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

La representación del espacio de estados se utiliza para simular sistemas físicos en ordenadores digitales. Para ello, primero hay que convertir la función de transferencia en espacio de estados.
Considere una ecuación diferencial lineal de nésimo orden con coeficientes constantes.
La salida y sus n-1 derivadas se eligen como variables de estado. Diferenciando estas series de ecuaciones y sustituyéndolas de nuevo en las ecuaciones originales, se obtienen las ecuaciones de estado.
Cada variable de estado subsiguiente se define como la derivada de la anterior.
Las ecuaciones resultantes se representan en forma de matriz vectorial, creando un patrón distinto de 1 y 0 junto con los coeficientes negativos de la ecuación diferencial. Esta estructura única es la forma de variable de fase de las ecuaciones de estado.
Considere una función de transferencia. La ecuación se multiplica de forma cruzada y la ecuación diferencial correspondiente se encuentra tomando la transformada inversa de Laplace, asumiendo condiciones iniciales cero.
Las variables de estado se eligen como derivadas sucesivas.
A continuación, la diferenciación se aplica a ambos lados de la ecuación, dando lugar a las ecuaciones de estado y a la ecuación de salida.
Estas ecuaciones se presentan en forma de matriz vectorial.

función de transferencia al espacio de estados

Transfer Function to State Space

La representación en el espacio de estados es una herramienta poderosa para simular sistemas físicos en computadoras digitales, lo que requiere la conversión de la función de transferencia a la forma de espacio de estados. Considere una ecuación diferencial lineal de orden n con coeficientes constantes, como las que se encuentran en un circuito RLC. Las variables de estado se seleccionan como la salida y sus derivadas n−1. Al diferenciar estas variables y sustituirlas nuevamente en la ecuación original se obtienen las ecuaciones de estado.
En un circuito RLC, el voltaje a través del capacitor y la corriente a través del inductor se pueden utilizar como variables de estado. Por ejemplo, en un sistema de segundo orden, como un circuito RLC en serie, podemos definir el voltaje a través del capacitor V_c como la primera variable de estado y la corriente a través del inductor i_L como la segunda variable de estado. La función de transferencia para un circuito RLC primero se multiplica de forma cruzada para obtener la ecuación diferencial. Al aplicar la transformada inversa de Laplace y suponer condiciones iniciales cero, se puede derivar la forma de dominio temporal de la ecuación.
<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;">
Luego, las variables de estado se eligen como derivadas sucesivas de la salida y se aplica la diferenciación a ambos lados de la ecuación para generar las ecuaciones de estado.
<img alt="Equation3" src="/files/ftp_upload/16101/16101_Equation_003.svg" style="height: 25px;">
Las ecuaciones diferenciales del sistema se representan luego en forma de matriz vectorial, lo que produce un patrón distintivo de 1 y 0, junto con los coeficientes negativos de la ecuación diferencial original.
<img alt="Equation4" src="/files/ftp_upload/16101/16101_Equation_004.svg" style="height: 90px;">
Esta forma se conoce como la forma variable de fase de las ecuaciones de estado. La estructura matricial proporciona un método claro y conciso para simular y analizar el comportamiento dinámico del sistema.

Función de transferencia al espacio de estados

La representación del espacio de estados se utiliza para simular sistemas físicos en ordenadores digitales. Para ello, la función de transferencia debe convertirse primero en ...

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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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Phase-Variable State Equations from Transfer Functions

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