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

La représentation de l’espace d’état est utilisée pour simuler des systèmes physiques sur des ordinateurs numériques. Pour cela, la fonction de transfert doit d’abord être convertie en espace d’état.
Considérons une équation différentielle linéaire d’ordre n avec des coefficients constants.
La sortie et ses dérivées n-1 sont choisies comme variables d’état. En dérivant ces séries d’équations et en les substituant dans les équations d’origine, on obtient les équations d’état.
Chaque variable d’état suivante est définie comme la dérivée de la précédente.
Les équations résultantes sont ensuite représentées sous forme de matrice vectorielle, créant un motif distinct de 1 et de 0 ainsi que les coefficients négatifs de l'équation différentielle. Cette structure unique est la forme à variable de phase des équations d’état.
Considérons une fonction de transfert. L’équation est multipliée par un croisement, et l’équation différentielle correspondante est ensuite trouvée en prenant la transformée de Laplace inverse, en supposant des conditions initiales nulles.
Les variables d’état sont choisies comme dérivées successives.
La dérivation est ensuite appliquée aux deux côtés de l’équation, ce qui donne les équations d’état et l’équation de sortie.
Ces équations sont ensuite présentées sous forme de matrice vectorielle.

fonction de transfert vers l'espace d'état

Transfer Function to State Space

La représentation de l'espace d'état est un outil puissant pour simuler des systèmes physiques sur des ordinateurs numériques, ce qui nécessite la conversion de la fonction de transfert sous forme d'espace d'état. Considérons une équation différentielle linéaire d'ordre n à coefficients constants, comme celles rencontrées dans un circuit RLC. Les variables d'état sont sélectionnées comme sortie et ses dérivées n−1. La différenciation de ces variables et leur remplacement dans l'équation d'origine produisent les équations d'état.

Dans un circuit RLC, la tension aux bornes du condensateur et le courant à travers l'inducteur peuvent être utilisés comme variables d'état. Par exemple, dans un système du second ordre, tel qu'un circuit RLC série, nous pouvons définir la tension aux bornes du condensateur V_c comme première variable d'état et le courant à travers l'inducteur i_L comme seconde variable d'état. La fonction de transfert d'un circuit RLC est d'abord multipliée par multiplication croisée pour obtenir l'équation différentielle. En appliquant la transformée de Laplace inverse et en supposant des conditions initiales nulles, on peut dériver la forme temporelle de l'équation.

<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;">

Les variables d’état sont ensuite choisies comme dérivées successives de la sortie, et la différenciation est appliquée aux deux côtés de l’équation pour générer les équations d’état.

<img alt="Equation3" src="/files/ftp_upload/16101/16101_Equation_003.svg" style="height: 25px;">

Les équations différentielles du système sont ensuite représentées sous forme de matrice vectorielle, produisant un motif distinct de 1 et de 0, ainsi que les coefficients négatifs de l'équation différentielle d'origine.

<img alt="Equation4" src="/files/ftp_upload/16101/16101_Equation_004.svg" style="height: 90px;">

Il s'agit de la forme à phase variable des équations d'état. La structure matricielle fournit une méthode claire et concise pour simuler et analyser le comportement dynamique du système.

Fonction de transfert vers l'espace d'état

La représentation de l’espace d’état est utilisée pour simuler des systèmes physiques sur des ordinateurs numériques. Pour cela, la fonction de transfert doit d’abord être convertie 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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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...

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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.

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

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