Chapter 6 Control systems analysis and design by the root-locus method. From the book (Ogata Modern Control Engineering 5th).
6-1 introduction.
6-2 Root locus plots.
6-5 root locus approach to control-system design.
This document discusses stability analysis of control systems using transfer functions and the Routh-Hurwitz criterion. It begins by defining stability and describing different types of system responses. The key points are:
1) The Routh-Hurwitz criterion can determine stability by analyzing the signs in the first column of a constructed Routh table, with changes in sign indicating right half-plane poles and instability.
2) Special cases like a zero only in the first column or an entire row of zeros require alternative methods like the epsilon method or reversing coefficients.
3) Examples demonstrate applying the Routh-Hurwitz criterion to determine stability for different polynomials, including handling special cases. Exercises also have readers practice stability analysis using
This document provides an overview of the ME 433 - State Space Control course. It introduces the course topics which include state-space modeling, observability and controllability, linear state feedback control, linear quadratic regulator, and Kalman filtering. It also lists relevant textbooks and describes various types of control problems including nonlinear, robust, adaptive, and distributed parameter systems control.
The document discusses polar plots, which graph the magnitude and phase of a transfer function G(jω)H(jω) as ω varies from 0 to infinity. It provides rules for drawing polar plots, such as substituting s=jω into the transfer function, finding the starting and ending magnitude and phase, and checking for intersections with the real and imaginary axes. An example is shown of creating a polar plot for a first order system, including determining the magnitude and phase expressions and values at specific ω points and drawing the resulting plot.
Root locus is a graphical representation of the closed-loop poles as a system parameter is varied.
It can be used to describe qualitatively the performance of a system as various parameters are changed.
It gives graphic representation of a system’s transient response and also stability.
We can see the range of stability, instability, and the conditions that cause a system to break into oscillation.
This presentation explains about the introduction of Polar Plot, advantages and disadvantages of polar plot and also steps to draw polar plot. and also explains about how to draw polar plot with an examples. It also explains how to draw polar plot with numerous examples and stability analysis by using polar plot.
1. The document discusses Lypunov stability and different types of stability including asymptotically stable, bounded-input bounded-output stable, and Lyapunov stability.
2. It provides conditions for asymptotic stability including having all eigenvalues of the system in the left half plane and defines an equilibrium state as a state where the system will not move from in the absence of input.
3. Lyapunov's method is introduced for analyzing stability using a Lyapunov function where the derivative must be negative semi-definite to guarantee asymptotic stability.
Digital control systems (dcs) lecture 18-19-20Ali Rind
This document discusses digital control systems and related topics such as difference equations, z-transforms, and mapping between the s-plane and z-plane. It begins with an outline of topics to be covered including difference equations, z-transforms, inverse z-transforms, and the relationship between the s-plane and z-plane. Examples are provided to illustrate difference equations, z-transforms, and mapping poles between the two planes. Standard z-transforms of discrete-time signals like the unit impulse and sampled step are also defined.
This document discusses stability analysis of control systems using transfer functions and the Routh-Hurwitz criterion. It begins by defining stability and describing different types of system responses. The key points are:
1) The Routh-Hurwitz criterion can determine stability by analyzing the signs in the first column of a constructed Routh table, with changes in sign indicating right half-plane poles and instability.
2) Special cases like a zero only in the first column or an entire row of zeros require alternative methods like the epsilon method or reversing coefficients.
3) Examples demonstrate applying the Routh-Hurwitz criterion to determine stability for different polynomials, including handling special cases. Exercises also have readers practice stability analysis using
This document provides an overview of the ME 433 - State Space Control course. It introduces the course topics which include state-space modeling, observability and controllability, linear state feedback control, linear quadratic regulator, and Kalman filtering. It also lists relevant textbooks and describes various types of control problems including nonlinear, robust, adaptive, and distributed parameter systems control.
The document discusses polar plots, which graph the magnitude and phase of a transfer function G(jω)H(jω) as ω varies from 0 to infinity. It provides rules for drawing polar plots, such as substituting s=jω into the transfer function, finding the starting and ending magnitude and phase, and checking for intersections with the real and imaginary axes. An example is shown of creating a polar plot for a first order system, including determining the magnitude and phase expressions and values at specific ω points and drawing the resulting plot.
Root locus is a graphical representation of the closed-loop poles as a system parameter is varied.
It can be used to describe qualitatively the performance of a system as various parameters are changed.
It gives graphic representation of a system’s transient response and also stability.
We can see the range of stability, instability, and the conditions that cause a system to break into oscillation.
This presentation explains about the introduction of Polar Plot, advantages and disadvantages of polar plot and also steps to draw polar plot. and also explains about how to draw polar plot with an examples. It also explains how to draw polar plot with numerous examples and stability analysis by using polar plot.
1. The document discusses Lypunov stability and different types of stability including asymptotically stable, bounded-input bounded-output stable, and Lyapunov stability.
2. It provides conditions for asymptotic stability including having all eigenvalues of the system in the left half plane and defines an equilibrium state as a state where the system will not move from in the absence of input.
3. Lyapunov's method is introduced for analyzing stability using a Lyapunov function where the derivative must be negative semi-definite to guarantee asymptotic stability.
Digital control systems (dcs) lecture 18-19-20Ali Rind
This document discusses digital control systems and related topics such as difference equations, z-transforms, and mapping between the s-plane and z-plane. It begins with an outline of topics to be covered including difference equations, z-transforms, inverse z-transforms, and the relationship between the s-plane and z-plane. Examples are provided to illustrate difference equations, z-transforms, and mapping poles between the two planes. Standard z-transforms of discrete-time signals like the unit impulse and sampled step are also defined.
This document provides an introduction to root locus analysis. It defines a root locus as a graphical representation of how closed-loop poles move in the s-plane as a system parameter, such as gain, is varied. The objectives are to learn how to sketch a root locus using five rules, including starting and ending points, symmetry, real axis behavior, and asymptotes. An example problem sketches the root locus for a system and calculates the gain value where the locus intersects a radial line representing a specific percent overshoot value. Calculating this intersection point accurately calibrates the root locus sketch.
Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively
This document summarizes the key rules and concepts for constructing root locus diagrams. It lists the names and student IDs of 5 group members working on a control systems engineering project. The document then explains that the root locus shows how the roots of the characteristic equation change in the s-plane as the system parameter K varies from 0 to infinity, and can be used to analyze a system's stability and transient response. It proceeds to describe 8 rules for constructing root loci, including rules about the locus originating from open-loop poles, terminating at open-loop zeros or infinity, determining breakaway points, and calculating the angle of departure.
This document summarizes key aspects of the discrete Fourier transform (DFT). It defines the DFT, provides the formula for calculating it, and explains that the DFT transforms a discrete-time signal from the time domain to the frequency domain. It also outlines several important properties of the DFT, including linearity, shift property, duality, symmetry, and circular convolution. Examples are provided to illustrate duality and symmetry. References for further information on the discrete Fourier transform are also included.
This document describes a student project to implement Friis transmission formula using MATLAB. The project report includes:
1) An objective to use MATLAB to calculate received power given various antenna parameters.
