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.
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.
The document discusses root locus techniques for analyzing control systems. It begins with an overview and objectives of root locus analysis. It then defines the root locus and describes how to sketch a root locus by determining the starting and ending points, branches, symmetry, behavior at infinity, and real axis segments. The document provides examples of using properties of root loci to find breakaway and break-in points, asymptotes, and the frequency and gain at imaginary axis crossings.
The document discusses the bipolar junction transistor (BJT), an important electronic device invented in 1947 at Bell Labs by Bardeen, Brattain, and Shockley. It summarizes the BJT's construction using either PNP or NPN semiconductor materials, its basic working involving forward and reverse biasing of the base-emitter and collector-emitter junctions, and its three main modes of operation - cutoff, saturation, and active. The document also covers BJT configurations like common base, common collector, and common emitter; and concludes with references.
This presentation explains about the introduction of Nyquist Stability criterion. It clearly shows advantages and disadvantages of Nyquist Stability criterion and also explains importance of Nyquist Stability criterion and steps required to sketch the Nyquist plot. It explains about the steps required to sketch Nyquist plot clearly. It also explains about sketching Nyquist plot and determines the stability by using Nyquist Stability criterion with an example.
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1. The document discusses Nyquist stability criteria and polar plots.
2. Nyquist stability criteria uses Cauchy's argument principle to relate the open-loop transfer function to the poles of the closed-loop characteristic equation.
3. For a system to be stable, the number of counter-clockwise encirclements of the Nyquist plot around the point -1 must equal the number of open-loop poles in the right half plane.
This Slide is made of many important information which are very easily discussed in this slide briefly. I hope, after watching this slide , you will get some analytical information on Alternative Current(AC).Actually, this slide was made for my University Presentation.
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
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.
The document discusses root locus techniques for analyzing control systems. It begins with an overview and objectives of root locus analysis. It then defines the root locus and describes how to sketch a root locus by determining the starting and ending points, branches, symmetry, behavior at infinity, and real axis segments. The document provides examples of using properties of root loci to find breakaway and break-in points, asymptotes, and the frequency and gain at imaginary axis crossings.
The document discusses the bipolar junction transistor (BJT), an important electronic device invented in 1947 at Bell Labs by Bardeen, Brattain, and Shockley. It summarizes the BJT's construction using either PNP or NPN semiconductor materials, its basic working involving forward and reverse biasing of the base-emitter and collector-emitter junctions, and its three main modes of operation - cutoff, saturation, and active. The document also covers BJT configurations like common base, common collector, and common emitter; and concludes with references.
This presentation explains about the introduction of Nyquist Stability criterion. It clearly shows advantages and disadvantages of Nyquist Stability criterion and also explains importance of Nyquist Stability criterion and steps required to sketch the Nyquist plot. It explains about the steps required to sketch Nyquist plot clearly. It also explains about sketching Nyquist plot and determines the stability by using Nyquist Stability criterion with an example.
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
1. The document discusses Nyquist stability criteria and polar plots.
2. Nyquist stability criteria uses Cauchy's argument principle to relate the open-loop transfer function to the poles of the closed-loop characteristic equation.
3. For a system to be stable, the number of counter-clockwise encirclements of the Nyquist plot around the point -1 must equal the number of open-loop poles in the right half plane.
This Slide is made of many important information which are very easily discussed in this slide briefly. I hope, after watching this slide , you will get some analytical information on Alternative Current(AC).Actually, this slide was made for my University Presentation.
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 discusses discrete-time signals and sequences. It defines discrete-time signals as sequences of numbers represented as x[n], where n is an integer. In practice, sequences arise from periodically sampling an analog signal. Linear time-invariant (LTI) systems are described by the convolution sum, where the impulse response h[n] completely characterizes the system. FIR systems have impulse responses of finite duration, while IIR systems can have impulse responses that extend to infinity.
Z trasnform & Inverse Z-transform in matlabHasnain Yaseen
This 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 systems as the Laplace transform does for continuous systems. The document covers the definition of the z-transform, its region of convergence in the z-plane, methods for taking the inverse z-transform, properties of the z-transform, and how to use MATLAB for z-transforms.
1) Effective current in an AC circuit is 0.707 times the maximum current. Effective voltage is 0.707 times the maximum voltage.
2) Inductive reactance is directly proportional to frequency and inductance. Capacitive reactance is inversely proportional to frequency and capacitance.
