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

A representação do espaço de estados é usada para simular sistemas físicos em computadores digitais. Para isso, a função de transferência deve primeiro ser convertida em espaço de estados.
Considere uma equação diferencial linear de nésima ordem com coeficientes constantes.
A saída e seus derivados n-1 são escolhidos como variáveis de estado. Diferenciar essas séries de equações e substituí-las de volta nas equações originais fornece as equações de estado.
Cada variável de estado subsequente é definida como a derivada da anterior.
As equações resultantes são então representadas na forma de matriz vetorial, criando um padrão distinto de 1 e 0 junto com os coeficientes negativos da equação diferencial. Essa estrutura única é a forma de variável de fase das equações de estado.
Considere uma função de transferência. A equação é multiplicada cruzada e a equação diferencial correspondente é então encontrada tomando a transformada de Laplace inversa, assumindo condições iniciais zero.
As variáveis de estado são escolhidas como derivadas sucessivas.
A diferenciação é então aplicada a ambos os lados da equação, produzindo as equações de estado e a equação de saída.
Essas equações são então apresentadas na forma de matriz vetorial.

função de transferência para espaço de estado

Transfer Function to State Space

A representação do espaço de estado é uma ferramenta poderosa para simular sistemas físicos em computadores digitais, necessitando da conversão da função de transferência para a forma de espaço de estado. Considere uma equação diferencial linear de ordem n com coeficientes constantes, como aqueles encontrados em um circuito RLC. As variáveis de estado são selecionadas como a saída e suas derivadas n−1. Diferenciar essas variáveis e substituí-las de volta na equação original produz as equações de estado.

Em um circuito RLC, a tensão através do capacitor e a corrente através do indutor podem ser usadas como variáveis de estado. Por exemplo, em um sistema de segunda ordem, como um circuito RLC em série, podemos definir a tensão através do capacitor V_c como a primeira variável de estado e a corrente através do indutor i_L como a segunda variável de estado. A função de transferência para um circuito RLC é primeiro multiplicada cruzadamente para obter a equação diferencial. Ao aplicar a transformada de Laplace inversa e assumir condições iniciais zero, pode-se derivar a forma de domínio de tempo da equação.

<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;">

As variáveis de estado são então escolhidas como derivadas sucessivas da saída, e a diferenciação é aplicada a ambos os lados da equação para gerar as equações de estado.

<img alt="Equation3" src="/files/ftp_upload/16101/16101_Equation_003.svg" style="height: 25px;"> 

As equações diferenciais do sistema são então representadas em forma de vetor-matriz, produzindo um padrão distinto de 1's e 0's, juntamente com os coeficientes negativos da equação diferencial original.

<img alt="Equation4" src="/files/ftp_upload/16101/16101_Equation_004.svg" style="height: 90px;">

Isso é conhecido como a forma de variável de fase das equações de estado. A estrutura da matriz fornece um método claro e conciso para simular e analisar o comportamento dinâmico do sistema.

Função de Transferência para Espaço de Estado

A representação do espaço de estados é usada para simular sistemas físicos em computadores digitais. Para isso, a função de transferência deve primeiro ser convertida em ...

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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 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

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

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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

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