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

비선형 시스템은 상태공간 표현에서 우수한 모델링을 찾습니다.
질량 중심이 절반 길이인 상태 공간으로 표시되는 간단한 진자를 생각해 보십시오. 무게와 적용된 토크는 중요한 요소입니다.
토크를 합산하면 미분 방정식이 생성됩니다. 상태 변수가 선택되고 상태 방정식이 공식화됩니다.
평형점에 대한 방정식을 선형화하려면 평형점에 대해 섭동된 상태 변수를 고려하십시오.
테일러 급수 근사치가 사용되고 해결되어 선형 상태 방정식이 생성됩니다.
비선형 스프링 시스템을 생각해 보십시오. 미분 방정식을 공식화하고 작은 섭동을 도입합니다.
그런 다음 이 방정식이 선형화됩니다. 이를 대체하고 미분을 수행하면 선형화된 중간 미분 방정식이 제공됩니다.
평형력은 xo를 계산하는 데 사용되며, 궁극적으로 최종 선형화된 미분 방정식을 형성합니다.
상태 변수가 선택되고 상태 및 출력 방정식이 작성되어 상태 공간에서 시스템의 완전한 모델을 제공합니다. 마지막으로 모든 방정식은 벡터-행렬 형식으로 변환됩니다.

시간 영역에서의 선형 근사

Linear Approximation in Time Domain

비선형 시스템은 정확한 모델링과 분석을 위해 정교한 접근 방식이 필요한 경우가 많으며, 상태 공간 표현이 특히 효과적입니다. 이 방법은 단순 진자나 비선형 스프링이 있는 이동 기계 시스템과 같이 변수와 매개변수가 시간이나 작동 조건에 따라 변하는 시스템에 특히 유용합니다.

질량이 길이에 따라 고르게 분포되어 있고 질량 중심이 진자 길이의 절반에 위치한 단순 진자의 경우, 피벗 지점 주변의 토크를 합산하여 동적 거동을 포착할 수 있습니다. 결과 미분 방정식은 중력과 적용된 토크의 효과를 통합합니다. 이를 상태 공간 형태로 표현하기 위해 시스템의 위치와 속도를 설명하는 상태 변수를 선택합니다. 이러한 상태 변수는 시스템의 시간적 진화를 설명하는 상태 방정식의 공식화로 이어집니다.

평형점 주변의 이러한 비선형 상태 방정식을 선형화하려면 작은 섭동을 고려해야 합니다. 평형 값 주변의 상태 변수를 섭동시키고 테일러 급수 확장을 적용하면 비선형 항을 선형 대응항으로 근사할 수 있습니다. 이 근사치는 선형 시스템 이론을 사용하여 분석할 수 있는 선형 상태 방정식을 생성합니다.

비선형 스프링이 있는 병진 기계 시스템의 경우, 시스템의 동역학은 비선형 스프링 힘을 설명하는 미분 방정식에 의해 유사하게 지배됩니다. 평형 위치 주변에 작은 섭동을 도입하면 미분 방정식의 선형화가 가능합니다. x_0(평형 위치)의 평형력이 계산되고, 섭동된 방정식이 미분되어 선형화된 미분 방정식을 얻습니다.

최종 선형화된 상태 공간 표현에는 적절한 상태 변수를 선택하는 것이 포함되며, 여기에는 종종 질량의 위치와 속도가 포함됩니다. 그런 다음 상태 방정식과 출력 방정식이 공식화됩니다. 이러한 방정식을 벡터 행렬 형태로 변환하면 시스템의 포괄적인 선형 모델이 제공됩니다. 이 모델은 다양한 선형 제어 및 추정 기술을 사용하여 분석할 수 있으며, 본질적으로 비선형인 시스템에 대한 제어 전략의 설계 및 구현을 용이하게 합니다.

비선형 시스템은 상태공간 표현에서 우수한 모델링을 찾습니다.
질량 중심이 에서 나타나는 상태 공간에 표시되는 간단한 진자를 고려하십시오.

Education

교육

JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

Modeling in Time and Frequency Domain

시간 및 주파수 영역 모델링

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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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