24 November 2023

Know and learn a little about linear systems It is of great importance in different fields of science and engineering. This is thanks to the fact that they are considered the basis for the understanding and analysis of a large number of processes and phenomena in the world that surrounds us.

We can find linear systems everywhere, as in the mathematical equations that describe the electrical circuit behavior. That is why, below, we will talk in more detail about linear systems, their characteristics, the types that exist and much more.

What does it mean for a system to be linear?

A system is considered to be linear when satisfies the superposition properties and is homogeneous. These properties are based on the principle of proportionality and additivity, this implies that the responses of the system are proportional to the inputs.

Likewise, it also implies that the sum of the responses to different inputs is equal to the response when adding the inputs separately. However, when a system does not comply with the main property of linearity, they are considered a nonlinear system.

On the other hand, when talking about technical terms, a system is linear when it meets the following properties:

  1. Overlay property: this occurs when, by mixing two or more elements, you introduce them into a system. Obtaining that, the total response of the system will be equal to the sum of the individual responses obtained by introducing each thing separately.
  2. Homogeneity Property: If increase or decrease the size of an element entering a system, the system’s response will also be increased or decreased by the same rate. In short, the system responds to size changes in the same way that the element does.

These properties are important because allow us to understand how linear systems work and behave before different inputs. In addition, they simplify the analysis and help to solve problems with the use of simpler or basic mathematics.

What is a linear system in control?

When we talk about a linear system in the control area, this refers to a dynamic system whose behavior is usually expressed by a system of linear equations. In general, these systems are used in control engineering for the analysis and modeling of physical and automated systems and industrial processes.

Linear systems in control are characterized by meet the properties of homogeneity and overlap, mentioned in the previous point. This means that, when applying a linear combination of different inputs to the system, it will give as a response the sum of the individual responses to each input.

On the other hand, if a system input scales by a constant factor, the system response will tend to scale by the same factor. In general, linear systems in control are represented by linear differential equations or in the frequency domain using transforms.

This type of representations allow the analysis of stability, the response in time and frequency, among other relevant aspects of the system. The advantage of working with linear systems in control means that you can apply well-developed design and analysis techniques, which are based on linear algebra and the theory of linear systems.

What is a linear system in economics?

In economics, a linear system refers to the mathematical model that describes the relationships and interactions between economic variables using systems of linear equations. These models are usually used for the analysis and prediction of the behavior of different aspects of the economy, such as investment, supply and demand, consumption, production and prices.

In this type of system, the economic variables are represented as the variable of interest or unknown and establish linear relationships between them through equations. Linear equations in economics are derived from economic theories and assumptions, and are often used to study how changes in one variable can affect other variables in a system.

The linearity in these economic models tells us allows for simplified analysis and troubleshooting. This is based on the idea that proportional changes in one economic variable produce proportional changes in the others, maintaining a constant of proportionality.

What is a time-invariant linear system?

When referring to a linear time-invariant system (LTI, or Linear Time-Invariant System), we are talking about a type of system that has two main properties, linearity and time invariance.

For its part, linearity implies that the system satisfies the properties of homogeneity and superposition. This means that if a linear combination of different inputs is applied to the system, the result will be the sum of the individual responses to each input.

While, the invariance in time speaks of that, the characteristics and behavior of the system do not vary over time. This means that the properties of the system, such as its response and behavior to different inputs, is constant and does not vary over time.

Said properties are fundamental in the design and analysis of systems in many areas, such as communication, control engineering, and signal and systems theory. In addition, it allows the use of well-established techniques and tools that are based on linear algebra and transforms to analyze and understand how the system behaves.

What are the characteristics of linear systems?

Linear systems usually have a series of characteristics that define and distinguish them from other types of systems. These are fundamental for the understanding and analysis of the behavior of said systems, among the main characteristics are:

  1. proportionality: These systems usually show proportional relationships between their inputs and outputs. This means that if the input is doubled, the output will double, and if the input is reduced, the output will also be reduced by the same rate.
  2. Overlap: these systems comply with superposition, this implies that the total response to a linear combination between several inputs is equal to the sum of the individual responses of each input. Thanks to this, the analysis and prediction of the behavior of the system is facilitated by breaking it down into more manageable parts.
  3. HomogeneityFor a system to be linear, it must satisfy the homogeneity property. This means that if the input of the system is scaled by a constant factor, its output is scaled by the same factor.
  4. predictable behavior: Linear systems are both predictable and consistent. This means that they retain linearity even during rotation.
  5. algebraic properties: Linear systems tend to be manipulable from an algebraic point of view. Which means that algebraic fractions, such as addition, subtraction or multiplication, can be applied to the equations that describe the system.

What are the types of linear systems?

There are several types of linear systems, each with specific properties and characteristicsamong which are the following:

  • Linear Time Invariant System (LTI): are linear systems with characteristics and behaviors that do not change over time.
  • continuous linear system: These are systems in which both the input and the output represent continuous signals over time. These are modeled using linear differential equations and analyzed in the discrete time domain.
  • discrete linear system: systems in which the input and output are signals discrete in time. They are modeled using linear difference equations and are analyzed in the discrete time domain.
  • Discrete-time linear system: speaks of systems in which the input represents a discrete signal in time, while the output represents a continuous signal in time. These systems are modeled using linear difference equations and are analyzed in the discrete time domain.
  • continuous time linear system: are known as systems where the input is a continuous signal in time and the output is a discrete signal in time. These types of systems are modeled using linear differential equations and are analyzed in the continuous time domain.

On the other hand, there is the existence of other linear systems that are not classified as types of systems, but to describe the equations that represent them. Among these are the homogeneous and heterogeneous linear systems.

What is a homogeneous linear system?

Linear systems are considered homogeneous when all the equations that describe them are homogeneous. An equation is homogeneous when all its terms have degree zerowhich means that no independent terms or constants are present.

In summary, if all the coefficients and variables in the equations of the system are multiplied by zero, the solution where all the variables are zero is valid. This type of systems is important in linear algebra and has well-defined mathematical properties.

What is a heterogeneous linear system?

It is understood that a linear system is heterogeneous when at least one of the four equations that define it is heterogeneous. An equation is considered to be heterogeneous when there are independent terms or constants presentthat is, they do not depend on the variables.

Thus, these constant terms define a “non-homogeneity” in the systems, which can have a significant impact on the behavior and solutions of the system.

Examples of linear systems

Linear systems have a wide range of uses and uses in various areas or sciences, which is why we will describe some examples below:

  1. During the electrical circuit analysis It is common to use linear equations to model the currents and voltages in the components. With the use of the Python programming language and the Sympy library, it is possible to solve and/or simulate complex linear systems efficiently.
  2. For process signals These systems are often used to filter, modulate, encode, and decode signals. An example of this is digital communication, where linear systems are used to modulate and demodulate signals in different formats, such as AM, FM and QAM.
  3. In Economy and Finance, economic models are represented by linear systems to analyze the interactions between different economic variables. These types of models help to carry out analysis and forecasts of economic policies.
  4. sciences like physics and mechanics they also often use linear systems to describe and predict the behavior of physical and mechanical systems. A clear example is the linear equations of motion applied to oscillating systems, such as pendulums or mass-springs.

These are just a few examples of the most common uses of the application of linear systems in various fields. Likewise, its use is wide and extends to many other areas, sciences or disciplines, such as acoustics, robotics, biology and chemistry.

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