24.6
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…
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。
View the full transcript and gain access to JoVE Core videos
Q1: What are breakaway and break-in points in root locus analysis?
Breakaway points are where the root locus leaves the real axis as system poles transition to the complex plane, occurring between open-loop poles where gain peaks. Break-in points are where the locus rejoins the real axis between two zeros, where minimum gain occurs. These critical points signal pole migration and are essential for understanding system behavior.
Q2: How do you determine if a point lies on the root locus?
A point lies on the root locus where the sum of total angles from all poles and zeros to that point equals an odd multiple of 180 degrees. Test points along a damping ratio line are selected and their angular sum is evaluated. Once a point satisfies this angle condition, the gain at that location can be calculated.
Q3: How is gain calculated at a specific point on the root locus?
Gain is calculated by dividing the product of distances from all poles to the point by the product of distances from all zeros to that point. This ratio determines the system gain required for the poles to occupy that specific location. The calculation uses vector representation complex numbers to measure these distances accurately.
Q4: What does it mean when poles cross into the right half-plane?
When increasing gain pushes system poles into the right half-plane, it signals potential instability. The jω-axis serves as the boundary between stable and unstable operations. Crossing this boundary indicates the system may become unstable, requiring careful gain selection during control system design.
Q5: What angle do root locus branches form with the real axis?
Root locus branches form an angle of 180/n degrees with the real axis, where n represents the number of branches at a breakaway or break-in point. This angular relationship is a fundamental property that helps predict how poles diverge as they transition from the real axis to the complex plane.
Q6: How do you find the exact coordinates where a root locus crosses a damping ratio line?
Multiple test points are selected along the damping ratio line, and the angular sum from poles and zeros to each point is evaluated. When the total angle equals an odd multiple of 180 degrees, that point lies on the root locus. The gain at the intersection is then calculated using the pole and zero distance ratio.
Q7: Why is the jω-axis crossing important in root locus analysis?
The jω-axis crossing marks the critical boundary between stable and unstable system operations. This crossing point indicates the gain value at which the system transitions from stability to instability. Identifying this crossing is essential for determining safe operating ranges and ensuring robust control system design.