23.6
ラウス・ハーウィッツ基準I
停電を防ぐために安定性が不可欠な電力網を考えてみましょう。ラウス・ハーウィッツ基準は、さまざまな負荷条件や障害下でのシステムの安定性を評価するための貴重なツールです。ラウス・ハーウィッツ基準は、閉ループ伝達関数を分析することで、システムが安定しているかどうかを判断するのに役立ちます。
ラウス・ハーウ…
停電を防ぐために安定性が重要な電力網を考えてみましょう。Routh-Hurwitz基準を使用すると、さまざまな負荷条件または故障下でのシステムの安定性を評価できます。
閉ループ伝達関数について考えます。Routh テーブルを作成するには、行に高いものから s の累乗でラベルを付けます。
最初の行には、分母の他のすべての係数が水平方向に入力され、最も高い累乗から開始されます。2 行目は、次に高い累乗の後に続き、スキップされた係数が一覧表示されます。
後続のエントリは、前の行の負の行列式を最初の列のエントリのすぐ上で除算して計算されます。
システムの場合、Routh テーブルの行が計算されます。正の定数は、行を個別にスケーリングします。
Routh-Hurwitz 基準では、右半平面の多項式の根の数は、Routh 表の最初の列の符号変更の数と等しく、システムが不安定であることを示しています。
すべての極が S 平面の左半分にある場合、システムは安定しており、これは Routh テーブルの最初の列に符号が変わらないことを意味します。
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Q1: What is the Routh-Hurwitz criterion used for in control systems?
The Routh-Hurwitz criterion is a mathematical tool for assessing system stability by analyzing the closed-loop transfer function without calculating pole locations explicitly. It determines whether all poles lie in the left-half s-plane, ensuring stable operation. This method is essential for evaluating electrical power grids and other complex systems under varying load conditions or faults.
Q2: How do you construct a Routh table?
A Routh table is built by labeling rows with powers of the complex frequency variable s, starting from the highest power. The first row contains every other coefficient of the denominator polynomial, while the second row holds the skipped coefficients. Subsequent entries are calculated using negative determinants of preceding rows divided by the first-column entry directly above.
Q3: What do sign changes in the Routh table's first column indicate?
Sign changes in the first column of the Routh table directly correspond to the number of polynomial roots in the right-half s-plane. Each sign change indicates an unstable pole. A stable system exhibits no sign changes, confirming all poles reside in the left-half s-plane.
Q4: Why is pole location important for system stability?
Pole location determines system stability: poles in the left-half s-plane produce stable responses that decay over time, while poles in the right-half s-plane cause unstable, growing responses. The Routh-Hurwitz criterion identifies pole locations by analyzing sign changes, enabling engineers to ensure reliable operation of critical systems like electrical power grids.
Q5: Can rows in a Routh table be scaled during calculation?
Yes, each row in the Routh table can be independently scaled by a positive constant to simplify calculations without affecting the stability conclusion. Scaling does not change the number of sign changes in the first column, so the final stability assessment remains valid regardless of scaling applied.
Q6: How does the Routh-Hurwitz criterion relate to transient and steady state response?
The Routh-Hurwitz criterion ensures system stability, which directly impacts both transient and steady state response characteristics. A stable system with poles in the left-half s-plane produces bounded transient responses that decay to steady-state values. Understanding stability through this criterion is fundamental to analyzing overall system behavior.
Q7: What happens if a Routh table shows multiple sign changes in the first column?
Multiple sign changes in the first column indicate multiple unstable poles in the right-half s-plane. Each sign change represents one unstable root, so a system with two sign changes has two unstable poles. This instability would cause the system response to grow unboundedly, making it unsuitable for practical applications like power grid operation.