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

状態空間表現は、システム解析で伝達関数に変換することもできます。
変換は、指定された状態方程式と出力方程式から始まります。
ここでは、初期条件がゼロであると仮定して、ラプラス変換が適用されます。これにより、方程式が時間領域から周波数領域に変換されます。
状態方程式は X(s) について解かれ、I は単位行列を表します。この解は、出力方程式に代入されます。
結果として得られる行列 (伝達関数行列) は、出力ベクトルを入力ベクトルにリンクします。
これらのベクトルがスカラーの場合、最終的な伝達関数を見つけることが可能になり、状態空間表現から伝達関数への変換が完了します。
状態方程式と出力方程式を形成するさまざまな次元の行列によって定義されるシステムについて考えてみます。
伝達関数方程式の他のすべての項はすでに定義されていますが、1つの項は不明なままです。
この項を見つけるために、状態方程式から既知の行列値が使用されます。さらに、逆数が計算されます。
置換時に、状態空間表現は伝達関数に変換されます。

状態空間から伝達関数へ

State Space to Transfer Function

状態空間表現から伝達関数への変換は、システム解析における基本的なプロセスです。これは、時間領域記述から周波数領域表現への移行方法を提供し、制御システムの解析と設計を簡素化するために重要です。

変換プロセスは、状態方程式と出力方程式によって特徴付けられる状態空間表現から始まります。これらの方程式は通常、次のように表されます。

<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;">

ここで、x(t) は状態ベクトル、u(t) は入力ベクトル、y(t) は出力ベクトル、A、B、C、D はシステムダイナミクスを定義する行列です。

これらの方程式を周波数領域に変換するには、初期条件がゼロであると仮定してラプラス変換を適用します。次に、状態方程式を X(s) について解きます。

与えられた行列 A、B、C、D を持つシステムを考えてみましょう。変換プロセスでは、(SI−A) の逆数を計算し、既知の値を代入し、式を簡略化して伝達関数を得ます。この変換は、システムの動作分析をし、制御設計をし、システムの周波数応答を理解するために極めて重要です。

結論として、状態空間表現を伝達関数に変換するには、ラプラス変換を適用し、周波数領域で状態方程式を解き、伝達関数行列を導出する必要があります。伝達関数行列は、単入力単出力 (SISO) システムのスカラー伝達関数に簡略化されます。

状態空間表現は、システム解析で伝達関数に変換することもできます。
変換は、指定された状態と出力から始まります...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Modeling in Time and Frequency Domain

時間領域および周波数領域におけるモデリング

KEY TERMS AND CONCEPTS

主要用語と概念

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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state space to transfer function

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...

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key terms and concepts

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

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electrical engineering

Finding a Transfer Function from State Space

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