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

In an open-loop system like a basic thermostat, the poles of the transfer function influence system response but don&#39;t govern stability.
When feedback is introduced - as in an advanced thermostat that adjusts heating based on room temperature, creating a closed-loop system - stability is dictated by the new poles.
Problems can arise if these poles cross into instability during closed-loop formation, causing potential temperature fluctuations.
The poles of the open-loop transfer function are relatively easy to identify and remain unaffected by changes in system gain.
However, finding the poles of the closed-loop transfer function, which changes with system gain adjustments, is more complex, necessitating factoring the denominator.
While the zeros and poles of transfer functions are typically known, identifying the poles of a specific function that vary with system gain is not straightforward.
The transient response and stability of a system hinge on its poles. Without factoring specific gain values, there&#39;s a lack of insight into the system&#39;s performance.
The root locus method visually depicts the variation of these poles with system gain changes.

control system problem

Control System Problem

In an open-loop system, such as a basic thermostat, the poles of the transfer function influence the system&#39;s response but do not determine its stability. However, when feedback is introduced to form a closed-loop system, such as an advanced thermostat that adjusts heating based on room temperature, stability is governed by the new poles of the closed-loop transfer function.
When forming a closed-loop system, issues can arise if the poles cross into the unstable region, leading to potential temperature fluctuations. Identifying the poles of the open-loop transfer function is relatively straightforward and remains constant despite changes in system gain. In contrast, the poles of the closed-loop transfer function vary with adjustments in system gain and require more complex calculations involving the factoring of the denominator.
Although the zeros and poles of transfer functions are generally known, pinpointing the poles of a specific function that changes with system gain is more challenging. The transient response and overall stability of a system are closely linked to these poles. Without considering specific gain values, the system&#39;s performance remains unclear.
The root locus method offers a visual approach to understanding how the poles of a system vary with changes in system gain. By plotting the possible locations of the closed-loop poles on the s-plane, the root locus method provides insights into how the system&#39;s stability and transient response will evolve as the gain changes. This method allows engineers to predict and adjust the system&#39;s behavior to ensure stability and desired performance.
In summary, while open-loop system poles are easily identified and stable, the poles of a closed-loop system depend on the system gain and require more detailed analysis. The root locus method is a valuable tool for visualizing these changes, aiding in the design and tuning of stable closed-loop systems.

In an open-loop system like a basic thermostat, the poles of the transfer function influence system response but don&#39;t govern stability.
When feedback ...

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JoVE Core

Electrical Engineering

Unit 1

Root-Locus Method

KEY TERMS AND CONCEPTS

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 fluctuate in temperature. The poles of the open-loop transfer function are easier to find and stay the same even when system gain changes.

The closed-loop poles are harder to calculate because they change as gain changes. Finding them usually requires factoring the denominator of the transfer function. The zeros and poles of transfer functions are often known, but the poles of a gain-dependent function are more difficult to pin down.

These poles matter because they shape transient response and overall stability. Without a specific gain value, it is not clear how the system will perform. The root locus method helps by showing, on the s-plane, where the closed-loop poles can move as gain changes.

This visual method lets engineers see how stability and transient response may change with gain. It supports system design and tuning so the closed-loop system stays stable and performs as desired. In that way, root locus helps predict and adjust control behavior before the system is put into use.

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 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 feedback changes closed-loop poles, why system gain affects stability, and how the root locus method maps pole movement on the s-plane.

Root Locus and Closed-Loop Stability | 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 and Closed-Loop Stability

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