3) Impedance is the total opposition to current flow in an AC circuit consisting of resistance and reactance. Power is consumed only by the resistive component of impedance and is proportional to the cosine of the phase angle.
Alternating current (AC), is an electric current in which the flow of electric charge periodically reverses direction, whereas in direct current (DC, also dc), the flow of electric charge is only in one direction.
This document discusses phase lead and lag compensators for digital control systems. It covers:
1. Designing a discrete-time phase lead/lag compensator by mapping the z-plane to the w-plane using bilinear transformation.
2. Defining phase lead and lag compensators based on the positions of poles and zeros in the w-domain transfer function.
3. A design approach using frequency response methods to meet a phase margin specification by determining the parameters of a first-order digital phase lead or lag compensator.
4. Examples of designing phase lead and lag compensators for different plant transfer functions to meet specifications on phase margin and steady state error.
O documento discute formas canônicas, controlabilidade, observabilidade e alocação de pólos para sistemas de controle. Aborda as formas canônicas controlável, observável e diagonal e como verificar controlabilidade e observabilidade. Também explica como alocar pólos de malha fechada em qualquer posição do plano complexo esquerdo para obter um sistema estável com desempenho desejado.
Two port network parameters, Z, Y, ABCD, h and g parameters, Characteristic impedance,
Image transfer constant, image and iterative impedance, network function, driving point and
transfer functions – using transformed (S) variables, Poles and Zeros.
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.
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.
EC8352-Signals and Systems - Laplace transformNimithaSoman
The document discusses the Laplace transform and its properties. It begins by introducing Laplace transform as a tool to transform signals from the time domain to the complex frequency (s-domain). It then provides the Laplace transforms of some elementary signals like impulse, step, ramp functions. It discusses properties like linearity, time shifting, frequency shifting. It also covers the region of convergence, causality, stability analysis using poles in the s-plane. The document provides examples of finding the Laplace transform and analyzing signals based on properties like time shifting and frequency shifting. In the end, it summarizes the convolution property and the initial and final value theorems.
Okay, here are the steps:
1) Given:
2) Transform into spherical unit vectors:
3) Write in terms of spherical components:
So the vector components in spherical coordinates are:
Thévenin's theorem states that any linear two-terminal circuit can be replaced by an equivalent circuit consisting of an ideal voltage source (VTh) in series with a resistor (RTh). VTh is equal to the open-circuit voltage at the terminals and RTh is the equivalent input resistance when independent sources are turned off. To find the Thevenin equivalent circuit, first the load is replaced with an open circuit to find VTh, then independent sources are turned off to find RTh, the resistance seen looking into the terminals. Once the Thevenin equivalent circuit is determined, it can be used to solve for voltages and currents in the original circuit.
State space analysis, eign values and eign vectorsShilpa Shukla
This document discusses state space analysis and the conversion of transfer functions to state space models. It covers:
1. The need to convert transfer functions to state space form in order to apply modern time domain techniques for system analysis and design.
2. Three possible representations for realizing a transfer function as a state space model: first companion form, second companion form, and Jordan canonical form.
3. The concepts of eigenvalues and eigenvectors, and how they relate to state space models.
4. Worked examples of converting transfer functions to state space models in first and second companion forms, as well as the Jordan canonical form for systems with repeated and non-repeated roots.
The document provides an overview
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.
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.
This document provides an overview of time domain analysis techniques for control systems. It discusses common test inputs like impulse, step, and ramp functions used to characterize system performance. It describes how to determine a system's poles and zeros from its transfer function and use a pole-zero plot to understand system dynamics. Standard forms are presented for first and second order systems. Transient performance metrics like rise time, peak time, settling time, and overshoot are defined for characterizing step responses. The effects of poles and zeros on the system response are explained.
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.
This document discusses star-delta transformations and their equivalence. It provides the formulas for converting between star and delta connections by equating the resistances between corresponding terminals. Examples are given of using the formulas to solve for unknown resistances in both directions. Conversion methods are demonstrated on sample networks, such as finding the resistance between two points or the current drawn from a battery. An important note is that network simplification may lose original points, so care must be taken to retain relevant information.
The document discusses root locus analysis, a technique for analyzing the stability and transient response of control systems. It provides rules for sketching root loci, including that branches represent closed-loop poles and the locus is symmetric about the real axis. The document also describes refining the root locus sketch by finding the imaginary axis crossing, angles of departure and arrival, and approximating higher-order systems as second-order. An example problem is given to apply these techniques.
