Name: cruise control stability with root locus
Uploaded: 2024-11-21T14:54:13.000Z
Duration: 1 min 19 s
Channel: JoVE

Consider a cruise control system in a car designed to maintain a set speed automatically. The control system measures the vehicle&#39;s speed and fine-tunes the accelerator.
The root locus method aids in understanding how the cruise control system behavior varies when there are changes, like going uphill, downhill, or strong wind resistance.
A block diagram can represent this system. The transfer function for this system can be given with the quadratic formula applied to its denominator to determine pole locations for different gas pedal forces.
As pedal force varies, one system pole moves right, the other left. They converge at a point, then diverge into the complex plane, altering the system&#39;s closed-loop poles.
The root locus shows the impact of the pedal force variation on the system response: overdamped at low forces, critically damped at a specific force, and underdamped at high forces.
Since the root locus never crosses into the right half-plane, the system remains stable, irrespective of the pedal force.
Root locus analysis proves valuable for analyzing and designing systems higher than second order.

root-locus method

A cruise control system in a car is designed to maintain a specified speed automatically by adjusting the gas pedal. The system continuously measures the vehicle&#39;s speed and makes fine adjustments to the pedal to achieve this goal. The root locus method is particularly useful for understanding how the cruise control system&#39;s behavior changes under varying conditions, such as when the car goes uphill, downhill, or faces strong wind resistance.
This system can be represented by a block diagram, with its transfer function providing a mathematical model. To determine the locations of the system&#39;s poles for different gas pedal forces, the quadratic formula is applied to the denominator of the transfer function. As the pedal force changes, one pole of the system moves to the right while the other moves to the left. These poles eventually converge at a specific point before diverging into the complex plane, influencing the closed-loop poles of the system.
The root locus method visually illustrates how variations in pedal force impact the system&#39;s response. At low pedal forces, the system is overdamped, meaning it returns to the desired speed without oscillating but may take longer. At a specific force, the system is critically damped, achieving the fastest return to the desired speed without overshooting. At high pedal forces, the system becomes underdamped, resulting in oscillations around the desired speed before settling.
Importantly, the root locus for this system never crosses into the right half-plane of the s-plane, ensuring that the system remains stable regardless of the pedal force applied. This stability is a crucial feature for the reliable operation of the cruise control system.
The root locus method is not only useful for analyzing second-order systems but also proves valuable for higher-order systems, providing insights into system behavior and aiding in the design of robust control mechanisms. By leveraging root locus analysis, engineers can optimize the performance of complex systems like cruise control, ensuring they remain stable and responsive under various operating conditions.

Root-Locus Method

Consider a cruise control system in a car designed to maintain a set speed automatically. The control system measures the vehicle&#39;s speed and ...

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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 denominator of the transfer function is solved with the quadratic formula.

As the pedal force changes, one pole moves to the right and the other moves to the left. The poles meet at one point and then move apart into the complex plane. This path shows how the closed-loop poles change as the force changes.

Root locus also describes how the speed response changes. At low pedal forces, the system is overdamped. That means it returns to the set speed without oscillating, but it may be slow. At one special force, it is critically damped, so it returns as fast as possible without overshooting.

At high pedal forces, the system becomes underdamped. Then it oscillates around the set speed before it settles. Even so, the root locus in this example never crosses into the right half-plane of the s-plane, so the cruise control stays stable for any pedal force applied.

This makes root locus useful for more than just second-order systems. It also helps analyze higher-order systems. Engineers use it to understand system behavior and design stable, responsive control systems like cruise control.

Cruise Control Stability with Root Locus

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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

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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...

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plotting and calibrating the root locus

Root Locus Breakaway and Stability Points

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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

See how root locus explains cruise control stability, pole movement, and damping changes as gas pedal force varies in a car speed-control system.

Cruise Control Stability with Root Locus | 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

Cruise Control Stability with Root Locus

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