Some experiments done on MATLAB for the course on Simulation and Modelling. Includes Model of Bouncing Ball, Model of Spring Mass System and Model of Traffic Flow.
The presentation summarizes algorithms topics including dynamic programming, greedy algorithms, and sorting. It covers dynamic programming approaches to matrix chain multiplication and polygon triangulation. It also discusses the recursive and memoized solutions to matrix chain multiplication, and compares their time complexities. Kruskal's minimum spanning tree algorithm is explained along with observations on its runtime with increasing edges or vertices. Quicksort is analyzed using least squares fitting to determine constants in its average time complexity formula.
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.
UNIT II DISCRETE TIME SYSTEM ANALYSIS 6+6
Z-transform and its properties, inverse z-transforms; difference equation – Solution by z transform,application to discrete systems - Stability analysis, frequency response –Convolution – Discrete Time Fourier transform , magnitude and phase representation.
D I G I T A L C O N T R O L S Y S T E M S J N T U M O D E L P A P E R{Wwwguest3f9c6b
This document contains questions from a digital control systems exam. It covers topics like digital to analog conversion, z-transforms, stability analysis of sampled data systems, time domain analysis of discrete time systems using block diagrams, root locus analysis in the z-domain, controller design techniques like lead compensation, state space modeling, and stability analysis using Lyapunov's theorem. The exam has 8 questions with multiple parts that can be answered in the allotted time of 3 hours.
UNIT II DISCRETE TIME SYSTEM ANALYSIS 6+6
Z-transform and its properties, inverse z-transforms; difference equation – Solution by ztransform,
application to discrete systems - Stability analysis, frequency response –Convolution – Discrete Time Fourier transform , magnitude and phase representation
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.
The presentation summarizes algorithms topics including dynamic programming, greedy algorithms, and sorting. It covers dynamic programming approaches to matrix chain multiplication and polygon triangulation. It also discusses the recursive and memoized solutions to matrix chain multiplication, and compares their time complexities. Kruskal's minimum spanning tree algorithm is explained along with observations on its runtime with increasing edges or vertices. Quicksort is analyzed using least squares fitting to determine constants in its average time complexity formula.
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.
UNIT II DISCRETE TIME SYSTEM ANALYSIS 6+6
Z-transform and its properties, inverse z-transforms; difference equation – Solution by z transform,application to discrete systems - Stability analysis, frequency response –Convolution – Discrete Time Fourier transform , magnitude and phase representation.
D I G I T A L C O N T R O L S Y S T E M S J N T U M O D E L P A P E R{Wwwguest3f9c6b
This document contains questions from a digital control systems exam. It covers topics like digital to analog conversion, z-transforms, stability analysis of sampled data systems, time domain analysis of discrete time systems using block diagrams, root locus analysis in the z-domain, controller design techniques like lead compensation, state space modeling, and stability analysis using Lyapunov's theorem. The exam has 8 questions with multiple parts that can be answered in the allotted time of 3 hours.
UNIT II DISCRETE TIME SYSTEM ANALYSIS 6+6
Z-transform and its properties, inverse z-transforms; difference equation – Solution by ztransform,
application to discrete systems - Stability analysis, frequency response –Convolution – Discrete Time Fourier transform , magnitude and phase representation
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 document discusses the z-transform, which is a mathematical tool used to analyze discrete-time control systems. It defines the one-sided and two-sided z-transform and provides examples of taking the z-transform of basic functions like unit step, ramp, polynomial and exponential functions. The document also covers important properties of the z-transform including linearity, shifting theorems, and the initial and final value theorems. It describes methods for finding the inverse z-transform including using tables, direct division, partial fraction expansion and inversion integrals.
Mathematics of nyquist plot [autosaved] [autosaved]Asafak Husain
This document provides a summary of a lecture on the mathematics of Nyquist plots. Some key topics covered include:
- Complex calculus concepts like Cauchy's theorem and the principle of argument.
- Derivation of the Cauchy-Riemann equations.
- Analytic functions and Cauchy's integral formula.
- Residue theorem and its application to determining stability using encirclements in the Nyquist plot.
- Construction of the Nyquist path and an example application to determine stability of a closed-loop system.
UNIT II DISCRETE TIME SYSTEM ANALYSIS 6+6
Z-transform and its properties, inverse z-transforms; difference equation – Solution by ztransform,
application to discrete systems - Stability analysis, frequency response –Convolution – Discrete Time Fourier transform , magnitude and phase representation
Z-transforms can be used to evaluate discrete functions, similar to how Laplace transforms are used for continuous functions. The z-transform of a discrete function f(n) is defined as the sum of f(n) multiplied by z to the power of -n, from n=0 to infinity. Some standard z-transform results include formulas for exponential, sinusoidal, and polynomial functions. Z-transforms have properties of linearity and shifting, and can be used to solve differential equations with constant coefficients and in applications of signal processing.