- Root-locus plots show how the roots of a system change with variations in a system parameter like gain.
- The plot determines if the system will become unstable as parameters vary by checking if roots cross to the right half of the complex plane.
- A document section provides examples of how real and complex roots correspond to different system responses and stability.
This document discusses discrete-time signals and sequences. It defines discrete-time signals as sequences of numbers represented as x[n], where n is an integer. In practice, sequences arise from periodically sampling an analog signal. Linear time-invariant (LTI) systems are described by the convolution sum, where the impulse response h[n] completely characterizes the system. FIR systems have impulse responses of finite duration, while IIR systems can have impulse responses that extend to infinity.
Z trasnform & Inverse Z-transform in matlabHasnain Yaseen
This 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 systems as the Laplace transform does for continuous systems. The document covers the definition of the z-transform, its region of convergence in the z-plane, methods for taking the inverse z-transform, properties of the z-transform, and how to use MATLAB for z-transforms.
1) Effective current in an AC circuit is 0.707 times the maximum current. Effective voltage is 0.707 times the maximum voltage.
2) Inductive reactance is directly proportional to frequency and inductance. Capacitive reactance is inversely proportional to frequency and capacitance.
3) Impedance is the total opposition to current flow in an AC circuit consisting of resistance and reactance. Power is consumed only by the resistive component of impedance and is proportional to the cosine of the phase angle.
Alternating current (AC), is an electric current in which the flow of electric charge periodically reverses direction, whereas in direct current (DC, also dc), the flow of electric charge is only in one direction.
This document discusses phase lead and lag compensators for digital control systems. It covers:
1. Designing a discrete-time phase lead/lag compensator by mapping the z-plane to the w-plane using bilinear transformation.
2. Defining phase lead and lag compensators based on the positions of poles and zeros in the w-domain transfer function.
3. A design approach using frequency response methods to meet a phase margin specification by determining the parameters of a first-order digital phase lead or lag compensator.
4. Examples of designing phase lead and lag compensators for different plant transfer functions to meet specifications on phase margin and steady state error.
O documento discute formas canônicas, controlabilidade, observabilidade e alocação de pólos para sistemas de controle. Aborda as formas canônicas controlável, observável e diagonal e como verificar controlabilidade e observabilidade. Também explica como alocar pólos de malha fechada em qualquer posição do plano complexo esquerdo para obter um sistema estável com desempenho desejado.
Two port network parameters, Z, Y, ABCD, h and g parameters, Characteristic impedance,
Image transfer constant, image and iterative impedance, network function, driving point and
transfer functions – using transformed (S) variables, Poles and Zeros.
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.
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.
EC8352-Signals and Systems - Laplace transformNimithaSoman
The document discusses the Laplace transform and its properties. It begins by introducing Laplace transform as a tool to transform signals from the time domain to the complex frequency (s-domain). It then provides the Laplace transforms of some elementary signals like impulse, step, ramp functions. It discusses properties like linearity, time shifting, frequency shifting. It also covers the region of convergence, causality, stability analysis using poles in the s-plane. The document provides examples of finding the Laplace transform and analyzing signals based on properties like time shifting and frequency shifting. In the end, it summarizes the convolution property and the initial and final value theorems.
Okay, here are the steps:
1) Given:
2) Transform into spherical unit vectors:
3) Write in terms of spherical components:
So the vector components in spherical coordinates are:
Thévenin's theorem states that any linear two-terminal circuit can be replaced by an equivalent circuit consisting of an ideal voltage source (VTh) in series with a resistor (RTh). VTh is equal to the open-circuit voltage at the terminals and RTh is the equivalent input resistance when independent sources are turned off. To find the Thevenin equivalent circuit, first the load is replaced with an open circuit to find VTh, then independent sources are turned off to find RTh, the resistance seen looking into the terminals. Once the Thevenin equivalent circuit is determined, it can be used to solve for voltages and currents in the original circuit.
State space analysis, eign values and eign vectorsShilpa Shukla
This document discusses state space analysis and the conversion of transfer functions to state space models. It covers:
1. The need to convert transfer functions to state space form in order to apply modern time domain techniques for system analysis and design.
2. Three possible representations for realizing a transfer function as a state space model: first companion form, second companion form, and Jordan canonical form.
