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

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.
2. 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.
• 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.
3. 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 $A$.
• 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].
4. 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 $s$ $\rightarrow$ $\infty$
5. 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 $\tau \, s.t. \, \tau F_{1} \leq F_{0}$

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