2) An introduction that describes Friis transmission formula and its ideal assumptions.
3) MATLAB code examples that allow the user to input antenna parameters like transmitted power, gains, wavelength, and distance to calculate received power in different unit systems.
4) Additional MATLAB code to calculate received power for a radar targeting an object by accounting for the object's cross-sectional area.
5) The project aims to predict received power given transmission conditions to aid antenna system design and power adjustment.
The document discusses Bode plots, which are used to analyze the frequency response of linear systems. Bode plots graphically depict the magnitude and phase of a system's frequency response. They are constructed by plotting the logarithm of frequency versus gain in decibels and phase in degrees. The document outlines how to construct Bode plots based on the poles and zeros of a system's transfer function, including the slopes and asymptotes of different component factors like integrators, differentiators, and first and second order terms. Examples are provided to demonstrate how to determine a transfer function based on a given Bode plot.
The document discusses root locus analysis and controller design techniques.
1) Root locus analysis involves plotting the trajectories of a system's closed-loop poles as a parameter (such as a controller gain) is varied. Properties of root loci like pole departure angles and asymptotes are examined.
2) Design specifications like overshoot and settling time can be translated to desired pole regions. A simple proportional controller can place poles near this region.
3) Additional controller types like integral and derivative are introduced to modify the root locus for improved steady-state response or damping. Combining controller types benefits stability.
The document discusses aircraft pitch control system modeling and controller design. It describes the pitch angle and angle of attack, principles of flight, and requirements for the pitch control system. It then models the aircraft pitch transfer function and analyzes the open-loop and closed-loop response. PID, root locus and state-space controllers are designed in MATLAB and Simulink to control the aircraft pitch and meet the requirements.
This document outlines the steps and procedure for performing a root locus analysis. It begins with an introduction to root locus analysis and its use in determining the stability of a closed-loop system. It then lists the general 8 steps for drawing a root locus as a parameter is varied. An example problem is worked through, showing the determination of poles, zeros, asymptotes, breakaway points, and drawing the overall root locus sketch. The document concludes with a brief MATLAB program for generating a root locus plot.
This document contains notes from lectures 23-24 on time response and steady state errors in discrete time control systems. It begins with an outline of the lecture topics, then provides introductions and examples related to time response, the final value theorem, and steady state errors. It defines concepts like position and velocity error constants and shows examples of calculating steady state error for different system transfer functions. The document contains MATLAB examples and homework problems related to analyzing discrete time systems.
Modern Control - Lec 02 - Mathematical Modeling of SystemsAmr E. Mohamed
This document provides an overview of mathematical modeling of physical systems. It discusses how to derive mathematical models from physical systems using differential equations based on governing physical laws. The key steps are: (1) defining the physical system, (2) formulating the mathematical model using differential equations, and (3) solving the equations. Common model types include differential equation, transfer function, and state-space models. The document also discusses modeling various physical elements like electrical circuits, mechanical translational/rotational systems, and electro-mechanical systems using differential equations. It covers block diagram representation and reduction of mathematical models. The overall goal is to realize the importance of deriving accurate mathematical models for analyzing and designing control systems.
The document provides example problems and solutions for mathematical modeling and analysis of control systems. It includes the following examples:
1) It derives the transfer function C(s)/R(s) for a system represented by a block diagram, obtaining the simplified closed-loop transfer function.
2) It models a simplified automobile suspension system as a mass-spring-damper system and derives the transfer function between the input and output displacements.
3) It obtains the transfer function Y(s)/U(s) for another simplified suspension system represented by a diagram relating displacements and forces.
4) It derives the state-space representation for a mechanical system represented by equations relating positions, velocities and
Crisp sets are classical sets defined in boolean logic that have only two membership values - an element either fully belongs or does not belong to the set. Crisp sets are fundamental to the study of fuzzy sets. Key concepts of crisp sets include the universe of discourse, set operations like union and intersection, and properties like commutativity, associativity, distributivity and De Morgan's laws. Crisp sets provide a definitive yes or no for membership, unlike fuzzy sets which allow partial membership.
This document provides an overview of signal flow graphs (SFG). It defines SFG as a graphical representation of linear systems, where each variable is represented by a node and transmissions are branches with arrows denoting signal flow direction. Key terms are defined, including input/output nodes, mixed nodes, transmittance, forward paths, loops, and path/loop gains. Properties and examples of SFG construction from equations and block diagrams are described. Mason's gain formula is introduced for determining overall transfer functions from SFGs. The effects of feedback on gain and stability are also briefly discussed.
In spherical coordinates, each point is represented by an ordered triple of a distance and two angles, similar to the latitude-longitude system used on Earth. A point P is specified by its coordinates P(r,θ,φ), where r is the distance from the origin and θ and φ are the angular coordinates. Orthogonal surfaces in the spherical coordinate system are generated by keeping r, θ, or φ constant, resulting in a sphere, circular cone, or semi-infinite plane, respectively.
The document discusses the z-transform, which is the discrete-time equivalent of the Laplace transform. It defines the z-transform and provides examples of calculating the z-transform for various sequences, including the unit impulse, unit step function, sinusoids, and exponential sequences. It also discusses properties of the z-transform such as the region of convergence and relationship to the discrete-time Fourier transform.
The document discusses Bode plots, which are frequency domain techniques used to analyze linear time-invariant systems. It covers poles and zeros, transfer functions, the S-plane, mechanics for constructing Bode plots, examples of plotting Bode plots by hand and using MATLAB, and designing a system to meet a target Bode plot specification. Key steps include identifying poles and zeros, approximating plots between break frequencies, and using MATLAB tools like Bode and Simulink to validate designs.
Modern Control - Lec 05 - Analysis and Design of Control Systems using Freque...Amr E. Mohamed
The document discusses frequency response analysis and Bode plots. It begins with an introduction to frequency response and how the steady state response of a linear time-invariant system to a sinusoidal input is another sinusoid at the same frequency with a different magnitude and phase. The complex ratio of the output to input is called the frequency response. It then discusses Bode plots which show the magnitude and phase of the frequency response on logarithmic scales. Key features of components in open-loop transfer functions and how they affect the Bode plot shapes are explained. An example demonstrates drawing the Bode plots for a sample transfer function.
The document discusses the z-transform, which is a mathematical tool used to analyze discrete-time control systems. The z-transform plays a similar role for discrete-time systems as the Laplace transform does for continuous-time systems. Some key properties of the z-transform discussed include the region of convergence, properties and theorems like the shifting theorem and initial/final value theorems, and applications to feedback control systems.
Analysis and Design of Control System using Root LocusSiyum Tsega Balcha
Root locus analysis is a powerful tool in control systems engineering used to analyze the behavior of a system's closed-loop poles as a function of a parameter, typically a controller gain. It provides engineers with valuable insights into how changing system parameters affect stability and performance, helping them design robust and stable control systems. Let's explore the key concepts, techniques, and practical implications of root locus analysis. At its core, root locus analysis focuses on the movement of the closed-loop poles in the complex plane as a control parameter varies. These poles represent the characteristic equation's roots, which determine the system's stability and transient response. By examining the pole locations as the parameter changes, engineers can gain a deeper understanding of the system's behavior and make informed design decisions.