3. The concepts of eigenvalues and eigenvectors, and how they relate to state space models.
4. Worked examples of converting transfer functions to state space models in first and second companion forms, as well as the Jordan canonical form for systems with repeated and non-repeated roots.
The document provides an overview
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.
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.
This document provides an overview of time domain analysis techniques for control systems. It discusses common test inputs like impulse, step, and ramp functions used to characterize system performance. It describes how to determine a system's poles and zeros from its transfer function and use a pole-zero plot to understand system dynamics. Standard forms are presented for first and second order systems. Transient performance metrics like rise time, peak time, settling time, and overshoot are defined for characterizing step responses. The effects of poles and zeros on the system response are explained.
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.
This document discusses star-delta transformations and their equivalence. It provides the formulas for converting between star and delta connections by equating the resistances between corresponding terminals. Examples are given of using the formulas to solve for unknown resistances in both directions. Conversion methods are demonstrated on sample networks, such as finding the resistance between two points or the current drawn from a battery. An important note is that network simplification may lose original points, so care must be taken to retain relevant information.
The document discusses root locus analysis, a technique for analyzing the stability and transient response of control systems. It provides rules for sketching root loci, including that branches represent closed-loop poles and the locus is symmetric about the real axis. The document also describes refining the root locus sketch by finding the imaginary axis crossing, angles of departure and arrival, and approximating higher-order systems as second-order. An example problem is given to apply these techniques.
- Root-locus plots show how the roots of a system change with variations in a system parameter like gain.
- The plot determines if the system will become unstable as parameters vary by checking if roots cross to the right half of the complex plane.
- A document section provides examples of how real and complex roots correspond to different system responses and stability.
This document provides background information on root locus analysis and techniques for sketching root loci. It discusses rules for determining the number and symmetry of root locus branches, where the locus begins and ends, and how to find breakaway/break-in points and jw-axis crossings. The document concludes by posing a problem to sketch the root locus for a given transfer function. The key information is that root locus analysis graphically shows how closed-loop poles vary with gain and provides insights into stability and transient response.
This document discusses using cascade compensation to improve control system performance. Cascade compensation involves adding additional poles and zeros to the open-loop transfer function. This can improve the transient response by placing poles farther out in the s-plane, and improve steady-state error by increasing the system type. An example shows designing a PI controller to reduce steady-state error to zero without affecting the 57.4% overshoot transient response. Pole-zero cancellation is used to maintain the original transient response while increasing the system type.
The document discusses various techniques for using root locus analysis to design cascade compensators to improve system performance, including:
1) Improving steady state error by adding an open loop pole at the origin using an integrator.
2) Improving transient response by inserting a differentiator in the forward path or augmenting the system with additional poles and zeros.
3) Using proportional, integral, derivative (PID) controllers and passive lag/lead networks to realize compensators that improve both steady state error and transient response.
The addition of poles to a system tends to shift the root locus towards the right side of the s-plane, lowering stability. Adding more poles further restricts stability by shifting the breakaway points more to the right. In contrast, the addition of zeros tends to pull the root locus left, improving stability by making the system less oscillatory and increasing the gain margin and range of k values.
The document discusses frequency response analysis of control systems. It defines frequency response as the amplitude and phase differences between the input and output sinusoids of a linear system subjected to a sinusoidal input. Frequency response consists of magnitude frequency response and phase frequency response. The document provides examples of using frequency response concepts like plotting Bode diagrams, calculating key points on Nyquist diagrams, and using the Nyquist criterion to determine stability.
A root locus plot is simply a plot of the s zero values and the s poles on a graph with real and imaginary coordinates.
This method is very powerful graphical technique for investigating the effects of the variation of a system parameter on the locations of the closed loop poles.
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.
This document discusses root locus analysis, which graphically shows how the closed-loop poles of a system change with a parameter. It outlines the general steps for drawing a root locus, which include determining open-loop poles and zeros, drawing the pole-zero plot, calculating asymptotes and breakaway points, and sketching the overall root locus. The document also briefly mentions that root locus analysis can provide insights into a system's stability and performance and be used to aid control system design.
The document outlines 7 rules for constructing a root locus diagram:
1. Draw the open-loop pole-zero plot and find the branches
2. Calculate the asymptotes and centroid
3. Find the breakaway points and check their validity
4. Determine the intercepts on the jw-axis
5. Calculate the angle of arrival and departure for complex roots
6. Predict the stability of the system based on where the dominant roots lie for different values of the gain parameter K.
