Name: phase-variable state equations from transfer functions
Uploaded: 2024-11-21T14:54:05.000Z
Duration: 1 min 23 s
Channel: JoVE

상태공간 표현은 디지털 컴퓨터에서 물리적 시스템을 시뮬레이션하는 데 사용됩니다. 이를 위해서는 먼저 전달 함수를 상태 공간으로 변환해야 합니다.
상수 계수를 갖는 n차 선형 미분 방정식이 있다고 가정하겠습니다.
출력과 n-1 도함수가 상태 변수로 선택됩니다. 이러한 일련의 방정식을 미분하고 원래 방정식으로 다시 대체하면 상태 방정식이 제공됩니다.
각 후속 상태 변수는 이전 상태 변수의 도함수로 정의됩니다.
그런 다음 결과 방정식은 벡터-매트릭스 형태로 표현되어 미분 방정식의 음의 계수와 함께 1과 0의 뚜렷한 패턴을 생성합니다. 이 독특한 구조는 상태 방정식의 위상 변수 형식입니다.
전달 함수가 있다고 가정하겠습니다. 방정식은 교차 곱셈되고, 해당 미분 방정식은 초기 조건이 0이라고 가정하여 역 라플라스 변환을 취하여 구합니다.
상태 변수는 연속적인 도함수로 선택됩니다.
그런 다음 방정식의 양쪽에 미분이 적용되어 상태 방정식과 출력 방정식을 산출합니다.
그런 다음 이러한 방정식은 벡터-행렬 형식으로 표시됩니다.

상태 공간으로의 전달 함수

Transfer Function to State Space

상태 공간 표현은 디지털 컴퓨터에서 물리적 시스템을 시뮬레이션하는 강력한 도구로, 전달 함수를 상태 공간 형태로 변환해야 합니다. RLC 회로에서 발생하는 것과 같은 상수 계수를 갖는 n차 선형 미분 방정식을 고려합니다. 상태 변수는 출력과 n-1 도함수로 선택됩니다. 이러한 변수를 미분하여 원래 방정식에 다시 대입하면 상태 방정식이 생성됩니다.

RLC 회로에서 커패시터 전압과 인덕터를 통과하는 전류는 상태 변수로 사용할 수 있습니다. 예를 들어, 직렬 RLC 회로와 같은 2차 시스템에서 커패시터 전압 V_c를 첫 번째 상태 변수로 정의하고 인덕터 i_L을 통과하는 전류를 두 번째 상태 변수로 정의할 수 있습니다. RLC 회로의 전달 함수는 먼저 교차 곱하여 미분 방정식을 얻습니다. 역 라플라스 변환을 적용하고 초기 조건이 0이라고 가정하면 방정식의 시간 영역 형태를 도출할 수 있습니다.

<img alt="Equation1" src="/files/ftp_upload/16101/16101_Equation_001.svg" style="height: 140px;">
<img alt="Equation2" src="/files/ftp_upload/16101/16101_Equation_002.svg" style="height: 60px;">

그런 다음 상태 변수를 출력의 연속적인 미분으로 선택하고 방정식의 양변에 미분을 적용하여 상태 방정식을 생성합니다.

<img alt="Equation3" src="/files/ftp_upload/16101/16101_Equation_003.svg" style="height: 25px;">

그런 다음 시스템의 미분 방정식을 벡터-행렬 형태로 표현하여 원래 미분 방정식의 음수 계수와 함께 1과 0의 고유한 패턴을 생성합니다.

<img alt="Equation4" src="/files/ftp_upload/16101/16101_Equation_004.svg" style="height: 90px;">

이것은 상태 방정식의 위상 변수 형태로 알려져 있습니다. 행렬 구조는 시스템의 동적 거동을 시뮬레이션하고 분석하기 위한 명확하고 간결한 방법을 제공합니다.

상태공간 표현은 디지털 컴퓨터에서 물리적 시스템을 시뮬레이션하는 데 사용됩니다. 이를 위해 전송 함수는 먼저 ...

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

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 first cross-multiplied to form the differential equation. Then the inverse Laplace transform is applied, with zero initial conditions, to get the time-domain form.

Next, the state variables are chosen from the output and its successive derivatives. For a second-order system such as a series RLC circuit, the voltage across the capacitor, Vc, can be used as the first state variable. The current through the inductor, iL, can be used as the second state variable.

After the variables are selected, each one is differentiated and substituted back into the original equation. This process generates the state equations. The equations are then written in vector-matrix form.

In phase-variable form, the matrix has a clear pattern of 1s and 0s. It also contains the negative coefficients from the original differential equation. This structure gives a compact way to simulate and analyze the system's dynamic behavior.

Phase-Variable State Equations from Transfer Functions

phase-variable-state-equations-from-transfer-functions

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 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

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 transfer functions are converted into phase-variable state equations using state variables, inverse Laplace transforms, and matrix form for RLC circuits.

Phase-Variable State Equations from Transfer Functions | 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

Phase-Variable State Equations from Transfer Functions

Description

Transcript

Tags