Name: root locus paths and asymptotes
Uploaded: 2024-11-21T14:54:13.000Z
Duration: 1 min 15 s
Channel: JoVE

根軌跡の構築には、閉ループ極の数に等しい枝の数を決定することが含まれます。根の軌跡は実軸に対して対称です。
根軌跡は、有限の開ループ極と左側のゼロの総数の合計が奇数である実軸の部分に存在します。
開ループ極を起点とする軌跡は、ゲインが増加する閉ループ システム ポールをトレースし、ゲインが上昇するとシステム ポールが安定する開ループ ゼロで終了します。
s が無限大に近づくと無限大に近づく関数は、無限大に極を持ちます。ゼロに近づくと、無限大にゼロになります。
根の軌跡が無限遠に向かって移動すると、漸近線と呼ばれる特定の直線に近づきます。特定のシステムについて、実軸の切片と角度が計算されます。ゲインが大きくなると、角度が繰り返され始めます。
彼らの方程式は、実軸の切片と角度に基づいており、極から無限大への軌跡の軌跡を明らかにします。
軌跡は開ループの極から始まり、開ループのゼロで終了します。

根軌跡の構築

根軌跡の構築には、さまざまなゲインを持つシステムの極の動作を解析および視覚化するためのいくつかの重要な手順が含まれます。根軌跡の分岐の数は、閉ループ極の数に等しく、実軸を中心に対称です。

ゲイン値が正の場合、ルート軌跡は実軸上の奇数個の有限な開ループ極またはゼロ点の左側に存在します。根軌跡は開ループ極から始まり、ゲインが増加するにつれて閉ループ極の経路を辿ります。ゲインが上昇し続けるにつれてシステム極が安定する開ループゼロ点で終了します。

s が無限大に近づくにつれて関数が無限大に近づく場合、関数の極は無限大に存在します。逆に、関数がゼロに近づく場合、関数のゼロは無限大に存在します。根軌跡が無限大に向かって伸びるにつれて、特定の漸近線をたどります。これらの漸近線の方程式は、実軸切片とその角度によって決定され、軌跡が極から無限大までたどる経路を示します。

与えられたシステムでこれらの経路を計算するには、実軸切片と漸近線の角度を決定します。角度は極とゼロ点の数に基づいて導出され、ゲインが増加すると繰り返します。漸近線の角度の式は次のように表されます。

<img alt="Equation1" src="/files/ftp_upload/16122/16122_PT_Equation_001.svg" style="height: 30px;">

ここで、k は整数、n は極の数、m はゼロ点の数です。

完全な根軌跡プロットは、開ループ極から始まり、開ループゼロ点で終わります。このプロットは根軌跡法の規則に準拠しており、システムの安定性と応答を徹底的に分析および予測できることを保証します。

これらの手順に従うことで、エンジニアは根軌跡法を効果的に使用して制御システムを設計および調整し、さまざまな動作条件で望ましいパフォーマンスと安定性を確保できます。このアプローチにより、ゲインの増加に伴ってシステム極がどのように移動するかが明確に視覚化され、制御戦略の堅牢な設計に役立ちます。

根軌跡の構築には、閉ループ極の数に等しい枝の数を決定することが含まれます。根の軌跡は対称的です...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Root-Locus Method

根軌跡法

KEY TERMS AND CONCEPTS

主要用語と概念

Construction of Root Locus

root locus paths and asymptotes

Root locus paths and asymptotes show how closed-loop poles move as gain changes. This method helps students track system stability by following the pole locations on a graph. It also shows where the poles start, where they may end, and how they travel in between.

The number of root locus branches equals the number of closed-loop poles. The plot is symmetric about the real axis. For positive gain, the root locus lies on the real axis to the left of an odd number of finite open-loop poles or zeros. The paths begin at the open-loop poles and move as gain increases.

As gain continues to rise, the root locus ends at the open-loop zeros. At those points, the system poles settle. If a function goes to infinity as s approaches infinity, it has a pole at infinity. If a function goes to zero, it has a zero at infinity.