This document summarizes a lecture on sketching root locus diagrams. It defines poles and zeros as values that make the denominator and numerator of a transfer function equal to zero. It then presents 10 rules for sketching root locus diagrams, such as the lines emanating from the poles and zeros, the paths roots take as a parameter K is varied, and how poles and zeros affect the shape and position of the locus. Examples are provided to demonstrate applying the rules to plot root loci from transfer functions.
Dsp U Lec06 The Z Transform And Its Applicationtaha25
This document discusses the Z-transform and its application in digital signal processing. It covers topics such as:
1) Defining the Z-transform and how it can characterize linear time-invariant (LTI) systems.
2) Properties of LTI systems in the Z-domain, including causal and stable systems.
3) How the frequency response of a system can be obtained from its Z-transform.
4) Methods for finding the inverse Z-transform, including power series and partial fraction expansion.
5) Examples of using these techniques to analyze simple discrete systems.
This document discusses root locus analysis, which is a graphical method for examining how the roots of a system change with variation of a gain parameter. It defines root locus as the path of the roots of the characteristic equation traced out in the s-plane as the gain K is varied from 0 to infinity. The root locus can show the range of stability, instability, and conditions that cause oscillation. It also provides an example of determining whether a point lies on the root locus and calculating the corresponding gain value.
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.
ppt on Time Domain and Frequency Domain Analysissagar_kamble
in this presentation, you will be able to know what is this freq. and time domain analysis.
At last one example is illustreted with video, which distinguishes these two analysis
Power System Modelling And Simulation LabSachin Airan
This document is a lab manual for a Power System Modeling and Simulation course. It provides instructions on how to simulate synchronous machines using MATLAB software. The first experiment introduces the swing equation, which models the dynamics of a synchronous generator's rotor motion. The second experiment describes how to model a synchronous machine in Simulink, including defining its electrical and mechanical parameters. The manual lists the synchronous machine model's equations and parameters that must be specified in the Simulink model block.
presentation on digital signal processingsandhya jois
The document discusses digital signal processing (DSP). It defines key terms like digital, signal, and processing. It explains how analog signals are converted to digital form by sampling and quantization. It also describes common digital modulation schemes and compares DSP processors to microprocessors. Finally, it discusses digital filters and their types as well as applications of DSP in areas like audio processing, communications, and imaging.
Frequency response techniques allow analysis of systems in the frequency domain. Key applications include modeling transfer functions from data, designing compensators, analyzing stability, and investigating transient and steady-state response. Frequency response is obtained by plotting the magnitude and phase of a system's transfer function evaluated at various frequencies. Bode plots provide asymptotic approximations of frequency response on logarithmic scales and are useful for analysis and design.
This presentation provides an overview of digital signal processing (DSP). It defines key terms like signal and signal processing and explains the basic principles and components of DSP systems. The presentation notes that DSP has advantages over analog processing like accuracy, flexibility, and ease of operation. It provides examples of DSP applications in areas like audio, communications, biomedicine, and more. In conclusion, the presentation emphasizes that DSP involves manipulating digital numbers using programmed instructions and is widely used in modern applications.
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.
Reduction of multiple subsystem [compatibility mode]azroyyazid
This document discusses techniques for reducing multiple subsystems to a single transfer function. It covers block diagram algebra and Manson's rule. Block diagram algebra can be used to reduce block diagrams representing cascaded, parallel, and feedback subsystems into equivalent single transfer functions. The key techniques are collapsing summing junctions and forming equivalent cascaded, parallel, and feedback systems. Signal-flow graphs also represent subsystems and can be reduced using Manson's rule by writing equations for each signal as the sum of incoming signals times their transfer functions. Examples demonstrate reducing various block diagrams and signal-flow graphs to equivalent single transfer functions.
The document discusses control system design using root locus and PID tuning. It introduces root locus analysis and how adjusting the system gain can affect transient and steady-state response. PID controllers are commonly used compensators that can improve performance. The effects of proportional, integral and derivative controllers on closed-loop systems are described. An example mass-spring-damper system is analyzed with P, PI, PD and PID controllers to meet different performance specifications. Design procedures and effects of adding poles and zeros to the open-loop transfer function are also covered. Lead and lag compensators are discussed as ways to reshape the root locus to achieve desired closed-loop poles.