This document discusses asymptotic notation which is used to describe the running time of algorithms. It introduces the Big O, Big Omega, and Theta notations. Big O notation represents the upper bound or worst case running time. Big Omega notation represents the lower bound or best case running time. Theta notation represents the running time between the upper and lower bounds. The document provides mathematical definitions and examples of how to determine if a function is Big O, Big Omega, or Theta notation.
This document provides an overview of Fourier series and Fourier transforms. It begins with examples of applying Fourier series to odd and even functions. It then defines the Fourier transform and provides examples of its application. Key properties of the Fourier transform are outlined, including linearity, symmetry, time shifting, and how differentiation in the time domain relates to multiplication in the frequency domain. Important applications of the Fourier transform mentioned include convolution, deconvolution, sampling of seismic time series, and filtering. The document is a lecture on seismic data processing that introduces Fourier analysis techniques.
This document summarizes a seminar report on discrete time systems and the Z-transform. It defines discrete time systems and different types of systems including causal/noncausal, linear/nonlinear, time-invariant/variant, static/dynamic. It then explains the Z-transform, its properties including region of convergence and time shifting. Some common Z-transform pairs are provided along with methods for the inverse Z-transform. Advantages of the Z-transform for analysis of discrete systems and signals are mentioned.
asymptotic analysis and insertion sort analysisAnindita Kundu
This document discusses asymptotic analysis of algorithms. It introduces key concepts like algorithms, data structures, best/average/worst case running times, and asymptotic notations like Big-O, Big-Omega, and Big-Theta. These notations are used to describe the long-term growth rates of functions and provide upper/lower/tight bounds on the running time of algorithms as the input size increases. Examples show how to analyze the asymptotic running time of algorithms like insertion sort, which is O(n^2) in the worst case but O(n) in the best case.
The document discusses various methods for finding the inverse z-transform, including inspection of z-transform pairs, partial fraction expansion, and power series expansion. It provides examples of using each method to find the inverse z-transform of given z-functions. It also discusses properties of the z-transform, such as time shifting and convolution, that can help in solving inverse problems. Sample problems demonstrate applying the techniques to compute inverse z-transforms and use properties to solve for sequences.
This document summarizes a numerical trajectory optimization method for path planning of multiple autonomous ground vehicles to avoid obstacles. It discretizes the continuous optimization problem into a nonlinear programming problem by approximating states with piecewise cubic polynomials and control with piecewise linear functions. The cost function accounts for minimizing time, distance to goal, and control effort. Constraints ensure obstacle and inter-vehicle distance avoidance. Simulation results show the effect of increasing node points in improving trajectory tracking accuracy.
Spline interpolation is a technique for generating new data points within the range of a discrete set of known data points. It uses piecewise polynomials, typically cubic polynomials, to fit curves to these data points. The document discusses linear and quadratic spline interpolation and provides an example of using quadratic splines to interpolate the velocity of a rocket at different times and calculate the velocity, distance, and acceleration at t=16 seconds.
This document discusses signals and systems. It defines signals as physical quantities that vary with respect to time, space, or another independent variable. Signals can be classified as discrete time or continuous time. It also defines unit impulse and unit step functions for discrete and continuous time. Periodic and aperiodic signals are discussed. The Fourier series and Fourier transform are introduced as ways to represent signals in the frequency domain. The Laplace transform, which generalizes the Fourier transform, is also mentioned. Key properties of linear time-invariant systems like superposition, time-invariance, and convolution are covered. Finally, sampling theory and the z-transform, which is analogous to the Laplace transform for discrete-time systems, are summarized at a high level
Z TRANSFORM PROPERTIES AND INVERSE Z TRANSFORMTowfeeq Umar
The document discusses various methods for computing the inverse z-transform including inspection, partial fraction expansion, and power series expansion. It provides examples to illustrate each method. The inverse z-transform finds the original time domain sequence from its z-transform. Key properties like linearity, time shifting, and convolution are also covered.
This presentation on Pseudo Random Number Generator enlists the different generators, their mechanisms and the various applications of random numbers and pseudo random numbers in different arenas.
The document discusses the Z-transform, which is a tool for analyzing and solving linear time-invariant difference equations. It defines the Z-transform, provides examples of common sequences and their corresponding Z-transforms, and discusses properties such as the region of convergence. Key topics covered include the difference between difference and differential equations, properties of linear time-invariant systems, and mapping between the s-plane and z-plane.
