Understanding Root Locus in Controller Design

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Questions and Answers

What does the root locus method show as the gain K is varied?

The number of zeros at infinity

The positions of the roots of the characteristic equation (correct)

The degree of stability of the system

The number of poles of G(s)

What does relative stability indicate?

The number of zeros at infinity

The degree of stability of the system (correct)

The number of poles of G(s)

The root of the characteristic equation

What does absolute stability refer to?

The degree of stability of the system (correct)

The positions of the roots of the characteristic equation

The number of zeros at infinity

The number of poles of G(s)

What is the purpose of using the root locus technique in controller design?

To find how a positive controller design parameter affects the resulting CL poles Signup and view all the answers

What does the number of zeros at infinity indicate in the root locus method?

The number of branches of the root locus that go to infinity (asymptotes) Signup and view all the answers

What does the root locus technique show by varying the gain K from zero to infinity?

The positions of the roots of the characteristic equation Signup and view all the answers

What does the number of zeros at infinity indicate in the root locus method?

The number of branches of the root locus that go to infinity Signup and view all the answers

In controller design, what does absolute stability refer to?

Whether the system is stable or unstable Signup and view all the answers

What is the significance of relative stability in controller design?

Gives the degree of stability or how close it is to instability Signup and view all the answers

How many branches must the root locus have if a system has n poles and m zeros (m < n)?

$n-m$ Signup and view all the answers

What does the root locus method define as the gain K is varied?

The positions of the poles of the characteristic equation Signup and view all the answers

In controller design, what does absolute stability refer to?

Stability regardless of the value of gain K Signup and view all the answers

What is the significance of relative stability in controller design?

Gives the degree of stability or how close it is to instability Signup and view all the answers

What does the number of zeros at infinity indicate in the root locus method?

The number of branches of the root locus Signup and view all the answers

How many branches must the root locus have if a system has n poles and m zeros (m < n)?

$n-m$ Signup and view all the answers

Study Notes

Root Locus Method and Gain Variation

The root locus method illustrates how the roots of the characteristic equation change as the gain K is adjusted.
It defines the paths of the poles in the complex plane as K transitions from zero to infinity.

Stability Concepts

Relative stability indicates the system's ability to maintain performance under varying conditions, measured by response time and overshoot.
Absolute stability refers to the system's overall stability, ensuring that all system poles are in the left half of the s-plane, leading to bounded output for bounded input.

Purpose of Root Locus in Controller Design

The root locus technique aids in analyzing and designing controllers by showing how system poles can be manipulated to achieve desired stability and performance characteristics.

Zeros at Infinity

The number of zeros at infinity in the root locus method reveals additional system dynamics; specifically, it correlates with the difference between the number of poles and zeros in the system.

Behavior of Root Locus with Gain Variation

By varying K from zero to infinity, the root locus highlights stability changes, showing where the system transitions from stable to unstable behavior.

Branch Count in Root Locus

The root locus must have a number of branches equal to the number of poles n; in cases where there are m zeros (m < n), there will be n - m branches that correspond to the remaining poles.

Significance of Stability in Controller Design

Relative stability is crucial for performance metrics such as rise time, settling time, and overshoot, which dictate how well the system responds under disturbances.
Understanding and ensuring absolute stability is essential to prevent system failure and ensure robust control.

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Description

Learn about root locus, the path of closed loop poles by varying the gain K from zero to infinity. Explore how the Root Locus technique is used to find the effect of a controller design parameter on the resulting closed loop poles, aiding in choosing the right value for the controller parameter.