This document introduces the root locus technique for analyzing how the closed-loop poles of a control system vary with changes in the controller gain. It provides 5 rules for constructing a root locus diagram:
1) Locate open-loop poles and zeros.
2) The number of root locus branches equals the greater of open-loop poles or zeros.
3) Points on the real axis are on the locus if open-loop poles/zeros to the right are odd.
4) Asymptotes radiate from the centroid at fixed angles depending on open-loop poles/zeros.
5) Branches depart breakaway points where multiple roots occur at angles of ±180/n degrees.
This document provides an introduction to root locus analysis. It defines a root locus as a graphical representation of how closed-loop poles move in the s-plane as a system parameter, such as gain, is varied. The objectives are to learn how to sketch a root locus using five rules, including starting and ending points, symmetry, real axis behavior, and asymptotes. An example problem sketches the root locus for a system and calculates the gain value where the locus intersects a radial line representing a specific percent overshoot value. Calculating this intersection point accurately calibrates the root locus sketch.
Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively
This document summarizes the key rules and concepts for constructing root locus diagrams. It lists the names and student IDs of 5 group members working on a control systems engineering project. The document then explains that the root locus shows how the roots of the characteristic equation change in the s-plane as the system parameter K varies from 0 to infinity, and can be used to analyze a system's stability and transient response. It proceeds to describe 8 rules for constructing root loci, including rules about the locus originating from open-loop poles, terminating at open-loop zeros or infinity, determining breakaway points, and calculating the angle of departure.
This document summarizes key aspects of the discrete Fourier transform (DFT). It defines the DFT, provides the formula for calculating it, and explains that the DFT transforms a discrete-time signal from the time domain to the frequency domain. It also outlines several important properties of the DFT, including linearity, shift property, duality, symmetry, and circular convolution. Examples are provided to illustrate duality and symmetry. References for further information on the discrete Fourier transform are also included.
This document describes a student project to implement Friis transmission formula using MATLAB. The project report includes:
1) An objective to use MATLAB to calculate received power given various antenna parameters.
2) An introduction that describes Friis transmission formula and its ideal assumptions.
3) MATLAB code examples that allow the user to input antenna parameters like transmitted power, gains, wavelength, and distance to calculate received power in different unit systems.
4) Additional MATLAB code to calculate received power for a radar targeting an object by accounting for the object's cross-sectional area.
5) The project aims to predict received power given transmission conditions to aid antenna system design and power adjustment.
The document discusses Bode plots, which are used to analyze the frequency response of linear systems. Bode plots graphically depict the magnitude and phase of a system's frequency response. They are constructed by plotting the logarithm of frequency versus gain in decibels and phase in degrees. The document outlines how to construct Bode plots based on the poles and zeros of a system's transfer function, including the slopes and asymptotes of different component factors like integrators, differentiators, and first and second order terms. Examples are provided to demonstrate how to determine a transfer function based on a given Bode plot.
The document discusses root locus analysis and controller design techniques.
1) Root locus analysis involves plotting the trajectories of a system's closed-loop poles as a parameter (such as a controller gain) is varied. Properties of root loci like pole departure angles and asymptotes are examined.
2) Design specifications like overshoot and settling time can be translated to desired pole regions. A simple proportional controller can place poles near this region.
3) Additional controller types like integral and derivative are introduced to modify the root locus for improved steady-state response or damping. Combining controller types benefits stability.
The document discusses aircraft pitch control system modeling and controller design. It describes the pitch angle and angle of attack, principles of flight, and requirements for the pitch control system. It then models the aircraft pitch transfer function and analyzes the open-loop and closed-loop response. PID, root locus and state-space controllers are designed in MATLAB and Simulink to control the aircraft pitch and meet the requirements.
This document outlines the steps and procedure for performing a root locus analysis. It begins with an introduction to root locus analysis and its use in determining the stability of a closed-loop system. It then lists the general 8 steps for drawing a root locus as a parameter is varied. An example problem is worked through, showing the determination of poles, zeros, asymptotes, breakaway points, and drawing the overall root locus sketch. The document concludes with a brief MATLAB program for generating a root locus plot.
This document contains notes from lectures 23-24 on time response and steady state errors in discrete time control systems. It begins with an outline of the lecture topics, then provides introductions and examples related to time response, the final value theorem, and steady state errors. It defines concepts like position and velocity error constants and shows examples of calculating steady state error for different system transfer functions. The document contains MATLAB examples and homework problems related to analyzing discrete time systems.
Modern Control - Lec 02 - Mathematical Modeling of SystemsAmr E. Mohamed
This document provides an overview of mathematical modeling of physical systems. It discusses how to derive mathematical models from physical systems using differential equations based on governing physical laws. The key steps are: (1) defining the physical system, (2) formulating the mathematical model using differential equations, and (3) solving the equations. Common model types include differential equation, transfer function, and state-space models. The document also discusses modeling various physical elements like electrical circuits, mechanical translational/rotational systems, and electro-mechanical systems using differential equations. It covers block diagram representation and reduction of mathematical models. The overall goal is to realize the importance of deriving accurate mathematical models for analyzing and designing control systems.
The document provides example problems and solutions for mathematical modeling and analysis of control systems. It includes the following examples:
1) It derives the transfer function C(s)/R(s) for a system represented by a block diagram, obtaining the simplified closed-loop transfer function.
2) It models a simplified automobile suspension system as a mass-spring-damper system and derives the transfer function between the input and output displacements.
3) It obtains the transfer function Y(s)/U(s) for another simplified suspension system represented by a diagram relating displacements and forces.
4) It derives the state-space representation for a mechanical system represented by equations relating positions, velocities and
Crisp sets are classical sets defined in boolean logic that have only two membership values - an element either fully belongs or does not belong to the set. Crisp sets are fundamental to the study of fuzzy sets. Key concepts of crisp sets include the universe of discourse, set operations like union and intersection, and properties like commutativity, associativity, distributivity and De Morgan's laws. Crisp sets provide a definitive yes or no for membership, unlike fuzzy sets which allow partial membership.
This document provides an overview of signal flow graphs (SFG). It defines SFG as a graphical representation of linear systems, where each variable is represented by a node and transmissions are branches with arrows denoting signal flow direction. Key terms are defined, including input/output nodes, mixed nodes, transmittance, forward paths, loops, and path/loop gains. Properties and examples of SFG construction from equations and block diagrams are described. Mason's gain formula is introduced for determining overall transfer functions from SFGs. The effects of feedback on gain and stability are also briefly discussed.
In spherical coordinates, each point is represented by an ordered triple of a distance and two angles, similar to the latitude-longitude system used on Earth. A point P is specified by its coordinates P(r,θ,φ), where r is the distance from the origin and θ and φ are the angular coordinates. Orthogonal surfaces in the spherical coordinate system are generated by keeping r, θ, or φ constant, resulting in a sphere, circular cone, or semi-infinite plane, respectively.
The document discusses the z-transform, which is the discrete-time equivalent of the Laplace transform. It defines the z-transform and provides examples of calculating the z-transform for various sequences, including the unit impulse, unit step function, sinusoids, and exponential sequences. It also discusses properties of the z-transform such as the region of convergence and relationship to the discrete-time Fourier transform.
