This document provides an overview of control systems and introduces key concepts that will be covered in the course. It discusses:
1) The class of systems that will be studied are single-input single-output, linear, time-invariant, causal dynamic systems modeled by ordinary differential equations with constant coefficients.
2) Control systems can be open-loop or closed-loop. Closed-loop systems use feedback to make them more robust to disturbances. This course will focus on closed-loop feedback control.
3) A basic example of a closed-loop control system is presented using a DC motor, with the goal of maintaining a desired output speed despite disturbances or changes. Feedback is used to calculate the error between
- The document provides an overview of feedback control systems and key concepts related to their design and analysis.
- It discusses the basic components of a closed-loop control system including a controller, plant/system, reference input, error calculation, and feedback loop. It also describes how sensor dynamics, actuator dynamics, and disturbances can impact the design.
- Two common mathematical representations - transfer function and state-space - are introduced. Stability and performance are identified as the two critical aspects to consider when designing a controller.
This document provides an introduction and overview of the course "Control Systems". It discusses the following key points in 3 sentences:
The course will cover classical control theory and designing controllers for dynamic systems that are linear, time-invariant, causal, and single-input single-output. A system is defined as an entity that receives an input u(t) and produces an output y(t) as functions of time. The approach taken in the course will be to first develop a mathematical model of the system, then analyze the system response, and finally design a controller to regulate the system's behavior as desired.
1) The document discusses first order systems and their characteristics. It defines a first order system model as having one pole and no zeros.
2) Students are assigned homework to calculate the unit step, ramp, and impulse responses of a first order system and observe characteristics from the responses.
3) In the next class, discussion will continue on analyzing first order system responses and extracting performance parameters that can be used to evaluate control systems.
This document provides an overview of feedback control systems through examples of mass-spring-damper systems. It discusses:
1) A linear time-invariant mass-spring-damper system as an example, with the governing equation being a second-order linear differential equation.
2) How the system could become linear time-varying if the spring constant changes over time, resulting in time-varying coefficients.
3) How a non-linear spring would result in a non-linear governing equation.
4) Simple simulations showing differences between free response, forced response, and response to a time-varying system.
The document discusses performance specifications and steady state errors in control systems.
It begins by explaining that performance can be specified by defining regions in the s-plane where the closed loop poles must lie. It then discusses analyzing steady state error by assuming the closed loop system is stable and rewriting the open loop transfer function G(s)H(s) in a generic form based on the number of open loop poles at the origin. This determines if it is a type 0, 1, 2 etc. system, which impacts the steady state error calculation.
This document is a lecture on modeling and control system design. The professor discusses modeling a mechanical system involving two shafts connected by gears as a second order system. He derives the transfer function of the system and shows it has two poles at the origin, making it unstable. The professor then poses two tasks: 1) derive the transfer function and 2) design a stable, negative feedback control system that meets settling time and overshoot requirements. Key points are that the open loop system is unstable, proportional control can stabilize it, and root locus can be used to meet performance goals.
The document discusses the effect of zeros on a control system's step response. Specifically:
- The example system has poles at -1 and -10, and a zero at 2, in the right half of the complex plane.
- The step response will start in one direction, then reverse and go in the opposite direction before settling to the final value of -2/10.
- A zero in the right half plane leads to a non-minimum phase response where the output changes signs during the step response.
This document discusses various control system strategies including on-off, proportional, derivative, modulating, and digital control. On-off control uses a switch to supply an on or off correcting signal, resulting in oscillations. Proportional control relates the controller output to the error size through a gain constant. Derivative control relates the controller output change to the error rate of change. Modulating control gradually adjusts the valve opening based on the level error. Digital control uses computers running control law programs to calculate control signals from sampled plant outputs and inputs.
- The document provides an overview of feedback control systems and key concepts related to their design and analysis.
- It discusses the basic components of a closed-loop control system including a controller, plant/system, reference input, error calculation, and feedback loop. It also describes how sensor dynamics, actuator dynamics, and disturbances can impact the design.
