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Construction of Root Locus

Construction of Root Locus

The construction of a root locus involves several key steps to analyze and visualize the behavior of a system's poles with varying gain. The number of branches in the root locus equals the number of closed-loop poles and is symmetrical about the real axis.
For positive gain values, the root locus exists on the real axis to the left of an odd number of finite open-loop poles or zeros. The root locus starts at the open-loop poles and traces the paths of the closed-loop poles as the gain increases.

Basic Plant Anatomy: Roots, Stems, and Leaves

Basic Plant Anatomy: Roots, Stems, and Leaves

The primary organs of vascular plants are roots, stems, and leaves, but these structures can be highly variable, adapted for the specific needs and environment of different plant species.

Radicals

Radicals

Roots, often written as radicals, identify the quantity that must be raised to a specific exponent to produce a given value. A radical expression consists of two main components: the radicand, which is the value placed inside the root symbol, and the index, which indicates the degree of the root being taken. The notation n√a indicates the principal nth root of a. If n equals 2, the operation is the square root, while n = 3 defines the cube root. When n is even, a negative radicand does not...

Properties of the Root Locus

Properties of the Root Locus

The root locus method is an invaluable tool for analyzing higher-order systems without needing to factor the denominator of the transfer function. A pole of the system is identified when the characteristic polynomial in the transfer function's denominator equals zero.
To determine if a point lies on the root locus, the criterion involves the sum of angles contributed by all poles and zeros to that point. Specifically, this sum must be an odd multiple of 180 degrees. The gain at any point on the...

Morphogenesis

Morphogenesis

Plant morphogenesis—the development of a plant’s form and structure—involves several overlapping developmental processes, including growth and cell differentiation. Precursor cells differentiate into specific cell types, which are organized into the tissues and organ systems that make up the functional plant.

Plotting and Calibrating the Root Locus

Plotting and Calibrating the Root Locus

Root loci often diverge as system poles shift from the real axis to the complex plane. Key points in this transition are the breakaway and break-in points, indicating where the root locus leaves and reenters the real axis. The branches of the root locus form an angle of 180/n degrees with the real axis, where n is the number of branches at a breakaway or break-in point.
The maximum gain occurs at the breakaway points between open-loop poles on the real axis, while the minimum gain is observed...

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Related Experiment Video

Updated: May 10, 2026

A Simple Protocol for Mapping the Plant Root System Architecture Traits

A Simple Protocol for Mapping the Plant Root System Architecture Traits

Published on: February 10, 2023

RootNav: navigating images of complex root architectures.

Michael P Pound¹, Andrew P French, Jonathan A Atkinson

¹Centre for Plant Integrative Biology, School of Biosciences, University of Nottingham, Sutton Bonington LE12 5RD, United Kingdom.

Plant Physiology

|June 15, 2013

Summary

This summary is machine-generated.

A new tool, RootNav, enables semiautomated analysis of plant root system architecture. This image analysis software quantifies complex root structures, aiding plant science research.

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Area of Science:

Plant Biology
Computational Biology
Image Analysis

Background:

Accurate quantification of plant root system architecture is crucial for understanding plant adaptation and development.
Existing methods for root system analysis can be labor-intensive and lack precision for complex architectures.

Purpose of the Study:

To introduce RootNav, a novel image analysis tool for semiautomated quantification of plant root system architectures.
To provide a user-friendly and intuitive platform for detailed root system analysis across various plant species.

Main Methods:

Utilizes an expectation maximization classification algorithm for initial root detection.
Employs an optimization approach to fit a root model to image data.
Incorporates a visual, interactive user interface for refining root path quantification.

Main Results:

Successfully evaluated on winter wheat (Triticum aestivum), with demonstrations on Arabidopsis, Brassica napus, and rice.
Quantified exemplar root traits, revealing significant differences between wheat accessions.
Demonstrated the capability to extract a wide range of biologically relevant root measures.

Conclusions:

RootNav offers an efficient and accurate method for analyzing complex root system architectures.
The tool provides valuable structural information for diverse plant science research applications.
A complementary viewer tool enhances the recovery of detailed architectural traits.