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

Los sistemas no lineales encuentran un modelado superior en la representación del espacio de estados.
Consideremos un péndulo simple representado en el espacio de estados con el centro de masa en su mitad de longitud. El peso y el par aplicado son factores importantes.
La suma de los pares conduce a una ecuación diferencial. Se seleccionan las variables de estado y se formulan las ecuaciones de estado.
Para linealizar la ecuación alrededor del punto de equilibrio, considere las variables de estado perturbadas alrededor del punto de equilibrio.
Se emplea y resuelve la aproximación de la serie de Taylor, lo que da como resultado ecuaciones de estado lineal.
Considere un sistema de resorte no lineal. Formule una ecuación diferencial e introduzca una pequeña perturbación.
A continuación, esta ecuación se linealiza. Sustituyendo esto y llevando a cabo la diferenciación se obtiene la ecuación diferencial intermedia linealizada.
La fuerza de equilibrio se utiliza para calcular xo, formando finalmente la ecuación diferencial linealizada final.
Se seleccionan las variables de estado y se escriben las ecuaciones de estado y salida, lo que proporciona un modelo completo del sistema en el espacio de estados. Finalmente, todas las ecuaciones se convierten en forma de matriz vectorial.

aproximación lineal en el dominio temporal.

Linear Approximation in Time Domain

Los sistemas no lineales suelen requerir métodos sofisticados para un modelado y análisis precisos, siendo la representación en el espacio de estados particularmente eficaz. Este método es especialmente útil para sistemas en los que las variables y los parámetros varían con el tiempo o las condiciones de funcionamiento, como en un péndulo simple o un sistema mecánico traslacional con resortes no lineales.
En el caso de un péndulo simple con una masa distribuida uniformemente a lo largo de su longitud y cuyo centro de masas se encuentra a la mitad de la longitud del péndulo, el comportamiento dinámico se puede captar sumando los pares de torsión alrededor del punto de pivote. La ecuación diferencial resultante incorpora los efectos de la fuerza gravitacional y el par de torsión aplicado. Para representar esto en forma de espacio de estados, se eligen variables de estado para describir la posición y la velocidad del sistema. Estas variables de estado conducen a la formulación de ecuaciones de estado, que describen la evolución temporal del sistema.
La linealización de estas ecuaciones de estado no lineales en torno a un punto de equilibrio implica considerar pequeñas perturbaciones. Al perturbar las variables de estado en torno a sus valores de equilibrio y aplicar una expansión en serie de Taylor, los términos no lineales pueden aproximarse mediante sus contrapartes lineales. Esta aproximación produce ecuaciones de estado lineales que pueden analizarse utilizando la teoría de sistemas lineales.
En el caso de un sistema mecánico traslacional con un resorte no lineal, la dinámica del sistema está gobernada de manera similar por una ecuación diferencial que tiene en cuenta la fuerza del resorte no lineal. La introducción de una pequeña perturbación alrededor de la posición de equilibrio permite la linealización de la ecuación diferencial. Se calcula la fuerza de equilibrio en x_0 (la posición de equilibrio) y la ecuación perturbada se deriva para obtener una ecuación diferencial linealizada.
La representación linealizada final del espacio de estados implica la selección de las variables de estado adecuadas, que a menudo incluyen la posición y la velocidad de la masa. A continuación, se formulan las ecuaciones de estado y las ecuaciones de salida. La conversión de estas ecuaciones en forma de matriz vectorial proporciona un modelo lineal integral del sistema. Este modelo se puede analizar utilizando diversas técnicas de control y estimación lineal, lo que facilita el diseño y la implementación de estrategias de control para sistemas que son inherentemente no lineales.

Aproximación lineal en el dominio temporal.

Los sistemas no lineales encuentran un modelado superior en la representación del espacio de estados.
Considere un péndulo simple representado en el espacio de estados con el centro de masa en ...

Education

Educación

JoVE Core

Electrical Engineering

Ingeniería Eléctrica

Unit 1

Unidad 1

Modeling in Time and Frequency Domain

Modelado en el dominio del tiempo y de la frecuencia

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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...

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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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