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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...

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...

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...

Root-Locus Method

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'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'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...

Root Loci for Positive-Feedback Systems

Root Loci for Positive-Feedback Systems

The Hartley oscillator is a positive feedback system that sustains oscillations by feeding the output back to the input in phase, thereby reinforcing the signal. Positive feedback systems can be viewed as negative feedback systems with inverted feedback signals. In these systems, the root locus encompasses all points on the s-plane where the angle of the system transfer function equals 360 degrees.
The construction rules for the root locus in positive feedback systems are similar to those in...

Stability

Stability

The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
The stability of an LTI system is determined by the roots of its characteristic equation, known as poles. A system is stable if it produces a bounded...

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Root locus-based stability analysis for biological systems.

¹Department of Electrical Engineering, Da-Yeh University, 168 University Rd., Dacun, Changhua 51591, Taiwan, R.O.C.

Journal of Bioinformatics and Computational Biology

|September 13, 2021

Summary

This summary is machine-generated.

Control technologies like root locus analysis enhance biological system understanding by predicting reaction limitations and stability for generalized Michaelis-Menten (MM) and S-systems. This method accurately analyzes system dynamics and operating principles.

Keywords:

System analysis biological system theory computational analysis computational biology mode analysis root locus

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Area of Science:

Biochemical Engineering
Systems Biology
Control Theory

Background:

Biological systems require suitable models for analysis, with generalized Michaelis-Menten (MM) and S-systems being prominent.
Analyzing generalized MM systems is challenging due to complex structures, while S-systems present difficulties with power-law structures.
Steady-state estimation and theoretical analysis remain complex for these biochemical models.

Purpose of the Study:

To apply control technologies for deeper analysis of biological systems.
To flexibly utilize root locus and mode analysis for generalized MM and S-systems.
To predict dynamic behavior, estimate steady states, and analyze stability margins.

Main Methods:

Root locus method and mode analysis for generalized MM systems to predict flux limitations, steady states, and stability.
Application of root locus, mode analysis, and converse theorem for S-systems to predict dynamics and stability.
Theoretical validation using simulations in Simulink/MATLAB.

Main Results:

Successfully predicted reaction limitations and steady states for reversible MM kinetics with high accuracy.
Demonstrated global semi-stability for systems with singular matrices and analyzed gain margins and setting times.
Confirmed stability analysis of S-systems via linearized models and root loci, showing infinite gain margins and varying setting times.

Conclusions:

Root locus-based analysis is applicable to diverse differential equation-based biological systems.
This research introduces a novel method for examining system dynamics and operating principles.
Provides a guideline for selecting independent variables to optimize system performance.