# [ACADEMIC STUDY] Terminologies and Concepts used in Control and Estimation Theory - 1

**Control methods**- Impedance control: A dynamic control method which relates force and position.
- Mechanical impedance: The ratio of force output to motion input
- Purpose of this method is to regulate the relationship between force and position. Thus, it requires a position, velocity or acceleration input to control the force output value.
- It is a conventional control method we use, which makes an actuator input using desired position, velocity or acceleration.

- Admittance control: A dynamic control method which relates force and position, but in the inverse way of impedance control.
- Mechanical admittance: The ratio of motion output to force input
- It requires a force input to control the position, velocity or acceleration.

- Bang-singular control: It is a control method which consists of both bang-bang portion and singular portion.
- Bang-bang control: A kind of feedback controller which abruptly switches its control law between two discrete phases.
- Singular control: An optimal control problem which cannot be solved by Pontryagin’s minimum principle.
- Pontyagin’s maximum principle is a sort of optimal control theory.
- This principle is a way to design the control law which enables a dynamical system to switch from one phase to another phase under some constraints on both state and input.

- According to the optimal control theory, it is proven that
**the time-optimal trajectory of the input-affined system is bang singular.**

- Event-triggered control: A control system which do not send any actuator signal unless the “event-triggering condition” is invoked.

- Impedance control: A dynamic control method which relates force and position.
**Control Systems**- Networked control system (NCS): The closed-loop system which is controlled by communication networks.
- There are four crucial elements: sensor + controller + actuator +
**communication network** - It facilitates the system to conduct some specific tasks which require the comminication between two places wihch are far from each other.
- It can reduce its communication load while using event-triggered control.

- There are four crucial elements: sensor + controller + actuator +
- Fuzzy control system: A control system based on fuzzy logic, which is a mathematical system that analyzes analog input values in terms of logical variables that take on discrete values, 0 or 1.

- Networked control system (NCS): The closed-loop system which is controlled by communication networks.
**Sets**- Convex hull: The smallest convex set which contains a dot or a region which are given in the form of set.
- Convex set: For a set $A$ in Euclidean space, $A$ is called a
**convex set**if we pick two arbitrary points inside $A$, then a segment which connects the two points is always the element of $A$.

- Convex set: For a set $A$ in Euclidean space, $A$ is called a
- Compact set
- $S$ is covered by
*a collection of open sets*, $O$ ($S \subset $ (at least one member of) $O$), and said to compact if $S$ is covered by some finite set of members of $O$ for every covering $O$ of $S$ by open sets. - In Euclidean space ($\mathbb{R}^{n}$), it is defined as a
**closed**and**bounded**subset of Euclidean space, e.g. closed interval, rectagnle, finite set of points. This property is proved in detail in [4].

- $S$ is covered by

- Convex hull: The smallest convex set which contains a dot or a region which are given in the form of set.
**Functions**- Class $K$ function
- a continuous function $\alpha$: [0,a) $\rightarrow$ [0,$\infty$)
- a strictly increasing function
- $\alpha(0)$ =$0$
- cf) class $K_{\infty}$ function: a class $K$ function which is radially unbounded

- Class $KL$ function
- a continuous function $\beta$: [0,a) x $[0,\infty]$ $\rightarrow$ [0,$\infty$)
- for each fixed $s$, the function $\beta(r,s)$ belongs to class $K$
- for each fixed $r$, the function $\beta(r,s)$ is decreasing with respect to $s$ and is s.t. $\beta(r,s)$ $\rightarrow$ 0 for $s$ $\rightarrow$ $\infty$

- Class $K$ function
**S-procedure**: The S-procedure, also called as “S-lemma” is defined as follows.**Definition**: A mathematical process to find the equivalent*linear matrix inequality (LMI)*that makes a*particular quadratic inequality*.**Procedure (w/o proof)***IF*$0 \leq z^TF_{0}z \, \rightarrow \, 0 \leq z^TF_{1}z$ and there exists $z_0 \, s.t. \, 0 \leq z_{0}^{T}F_{0}z_{0}$*THEN*there exists a nonnegative $\tau \, s.t. \, \tau F_{1} \leq F_{0}$

## Reference

[1] https://en.wikipedia.org/wiki/Convex_set

[2] https://en.wikipedia.org/wiki/Impedance_control

[3] https://en.wikipedia.org/wiki/Bang%E2%80%93bang_control

[4] https://en.wikipedia.org/wiki/Singular_control

[5] https://en.wikipedia.org/wiki/Pontryagin%27s_maximum_principle

[6] https://en.wikipedia.org/wiki/Networked_control_system

[7] https://en.wikipedia.org/wiki/Fuzzy_control_system#Fuzzy_control_in_detail

[8] Heemels, W. P. M. H., Karl Henrik Johansson, and Paulo Tabuada. “An introduction to event-triggered and self-triggered control.” 2012 ieee 51st ieee conference on decision and control (cdc). IEEE, 2012.

[9] https://en.wikipedia.org/wiki/Class_kappa_function

[10] https://en.wikipedia.org/wiki/Class_kappa-ell_function

[11] https://en.wikipedia.org/wiki/Compact_space

[12] http://www-math.mit.edu/~djk/calculus_beginners/chapter16/section02.html

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