Linear Control Hard-Disk Read/Write Controller AssignmentIsham Rashik
Classic Hard-Disk Read/Write Head Controller Assignment completed using MATLAB and SIMULINK. To see the diagrams in detail, please download first and zoom it.
DC Motor Modelling & Design Fullstate Feedback Controller Idabagus Mahartana
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1. Chapter 5: Root Locus Analysis
Outline
Motivational Example
desired pole region
Simple controller design using desired pole
region
Construction of Root Loci
Magnitude condition
• Stability range from Root Loci
phase condition
• properties of Root Loci
Effects of addition of pole(s) and zero(s)
Classical Dynamic Compensation
1
2. Motivational Example
Consider a plant with TF
(5.1)
• The problem is to design an overall system to meet
the following specifications:
position error = 0
overshoot ≤ 5%
Settling time ≤ 8 seconds
Rise time as small as possible
• Before carrying out the design, we must first choose
a configuration and a compensator ( controller) with
one open parameter. The simplest possible FB
configuration & compensator are shown in Figure 5.1
2
1
2
3
4
3. Motivational Example
5.1 Unity-feedback control system
• The overall TF ( for system in Fig 5.1) is
(5.2)
Note:
• is stable if and only if k>0
• Because , the system has zero position
error 3
+ K
-
Ctrl Plant
u yr e
4. Motivational Example
Thus, the design reduces to the search for a positive
k to meet requirements (2) through (4)
Remark:
Design specifications are given in the time domain ,
whereas the design is carried out using TF’s , or in
the s-plane. Hence, a mapping relating transient
performance specifications and desired pole region
must be done!
Consider a control system with a TF
(5.3)
The overshoot is
4
5. Desired pole region
• From Fig 5.2(b), the range of for a given
overshoot can be obtained.
5
5. 2 Damping ratio &
Overshoot
(a) (b)
6. Desired pole region
• ( as defined in Fig 5.2(b))
• is a decreasing function of in
i.e., if , then
Example on translating overshoot specif. into a pole
region in the s-plane
• Overshoot ≤ 10%
• Overshoot ≤ 5%
6
Figure 5.3 Overshoot & pole
region
7. Desired pole region
Translating settling time into a pole region in the s-
plane
7
Figure 5.4 Desired pole
region
8. Desired pole region
Remarks:
• The translation of the rise time into a pole region can
not be done quantitatively as in the case of overshoot
and settling time.
• Generally, the farther away the closest pole from the
origin of the s-plane, the smaller the rise time.
8
Summary
Overshoot Sector with & is determined from Fig 5.2b
Settling
Time
Rise Time
9. Simple controller design using desired pole region
Consider the design problem in Figure 5.1
9
Figure 5.5 Poles of
Gcl(s)
-2
2
1
-1
-2-3 -1 1
Re(s
)
Im(s)
K=0
K=.36K=.75
K=5
K=1
K=2
10. Simple controller design using desired pole region
Consider the design problem in Figure 5.1…..
Q. How to choose k from 0.75, 1 and 2, so that the
rise time will be the smallest?
• the poles corresponding to k=0.75 are -0.5 & -1.5;
therefore, the distance of the closer pole from the
origin is 0.5
10
Gain Poles Comments
K=0.36 -0.2, -1.8 Meet (2) but not (3)
K=0.75 -0.5, -1.5 Meet both (2) & (3)*
K=1 -1,-1 Meet both (2) & (3)*
K=2 -1 j1 Meet both (2) & (3)*
K=5 -1 2j Meet (3) but not (2)
11. Simple controller design using desired pole region
• the poles corresponding to k=1 are -1 and -1. Their
distance from the origin is 1 and is larger than 0.5.
• the poles corresponding to k=2 are -1± j1. their distance
from the origin is , which is the largest among
k=0.75, 1 and 2.
• Therefore, the system with k=2 has the smallest rise time
or, equivalently, responds faster.
Remark
If some of the specifications are more stringent, then no k
may exist. For example , if Ts< 2 sec, then all poles of
Gcl(s) must lie on the left-hand side of the vertical line
passing through -4/2=-2. From Figure 5.5 no poles meet
the requirement. In this case, we must choose a different
configuration and/or a more complicated compensator and
redesign. 11
Verify via Matlab Simulation!