The document discusses Bode plots, which are frequency domain techniques used to analyze linear time-invariant systems. It covers poles and zeros, transfer functions, the S-plane, mechanics for constructing Bode plots, examples of plotting Bode plots by hand and using MATLAB, and designing a system to meet a target Bode plot specification. Key steps include identifying poles and zeros, approximating plots between break frequencies, and using MATLAB tools like Bode and Simulink to validate designs.
Modern Control - Lec 05 - Analysis and Design of Control Systems using Freque...Amr E. Mohamed
The document discusses frequency response analysis and Bode plots. It begins with an introduction to frequency response and how the steady state response of a linear time-invariant system to a sinusoidal input is another sinusoid at the same frequency with a different magnitude and phase. The complex ratio of the output to input is called the frequency response. It then discusses Bode plots which show the magnitude and phase of the frequency response on logarithmic scales. Key features of components in open-loop transfer functions and how they affect the Bode plot shapes are explained. An example demonstrates drawing the Bode plots for a sample transfer function.
The document discusses the z-transform, which is a mathematical tool used to analyze discrete-time control systems. The z-transform plays a similar role for discrete-time systems as the Laplace transform does for continuous-time systems. Some key properties of the z-transform discussed include the region of convergence, properties and theorems like the shifting theorem and initial/final value theorems, and applications to feedback control systems.
Analysis and Design of Control System using Root LocusSiyum Tsega Balcha
Root locus analysis is a powerful tool in control systems engineering used to analyze the behavior of a system's closed-loop poles as a function of a parameter, typically a controller gain. It provides engineers with valuable insights into how changing system parameters affect stability and performance, helping them design robust and stable control systems. Let's explore the key concepts, techniques, and practical implications of root locus analysis. At its core, root locus analysis focuses on the movement of the closed-loop poles in the complex plane as a control parameter varies. These poles represent the characteristic equation's roots, which determine the system's stability and transient response. By examining the pole locations as the parameter changes, engineers can gain a deeper understanding of the system's behavior and make informed design decisions.
This document introduces the root locus technique for analyzing how the closed-loop poles of a control system vary with changes in the controller gain. It provides 5 rules for constructing a root locus diagram:
1) Locate open-loop poles and zeros.
2) The number of root locus branches equals the greater of open-loop poles or zeros.
3) Points on the real axis are on the locus if open-loop poles/zeros to the right are odd.
4) Asymptotes radiate from the centroid at fixed angles depending on open-loop poles/zeros.
5) Branches depart breakaway points where multiple roots occur at angles of ±180/n degrees.
ME-314 Introduction to Control Engineering is a course taught to Mechanical Engineering senior undergrads. The course is taught by Dr. Bilal Siddiqui at DHA Suffa University. This lecture is about basic rules of sketching root locus.
Root locus description in lucid way by ME IITBAmreshAman
The document discusses root locus analysis, which is a graphical method to analyze how the poles of a closed-loop system change with variations in system parameters like gain. It provides rules for plotting the root locus, including that branches originate from open-loop poles and terminate at zeros or infinity. The number of branches ending at infinity equals the number of open-loop poles minus zeros. Asymptotes indicate how branches approach infinity. The root locus helps understand how stability changes with varying parameters without recalculating closed-loop poles each time.
This document provides an introduction to root locus analysis and how to sketch and analyze root loci. It defines a root locus as a graphical representation of how closed-loop poles move in the s-plane as a system parameter, such as gain, is varied. It then describes how to sketch a root locus using 5 rules regarding the number of branches, symmetry, real axis segments, starting/ending points, and behavior at infinity. The document demonstrates how to sketch a root locus, find asymptotes, and calibrate the sketch to locate specific points representing system characteristics like percent overshoot. The goal of root locus analysis is to understand how changes in gain affect system stability and transient response qualitatively.
ROOT-LOCUS METHOD, Determine the root loci on the real axis /the asymptotes o...Waqas Afzal
Angle and Magnitude Conditions
Example of Root Locus
Steps
constructing a root-locus plot is to locate the open-loop poles and zeros in s-plane.
Determine the root loci on the real axis
Determine the asymptotes of the root loci
Determine the breakaway point.
Closed loop stability via root locus
The document discusses root locus analysis and stability analysis in the frequency domain. It begins with an overview of absolute and relative stability. It then discusses Routh's stability criterion for analyzing the stability of linear time-invariant systems using the coefficients of the characteristic equation. The document provides examples of applying Routh's criterion. It also discusses the root locus method for analyzing how the location of closed-loop poles varies with changes in a system parameter like gain. Key concepts of the root locus method like angle and magnitude conditions are explained. An example demonstrates how to construct a root locus plot.
This document provides an overview of stability analysis in the frequency domain, including absolute stability, relative stability, Routh's stability criterion, and the root locus method. It defines absolute and relative stability and describes how Routh's stability criterion can be used to determine absolute stability by analyzing the signs of coefficients in the characteristic equation. The document also introduces the root locus method for analyzing how closed-loop poles move in the s-plane as the loop gain is varied.
This document provides an overview of stability analysis in the frequency domain, including absolute stability, relative stability, Routh's stability criterion, and the root locus method. It defines absolute and relative stability and describes how Routh's stability criterion can be used to determine absolute stability by analyzing the signs of coefficients in a characteristic equation. The document also introduces the root locus method for analyzing how closed-loop poles move in the s-plane as the loop gain is varied.
1. Root locus analysis graphically displays the location of closed-loop poles as a function of the open-loop gain K.
2. Root loci start at open-loop poles when K=0 and end at open-loop zeros or infinity as K approaches infinity. There are an equal number of loci and system order.
3. Breakaway points occur where loci depart or arrive at the real axis between poles or zeros. These may be complex.
The document discusses root locus analysis and provides examples of solving for root loci of different control systems. It contains:
1) An example problem walking through the steps to sketch the root loci of a control system, including locating poles and zeros, finding breakaway/break-in points, and determining where loci cross the imaginary axis.
2) Additional example problems providing the solutions for sketching various other control system root loci, using the same procedure and noting how the system response depends on the location of closed-loop poles.
3) Details on how to determine asymptotes, departure angles, and intersection points used in the root locus analysis of different order systems.
This document discusses the design of digital controllers using root locus analysis. It provides examples of designing proportional controllers for first and second order systems to meet specifications on damping ratio, natural frequency, and settling time. The procedures involve constructing root loci, determining breakaway points and critical gains, and using the MATLAB root locus tool to plot contours and obtain design values for proportional gain.
Modern Control - Lec 04 - Analysis and Design of Control Systems using Root L...Amr E. Mohamed
The document provides an overview of root locus analysis and design of control systems. It begins with an introduction to root locus including motivation, definition, and the basic feedback control system model. It then covers the key rules and steps for constructing and interpreting root loci, including determining asymptotes, breakaway/break-in points, and imaginary axis crossings. Three examples are worked through step-by-step to demonstrate how to apply the rules and steps to sketch root loci for different open-loop transfer functions. The document emphasizes that root locus allows choosing controller parameters to place closed-loop poles in desired performance regions.
