[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 $\tau \, s.t. \, \tau F_{1} \leq F_{0}$

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