DSP_FOEHU - MATLAB 01 - Discrete Time Signals and SystemsAmr E. Mohamed
This document provides an overview of discrete-time signals and systems in MATLAB. It defines discrete signals as sequences represented by x(n) and how they can be implemented as vectors in MATLAB. It describes various types of sequences like unit sample, unit step, exponential, sinusoidal, and random. It also covers operations on sequences like addition, multiplication, scaling, shifting, folding, and correlations. Discrete time systems are defined as operators that transform an input sequence x(n) to an output sequence y(n). Key properties discussed are linearity, time-invariance, stability, causality, and the use of convolution to represent the output of a linear time-invariant system. Examples are provided to demonstrate various concepts.
The document discusses the Nyquist stability criterion for analyzing the stability of sampled-data control systems. It begins by defining the Nyquist criterion and contour, and how they can be used to determine the number of closed-loop poles outside the unit circle (Z). It then provides an example showing how to apply the Nyquist criterion by plotting the loop gain and counting encirclements of the critical point. The document also discusses modifications to the Nyquist contour when open-loop poles are on the unit circle and defines the Nyquist criterion theorem.
Generally it has been noticed that differential equation is solved typically. The Laplace transformation makes it easy to solve. The Laplace transformation is applied in different areas of science, engineering and technology. The Laplace transformation is applicable in so many fields. Laplace transformation is used in solving the time domain function by converting it into frequency domain. Laplace transformation makes it easier to solve the problems in engineering applications and makes differential equations simple to solve. In this paper we will discuss how to follow convolution theorem holds the Commutative property, Associative Property and Distributive Property. Dr. Dinesh Verma"Application of Convolution Theorem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-4 , June 2018, URL: http://www.ijtsrd.com/papers/ijtsrd14172.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/14172/application-of-convolution-theorem/dr-dinesh-verma
Controller design of inverted pendulum using pole placement and lqreSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Controller design of inverted pendulum using pole placement and lqreSAT Journals
Abstract In this paper modeling of an inverted pendulum is done using Euler – Lagrange energy equation for stabilization of the pendulum. The controller gain is evaluated through state feedback and Linear Quadratic optimal regulator controller techniques and also the results for both the controller are compared. The SFB controller is designed by Pole-Placement technique. An advantage of Quadratic Control method over the pole-placement techniques is that the former provides a systematic way of computing the state feedback control gain matrix.LQR controller is designed by the selection on choosing. The proposed system extends classical inverted pendulum by incorporating two moving masses. The motion of two masses that slide along the horizontal plane is controllable .The results of computer simulation for the system with Linear Quardatic Regulator (LQR) & State Feedback Controllers are shown in section 6. Keyword-Inverted pendulum, Mathematical modeling Linear-quadratic regulator, Response, State Feedback controller, gain formulae.
This document discusses the z-transform, which is a mathematical tool used to analyze discrete-time control systems. It defines the one-sided and two-sided z-transform and provides examples of taking the z-transform of basic functions like unit step, ramp, polynomial and exponential functions. The document also covers important properties of the z-transform including linearity, shifting theorems, and the initial and final value theorems. It describes methods for finding the inverse z-transform including using tables, direct division, partial fraction expansion and inversion integrals.
Mathematics of nyquist plot [autosaved] [autosaved]Asafak Husain
This document provides a summary of a lecture on the mathematics of Nyquist plots. Some key topics covered include:
- Complex calculus concepts like Cauchy's theorem and the principle of argument.
- Derivation of the Cauchy-Riemann equations.
- Analytic functions and Cauchy's integral formula.
- Residue theorem and its application to determining stability using encirclements in the Nyquist plot.
- Construction of the Nyquist path and an example application to determine stability of a closed-loop system.
UNIT II DISCRETE TIME SYSTEM ANALYSIS 6+6
Z-transform and its properties, inverse z-transforms; difference equation – Solution by ztransform,
application to discrete systems - Stability analysis, frequency response –Convolution – Discrete Time Fourier transform , magnitude and phase representation
Z-transforms can be used to evaluate discrete functions, similar to how Laplace transforms are used for continuous functions. The z-transform of a discrete function f(n) is defined as the sum of f(n) multiplied by z to the power of -n, from n=0 to infinity. Some standard z-transform results include formulas for exponential, sinusoidal, and polynomial functions. Z-transforms have properties of linearity and shifting, and can be used to solve differential equations with constant coefficients and in applications of signal processing.
This document discusses asymptotic notation which is used to describe the running time of algorithms. It introduces the Big O, Big Omega, and Theta notations. Big O notation represents the upper bound or worst case running time. Big Omega notation represents the lower bound or best case running time. Theta notation represents the running time between the upper and lower bounds. The document provides mathematical definitions and examples of how to determine if a function is Big O, Big Omega, or Theta notation.
