Modified 2018-06-24 by Andrea Censi
Theory chapters benefit from a standardized exposition. Here, we define the template for these chapters. Remember to check Part B - Markduck format for a comprehensive and up-to-date list of Duckiebook supported features.
Modified 2018-06-24 by Andrea Censi
Start with a brief introduction of the discussed topic, describing its place in the bigger picture, justifying the reading constraints/guidelines below. Write it as if the reader knew the relevant terminology. For example:
PID control is the simplest approach to making a system behave in a desired way rather than how it would naturally behave. It is simple because the measured output is directly feedbacked, as opposed to, e.g., the system’s states. The control signal is obtained as a weighted sum of the tracking error (_P_roportional term), its integral over time (_I_ntegrative term) and its instantaneous derivative (_D_erivative term), from which the appellative of PID control. The tracking error is defined as the instantaneous difference between a reference and a measured system output.
Knowledge necessary:
Required Reading: Insert here a list of topics and suggested resources related to necessary knowledge in order to understand the content presented. Example:
Terminology: autonomy overview
System Modeling: basic kinematics, basic dynamics, linear algebra, State space representations, Linear Time Invariant Systems
Suggested Reading: Insert here a list of topics and suggested resources related to recommended knowledge in order to better understand the content presented. Example:
Definitions of Stability, Performances and Robustness: [11], …
observability/detectability and controllability/reachability: [11]
Discrete time PID: [11]
Bode diagrams: [11]
Nyquist plots: [11]
[…]
Modified 2018-06-22 by Andrea Censi
In this section we crisply define the problem object of this chapter. It serves as a very brief recap of exactly what is needed from previous atoms as well. E.g.
Let:
\begin{align} \dot{\state}_t = A\state_t+Bu_t \\ y = C\state_t+Du_t \label{eq:system}\tag{1} \end{align}
be the LTI model of the Duckiebot’s plant, with $x \in \statesp$, $y \in \mathbb{R}^p$ and $u \in \mathbb{R}^m$. We recall (Duckiebot Modeling) that:
\begin{align} A &= \left[ \begin{array}{ccc} a_{11} & \dots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \dots & a_{nn} \end{array} \right] \\ B &= \left[ b_1 \,\, \dots \,\, b_m \right]^T \\ C &= \left[ c_1 \ \,\, \dots \,\, c_p \right] \\ D &= 0. \end{align}
[…]
Remember you can use the problem
environment of $\LaTeX$ to formally state a problem:
PID Given a system \eqref{eq:system} and measurements of the output $\tilde{y}_t = y_t + n_t, n_t \sim \cal{N}(0,\sigma)$, find a set of PID coefficients that meet the specified requirements for: - stability, - performance, - robustness.
as shown in (Figure 1.2).
Modified 2018-06-22 by Andrea Censi
Modified 2018-06-22 by Andrea Censi
Reference signals A reference signal $\tilde{y}_t \in \mathcal{L}_2(\mathcal{T})$ is …
Definition 6 - Reference signals is very important.
Insert ‘random’ checks to keep the reader’s attention up:
if you can’t be woken up in the middle of the night and remember the definition of $\mathcal{L}_2(\cdot)$, read: [11]
Another definition Lorem
Modified 2018-06-22 by Andrea Censi
Now that we know what we’re talking about, lets get in the meat of the problem. Here is what is happening:
$$ \cal{Lorem} $$
Modified 2018-06-22 by Andrea Censi
Introduce the ‘synthesis through attempts’ methodology (a.k.a. tweak until death)
Modified 2018-06-22 by Andrea Censi
How do we know if the PID controller designed above is doing well? We need to define some performance metrics first:
Overshoot, Module at resonance, Settling Time, Rising Time
[…]
This is a ‘think about it’ interrupt, used as attention grabber:
When a Duckiebot ‘overshoots’, it means that […] and the following will happen […].
Modified 2018-06-22 by Andrea Censi
And finally, this is how you save the world, in theory.
Modified 2018-06-22 by Andrea Censi
This section serves as a collection of theoretical and practical examples that can clarify part or all of the above.
Modified 2018-06-22 by Andrea Censi
More academic examples
Immagine a spring-mass-damper system…
[…]
Modified 2018-06-22 by Andrea Censi
More Duckiebot related examples
[…]
Modified 2018-06-24 by Andrea Censi
Here we just add references to the suggested exercises, defined in the appropriate exercise chapters.
Modified 2018-06-22 by Andrea Censi
Modified 2018-06-22 by Andrea Censi
Strong of this new knowledge (what have we learned), we can now […].
Further Reading: insert here reference resources for the interested reader:
learn all there is to know about PID: [11]
become a linear algebra master: Matrix cookbook
Modified 2019-04-28 by tanij
Do not include a reference chapter. References are automatically compiled to the Bibliography Section.
Jacopo Tani
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