Name: linearizing nonlinear state-space models
Uploaded: 2024-11-21T14:54:05.000Z
Duration: 1 min 21 s
Channel: JoVE

Os sistemas não lineares encontram uma modelagem superior na representação do espaço de estados.
Considere um pêndulo simples representado no espaço de estados com o centro de massa em seu meio comprimento. O peso e o torque aplicado são fatores significativos.
A soma dos torques leva a uma equação diferencial. As variáveis de estado são selecionadas e as equações de estado são formuladas.
Para linearizar a equação em torno do ponto de equilíbrio, considere as variáveis de estado perturbadas em torno do ponto de equilíbrio.
A aproximação da série de Taylor é empregada e resolvida, resultando em equações de estado lineares.
Considere um sistema de molas não linear. Formule uma equação diferencial e introduza uma pequena perturbação.
Esta equação é então linearizada. Substituindo isso e realizando a diferenciação, obtém-se a equação diferencial intermediária linearizada.
A força de equilíbrio é usada para calcular xo, formando a equação diferencial linearizada final.
As variáveis de estado são selecionadas e as equações de estado e saída são escritas, fornecendo um modelo completo do sistema no espaço de estados. Finalmente, todas as equações são convertidas em forma de matriz vetorial.

aproximação linear no domínio do tempo

Linear Approximation in Time Domain

Sistemas não lineares geralmente exigem abordagens sofisticadas para modelagem e análise precisas, com representação de espaço de estado sendo particularmente eficaz. Este método é especialmente útil para sistemas onde variáveis ​​e parâmetros variam com o tempo ou condições operacionais, como em um pêndulo simples ou um sistema mecânico translacional com molas não lineares.


 Para um pêndulo simples com uma massa uniformemente distribuída ao longo de seu comprimento e o centro de massa localizado na metade do comprimento do pêndulo, o comportamento dinâmico pode ser capturado pela soma de torques em torno do ponto de pivô. A equação diferencial resultante incorpora os efeitos da força gravitacional e do torque aplicado. Para representar isso na forma de espaço de estado, variáveis ​​de estado são escolhidas para descrever a posição e a velocidade do sistema. Essas variáveis ​​de estado levam à formulação de equações de estado, que descrevem a evolução temporal do sistema.


 A linearização dessas equações de estado não lineares em torno de um ponto de equilíbrio envolve considerar pequenas perturbações. Ao perturbar as variáveis ​​de estado em torno de seus valores de equilíbrio e aplicar uma expansão da série de Taylor, os termos não lineares podem ser aproximados por suas contrapartes lineares. Esta aproximação produz equações de estado lineares que podem ser analisadas usando a teoria do sistema linear.


 No caso de um sistema mecânico translacional com uma mola não linear, a dinâmica do sistema é governada de forma semelhante por uma equação diferencial que considera a força da mola não linear. A introdução de uma pequena perturbação em torno da posição de equilíbrio permite a linearização da equação diferencial. A força de equilíbrio em x_0 (a posição de equilíbrio) é calculada, e a equação perturbada é diferenciada para obter uma equação diferencial linearizada.


 A representação final do espaço de estado linearizado envolve a seleção de variáveis ​​de estado apropriadas, que geralmente incluem a posição e a velocidade da massa. Equações de estado e equações de saída são então formuladas. A conversão dessas equações em forma de vetor-matriz fornece um modelo linear abrangente do sistema. Este modelo pode ser analisado usando várias técnicas de controle e estimativa linear, facilitando o design e a implementação de estratégias de controle para sistemas que são inerentemente não lineares.

Aproximação Linear no Domínio do Tempo

Os sistemas não lineares encontram uma modelagem superior na representação do espaço de estados.
Considere um pêndulo simples representado no espaço de estados com o centro de massa em ...

Education

JoVE Core

Electrical Engineering

Engenharia Elétrica

Unit 1

Unidade 1

Modeling in Time and Frequency Domain

Modelagem no Domínio do Tempo e da Frequência

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

linearizing nonlinear state-space models

Linearizing nonlinear state-space models helps describe systems that change with time or operating conditions. State-space representation is often effective for this kind of modeling. It is especially useful for systems such as a simple pendulum or a translational mechanical system with nonlinear springs.

A simple pendulum can be modeled by summing torques around the pivot point. In this case, the mass is evenly distributed along the pendulum, and the center of mass is located at half its length. The resulting differential equation includes gravitational force and any applied torque. State variables are then chosen to represent position and velocity.

These state variables lead to state equations, which describe how the system changes over time. To study the system near an equilibrium point, small perturbations are added to the state variables. A Taylor series expansion is then used to approximate the nonlinear terms with linear ones. The result is a set of linear state equations that can be analyzed with linear system theory.