- Two common mathematical representations - transfer function and state-space - are introduced. Stability and performance are identified as the two critical aspects to consider when designing a controller.
This document provides an introduction and overview of the course "Control Systems". It discusses the following key points in 3 sentences:
The course will cover classical control theory and designing controllers for dynamic systems that are linear, time-invariant, causal, and single-input single-output. A system is defined as an entity that receives an input u(t) and produces an output y(t) as functions of time. The approach taken in the course will be to first develop a mathematical model of the system, then analyze the system response, and finally design a controller to regulate the system's behavior as desired.
1) The document discusses first order systems and their characteristics. It defines a first order system model as having one pole and no zeros.
2) Students are assigned homework to calculate the unit step, ramp, and impulse responses of a first order system and observe characteristics from the responses.
3) In the next class, discussion will continue on analyzing first order system responses and extracting performance parameters that can be used to evaluate control systems.
This document provides an overview of feedback control systems through examples of mass-spring-damper systems. It discusses:
1) A linear time-invariant mass-spring-damper system as an example, with the governing equation being a second-order linear differential equation.
2) How the system could become linear time-varying if the spring constant changes over time, resulting in time-varying coefficients.
3) How a non-linear spring would result in a non-linear governing equation.
4) Simple simulations showing differences between free response, forced response, and response to a time-varying system.
The document discusses performance specifications and steady state errors in control systems.
It begins by explaining that performance can be specified by defining regions in the s-plane where the closed loop poles must lie. It then discusses analyzing steady state error by assuming the closed loop system is stable and rewriting the open loop transfer function G(s)H(s) in a generic form based on the number of open loop poles at the origin. This determines if it is a type 0, 1, 2 etc. system, which impacts the steady state error calculation.
This document is a lecture on modeling and control system design. The professor discusses modeling a mechanical system involving two shafts connected by gears as a second order system. He derives the transfer function of the system and shows it has two poles at the origin, making it unstable. The professor then poses two tasks: 1) derive the transfer function and 2) design a stable, negative feedback control system that meets settling time and overshoot requirements. Key points are that the open loop system is unstable, proportional control can stabilize it, and root locus can be used to meet performance goals.
The document discusses the effect of zeros on a control system's step response. Specifically:
- The example system has poles at -1 and -10, and a zero at 2, in the right half of the complex plane.
- The step response will start in one direction, then reverse and go in the opposite direction before settling to the final value of -2/10.
- A zero in the right half plane leads to a non-minimum phase response where the output changes signs during the step response.
This document discusses various control system strategies including on-off, proportional, derivative, modulating, and digital control. On-off control uses a switch to supply an on or off correcting signal, resulting in oscillations. Proportional control relates the controller output to the error size through a gain constant. Derivative control relates the controller output change to the error rate of change. Modulating control gradually adjusts the valve opening based on the level error. Digital control uses computers running control law programs to calculate control signals from sampled plant outputs and inputs.
- The document discusses frequency response analysis of control systems.
- It begins by reviewing state space representation and relating transfer functions to state matrices. Two homework problems are assigned involving checking these relationships for a mass-spring-damper system.
- The lecture then introduces frequency response, which analyzes the response of a system to a sinusoidal input. It considers providing a sinusoidal input u(t) to a stable linear time-invariant system and derives the output y(t). This lays the foundation for discussing how frequency response can be used for control design.
1) The document defines the transfer function of a system as the ratio of the Laplace transform of the output to the Laplace transform of the input, under zero initial conditions.
2) For a mass-spring-damper system, the transfer function is derived as F(s)/(Ms^2 + Cs + K), where F(s) is the input force and the denominator terms represent mass, damping and spring constants.
3) The transfer function allows predicting the system output for any input, or determining the needed input to achieve a desired output. The impulse response of a system is equal to its transfer function.
The document discusses the describing function, a tool for predicting oscillations in nonlinear systems. It introduces the concept through an example system with a nonlinear "bang-bang" controller driving a linear plant. By assuming the output oscillates sinusoidally, it shows how the describing function technique can determine the frequency and amplitude of limit cycles in the system. Specifically, it finds the frequency where the plant's transfer function induces a -180 degree phase shift, meeting the conditions for oscillation in the nonlinear closed loop.
