Name: finding a transfer function from state space
Uploaded: 2024-11-21T14:54:05.000Z
Duration: 1 min 21 s
Channel: JoVE

상태공간 표현은 시스템 분석에서 전달 함수로 변환될 수도 있습니다.
변환은 주어진 상태 방정식과 출력 방정식으로 시작됩니다.
여기에 라플라스 변환이 적용되며, 초기 조건이 0이라고 가정합니다. 이렇게 하면 방정식이 시간 영역에서 주파수 영역으로 변환됩니다.
상태 방정식은 X(s)에 대해 해를 구하며, 여기서 I는 단위 행렬을 나타냅니다. 그런 다음 이 솔루션은 출력 방정식에 대체됩니다.
결과 행렬인 전달 함수 행렬(transfer function matrix)은 출력 벡터를 입력 벡터에 연결합니다.
이러한 벡터가 스칼라인 경우 최종 전달 함수를 구할 수 있게 되며, 이에 따라 상태공간 표현에서 전달 함수로의 변환이 완료됩니다.
상태 방정식과 출력 방정식을 형성하는 서로 다른 차원의 행렬로 정의되는 시스템이 있다고 가정해 보겠습니다.
전달 함수 방정식의 다른 모든 항은 이미 정의되어 있지만 하나의 항은 알려지지 않은 상태로 남아 있습니다.
이 항을 찾기 위해 상태 방정식에서 알려진 행렬 값이 사용됩니다. 또한 역이 계산됩니다.
대입 시 상태공간 표현은 전달 함수로 변환됩니다.

상태 공간에서 함수 전달

State Space to Transfer Function

상태 공간 표현을 전달 함수로 변환하는 것은 시스템 분석에서 기본적인 프로세스입니다. 시간 영역 설명에서 주파수 영역 표현으로 전환하는 방법을 제공하며, 이는 제어 시스템의 분석 및 설계를 단순화하는 데 중요합니다.

변환 과정은 상태 공간 표현으로 시작하며, 상태 방정식과 출력 방정식으로 특징지어집니다. 이러한 방정식은 일반적으로 다음과 같이 표현됩니다.

<img alt="Equation1" src="/files/ftp_upload/16102/16102_PT_equation_001.svg" style="height: 14px;">
<img alt="Equation2" src="/files/ftp_upload/16102/16102_PT_equation_002.svg" style="height: 14px;">

여기서 x(t)는 상태 벡터, u(t)는 입력 벡터, y(t)는 출력 벡터이며, A, B, C, D는 시스템 동역학을 정의하는 행렬입니다.

이러한 방정식을 주파수 영역으로 변환하기 위해, 초기 조건이 0이라고 가정하고 라플라스 변환을 적용합니다. 그런 다음 상태 방정식을 X(s)에 대해 풉니다.

주어진 행렬 A, B, C, D를 갖는 시스템을 고려합니다. 변환 프로세스는 (SI-A)의 역수를 계산하고, 알려진 값을 대입하고, 표현식을 단순화하여 전달 함수를 얻는 것을 포함합니다. 이 변환은 시스템 동작을 분석하고, 컨트롤러를 설계하고, 시스템의 주파수 응답을 이해하는 데 중요합니다.

결론적으로, 상태 공간 표현을 전달 함수로 변환하려면 라플라스 변환을 적용하고, 주파수 영역에서 상태 방정식을 풀고, 전달 함수 행렬을 도출해야 합니다. 이는 단일 입력, 단일 출력(SISO) 시스템에 대한 스칼라 전달 함수로 단순화됩니다.

상태공간 표현은 시스템 분석에서 전달 함수로 변환될 수도 있습니다.
변환은 주어진 상태와 출력으로 시작됩니다 ...

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

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 transform is applied. This step assumes zero initial conditions. The state equation is then solved for X(s), the state vector in the Laplace domain.

For a system with known A, B, C, and D matrices, the next step is to calculate the inverse of (sI−A). The known values are substituted into the expression, and the result is simplified to obtain the transfer function. This result is especially useful for studying system behavior and frequency response.

The same method gives a transfer function matrix for multi-input, multi-output systems. For single-input, single-output, or SISO systems, that matrix reduces to a scalar transfer function. This conversion is an important tool for control design and system analysis.

Finding a Transfer Function from State Space

finding-a-transfer-function-from-state-space

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 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...

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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...

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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 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

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 state-space equations are converted into a transfer function using the Laplace transform, zero initial conditions, and the inverse of (sI−A).

Finding a Transfer Function from State Space | 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

Finding a Transfer Function from State Space

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