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

状態空間表現は、デジタル コンピューター上の物理システムをシミュレートするために使用されます。このためには、伝達関数を最初に状態空間に変換する必要があります。
係数が一定の n 次線形微分方程式を考えます。
出力とその n-1 導関数が状態変数として選択されます。これらの一連の方程式を微分し、それらを元の方程式に代入すると、状態方程式が得られます。
後続の各状態変数は、前の変数の導関数として定義されます。
次に、結果の方程式はベクトル行列形式で表され、微分方程式の負の係数とともに 1 と 0 の明確なパターンが作成されます。このユニークな構造は、状態方程式の位相変数形式です。
伝達関数を考えてみましょう。方程式は交差乗算され、対応する微分方程式は、初期条件がゼロであると仮定して逆ラプラス変換を行うことによって求められます。
状態変数は、連続する導関数として選択されます。
次に、方程式の両側に微分が適用され、状態方程式と出力方程式が得られます。
これらの方程式は、ベクトル行列形式で表されます。

伝達関数から状態空間への変換

Transfer Function to State Space

状態空間表現は、デジタルコンピューター上で物理システムをシミュレートするための強力なツールであり、伝達関数を状態空間形式に変換する必要があります。RLC 回路で見られる定数係数を持つ n 次線形微分方程式を考えてみましょう。状態変数は出力とその n−1 導関数として選択されます。これらの変数を微分し、元の方程式に代入すると、状態方程式が生成されます。

RLC 回路では、コンデンサの両端の電圧とインダクタを流れる電流を状態変数として使用できます。たとえば、直列 RLC 回路などの 2 次システムでは、コンデンサの両端の電圧 V_c を最初の状態変数として定義し、インダクタを流れる電流 i_L を 2 番目の状態変数として定義できます。RLC 回路の伝達関数は、最初にクロス乗算されて微分方程式を取得します。逆ラプラス変換を適用し、初期条件がゼロであると仮定すると、方程式の時間領域形式を導出できます。

<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;">

次に、状態変数が出力の連続導関数として選択され、方程式の両辺に微分が適用されて状態方程式が生成されます。

<img alt="Equation3" src="/files/ftp_upload/16101/16101_Equation_003.svg" style="height: 25px;">

次に、システムの微分方程式がベクトル行列形式で表現され、元の微分方程式の負の係数とともに、1 と 0 の明確なパターンが生成されます。

<img alt="Equation4" src="/files/ftp_upload/16101/16101_Equation_004.svg" style="height: 90px;">

これは状態方程式の位相変数形式として知られています。行列構造は、システムの動的動作をシミュレートおよび分析するための明確で簡潔な方法を提供します。

状態空間表現は、デジタル コンピューター上の物理システムをシミュレートするために使用されます。このためには、伝達関数を最初に...

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

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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Linearizing Nonlinear RL Circuits

Linearizing nonlinear RL circuits helps turn a hard circuit problem into a simpler linear one. Linear systems follow superposition, which means the total response equals the sum of the responses to separate inputs. They also follow homogeneity, which means scaling the input scales the output by the same amount. Nonlinear systems do not automatically have those properties. For small changes around an operating point, though, a nonlinear system can often be treated as linear. This is done with a...

state space representation

RLC Circuit State Equations

RLC circuit state equations show how a circuit changes over time using state variables. This time-domain method is useful for linear, time-invariant systems, but it also helps when frequency-domain analysis is not enough. It is especially useful for nonlinear systems, time-varying systems, and multiple-input multiple-output systems. The state-space approach turns an nth-order system into a set of simultaneous first-order differential equations called state equations. For a common second-order...

transfer function to state space

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...

state space to transfer function

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...

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Linearizing Nonlinear State-Space Models

Linearizing nonlinear state-space models helps describe systems that change with time or operating conditions. State-space representation is often effective for this kind of modeling. It is especially useful for systems such as a simple pendulum or a translational mechanical system with nonlinear springs. A simple pendulum can be modeled by summing torques around the pivot point. In this case, the mass is evenly distributed along the pendulum, and the center of mass is located at half its...

key terms and concepts

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 電気工学 は、電気回路の基本概念と回路解析テクニックを網羅したアニメーションビデオシリーズです。オペアンプと回路変換および 1 次・ 2 次回路の概念を網羅しています。実験を撮影した実写動画では、設計例や現代社会での応用が含まれており、複雑な概念も分かりやすく学習いただけます。

electrical engineering

Phase-Variable State Equations from Transfer Functions

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