the author had at minimum taken some science labs in college (and thus is probably aware of the importance of the scientific method all that stuff).  If/when I do find it I’ll post a link here 🙂
  • I was also thinking of trying out this experiment with growing crystals instead of plants, but I’m not quite as sure what differences to look for in the results (I’m thinking it would at least require a microscope). If anyone has any ideas please let me know!

Overview of ideas

Note: I don’t know gardening “lingo” (I’ve never gardened before), so I am going to make up some terms for now

  • Hypothesis: Seeds sown in soil placed under a pyramid will have better “growth yields” than seeds sown in control soil.
  • Materials:
    • Copper wires connected in the shape of a pyramid
    • Copper wires connected in the shape of a cube (one of the control conditions)
    • Three plant pots
    • Soil/fertilizer
    • Seeds
    • (?) A room with a south- facing window (for optimal sunlight exposure each day, I don’t really know, I just made this one up)
      • At minimum, don’t pick a room without windows or that only has North-facing windows

Things To Figure Out

  • Is there a way to quantify growth “yield”?
  • Should I try the experiment with crystals? If so, what differences in the results would I be looking for?
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Synchronization of Coupled Oscillators

[latexpage]

To-Do:

  • Make adjacency matrices for different networks

Reference

  • Mason and Verwoerd (2006)

Applications

  • Spike-timing dependent plasticity and learning/memory
  • Permanent changes in synchronization after novel experiences
    • e.g., fear-based learning (hippocampus and amygdala)
  • Spatiotemporal patterns
    • “Language” of neurons
    • Mechanism of information flow throughout the brain
    • Dynamics of interactions between different regions
      • Dynamic coupling between brain regions (e.g., active or attentive behavior) gives rise to perception, movement, consciousness, etc.

Main Issue: Scale at which synchronization occurs

My own thoughts:

  • Scales with respect to size: “universal” → “galactic” → solar systems → planets, stars, asteroids, etc. → anatomical → cellular → subcellular → atomic → subatomic (quantum)

Kuramoto Model

\dot { \theta _{ i } } =\omega_i+\frac{K}{N}\sum _{j=1}^{N}{ \sin(\theta_j-\theta_i) }