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# Chapter template

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.

## Example Title: PID control

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

Suggested Reading: Insert here a list of topics and suggested resources related to recommended knowledge in order to better understand the content presented. Example:

## Problem Definition

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). A classical block diagram for PID control. We like to use a lot of clear figures in the Duckiebook.

## Introduced Notions

Modified 2018-06-22 by Andrea Censi

## Examples

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.

### Theoretical Examples

Modified 2018-06-22 by Andrea Censi

Immagine a spring-mass-damper system…

[…]

### Implementation Examples

Modified 2018-06-22 by Andrea Censi

More Duckiebot related examples

[…]

## Pointers to Exercises

Modified 2018-06-24 by Andrea Censi

Here we just add references to the suggested exercises, defined in the appropriate exercise chapters.

## References

Modified 2019-04-28 by tanij

Do not include a reference chapter. References are automatically compiled to the Bibliography Section.

Jacopo Tani

Jacopo Tani

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