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

La représentation de l’espace d’état peut également être convertie en fonction de transfert dans l’analyse du système.
La transformation commence par l’état donné et les équations de sortie.
La transformée de Laplace est appliquée ici, en supposant que les conditions initiales sont nulles. Cela transforme les équations du domaine temporel au domaine fréquentiel.
L’équation d’état est résolue pour X(s), où I représente la matrice identité. Cette solution est ensuite substituée dans l’équation de sortie.
La matrice résultante, connue sous le nom de matrice de fonction de transfert, relie le vecteur de sortie au vecteur d’entrée.
Lorsque ces vecteurs sont scalaires, il devient possible de trouver la fonction de transfert finale, complétant ainsi la transformation de la représentation de l’espace d’état en fonction de transfert.
Considérons un système défini par des matrices de différentes dimensions qui forment les équations d’état et de sortie.
Bien que tous les autres termes de l’équation de la fonction de transfert soient déjà définis, un terme reste inconnu.
Pour trouver ce terme, les valeurs de matrice connues sont utilisées à partir de l’équation d’état. De plus, l’inverse est calculé.
Lors de la substitution, la représentation de l’espace d’état est convertie en une fonction de transfert.

espace d'état pour fonction de transfert

State Space to Transfer Function

La conversion d'une représentation de l'espace d'état en fonction de transfert est un processus fondamental dans l'analyse des systèmes. Elle fournit une méthode de transition d'une description du domaine temporel à une représentation du domaine fréquentiel, ce qui est essentiel pour simplifier l'analyse et la conception des systèmes de contrôle.

Le processus de transformation commence par la représentation de l'espace d'état, caractérisée par l'équation d'état et l'équation de sortie. Ces équations sont généralement représentées comme suit:

<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;">

Où x(t) est le vecteur d'état, u(t) est le vecteur d'entrée, y(t) est le vecteur de sortie et A, B, C et D sont des matrices définissant la dynamique du système.

Pour convertir ces équations dans le domaine fréquentiel, on applique la transformée de Laplace, en supposant des conditions initiales nulles. L'équation d'état est ensuite résolue pour X(s).

Considérons un système avec les matrices A, B, C et D données. Le processus de transformation consiste à calculer l'inverse de (SI−A), à remplacer les valeurs connues et à simplifier l'expression pour obtenir la fonction de transfert. Cette transformation est essentielle pour analyser le comportement du système, concevoir des contrôleurs et comprendre la réponse en fréquence du système.

En conclusion, la conversion de la représentation de l'espace d'état en une fonction de transfert implique l'application de la transformée de Laplace, la résolution de l'équation d'état dans le domaine fréquentiel et la dérivation de la matrice de fonction de transfert, qui se simplifie en une fonction de transfert scalaire pour les systèmes à entrée unique et sortie unique (SISO).

Espace d'état pour fonction de transfert

La représentation de l’espace d’état peut également être convertie en fonction de transfert dans l’analyse du système.
La transformation commence par l’état donné et la sortie ...

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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

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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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