12. Simple controller design using desired pole region
A more systematic design method consists two major
components ;
(1) Translation of the transient performance into a
desired pole region. We then try to place the poles
of the overall system inside the region by
choosing a parameter.
(2) In order to facilitate the choice of the parameter,
the poles of the overall system as a function of the
parameter will be plotted graphically. Such plot is
called ROOT-LOCUS.
Reading assignment
Effects of introducing a zero/pole on the quadratic TF
with constant numerator(the details).
12
13. Construction of Root-Locus
Consider the unity-feedback control system shown in
Fig 5.6, where G(s) is a proper rational function* and
k is a real constant.
Let
Then the overall TF is
13
Figure 5.6 Unity feedback
system
+ K G(s)
-
Ctrl Plant
u yr e
14. Construction of Root-Locus
The poles of Gcl(s) are the zeros of the rational
function
(5.4) or the solution of the
eqn.
(5.5)
Definition 5.1
To simplify discussion, let assume
(5.6) ; q- real constant
Re-writing (5.5) as
14
The roots of D(s)+kN(s) or the poles of Gcl(s) as a
function of a real k are called the root loic.
15. Construction of Root-Locus
Then the roots of D(s)+kN(s) are those s,
real/complex which satisfy (5.7) for some real k.
From Figure 5.7, we have
15
Figure 5.7 Vector in s-
plane
16. Construction of Root-Locus
Substituting the above relations into (5.7) gives
(5.8)
Eqn.(5.8) consists of two parts; the magnitude
condition
(5.9)
And the phase condition
(5.10) 16
17. Construction of Root-Locus
Note
• equals 0 if , if
• can be positive ( if measured
counterclockwise)
Phase Condition
Because k is real , we have
Note two angles will be considered the same if they
differ by ±2π radians ( or 3600) or their multiples.
Hence, (5.10) becomes
(5.11) Total phase:=
= 17
18. Construction of Root-Locus
Conclusion
If s1 satisfy (5.11), then there exists a real k1 such that
D(s1)+k1N(s1)=0. This k1 can be computed from (5.9)
BIG QUESTION: How to search for s1?
ANSWER: Using the properties to be discussed
shortly, we can obtain a rough sketch of root loci
without any measurement.
Properties of Root Loic - Phase condition
Consider a TF with real coefficients expressed as
(5.12)
with
18
19. Properties of Root Loci - Phase Condition
To simplify discussion, assume and in
(5.13)
Proof:
The polynomial in (5.13) has degree n. Thus for each
real k, there are n-roots. Because the roots of a
polynomial are continuous function of its coefficients,
the n-roots form n continuous trajectories as k varies
from 0 to .
19
PROPERTY -1
The root loci consists of n-continuous trajectories as k
varies continuously from 0 to . The trajectories are
symmetric w.r.t the real axis.
20. Properties of Root Loci - Phase Condition
Example 5.1
For the systems described by the TFs given below,
find the root loci on the real axis for using
property 1 & 2. And verify the correctness using
Phase Condition.
20
PROPERTY -2
Every section of the real axis with an odd number of
real poles and zeros ( counting together) on its right
side is a part of the root loci for
21. Properties of Root Loci - Phase Condition
Example 5.1…
21
Figure 5.8 Root Loci on real-
axis
S2=
0
2.
5
s1
-4 -2
1
Re(s
)
Im(s)
Asymptot
e
(a
)
600
-
600
-2-3 21
3.
5
Asymptot
e
Asymptot
e
Re(s
)
Im(s)
(b)
22. Properties of Root Loci - Phase Condition
Example 5.1…
Remark:
The net phase due to the pair to any point on the real
axis equals 0 or 2π as shown in Figure 5.8(b).
Therefore, in applying property 2, complex-conjugate
poles and zeros can be disregarded.
Proof:
The roots of are the same as the
roots of .
22
PROPERTY -3
The n-trajectories migrate from the poles of G(s) to the
zeros of G(s) as k increases from 0 to .
23. Properties of Root Loci - Phase Condition
Remark:
If n( # of poles) > m ( # of zeros) , then m trajectories
will enter the m zeros. The remaining (n-m)
trajectories will approach (n-m) asymptotes, as
discussed in property 4.
23
PROPERTY -4
For large s, the root loci will approach (n-m) number of
straight lines called asymptotes, emitting from
(5.14a)
called the centroid. These (n-m) asymptotes have
angles
(5.14b)
These formulas will give only (n-m) distinct angle.
