Name: root locus breakaway and stability points
Uploaded: 2024-11-21T14:54:13.000Z
Duration: 1 min 19 s
Channel: JoVE

根遺伝子座は、通常、システム極が実数平面から複素平面に遷移するときに分岐します。
離脱点と慣らし位置は、軌跡が実際の軸を離れて再結合する場所を通知します。根の軌跡は、実軸に対して180/n度の角度を形成します。
ゲインは実軸上の開ループ極間の離脱点でピークに達し、最小ゲインは2つのゼロ間のブレークインポイントで発生します。
ゲインを大きくすると、一部のシステム極が右半平面に押し込まれ、不安定になる可能性があることを示します。jω軸の交差は、安定したシステム動作と不安定なシステム動作の境界を示しています。
根軌跡解析では、特定の点を特定し、それらに関連するゲインを計算します。
ルート軌跡が特定の減衰比ラインと交差するときのルート軌跡の正確な座標を知るために、ラインに沿ったいくつかのテスト点が選択され、それらの角度和が評価されます。
根軌跡は、合計角度の合計が 180 度の奇数倍に等しい場合に存在します。
その特定のポイントでのゲインは、ポールの長さの積をそのポイントまでの長さが 0 の積で割ることによって計算されます。

根軌跡のプロットと較正

根軌跡は、システムの極が実軸から複素平面にシフトするにつれて、発散することがよくあります。この遷移の重要な点は、根軌跡が実軸から離れ、再び実軸に入る場所を示す、離脱点（breakaway point）と結合点（break-in point）です。根軌跡のブランチは、実軸と 180/n 度の角度を形成します。ここで、n は離脱点または結合点でのブランチの数です。

最大ゲインは実軸上の開ループ極間の離脱点で発生し、最小ゲインは 2 つのゼロ間の結合点で観測されます。ゲインが増加すると、一部のシステム極が右半平面に押し込まれ、潜在的な不安定性を示す場合があります。jω 軸の交差は、安定した動作と不安定な動作の境界を描きます。

根軌跡解析では、これらの重要な点を識別し、対応するゲインを計算します。特定の減衰比線と交差する根軌跡の正確な座標を決定するために、この線に沿ってさまざまなテスト点を選択し、システムの極とゼロからこれらの点までの角度の合計を評価します。根軌跡は、合計角度が 180 度の奇数倍になる場所に存在します。

根軌跡上の点が確認されると、その点のゲインを計算できます。これは、極から点までの距離の積を、ゼロから点までの距離の積で割ることによって行われます。このアプローチは、ゲインの変化に伴ってシステムの極がどのように移動するかを判断し、安定性と過渡応答を評価するのに役立ちます。

要約すると、根軌跡解析は、ゲインが変化するときのシステムの極の動作を視覚化して理解するための包括的な方法を提供します。離脱点と結合点を調べ、特定の点でのゲインを計算することで、エンジニアは制御システムを設計して微調整し、安定性と必要なパフォーマンス特性を確保できます。この方法は、システム極の動きを予測し、堅牢な制御システム設計を確保するために不可欠です。

根遺伝子座は、通常、システム極が実数平面から複素平面に遷移するときに分岐します。
ブレイクアウェイポイントとブレークインポイントは、軌跡が...

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

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 180/n degrees, where n is the number of branches at that point.

Gain also changes at these points. The maximum gain occurs at breakaway points between open-loop poles on the real axis. The minimum gain occurs at break-in points between two zeros. As gain increases, some poles may move into the right half-plane, which can signal instability.

The jω-axis crossings mark the boundary between stable and unstable operation. Root locus analysis uses these crossings to show how far the system can move before stability changes. It also helps find the corresponding gain at a chosen point on the locus.

To locate a point on a damping ratio line, several test points are checked along that line. The angles from the system poles and zeros to each point are added. The point lies on the root locus when the total angle is an odd multiple of 180 degrees.

Once a point is confirmed, the gain can be calculated from distances. The product of the distances from the poles to the point is divided by the product of the distances from the zeros to the point. This method helps predict how poles migrate with changing gain and supports control system design with the desired stability and transient response.

Root Locus Breakaway and Stability Points

root-locus-breakaway-and-stability-points

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

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 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 breakaway and break-in points, jω-axis crossings, and gain calculations help predict pole motion, stability, and response in control systems.

Root Locus Breakaway and Stability Points | JoVE K12

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electrical engineering

Root Locus Breakaway and Stability Points

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