This document provides an overview of Fourier series and Fourier transforms. It begins with examples of applying Fourier series to odd and even functions. It then defines the Fourier transform and provides examples of its application. Key properties of the Fourier transform are outlined, including linearity, symmetry, time shifting, and how differentiation in the time domain relates to multiplication in the frequency domain. Important applications of the Fourier transform mentioned include convolution, deconvolution, sampling of seismic time series, and filtering. The document is a lecture on seismic data processing that introduces Fourier analysis techniques.
This document summarizes a seminar report on discrete time systems and the Z-transform. It defines discrete time systems and different types of systems including causal/noncausal, linear/nonlinear, time-invariant/variant, static/dynamic. It then explains the Z-transform, its properties including region of convergence and time shifting. Some common Z-transform pairs are provided along with methods for the inverse Z-transform. Advantages of the Z-transform for analysis of discrete systems and signals are mentioned.
asymptotic analysis and insertion sort analysisAnindita Kundu
This document discusses asymptotic analysis of algorithms. It introduces key concepts like algorithms, data structures, best/average/worst case running times, and asymptotic notations like Big-O, Big-Omega, and Big-Theta. These notations are used to describe the long-term growth rates of functions and provide upper/lower/tight bounds on the running time of algorithms as the input size increases. Examples show how to analyze the asymptotic running time of algorithms like insertion sort, which is O(n^2) in the worst case but O(n) in the best case.
The document discusses various methods for finding the inverse z-transform, including inspection of z-transform pairs, partial fraction expansion, and power series expansion. It provides examples of using each method to find the inverse z-transform of given z-functions. It also discusses properties of the z-transform, such as time shifting and convolution, that can help in solving inverse problems. Sample problems demonstrate applying the techniques to compute inverse z-transforms and use properties to solve for sequences.
This document summarizes a numerical trajectory optimization method for path planning of multiple autonomous ground vehicles to avoid obstacles. It discretizes the continuous optimization problem into a nonlinear programming problem by approximating states with piecewise cubic polynomials and control with piecewise linear functions. The cost function accounts for minimizing time, distance to goal, and control effort. Constraints ensure obstacle and inter-vehicle distance avoidance. Simulation results show the effect of increasing node points in improving trajectory tracking accuracy.
Spline interpolation is a technique for generating new data points within the range of a discrete set of known data points. It uses piecewise polynomials, typically cubic polynomials, to fit curves to these data points. The document discusses linear and quadratic spline interpolation and provides an example of using quadratic splines to interpolate the velocity of a rocket at different times and calculate the velocity, distance, and acceleration at t=16 seconds.
This document discusses signals and systems. It defines signals as physical quantities that vary with respect to time, space, or another independent variable. Signals can be classified as discrete time or continuous time. It also defines unit impulse and unit step functions for discrete and continuous time. Periodic and aperiodic signals are discussed. The Fourier series and Fourier transform are introduced as ways to represent signals in the frequency domain. The Laplace transform, which generalizes the Fourier transform, is also mentioned. Key properties of linear time-invariant systems like superposition, time-invariance, and convolution are covered. Finally, sampling theory and the z-transform, which is analogous to the Laplace transform for discrete-time systems, are summarized at a high level
Z TRANSFORM PROPERTIES AND INVERSE Z TRANSFORMTowfeeq Umar
The document discusses various methods for computing the inverse z-transform including inspection, partial fraction expansion, and power series expansion. It provides examples to illustrate each method. The inverse z-transform finds the original time domain sequence from its z-transform. Key properties like linearity, time shifting, and convolution are also covered.
This presentation on Pseudo Random Number Generator enlists the different generators, their mechanisms and the various applications of random numbers and pseudo random numbers in different arenas.
The document discusses the Z-transform, which is a tool for analyzing and solving linear time-invariant difference equations. It defines the Z-transform, provides examples of common sequences and their corresponding Z-transforms, and discusses properties such as the region of convergence. Key topics covered include the difference between difference and differential equations, properties of linear time-invariant systems, and mapping between the s-plane and z-plane.
DSP_FOEHU - MATLAB 01 - Discrete Time Signals and SystemsAmr E. Mohamed
This document provides an overview of discrete-time signals and systems in MATLAB. It defines discrete signals as sequences represented by x(n) and how they can be implemented as vectors in MATLAB. It describes various types of sequences like unit sample, unit step, exponential, sinusoidal, and random. It also covers operations on sequences like addition, multiplication, scaling, shifting, folding, and correlations. Discrete time systems are defined as operators that transform an input sequence x(n) to an output sequence y(n). Key properties discussed are linearity, time-invariance, stability, causality, and the use of convolution to represent the output of a linear time-invariant system. Examples are provided to demonstrate various concepts.