24. Properties of Root Loci - Phase Condition
Property 4 continued…
24
n-m Angles of
asymptotes
1 1800
2 ±900
3 ±600, 1800
4 ±450, ±1350
5 ±360, ±1080,1800
PROPERTY -5
Breakaway points- solution of
A breakaway point is where two roots collide and break
away; therefore, there are at least two roots at every
breakaway point.
25. Properties of Root Loci - Phase Condition
Property 5 continued…
Proof:
Let s0 be a breakaway point of D(s)+kN(s). Then it is
a repeated root of D(s)+kN(s). Consequently, we
have
(5.15a) and
(5.15b)
Solving for k from (5.15b) and substituting into (5.15a)
gives
(5.16) 25
26. Properties of Root Loci - Phase Condition
Remark:
In general, not every solution of (5.16) is necessarily
a breakaway point for k ≥ 0. Although breakaway
points occur mostly on the real axis, they may appear
elsewhere.
Bottom line: using the properties discussed so far, we can
often obtain a rough sketch of root loci with min.26
PROPERTY -6*
Angle of departure or arrival
Every trajectory will depart from a pole. If the pole is
real and distinct, the direction of the departure is
usually 00 or 1800.
If the pole is complex , then the direction of the
departure may assume any degree between 00 and
3600.
27. Effects of addition of pole(s) and zero(s)
Effects of 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 relative stability.
27
Figure 5.9 (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
(a
)
(b
)
(c)
28. Effects of addition of pole(s) and zero(s)
Effects of 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.
28
Figure 5.10 (a)Root Locus plot of a three-pole system; (b),(c) , and (d)
Root
(a
)
(b
)
(c) (d
)
29. Dynamic Compensation
• So far we studied how to draw a Root-Locus for the
given plant dynamics.
Q. What if our desired pole locations are not on this
locus?
A. we need to modify the locus itself by adding Extra
Dynamics in Nc, Dc
Dynamic compensation
29
Figure 5.11 Basic Feedback
System
+
-
r e u y
30. Dynamic Compensation
BIG QUESTIONS
• What type of compensator should we use?
• How do we figure out where to put the additional
dynamics?
TYPES OF CONTROL DYNAMICS
There are 3 classical types of controllers:
Controller:
(i). Proportional Feedback: ( A constant)
i.e., Nc=Dc=1
• Here, we take the locus “as given” since we have no
extra dynamics to modify it.
• Usually very limited approach, but a good place to start.
30
31. Dynamic Compensation
(ii). Integral Feedback:
• Used to Reduce/Eliminate Steady-State Errors.
If , will become very large and
thus hopefully correct the error.
Example 5.1:
• With Proportional Feedback, (for step-input)
can make ess small but need
large k
• With integral control, ess=0 since
31
32. Dynamic Compensation
(ii). Integral Feedback…
• Integral Feedback improves the steady-state
response, but this is often at the expense of the
transient response(This get worse not as well
damped)
• Referring Example 5.1, we can observe increasing
KI to increase the speed of the response pushes the
poles towards the imaginary axis OSCILLATORY
32Figure 5.12 RL after adding integral
Im(s
)
Re(
s)
33. Dynamic Compensation
Proportional-Integral
Combine proportional and integral (PI) feedback:
Which introduce a pole at the origin and zero at s= -
k2/k1
• PI solves many of the problems of with just integral
control
33
Figure 5.13 RL with proportional & Integral
• #
ASYMPTOTES?
• CENTROID
Im(s)
Re(s
)
34. Dynamic Compensation
(iii). Derivative Feedback: so that
• Doesn’t help with the steady-state
• Provides feedback on the rate of change of e(t) so
that the control can anticipate future errors.
Example 5.2 consider
With
34
Figure 5.15 RL with Derivative
35. Dynamic Compensation
(iii). Derivative Feedback…
• Derivative feedback is very useful for pulling the
root locus into the LHP – increase damping and
more stable response
• Typical used in combination with Proportional
feedback to from Proportional-derivative feedback
PD
which moves the zero from the origin.
• Unfortunately, pure PD is not realizable in the lab as
pure differentiation of a measured signal is typically
a bad idea.
use band-limited differentiation instead, by
rolling-off the PD control with a high-frequency pole.35