Modern Control - Lec 04 - Analysis and Design of Control Systems using Root L...Amr E. Mohamed
The document provides an overview of root locus analysis and design of control systems. It begins with an introduction to root locus including motivation, definition, and the basic feedback control system model. It then covers the key rules and steps for constructing and interpreting root loci, including determining asymptotes, breakaway/break-in points, and imaginary axis crossings. Three examples are worked through step-by-step to demonstrate how to apply the rules and steps to sketch root loci for different open-loop transfer functions. The document explains how root locus can be used to choose controller parameters to satisfy transient performance requirements.
Schelkunoff Polynomial Method for Antenna SynthesisSwapnil Bangera
The document discusses the Schelkunoff polynomial method for antenna synthesis. It involves designing an antenna array to produce a desired radiation pattern with nulls in specific directions. The method models the array factor as a polynomial and solves for the roots, which correspond to null locations. Array coefficients are then determined to produce the required roots within the visible region of the unit circle based on the element spacing and progressive phase shifts. As an example, a 4 element linear array is designed with nulls at 0, 90, and 180 degrees.
1) The document discusses root locus analysis for a control system. It provides definitions of root locus, direct root locus, and inverse root locus.
2) Seven rules for drawing root loci are described, including the number of loci branches, real axis loci, asymptotes, breakaway points, and angle of arrival/departure.
3) An example problem demonstrates applying the rules to determine the root locus for a given open-loop transfer function. The poles and zeros are identified and the root locus is drawn.
27.docking protein-protein and protein-ligandAbhijeet Kadam
This document discusses protein-protein and protein-ligand docking. It begins by defining docking as determining whether two biological molecules interact and finding their lowest energy orientation if so. The document then discusses challenges like the large number of possible conformations and small energy changes. It describes different docking study types and techniques used, including surface representation methods and algorithms like DOCK and RosettaDOCK. Finally, it summarizes a protein-protein docking algorithm and notes current problems in docking relate to limited flexibility handling and scoring function efficiency.
This presentation gives complete idea about definitions of stability, BIBO, Absolute and relative stability, Routh-Hurwitz Criterion, Special Cases and numerical examples.
Getting date and time from ntp server with esp8266 node mcuElaf A.Saeed
Getting Date & Time From NTP Server With ESP8266 NodeMCU
-----------------------------------------------------------------------------------
Email: elafe1888@gmail.com
linkden: www.linkedin.com/in/elaf-a-saeed-97bbb6150
facebook: https://www.facebook.com/profile.php?id=100004305557442
twitter: https://twitter.com/ElafASaeed1
github: https://github.com/ElafAhmedSaeed
youtube: https://youtube.com/channel/UCE_RiXkyqREUdLAiZcbBqSg
slideshare: https://www.slideshare.net/ElafASaeed
Slideplayer: https://slideplayer.com/search/?q=Elaf+A.Saeed
Google Scholar: https://scholar.google.com/citations?user=VIpVZKkAAAAJ&hl=ar&gmla=AJsN-F7PIgAjWJ44Hzb18fwPqJaaUmG0XzbLdzx09
Lesson 10- NodeMCU with LCD I2C
كورس تعلم شريحة NodeMCU وكيفيه استخدامها مع الحساسات.
Course Contents:
1- What is NodeMCU.
2- NodeMCU Instillation in Arduino IDE.
3- Simple Projects with NodeMCU (Sensors & Actuators)
4- NodeMCU with Communication protocols.
5- Connection NodeMCU with Wi-Fi.
6- Use NodeMCU as Clients & Server.
7- Different Platform uses with IOT application.
-------------------------------------------------------------------------------------
https://www.youtube.com/watch?v=eHJhuHb_bIc&list=PLknlsmxJbiDAdp1HJub7-RNcAZg9Sbj3s
Email: elafe1888@gmail.com
linkden: www.linkedin.com/in/elaf-a-saeed-97bbb6150
facebook: https://www.facebook.com/profile.php?id=100004305557442
github: https://github.com/ElafAhmedSaeed
youtube: https://youtube.com/channel/UCE_RiXkyqREUdLAiZcbBqSg
slideshare: https://www.slideshare.net/ElafASaeed
Slideplayer: https://slideplayer.com/search/?q=Elaf+A.Saeed
Google Scholar: https://scholar.google.com/citations?user=VIpVZKkAAAAJ&hl=ar&gmla=AJsN-F7PIgAjWJ44Hzb18fwPqJaaUmG0XzbLdzx09
Lesson 9- NodeMCU with Arduino UNO (UART)Elaf A.Saeed
Lesson 9- NodeMCU with Arduino UNO (UART)
Course Contents:
1- What is NodeMCU.
2- NodeMCU Instillation in Arduino IDE.
3- Simple Projects with NodeMCU (Sensors & Actuators)
4- NodeMCU with Communication protocols.
5- Connection NodeMCU with Wi-Fi.
6- Use NodeMCU as Clients & Server.
7- Different Platform uses with IOT application.
Lesson 8- NodeMCU with Servo Motor
كورس تعلم شريحة NodeMCU وكيفيه استخدامها مع الحساسات.
Course Contents:
1- What is NodeMCU.
2- NodeMCU Instillation in Arduino IDE.
3- Simple Projects with NodeMCU (Sensors & Actuators)
4- NodeMCU with Communication protocols.
5- Connection NodeMCU with Wi-Fi.
6- Use NodeMCU as Clients & Server.
7- Different Platform uses with IOT application.
Lesson 7- NodeMCU with DC Motor
كورس تعلم شريحة NodeMCU وكيفيه استخدامها مع الحساسات.
Course Contents:
1- What is NodeMCU.
2- NodeMCU Instillation in Arduino IDE.
3- Simple Projects with NodeMCU (Sensors & Actuators)
4- NodeMCU with Communication protocols.
5- Connection NodeMCU with Wi-Fi.
6- Use NodeMCU as Clients & Server.
7- Different Platform uses with IOT application.
Lesson 6 - NodeMCU with PWM Pin
كورس تعلم شريحة NodeMCU وكيفيه استخدامها مع الحساسات.
Course Contents:
1- What is NodeMCU.
2- NodeMCU Instillation in Arduino IDE.
3- Simple Projects with NodeMCU (Sensors & Actuators)
4- NodeMCU with Communication protocols.
5- Connection NodeMCU with Wi-Fi.
6- Use NodeMCU as Clients & Server.
7- Different Platform uses with IOT application.
كورس تعلم شريحة NodeMCU وكيفيه استخدامها مع الحساسات.
Course Contents:
1- What is NodeMCU.
2- NodeMCU Instillation in Arduino IDE.
3- Simple Projects with NodeMCU (Sensors & Actuators)
4- NodeMCU with Communication protocols.
5- Connection NodeMCU with Wi-Fi.
6- Use NodeMCU as Clients & Server.