The document discusses the Nyquist stability criterion for analyzing the stability of sampled-data control systems. It begins by defining the Nyquist criterion and contour, and how they can be used to determine the number of closed-loop poles outside the unit circle (Z). It then provides an example showing how to apply the Nyquist criterion by plotting the loop gain and counting encirclements of the critical point. The document also discusses modifications to the Nyquist contour when open-loop poles are on the unit circle and defines the Nyquist criterion theorem.
Generally it has been noticed that differential equation is solved typically. The Laplace transformation makes it easy to solve. The Laplace transformation is applied in different areas of science, engineering and technology. The Laplace transformation is applicable in so many fields. Laplace transformation is used in solving the time domain function by converting it into frequency domain. Laplace transformation makes it easier to solve the problems in engineering applications and makes differential equations simple to solve. In this paper we will discuss how to follow convolution theorem holds the Commutative property, Associative Property and Distributive Property. Dr. Dinesh Verma"Application of Convolution Theorem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-4 , June 2018, URL: http://www.ijtsrd.com/papers/ijtsrd14172.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/14172/application-of-convolution-theorem/dr-dinesh-verma
Controller design of inverted pendulum using pole placement and lqreSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Controller design of inverted pendulum using pole placement and lqreSAT Journals
Abstract In this paper modeling of an inverted pendulum is done using Euler – Lagrange energy equation for stabilization of the pendulum. The controller gain is evaluated through state feedback and Linear Quadratic optimal regulator controller techniques and also the results for both the controller are compared. The SFB controller is designed by Pole-Placement technique. An advantage of Quadratic Control method over the pole-placement techniques is that the former provides a systematic way of computing the state feedback control gain matrix.LQR controller is designed by the selection on choosing. The proposed system extends classical inverted pendulum by incorporating two moving masses. The motion of two masses that slide along the horizontal plane is controllable .The results of computer simulation for the system with Linear Quardatic Regulator (LQR) & State Feedback Controllers are shown in section 6. Keyword-Inverted pendulum, Mathematical modeling Linear-quadratic regulator, Response, State Feedback controller, gain formulae.
10 Discrete Time Controller Design.pptxSaadAzhar15
This document discusses digital control system design. It begins with an overview of discretization methods and the effect of zero-order hold. Examples are provided to illustrate discretization and digital controller design. Design of PI and PID digital controllers via pole placement is covered. An example designs a cruise control system for a car using a digital PI controller. The controller is designed by deriving specifications from the design problem, discretizing the plant, determining controller parameters, and simulating the closed-loop response. The controller meets specifications when applied to both the discretized and actual continuous-time plant.
Troubleshooting and Enhancement of Inverted Pendulum System Controlled by DSP...Thomas Templin
An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is often implemented with the pivot point mounted on a cart that can move horizontally and may be called a cart-and-pole system. A normal pendulum is always stable since the pendulum hangs downward, whereas the inverted pendulum is inherently unstable and trivially underactuated (because the number of actuators is less than the degrees of freedom). For these reasons, the inverted pendulum has become one of the most important classical problems of control engineering. Since the 1950s, the inverted-pendulum benchmark, especially the cart version, has been used for the teaching and understanding of the use of linear-feedback control theory to stabilize an open-loop unstable system.
The objectives of this project are to:
• Focus on hardware and software troubleshooting and enhancement of an inverted-pendulum system controlled by a DSP28355 microprocessor and CCSv7.1 software.
• Use the swing-up strategy to move the pendulum into the unstable upward position (‘saddle’). The cart/pole system employs linear bearings for back-and-forward motion. The motor shaft has a pinion gear that rides on a track permitting the cart to move in a linear fashion. Both rack and pinion are made of hardened steel and mesh with a tight tolerance. The rack-and-pinion mechanism eliminates undesirable effects found in belt-driven and free-wheel systems, such as slippage or belt stretching, ensuring consistent and continuous traction.
• The motor shaft is coupled to a high-resolution optical encoder that accurately measures the position of the cart. The angle of the pendulum is also measured by an optical encoder, and the system employs an LQR controller to stabilize the pendulum rod at the unstable-equilibrium position.
• Addition of real-time status reporting and visualization of the system.
For the project, the Quanser High Frequency Linear Cart (HFLC) was used. The HFLC system consists of a precisely machined solid aluminum cart driven by a high-power 3-phase brushless DC motor. The cart slides along two high-precision, ground-hardened stainless steel guide rails, allowing for multiple turns and continuous measurement over the entire range of motion.
