Computer-aided root -locus numerical technique (original) (raw)

A Method for Plotting the Complementary Root Locus Using the Root-Locus (Positive Gain) Rules

IEEE Transactions on Education, 2004

The root-locus method is a well-known and commonly used tool in control system analysis and design. It is an important topic in introductory undergraduate engineering control disciplines. Although complementary root locus (plant with negative gain) is not as common as root locus (plant with positive gain) and in many introductory textbooks for control systems is not presented, it has been shown a valuable tool in control system design. This paper shows that complementary root locus can be plotted using only the well-known construction rules to plot root locus. It can offer for the students a better comprehension on this subject. These results present a procedure to avoid problems that appear in root-locus plots for plants with the same number of poles and zeros.

Morphometric analysis of root shape

New Phytologist, 2004

Alterations in the root shape in plant mutants indicate defects in hormonal signalling, transport and cytoskeleton function. • To quantify the root shape, we introduced novel parameters designated vertical growth index ( VGI ) and horizontal growth index ( HGI ). VGI was defined as a ratio between the root tip ordinate and the root length. HGI was the ratio between the root tip abscissa and the root length.

Rooting Other Characteristics of a Trmgenic~ zsa

ADD~~ONAL INDEX WORDS. Paradox, rooting, difficult-to-root, cuttings, tissue culture AJEZRA~~. Walnuts Vt~glbns spp.) are =cult-to-mt woody plants. The roZABC genes (roM i -r& + rolC), derived from the bacteria Agr0b~:tenum rhizogenes, have been shown to increase the rooting potential of other difficdt-to-root woody plants. We inserted the roZABC genes into somatic embryos of a 'Paradox' hybrid (J. kindsii x J. reg&) clone PXI using the A. hrmefaciens gene transfer system. A transgenic sub-clone, designated PX1 roXkBC2-2 was selected and compared to the mtransfomed clone for a variety of phenotypic characteristics, including rooting potential. Transformed and untradormed shoots were budded onto seedling J. regia mobtock in the greenhouse and established in the field. Transformed trees d'ilayed reduced internode length, an increase in lateral branch'ig, and wrinkled leaves. In another test, a commercial persian walnut cultivar J. regid 'Chandler' was gmlled onto rooted cuttings of both the untmnsPomed and transformed plants.The presence of themljlBC genes in the rootstock had no visible eBects on the grafted scion. Several of these trees were excavated from the field and the mot systems of each genotype were examined for root number, diameter, and biomassmasS Trees with the r o W C rootstock had si-cantfy more sinall diameter roots compared to the controls and less recovered biomass.

Accuracy and bias of methods used for root length measurements in functional root research

Methods in Ecology and Evolution

1. Functional root traits are becoming a key measure in plant ecology, and root length measurements are needed for the calculation of root traits. Several methods are used to estimate the total root length (TRL) of washed root samples [e.g. modified line intersect (LI) method, WinRHIZO TM and IJ_Rhizo], but no standardized comparison of methods exists. 2. We used a set of digital images of unstained root samples to compare measurements given by the LI method and automated methods provided by WinRHIZO TM and IJ_Rhizo. Linear regression models were used to detect bias. Both linear regression models and the Bland-Altman's method of differences were used to evaluate the accuracy of eight methods (1 manual, 2 semi-automated and 5 automated) in comparison with a reference method that avoided root detection errors. 3. Length measurements were highly correlated, but did not exactly agree with each other in 11 of 12 method comparisons. All tested methods tended to underestimate the TRL of unstained root samples. The accuracy of WinRHIZO TM was influenced by the thresholding method and the root length density (RLD) in the pictures. For the other methods, no linear relationship was found between accuracy and RLD. With WinRHIZO TM (global thresholding + pixel reclassification; RLD = 1 cm cm À2), the Regent's method and the Tennant's method underestimated the TRL by 7Á0 AE 6Á2% and 4Á7 AE 7Á9%, respectively. The LI method gave satisfactory results on average (underestimation: 4Á2 AE 6Á0%), but our results suggest that it can lead to inaccurate estimations for single images. In IJ_Rhizo, the Kimura method was the best and underestimated the TRL by 5Á4 AE 6Á1%. 4. Our results showed that care must be taken when comparing measurements acquired with different methods because they can lead to different results. When acquiring root images, we advise to (i) increase the contrast between fine roots and background by staining the roots, and (ii) avoid overlapping roots by not exceeding a RLD of 1 cm cm À2. Under these conditions, good length estimates can be obtained with WinRHIZO TM (global thresholding + pixel reclassification). The Kimura method in IJ_Rhizo can be an alternative to WinRHIZO TM .

A Linear Model to Describe Branching and Allometry in Root Architecture

Plants, 2019

Root architecture is a complex structure that comprises multiple traits of the root phenotype. Novel platforms and models have been developed to better understand root architecture. In this methods paper, we introduce a novel allometric model, named rhizochron index (m), which describes lateral root (LR) branching and elongation patterns across the primary root (PR). To test our model, we obtained data from 16 natural accessions of Arabidopsis thaliana at three stages of early root development to measure conventional traits of root architecture (e.g., PR and LR length), and extracted the rhizochron index (m). In addition, we tested previously published datasets to assess the utility of the rhizochron index (m) to distinguish mutants and environmental effects on root architecture. Our results indicate that rhizochron index (m) is useful to distinguish the natural variations of root architecture between A. thaliana accessions, but not across early stages of root development. Correlati...

Multivariable root loci on the real axis†

International Journal of Control, 1982

Some methods for determining the number of branches of multivariable root loci which are located on the real axis at a given point are obtained by using frequency domain methods. An equation for the number of branches is given for the general case, and simpler results for the special cases when the transfer function G(s) has size 2 x 2, and when G(s) is symmetric, are also presented.