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

根轨迹的绘制包括确定分支的数量，其数量等于闭环极点的数量。根轨迹关于实轴对称。
根轨迹存在于实轴的某一部分，该部分满足其左侧有限开环极点和零点的总数为奇数。
根轨迹起始于开环极点，随着增益增大而描绘出闭环系统的极点变化路径，最终终止于开环零点，此时系统极点随增益升高而趋于稳定。
当 s 趋近于无穷大时，若一个函数趋近于无穷大，则该函数在无穷远处有一个极点；若函数趋近于零，则该函数在无穷远处有一个零点。
当根轨迹趋向无穷远时，会逐渐接近某些称为渐近线的直线。对于给定系统，可计算出其在实轴上的交点及渐近线角度。随着增益的增加，这些角度开始重复出现。
他们的方程基于实轴截距和角度，揭示了根轨迹从极点延伸至无穷远的路径。
该轨迹起始于开环极点，并终止于开环零点。

根轨迹的构建

Construction of Root Locus

根轨迹的构建涉及几个关键步骤，用于分析和可视化系统极点随增益变化的行为。根轨迹中的分支数等于闭环极点数，并且关于实轴对称。
对于正增益值，根轨迹位于实轴上奇数个有限开环极点或零点的左侧。根轨迹从开环极点开始，随着增益的增加，延着闭环极点的路径移动。它结束于开环零点，随着增益的继续上升，系统极点在此稳定下来。
如果函数随着 s 趋近于无穷大而趋近于无穷大，则它在无穷大处有一个极点。相反，如果函数趋近于零，则它在无穷大处有一个零点。当根轨迹向无穷大延伸时，它会遵循特定的渐近线。这些渐近线的方程由实轴截距及其角度决定，它们表示轨迹从极点到无穷大的路径。
要计算给定系统的这些路径，需要确定实轴截距和渐近线的角度。这些角度是根据极点和零点的数量得出的，并随着增益的增加而开始重复。渐近线角度的公式如下:
<img alt="Equation1" src="/files/ftp_upload/16122/16122_PT_Equation_001.svg" style="height: 30px;">
其中 k 是整数，n 是极点的数量，m 是零点的数量。
完整的根轨迹图始于开环极点，终止于开环零点。此图遵循根轨迹法的规则，确保可以彻底分析和预测系统的稳定性和响应。
通过遵循这些步骤，工程师可以有效地使用根轨迹法来设计和调整控制系统，确保在各种操作条件下实现所需的性能和稳定性。这种方法可以清晰地显示系统极点如何随着增益的增加而迁移，有助于控制策略的稳健设计。

根轨迹的绘制包括确定等于闭环极点数量的分支数。根轨迹关于……对称。

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Root-Locus Method

根轨迹法

KEY TERMS AND CONCEPTS

关键术语与概念

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 Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Root Locus Paths and Asymptotes

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