Our team implemented a control strategy that consists of a linear stabilizing LQR controller, proportional-integral swing-up control, and a supervisory coordinator that determines the control strategy (LQR or swing-up) to be used at any given time. The function of the linear stabilizer is to stabilize the system when it is in the vicinity of the unstable equilibrium. When the pendulum is in its natural state (straight-down stable-equilibrium node), the swing-up controller provides the cart/pendulum system with adequate energy to move the pendulum to the unstable equilibrium inside the “region of attraction” in which the linearized LQR controller is functional.
- The document details a state space solver approach for analog mixed-signal simulations using SystemC. It models analog circuits as sets of linear differential equations and solves them using the Runge-Kutta method of numerical integration.
- Two examples are provided: a digital voltage regulator simulation and a digital phase locked loop simulation. Both analog circuits are modeled in state space and simulated alongside a digital design to verify mixed-signal behavior.
- The state space approach allows modeling analog circuits without transistor-level details, improving simulation speed over traditional mixed-mode simulations while still capturing system-level behavior.
Transient three dimensional cfd modelling of ceilng fanLahiru Dilshan
Ceiling fans are used to get thermal comfort, especially in tropical countries. With the increment of the usage of air conditioners, the emission of CO2 is increased. But ceiling fans are a limited solution, that saves much energy compared to air conditioners. Ceiling fans generate a non-uniform velocity profile, so that, there is a non-uniform thermal environment. That non-uniform environment does not imply lower thermal comfort, that will give enough thermal comfort with low energy cost by air velocity. Hence, there will be difficulties of analysing with simple modelling techniques in that environment. So, to predict the performance of the ceiling fan required more accurate models.
The accurate model of a ceiling fan will generate complex geometry that makes difficulties for the simulation process and requires higher computational power. Because of that, there are several methods used to predict the performance of the ceiling fan using mathematical techniques but that will give an estimated value of properties in the surrounding.
This document provides an overview of basic mechanical systems, including translational and rotational systems. For translational systems, it describes common elements like springs, masses, and dampers. It provides examples of how to model simple systems using equations of motion. For rotational systems, it similarly outlines common elements like rotational springs and dampers, as well as moment of inertia. Examples are provided for modeling rotational systems. The document also provides an introduction to gears, including fundamental properties, gear ratios, and examples of gear trains.
This document provides an overview of mathematical modeling of mechanical systems, including:
- Translational systems with springs, masses, and dampers and examples of modeling simple systems.
- Rotational systems with rotational springs, dampers, and moments of inertia along with examples.
- Mechanical linkages including gears and gear trains. Properties of gears are discussed and gear ratios explained. Examples of modeling gear trains mathematically are provided.
The document covers the basic elements and concepts for modeling both translational and rotational mechanical systems, along with examples, and also introduces mechanical linkages focused on gears and gear trains.
This document summarizes a lab experiment where students modeled a Furuta pendulum system and designed a state-space controller to control the pendulum's motion. Students first created a mathematical model of the system and calculated theoretical controller gains. They then implemented the controller on the physical pendulum system and manually tuned the gains to improve performance. Manual tuning produced better tracking of the pendulum and swing arm positions compared to using the theoretical gains. This demonstrated the challenges of applying a mathematical model to a physical system that can vary over time.
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.
This document describes the modeling and simulation of an inverted pendulum system. It begins with deriving the nonlinear equations of motion for an inverted pendulum mounted on a moving cart. It then linearizes the model around the equilibrium point and simulates both the linear and nonlinear models. Various controller designs are tested, including state feedback, PID control, and using position of the cart and pendulum as feedback. The linear model is shown to approximate the nonlinear model well. Increased mass or length are found to decrease stability. PID control is optimized by tuning gains.
This document discusses a control systems module presented by Dr. Devaraj Somasundaram. It introduces various types of control systems and their differential equations. It describes open loop and closed loop control systems, and the effects of feedback on overall gain, sensitivity, stability, and noise. Mathematical models including differential equation, transfer function, and state space models are presented. Examples are provided to demonstrate modeling of mechanical systems including springs, masses, dampers and their combinations in both translational and rotational systems. Gear ratios and modeling of gear trains are also discussed.
Experimental verification of SMC with moving switching lines applied to hoisti...ISA Interchange
In this paper we propose sliding mode control strategies for the point-to-point motion control of a hoisting crane. The strategies employ time-varying switching lines (characterized by a constant angle of inclination) which move either with a constant deceleration or a constant velocity to the origin of the error state space. An appropriate design of these switching lines results in non-oscillatory convergence of the regulation error in the closed-loop system. Parameters of the lines are selected optimally in the sense of two criteria, i.e. integral absolute error (IAE) and integral of the time multiplied by the absolute error (ITAE). Furthermore, the velocity and acceleration constraints are explicitly taken into account in the optimization process. Theoretical considerations are verified by experimental tests conducted on a laboratory scale hoisting crane.
