Lecture

Discrete Systems Theory

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In course

ME-324: Discrete-time control of dynamical systems

On introduit les bases de l'automatique linéaire discrète qui consiste à appliquer une commande sur des intervalles uniformément espacés. La cadence de l'échantillonnage qui est associée joue un rôle primordial pour assurer la stabilité et la performance d ...

Description

This lecture introduces discrete systems theory, focusing on difference equations, dynamic systems, linearity, causality, and stationary systems. It covers the concepts of impulse response, convolution product, and initial conditions in the context of discrete systems.

Instructor

Philippe Müllhaupt

Expertise Nonlinear Control (Geometry and Algebra)Classical DynamicsRoboticsStatistics Mission Provide rigorous technical and scientific training in nonlinear control, classical mechanics, and statistical theory Current Work Research and teaching in the fields of control theory, statistics, dynamical systems, and general mechanics. Philippe Mullhaupt received his Diplôme d Education Ingï¿½nieur | Electricitï¿½ 1993 – 1993 EPFL DEA | Automatique et Traitement du Signal 199

Official source

Related concepts (32)

Recurrence relation

In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation. If the values of the first numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In linear recurrences, the nth term is equated to a linear function of the previous terms.

Initial condition

In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). For a system of order k (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension n (that is, with n different evolving variables, which together can be denoted by an n-dimensional coordinate vector), generally nk initial conditions are needed in order to trace the system's variables forward through time.

Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured.

Convolution

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function () that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity).

Impulse response

In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (δ(t)). More generally, an impulse response is the reaction of any dynamic system in response to some external change. In both cases, the impulse response describes the reaction of the system as a function of time (or possibly as a function of some other independent variable that parameterizes the dynamic behavior of the system).

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