7- Different Platform uses with IOT application
lesson2 - Nodemcu course - NodeMCU dev BoardElaf A.Saeed
This document provides an overview of the NodeMCU course and ESP8266 development boards. It discusses the ESP8266 chip, the NodeMCU 1.0 development board, and the ESP8266 12-E NodeMCU kit. The ESP8266 is a low-cost WiFi-enabled microcontroller that is commonly used in IoT projects. It has an integrated TCP/IP stack and supports various protocols. The NodeMCU boards make it easy to program the ESP8266 chip and interface with inputs and outputs. The document describes the components and pinouts of the NodeMCU boards and how to interface with GPIO, I2C, SPI, PWM and analog pins.
lesson1 - Getting Started with ESP8266Elaf A.Saeed
lesson1 - Getting Started with ESP8266
1- What is NodeMCU.
2- NodeMCU Instillation in Arduino IDE.
3- Simple Projects with NodeMCU (Sensors & Actuators)
4- NodeMCU with Communication protocols.
5- Connection NodeMCU with Wi-Fi.
6- Use NodeMCU as Clients & Server.
7- Different Platform uses with IOT application.
Embedded system course projects - Arduino CourseElaf A.Saeed
• Arduino IDE.
• P1-Arduino with led.
• P2-Arduino with push button.
• P3-Arduino with potentiometer.
• P4-Arduino with PWM.
• P5-Arduino with LCD.
• P6-Arduino with PIR.
• P7-Arduino with DHT11
• P8-Arduino with LM35.
• P9-Arduino with gas sensor.
• P10-Arduino with dc motor.
• P11-Arduino with Servo Motor.
• P12-Arduino with Bluetooth.
• P13-Arduino with ultrasonic.
• P14-Arduino with IR sensor.
--------------------------------------------------------
Email: elafe1888@gmail.com
linkden: www.linkedin.com/in/elaf-a-saeed-97bbb6150
facebook: https://www.facebook.com/profile.php?id=100004305557442
github: https://github.com/ElafAhmedSaeed
youtube: https://youtube.com/channel/UCE_RiXkyqREUdLAiZcbBqSg
slideshare: https://www.slideshare.net/ElafASaeed
Slideplayer: https://slideplayer.com/search/?q=Elaf+A.Saeed
Google Scholar: https://scholar.google.com/citations?user=VIpVZKkAAAAJ&hl=ar&gmla=AJsN-F7PIgAjWJ44Hzb18fwPqJaaUmG0XzbLdzx09
programming languages
The programming languages supported by Raspberry Pi are all the languages supported by Linux, such as Python, Pascal, Java, and many other languages. We will use the python language in this course to program the GPIO control ports for the features that are available in this language over other languages which we will list when introducing Python language.
Matlab is basically a high level language which has many specialized toolboxes for making things easier for us.
Matlab stands for MATrix LABoratory.
The first version of MATLAB was produced in the mid 1970s as a teaching tool. MATLAB started as an interactive program for doing matrix calculations.
MATLAB has now grown to a high level mathematical language that can solve integrals and differential equations numerically and plot a wide variety of two and three Dimensional graphs.
The expanded MATLAB is now used for calculations and simulation in companies and government labs ranging from aerospace, car design, signal analysis through to instrument control and financial analysis.
In practice, it provides a very nice tool to implement numerical method.
- The desktop includes these panels:
Current Folder — Access your files.
Command Window — Enter commands at the command line, indicated by the prompt (>>).
Workspace — Explore data that you create or import from files.
- what we learn:
1- Introduction to Matlab.
2- MATLAB InstallationVersion 2018.
3- Assignment.
4- Operations in MATLAB.
5- Vectors and Matrices in MATLAB.
ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
HEAP SORT ILLUSTRATED WITH HEAPIFY, BUILD HEAP FOR DYNAMIC ARRAYS.
Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to the selection sort where we first find the minimum element and place the minimum element at the beginning. Repeat the same process for the remaining elements.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
3. 6-1 INTRODUCTION
• The basic characteristic of the transient response of a closed-loop system is closely
related to the location of the closed-loop poles.
• If the system has a variable loop gain, then the location of the closed-loop poles depends
on the value of the loop gain chosen. It is important, therefore, that the designer know
how the closed-loop poles move in the s plane as the loop gain is varied.
• From the design viewpoint, in some systems simple gain adjustment may move the
closed-loop poles to desired locations. Then the design problem may become the
selection of an appropriate gain value. If the gain adjustment alone does not yield a
desired result, addition of a compensator to the system will become necessary.
4. 6-1 INTRODUCTION
• The closed-loop poles are the roots of the characteristic equation. Finding the roots of the
characteristic equation of degree higher than 3 is laborious and will need computer solution.
However, just finding the roots of the characteristic equation may be of limited value,
because as the gain of the open-loop transfer function varies, the characteristic equation
changes and the computations must be repeated.
• A simple method for finding the roots of the characteristic equation has been developed by
W. R. Evans and used extensively in control engineering. This method, called the root-locus
method, is one in which the roots of the characteristic equation are plotted for all values of
a system parameter.
• The roots corresponding to a particular value of this parameter can then be located on the
resulting graph.
5. 6-1 INTRODUCTION
• Note that the parameter is usually the gain, but any other variable of the open-loop transfer
function may be used. Unless otherwise stated, we shall assume that the gain of the open-loop
transfer function is the parameter to be varied through all values, from zero to infinity.
• By using the root-locus method the designer can predict the effects on the location of the
closed-loop poles of varying the gain value or adding open-loop poles and/or open-loop zeros.
• Therefore, it is desired that the designer have a good understanding of the method for
generating the root loci of the closed-loop system, both by hand and by use of a computer
software program like MATLAB.
• In designing a linear control system, we find that the root-locus method proves to be quite
useful, since it indicates the manner in which the open-loop poles and zeros should be modified
so that the response meets system performance specifications. This method is particularly suited
to obtaining approximate results very quickly.
6. 6-1 INTRODUCTION
• Because generating the root loci by use of MATLAB is very simple, one may think sketching
the root loci by hand is a waste of time and effort. However, experience in sketching the
root loci by hand is invaluable for interpreting computer-generated root loci, as well as for
getting a rough idea of the root loci very quickly.
8. 6-2 ROOT LOCUS PLOTS
• Consider the negative feedback system shown below.The closed-loop transfer function is
• The characteristic equation for this closed-loop system is obtained by setting the
denominator of the right-hand side of Equation equal to zero.That is
or
9. 6-2 ROOT LOCUS PLOTS
• Here we assume that G(s)H(s) is a ratio of polynomials in s. Since G(s)H(s) is a complex
quantity, Equation can be split into two equations by equating the angles and magnitudes of
both sides, respectively, to obtain the following:
Angle condition:
Magnitude condition:
• The values of s that fulfill both the angle and magnitude conditions are the roots of the
characteristic equation, or the closed-loop poles.
• A locus of the points in the complex plane satisfying the angle condition alone is the root
locus.
10. 6-2 ROOT LOCUS PLOTS
• The roots of the characteristic equation (the closed-loop poles) corresponding to a given
value of the gain can be determined from the magnitude condition. The details of applying
the angle and magnitude conditions to obtain the closed-loop poles are presented later in
this section.