This document discusses basic differentiation rules and rates of change. It introduces the constant rule, power rule, constant multiple rule, sum and difference rules, derivatives of sine and cosine functions, and using derivatives to find rates of change such as velocity and acceleration. Examples are provided to demonstrate finding derivatives of various functions and using derivatives to calculate average velocity and velocity from a position function.
1) The document describes the equation of motion of an inverted pendulum attached to a moving cart. It provides background on control theory and pendulums.
2) A simulation was created using Simulink to model an inverted pendulum with and without a controller. In the open loop model, a disturbance force caused the pendulum to fall over.
3) In the closed loop model with a controller, two disturbance forces were applied but the controller kept the pendulum balanced by moving the cart, showing the equation of motion works.
Iaetsd design of a robust fuzzy logic controller for a single-link flexible m...Iaetsd Iaetsd
This document describes the design of a fuzzy logic controller for a single-link flexible manipulator. A fuzzy-PID controller is used to control an uncertain flexible robotic arm and its internal motor dynamics parameters. The controller is tested against conventional integral and PID controllers in simulations. The results show the proposed fuzzy PID controller has better robustness under variations in motor dynamics compared to the other controllers.
This document describes research into controlling the motion of a tall tower with a tuned mass damper system after it has been disturbed. The researcher will create an open-loop and closed-loop controller for a linearized model of the system. An observer will be paired with the closed-loop controller and applied to the original nonlinear system. The nonlinear dynamics of the system are modeled using differential equations. Numerical analysis and simulation show that the controller and observer are effective at damping vibrations in the linearized and nonlinear systems, even with imperfect parameter estimation.
This document describes the design and implementation of a controller for an inverted pendulum on a cart system. It provides the nonlinear and linearized models of the system and designs a PID controller using root locus analysis. Simulation results show the uncompensated system is unstable but the controlled system with PID controller and pre-compensator meets design specifications with less than 0.2 seconds settling time and 8% overshoot for a unit step input.
A Detailed Comparative Study between Reduced Order Cumming Observer & Reduced...IJERD Editor
In this article a detailed comparative study between two well known observer design methodologies
namely, reduced order Cumming observer & reduced order Das &Ghoshal observer has been presented. The
necessary equations & conditions corresponding to these two types of observers are discussed in brief.
Thereafter with the help of a structure wise comparison the similarities & dissimilarities between the above
mentioned methods are explained in details. Finally a performance wise comparison between these two is shown
using proper numerical example & illustrations in open loop as well as closed loop.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
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The chapter Lifelines of National Economy in Class 10 Geography focuses on the various modes of transportation and communication that play a vital role in the economic development of a country. These lifelines are crucial for the movement of goods, services, and people, thereby connecting different regions and promoting economic activities.
This presentation was provided by Rebecca Benner, Ph.D., of the American Society of Anesthesiologists, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.pptHenry Hollis
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
4. System, Model, and Simulation
● System is the articulate object, which exists in the real world and operates under definite
conditions of time and space.
● A model is a simplified representation of a system at some particular point in time or space
intended to promote understanding of the real system.
● Simulation of a system is the operation of a model in terms of time or space, which helps
analyze the performance of an existing or a proposed system. In other words, simulation is the
process of using a model to study the performance of a system.
5. System State Variables
The system state variables are a set of data, required to define the internal process within the
system at a given point of time.
● In a discrete-event model, the system state variables remain constant over intervals of time
and the values change at defined points called event times.
● In continuous-event model, the system state variables are defined by differential equation
results whose value changes continuously over time.
7. Ball Bouncing Model
● The bouncing ball problem is a classic
example of a hybrid dynamic problem,
involving :
○ Continuous dynamics
○ Discrete transitions where the
system dynamics can change and
the state values can jump
10. Zeno Phenomenon
● Characterized by an infinite number of events occurring in a finite time interval for
certain hybrid systems.
● As the ball loses energy in the bouncing ball model, a large number of collisions with
the ground start occurring in successively smaller intervals of time. Hence the
model experiences Zeno behavior.
11. Using Two Integrator Blocks to Model a Bouncing Ball
Bouncing Ball Model
With two integrator blocks
✓ The state port of the position integrator and the corresponding comparison result is used
to detect when the ball hits the ground and to reset both integrators.
✓ The state port of the velocity integrator is used for the calculation of v+
12. ● To observe the Zeno behavior, confirm that 'Algorithm' is set
to 'Non-adaptive' and that the simulation 'Stop time' is set
to 25 seconds, in the 'Zero-crossing options' section of the
Solver pane of the Configuration Parameters dialog box.