The same idea applies to a translational mechanical system with a nonlinear spring. Its motion is governed by a differential equation that includes the nonlinear spring force. A small perturbation around the equilibrium position allows the equation to be linearized. The equilibrium force at x0, the position of equilibrium, is calculated, and the perturbed equation is differentiated to obtain a linearized differential equation.

The final linearized state-space model uses appropriate state variables, often position and velocity of the mass. State equations and output equations are written in vector-matrix form. This linear model makes it easier to apply linear control and estimation methods to systems that are inherently nonlinear.

Linearizing Nonlinear State-Space Models

transfer function in control systems

Transfer Function from Input to Output

The transfer function describes how a linear time-invariant, or LTI, system maps an input to an output in the frequency domain. It gives a compact way to study system behavior without working only in the time domain. It also connects the differential equations of the system to a form that is easier to handle mathematically. For an nth-order LTI system, the behavior is written as a differential equation with constant coefficients. In that equation, r(t) is the input and c(t) is the output. The...

electrical systems

RLC Circuit Transfer Functions

RLC circuits use resistors, capacitors, and inductors to show how electrical signals change over time. In circuit analysis, these passive linear components are arranged so the link between an input and an output can be described with a transfer function. For this topic, the output is the voltage across the capacitor, compared with the input voltage. A series RLC circuit can be analyzed with Kirchhoff's Voltage Law, or KVL. KVL says that the sum of all voltages around a closed loop is zero.

mechanical systems

Transfer Functions in Mechanical Motion

Mechanical motion can be modeled with springs, masses, and viscous dampers. In these models, springs behave like inductors and masses act like capacitors in an electrical network. A viscous damper works like a resistor because it removes energy from the system. A force acts on the mass in the direction of motion. It is balanced by the spring force, the damping force, and the force from the mass’s acceleration. Newton’s second law is used to describe this balance, and the sum of the forces on...

electro-mechanical systems

DC Motor Modeling in Electromechanical Systems

DC motor modeling is a key part of electromechanical systems. These systems combine electrical parts and mechanical parts to produce a useful output. A DC motor is one common example because it turns electrical energy into mechanical motion. It is used in many places, from simple fans to complex robotic mechanisms. Inside the DC motor, the armature is a rotating circuit placed in a magnetic field. When current flows through the armature, it experiences a force from the magnetic field. That...

linear approximation in frequency domain

Linearizing Nonlinear RL Circuits

Linearizing nonlinear RL circuits helps turn a hard circuit problem into a simpler linear one. Linear systems follow superposition, which means the total response equals the sum of the responses to separate inputs. They also follow homogeneity, which means scaling the input scales the output by the same amount. Nonlinear systems do not automatically have those properties. For small changes around an operating point, though, a nonlinear system can often be treated as linear. This is done with a...

state space representation

RLC Circuit State Equations

RLC circuit state equations show how a circuit changes over time using state variables. This time-domain method is useful for linear, time-invariant systems, but it also helps when frequency-domain analysis is not enough. It is especially useful for nonlinear systems, time-varying systems, and multiple-input multiple-output systems. The state-space approach turns an nth-order system into a set of simultaneous first-order differential equations called state equations. For a common second-order...

transfer function to state space

Phase-Variable State Equations from Transfer Functions

Phase-variable state equations show how a transfer function can be rewritten for computer simulation. In state-space form, the system is described with state variables, which are the output and its first several derivatives. This makes it easier to model a physical system on a digital computer. The starting point is an nth-order linear differential equation with constant coefficients. Systems like an RLC circuit can be written this way. To build the state equations, the transfer function is...

state space to transfer function

Finding a Transfer Function from State Space

Finding a transfer function from state-space form is a key step in system analysis. It changes a time-domain model into a frequency-domain form. That makes it easier to study and design control systems. The process starts with the state-space equations. These include the state equation and the output equation. In these equations, x(t) is the state vector, u(t) is the input vector, y(t) is the output vector, and A, B, C, and D are the matrices that define the system dynamics. Next, the Laplace...

linear approximation in time domain

Linearizing nonlinear state-space models helps describe systems that change with time or operating conditions. State-space representation is often effective for this kind of modeling. It is especially useful for systems such as a simple pendulum or a translational mechanical system with nonlinear springs. A simple pendulum can be modeled by summing torques around the pivot point. In this case, the mass is evenly distributed along the pendulum, and the center of mass is located at half its...

key terms and concepts

Learn how nonlinear systems like a pendulum or spring-mass setup are linearized with small perturbations, Taylor series, and state-space equations.

Linearizing Nonlinear State-Space Models | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Linearizing Nonlinear State-Space Models

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