• In many cases, G(s)H(s) involves a gain parameter K, and the characteristic equation may be
written as
• Then the root loci for the system are the loci of the closed-loop poles as the gain K is
varied from zero to infinity.
11. 6-2 ROOT LOCUS PLOTS
• Note that to begin sketching the root loci of a system by the root-locus method we must
know the location of the poles and zeros of G(s)H(s). Remember that the angles of the
complex quantities originating from the open-loop poles and open-loop zeros to the test
point s are measured in the counterclockwise direction. For example, if G(s)H(s) is given by
13. 6-2 ROOT LOCUS PLOTS
• where –p2 and –p3 are complex-conjugate poles, then the angle of G(s)H(s) is
• where ∅1 , 𝜃1, 𝜃2 , 𝜃3 , and 𝜃4 are measured counterclockwise as shown in previous Figures
(a) and (b).The magnitude of G(s)H(s) for this system is
• where A1 , A2 , A3 , A4 , and B1 are the magnitudes of the complex quantities s+p1 , s+p2 ,
s+p3 , s+p4 , and s+z1 , respectively, as shown in previous Figure (a).
14. 6-2 ROOT LOCUS PLOTS
General Rules for Constructing Root Loci.
• For a complicated system with many open-loop poles and zeros, constructing a root-locus
plot may seem complicated, but actually it is not difficult if the rules for constructing the
root loci are applied.
• By locating particular points and asymptotes and by computing angles of departure from
complex poles and angles of arrival at complex zeros, we can construct the general form of
the root loci without difficulty.
• We shall now summarize the general rules and procedure for constructing the root loci of
the negative feedback control system.
15. 6-2 ROOT LOCUS PLOTS
General Rules for Constructing Root Loci.
• First, obtain the characteristic equation
• Then rearrange this equation so that the parameter of interest appears as the multiplying
factor in the form
• In the present discussions, we assume that the parameter of interest is the gain K, where
K>0. In the present discussions, we assume that the parameter of interest is the gain K,
where K>0. (If K < 0, which corresponds to the positive-feedback case, the angle condition
must be modified.).
16. 6-2 ROOT LOCUS PLOTS
General Rules for Constructing Root Loci.
• 1. Locate the poles and zeros of G(s)H(s) on the s plane.
The root-locus branches start from open-loop poles and terminate at zeros (finite zeros or
zeros at infinity). From the factored form of the open-loop transfer function, locate the open-
loop poles and zeros in the s plane. e. [Note that the open-loop zeros are the zeros of
G(s)H(s), while the closed-loop zeros consist of the zeros of G(s) and the poles of H(s).]
17. 6-2 ROOT LOCUS PLOTS
General Rules for Constructing Root Loci.
• Note that the root loci are symmetrical about the real axis of the s plane, because the
complex poles and complex zeros occur only in conjugate pairs.
• A root-locus plot will have just as many branches as there are roots of the characteristic
equation. Since the number of open-loop poles generally exceeds that of zeros, the number
of branches equals that of poles.
• If the number of closed-loop poles is the same as the number of open-loop poles, then the
number of individual root-locus branches terminating at finite open-loop zeros is equal to
the number m of the open-loop zeros.
18. 6-2 ROOT LOCUS PLOTS
General Rules for Constructing Root Loci.
• The remaining n-m branches terminate at infinity (n-m implicit zeros at infinity) along
asymptotes.
2. Root loci on the real axis are determined by open-loop poles and zeros lying on it.
The complex-conjugate poles and complex conjugate zeros of the open-loop transfer
function have no effect on the location of the root loci on the real axis because the angle
contribution of a pair of complex-conjugate poles or complex-conjugate zeros is 360° on the
real axis.
• Each portion of the root locus on the real axis extends over a range from a pole or zero to
another pole or zero.
19. 6-2 ROOT LOCUS PLOTS
General Rules for Constructing Root Loci.
• In constructing the root loci on the real axis, choose a test point on it. If the total number of
real poles and real zeros to the right of this test point is odd, then this point lies on a root
locus. If the open-loop poles and open-loop zeros are simple poles and simple zeros, then
the root locus and its complement form alternate segments along the real axis.
3. Determine the asymptotes of root loci.
If the test point s is located far from the origin, then the angle of each complex quantity may
be considered the same. One open-loop zero and one open-loop pole then cancel the effects
of the other. Therefore, the root loci for very large values of s must be asymptotic to straight
lines whose angles (slopes) are given by
20. 6-2 ROOT LOCUS PLOTS
General Rules for Constructing Root Loci.
• where n= number of finite poles of G(s)H(s) m= number of finite zeros of G(s)H(s)
• Here, k=0 corresponds to the asymptotes with the smallest angle with the real axis.
Although k assumes an infinite number of values, as k is increased the angle repeats itself,
and the number of distinct asymptotes is n-m.
4. Find the breakaway and break-in points.
Because of the conjugate symmetry of the root loci, the breakaway points and break-in points
either lie on the real axis or occur in complex-conjugate pairs.
If a root locus lies between two adjacent open-loop poles on the real axis, then there exists at
least one breakaway point between the two poles.
21. 6-2 ROOT LOCUS PLOTS
General Rules for Constructing Root Loci.
• Similarly, if the root locus lies between two adjacent zeros (one zero may be located at –q)
on the real axis, then there always exists at least one break-in point between the two zeros.
• If the root locus lies between an open-loop pole and a zero (finite or infinite) on the real
axis, then there may exist no breakaway or break-in points or there may exist both
breakaway and break-in points.
• Suppose that the characteristic equation is given by
• The breakaway points and break-in points correspond to multiple roots of the characteristic
equation. the breakaway and break-in points can be determined from the roots of
22. 6-2 ROOT LOCUS PLOTS
General Rules for Constructing Root Loci.
5. Determine the angle of departure (angle of arrival) of the root locus from a complex pole (at a
complex zero).
To sketch the root loci with reasonable accuracy, we must find the directions of the root loci near
the complex poles and zeros.
If a test point is chosen and moved in the very vicinity of a complex pole (or complex zero), the
sum of the angular contributions from all other poles and zeros can be considered to remain the
same.
Therefore, the angle of departure (or angle of arrival) of the root locus from a complex pole (or at
a complex zero) can be found by subtracting from 180° the sum of all the angles of vectors from all
other poles and zeros to the complex pole (or complex zero) in question, with appropriate signs
included.
23. 6-2 ROOT LOCUS PLOTS
General Rules for Constructing Root Loci.
Angle of departure from a complex pole=180°
– (sum of the angles of vectors to a complex pole in question from other poles)
+ (sum of the angles of vectors to a complex pole in question from zeros)
Angle of arrival at a complex zero=180°
– (sum of the angles of vectors to a complex zero in question from other zeros)
+ (sum of the angles of vectors to a complex zero in question from poles)
24. 6-2 ROOT LOCUS PLOTS
General Rules for Constructing Root Loci.
6. Find the points where the root loci may cross the imaginary axis.
The points where the root loci intersect the jw axis can be found easily by (a) use of Routh’s
stability criterion or (b) letting s=jw in the characteristic equation, equating both the real part
and the imaginary part to zero, and solving for w and K.
The values of w thus found give the frequencies at which root loci cross the imaginary axis.