● Observation:
○ The simulation errors out as the ball hits the ground
more and more frequently and loses energy.
○ Consequently, the simulation exceeds the default
limit of 1000 for the 'Number of consecutive zero
crossings' allowed.
● Now ,in the 'Zero-crossing options' section, set the
'Algorithm' to 'Adaptive'. This algorithm introduces a
sophisticated treatment of such chattering behavior.
Therefore, we can now simulate the system beyond 20
seconds.
Results:
13. Using One Second Order Integrator Block to Model a Bouncing Ball
14. Using One Second Order Integrator Block to Model a Bouncing Ball
Bouncing Ball Model
With one second order integrator block
15. Results:
● Confirm that 'Algorithm' is set to
'Non-adaptive' in the 'Zero-crossing
options' section and the simulation 'Stop
Time' is set to 25 seconds. Simulate the
model.
● Observation:
○ The simulation encountered no problems. We
were able to simulate the model without
experiencing excessive chatter after t = 20
seconds and without setting the 'Algorithm' to
'Adaptive'.
16. Comparison between two Models
● Second-Order Integrator Model is the Preferable Approach to Modeling Bouncing Ball.
● We can analytically calculate the exact time t* when the ball settles down to the
ground with zero velocity by summing the time required for each bounce. This time is
the sum of an infinite geometric series given by:
● Here x0
and v0
are initial conditions for position and velocity respectively. The velocity
and the position of the ball must be identically zero for t > t*.
17. Comparison(Contd):
• Alongside figure conclusively shows that
the second model has superior
numerical characteristics as compared
to the first model.
• In the figure, results from both
simulations are plotted near t*. The
vertical red line in the plot is t* for the
given model parameters.
• For t<t* and far away from t*, both
models produce accurate and identical
results.
• However, the simulation results from
the first model are inexact after t*; it
continues to display excessive
chattering behavior for t>t*. In contrast,
the second model using the
Second-Order Integrator block settles to
exactly zero for t>t*.
• The second differential equation dx/dt = v is internal to the
Second-Order Integrator block. Therefore, the block
algorithms can leverage this known relationship between the
two states and deploy heuristics to clamp down the
undesirable chattering behavior for certain conditions.
19. Spring Mass System
● Objective : To study the motion of a body under
the constraint of a spring and damping effect
● Basic Model :
○ m : mass of the body under consideration
○ F : Force experienced by the body
○ k : stiffness constant of the spring
○ b : damping constant
26. SHM in Spring Mass Damper System
● Equation of Motion (Free Oscillations) and Solution:
● The motion of the system is affected by the magnitude of damping: under-damped,
critically-damped or over-damped.
34. Car Following Model
● Describes how one vehicle follows
another vehicle in an uninterrupted flow.
● Based on two assumptions;
○ higher the speed of the vehicle,
higher will be the spacing between
the vehicles and
○ to avoid collision, driver must
maintain a safe distance with the
vehicle ahead
35. Mathematical Model
Let ∆xn
t
is the gap available for nth
vehicle, and let ∆xsafe
is the safe distance, vn
t
and vn-1
t
are the
velocities, the gap required is given by,
∆xn
t
= ∆xsafe
+ τ.vn
t
…(1)
where τ is a sensitivity coefficient. The above equation can be written as :
xn-1
t
− xn
t
= ∆xsafe
+ τ.vn
t
…(2)
Differentiating the above equation with respect to time, we get
vn-1
t
− vn
t
= τ.an
t
an
t
= (vn-1
t
− vn
t
)/τ …(3)
36. Mathematical Model (contd.)
Now assume that T is the reaction time, or time taken by a driver to observe what is going on around
them and to update his velocity, then :
an
t + T
= λ(vn-1
t
− vn
t
) …(4)
where λ = 1/τ. The solution of (4) can be a steady-state solution, where all vehicles move at a
constant velocity, or it can be a superposition of exponential solutions of the form
vn
t
= eαt
.u …(5)
where u and α are constants. Differentiating, we get :
an
t + T
= αeαT
.eαt
.u …(6)
37. Mathematical Model (contd.)
Substituting in (3):
αeαT
.eαt
.u = λ(S.eαt
.u − I.eαt
.u) …(4)
where I is the identity matrix of degree equal to the dimensions of the system and S is the “shift”
matrix that, when it multiplies a vector on the left, cyclically permutes the entries of the vector.
[ S − [1 + (α/λ)eαT
] I ] u = 0 …(5)
Thus u is an eigenvector for S with eigenvalue 1 + (α/λ)eαT
.