When the root locus extends toward infinity, it follows asymptotes. An asymptote is a line that shows the direction of a branch far from the poles and zeros. The asymptotes are found from the real-axis intercept and their angles. The angle formula depends on the number of poles, n, the number of zeros, m, and an integer k. These angles repeat as gain increases.

A complete root locus plot starts at the open-loop poles and ends at the open-loop zeros. It follows the rules of the root locus method so stability and response can be studied and predicted. Engineers use this plot to design and tune control systems for the desired performance across different operating conditions. The result is a clear picture of how system poles migrate with increasing gain.

Root Locus Paths and Asymptotes

control system problem

Root Locus and Closed-Loop Stability

Root locus and closed-loop stability are key ideas in control systems. A basic thermostat is an open-loop system, so its transfer function poles affect the response but do not set stability. When feedback is added, the system becomes closed-loop, like an advanced thermostat that changes heating based on room temperature. In a closed-loop system, stability depends on the new poles of the closed-loop transfer function. If those poles move into the unstable region, the system can start to...

vector representation of complex numbers

Complex Numbers as Vectors in Polar Form

Complex numbers can be shown as vectors, not just as points in Cartesian coordinates. In polar form, a vector is described by its magnitude, or length, and its angle. This makes complex numbers easier to compare by size and direction. The same idea also helps describe complex functions. When a complex number goes into a function, the output is another complex number. The function can be viewed as having a zero point, and the vector picture starts from that point. For a function written as a...

root-locus method

Cruise Control Stability with Root Locus

Root locus helps show how cruise control stays stable as conditions change. In a car, cruise control keeps a set speed by adjusting the gas pedal automatically. It keeps checking the car’s speed and makes small pedal changes to hold that target speed. This system can be drawn as a block diagram, and its transfer function gives the math model. The model is useful when the car goes uphill, downhill, or faces strong wind resistance. To find the system poles for different pedal forces, the...

properties of the root locus

Root Locus Angle and Gain Rules

Root locus angle and gain rules help analyze higher-order control systems. The method works without factoring the denominator of the transfer function. It is especially useful for unity feedback systems, where the transfer function can be studied by tracking how poles and zeros shape the response. A pole is a point where the characteristic polynomial in the denominator of the transfer function equals zero. To check whether a point lies on the root locus, use the angle criterion. Add the angles...

construction of root locus

Root locus paths and asymptotes show how closed-loop poles move as gain changes. This method helps students track system stability by following the pole locations on a graph. It also shows where the poles start, where they may end, and how they travel in between. The number of root locus branches equals the number of closed-loop poles. The plot is symmetric about the real axis. For positive gain, the root locus lies on the real axis to the left of an odd number of finite open-loop poles or...

plotting and calibrating the root locus

Root Locus Breakaway and Stability Points

Root locus analysis tracks how system poles move as gain changes. It shows when poles leave the real axis, move into the complex plane, and affect stability. For high school students studying control systems, the key ideas are breakaway points, break-in points, and jω-axis crossings. Breakaway points are locations where the root locus leaves the real axis. Break-in points are locations where it returns to the real axis. At either type of point, the branches meet the real axis at an angle of...

root loci for positive-feedback systems

Root Locus Rules for Oscillator Feedback

Root locus rules help describe how oscillator feedback systems behave as gain changes. The Hartley oscillator is one example of a positive feedback system. It keeps oscillating by feeding the output back to the input in phase, which reinforces the signal. Positive feedback systems can also be treated as negative feedback systems with the feedback signal inverted. In root locus analysis, the locus is the set of points on the s-plane where the transfer function angle equals 360 degrees. The...

key terms and concepts

Learn how root locus branches move from open-loop poles to zeros, use asymptotes, and show how gain affects stability in control systems.

Root Locus Paths and Asymptotes | 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

Root Locus Paths and Asymptotes

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