The K value corresponding to each crossing frequency gives the gain at the crossing point.
25. 6-2 ROOT LOCUS PLOTS
General Rules for Constructing Root Loci.
7. Taking a series of test points in the broad neighborhood of the origin of the s plane, sketch
the root loci.
Determine the root loci in the broad neighborhood of the jw axis and the origin. The most
important part of the root loci is on neither the real axis nor the asymptotes but is in the
broad neighborhood of the jw axis and the origin.The shape
26. 6-2 ROOT LOCUS PLOTS
General Rules for Constructing Root Loci.
27. 6-2 ROOT LOCUS PLOTS
General Rules for Constructing Root Loci.
• of the root loci in this important region in the s plane must be obtained with reasonable
accuracy.
8. Determine closed-loop poles.
A particular point on each root-locus branch will be a closed-loop pole if the value of K at
that point satisfies the magnitude condition.
Conversely, the magnitude condition enables us to determine the value of the gain K at any
specific root location on the locus.
The value of K corresponding to any point s on a root locus can be obtained using the
magnitude condition, or
28. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN
Preliminary Design Consideration.
• In building a control system, we know that proper modification of the plant dynamics may
be a simple way to meet the performance specifications.
• This, however, may not be possible in many practical situations because the plant may be
fixed and not modifiable
• Then we must adjust parameters other than those in the fixed plant. In this book, we
assume that the plant is given and unalterable.
• In practice, the root-locus plot of a system may indicate that the desired performance
cannot be achieved just by the adjustment of gain (or some other adjustable parameter).
29. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Preliminary Design Consideration.(Cont.)
• In fact, in some cases, the system may not be stable for all values of gain (or other adjustable
parameter). Then it is necessary to reshape the root loci to meet the performance
specifications.
• The design problems, therefore, become those of improving system performance by
insertion of a compensator. Compensation of a control system is reduced to the design of a
filter whose characteristics tend to compensate for the undesirable and unalterable
characteristics of the plant.
30. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Design by Root-Locus Method.
• The design by the root-locus method is based on reshaping the root locus of the system by
adding poles and zeros to the system’s open-loop transfer function and forcing the root loci
to pass through desired closed-loop poles in the s plane.
• The characteristic of the root-locus design is its being based on the assumption that the
closed-loop system has a pair of dominant closed-loop poles. This means that the effects of
zeros and additional poles do not affect the response characteristics very much.
• In designing a control system, if other than a gain adjustment (or other parameter
adjustment) is required, we must modify the original root loci by inserting a suitable
compensator.
31. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Design by Root-Locus Method.(Cont.)
• Once the effects on the root locus of the addition of poles and/or zeros are fully
understood, we can readily determine the locations of the pole(s) and zero(s) of the
compensator that will reshape the root locus as desired.
• In essence, in the design by the root locus method, the root loci of the system are reshaped
through the use of a compensator so that a pair of dominant closed-loop poles can be
placed at the desired location.
32. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Series Compensation and Parallel (or Feedback) Compensation.
• Figures (a) and (b) show compensation schemes commonly used for feedback control
systems.
• Figure (a) shows the configuration where the compensator Gc(s) is placed in series with the
plant.This scheme is called series compensation.
• An alternative to series compensation is to feed back the signal(s) from some element(s)
and place a compensator in the resulting inner feedback path, as shown in Figure (b). Such
compensation is called parallel compensation or feedback compensation.
• In compensating control systems, we see that the problem usually boils down to a suitable
design of a series or parallel compensator.
33. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Series Compensation and Parallel (or Feedback)
Compensation.(Cont.)
• The choice between series compensation and parallel compensation depends on the nature
of the signals in the system, the power levels at various points, available components, the
designer’s experience, economic considerations, and so on.
• In general, series compensation may be simpler than parallel compensation; however, series
compensation frequently requires additional amplifiers to increase the gain and/or to
provide isolation. (To avoid power dissipation, the series compensator is inserted at the
lowest energy point in the feedforward path.).
34. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Series Compensation and Parallel (or Feedback)
Compensation.(Cont.)
• Note that, in general, the number of components required in parallel compensation will be
less than the number of components in series compensation, provided a suitable signal is
available, because the energy transfer is from a higher power level to a lower level. (This
means that additional amplifiers may not be necessary.).
35. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Series Compensation and Parallel (or Feedback)
Compensation.(Cont.)
36. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Commonly Used Compensators.
• If a compensator is needed to meet the performance specifications, the designer must
realize a physical device that has the prescribed transfer function of the compensator.
• Numerous physical devices have been used for such purposes. In fact, many noble and useful
ideas for physically constructing compensators may be found in the literature.
• If a sinusoidal input is applied to the input of a network, and the steady-state output (which
is also sinusoidal) has a phase lead, then the network is called a lead network.
• If the steady-state output has a phase lag, then the network is called a lag network.
• In a lag–lead network, both phase lag and phase lead occur in the output but in different
frequency regions; phase lag occurs in the low-frequency region and phase lead occurs in
the high-frequency region.
37. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Commonly Used Compensators.(Cont.)
• A compensator having a characteristic of a lead network, lag network, or lag–lead network
is called a lead compensator, lag compensator, or lag–lead compensator.
• Among the many kinds of compensators, widely employed compensators are the lead
compensators, lag compensators, lag–lead compensators, and velocity-feedback
(tachometer) compensators.
38. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Commonly Used Compensators.(Cont.)
• It is noted that in designing control systems by the root-locus or frequency-response
methods the final result is not unique, because the best or optimal solution may not be
precisely defined if the time-domain specifications or frequency-domain specifications are
given.
39. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Effects of the Addition of Poles.
• The addition of a pole to the open-loop transfer function has the effect of pulling the root
locus to the right, tending to lower the system’s relative stability and to slow down the
settling of the response. (Remember that the addition of integral control adds a pole at the
origin, thus making the system less stable.) Figure shows examples of root loci illustrating
the effects of the addition of a pole to a single-pole system and the addition of two poles to
a single-pole system.
40. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Effects of the Addition of Poles.(Cont.)
• (a) Root-locus plot of a single-pole system; (b) root-locus plot of a two-pole system; (c)
root-locus plot of a three-pole system
41. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Effects of the Addition of Zeros.
• The addition of a zero to the open-loop transfer function has the effect of pulling the root
locus to the left, tending to make the system more stable and to speed up the settling of the
response. (Physically, the addition of a zero in the feedforward transfer function means the
addition of derivative control to the system. The effect of such control is to introduce a
degree of anticipation into the system and speed up the transient response.) Figure (a)
shows the root loci for a system
42. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Effects of the Addition of Zeros.(Cont.)
• (a) Root-locus plot of a three-pole system; (b), (c), and (d) root-locus plots showing effects
of addition of a zero to the three-pole system.
43. 6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN.(CONT.)
Effects of the Addition of Zeros.(Cont.)
• that is stable for small gain but unstable for large gain. Figures(b), (c), and (d) show root-
locus plots for the system when a zero is added to the open-loop transfer function. Notice
that when a zero is added to the system of Figure(a), it becomes stable for all values of gain.