1) The document discusses different variants of first order systems including: a) systems with non-unity steady state gain, and b) systems with time delay.
2) It then introduces second order systems, defined by a governing differential equation involving natural frequency, damping ratio, and input.
3) Based on the damping ratio value, second order systems can have complex conjugate poles (underdamped, ratio between 0-1), repeated real poles (critically damped, ratio equals 1), or distinct real poles (overdamped, ratio above 1). The implications of these cases will be explored next.
The document provides an introduction to control systems. It defines key terms like systems, control systems, open loop and closed loop systems. It explains that a system is a combination of components that work together, while a control system includes feedback to achieve a desired output. Open loop systems operate independently of feedback, while closed loop systems use feedback to adjust. Common examples of open and closed loop systems are also provided like electric hand driers and automatic washing machines. The basic elements of control systems like resistors, inductors, and capacitors are also introduced in the context of electrical systems.
1) The document discusses the mathematical preliminaries needed for studying control systems, including linear ordinary differential equations (ODEs) with constant coefficients.
2) It describes how the exponential function is the unique solution to linear ODEs, and presents the solutions to both homogeneous and inhomogeneous first order ODEs.
3) Laplace transforms are introduced as a tool to convert ODEs to algebraic equations, and standard inputs like impulse, step, ramp and sinusoidal functions are described that are used to analyze system responses.
This document provides an introduction to control engineering. It discusses several key points:
1) Control engineering deals with designing systems to control dynamic processes and improve response speed, accuracy, and stability. This includes analyzing both classical and modern control methods.
2) Modern control engineering uses state-space and eigenvector approaches to model multi-input multi-output systems as sets of first-order differential equations.
3) Automatic control systems are commonly used, where a controlled variable is measured and compared to a setpoint to generate an output that achieves the desired result. This reduces costs and improves quality and productivity over manual control.
This document provides an overview of the syllabus for a course on advanced electric drives. The course will begin with a generalized theory of electric machines to analyze different machine types within a common framework. It will then cover reference frame theory and modeling of machines like DC, induction, and synchronous motors. A significant portion of the course will focus on control strategies for induction motors like scalar, vector, and sensorless control. The course will also cover advanced control methods for synchronous motors and special drive systems like switched reluctance motors.
This document provides an overview of gas turbine power cycles. It begins with an animation showing the basic components and working of a gas turbine, including compression of air, combustion, expansion through a turbine, and heat recovery from exhaust gases. It then discusses the ideal Brayton cycle used to model gas turbines, showing pressure-volume and temperature-entropy diagrams. The document concludes by deriving an expression for the thermal efficiency of an ideal Brayton cycle in terms of pressure ratios.
This document provides definitions and concepts related to thermodynamics. It defines thermodynamics as the science relating heat, work, and properties of systems. A system is the part of the universe being studied, while the surroundings are what is outside the system boundaries. Systems can be closed, open, or isolated depending on if mass and energy cross the boundaries. Properties, state, path, process, and equilibrium are also defined. Temperature is a measure of how heat transfers between bodies in contact until thermal equilibrium is reached.
The document provides an introduction to automatic control systems. It discusses:
1. The objectives of understanding basic control concepts, mathematical modeling using block diagrams, and studying systems in time and frequency domains.
2. The differences between manual and automatic control systems, with examples of driverless cars versus manual driving.
3. A brief history of automatic control, including James Watt's flyball governor and Ivan Polzunov's water-level regulator.
4. An overview of control system components and their representation in block diagrams.
This document provides an overview and introduction to the study of machine dynamics and vibrations. It discusses:
1) Why the study of vibrations is important, as dynamic loading can cause higher stresses than static loading, and increasing thickness to strengthen a design may paradoxically increase stresses under dynamic loads.
2) Common types of vibrations classified by their source of energy - free vibrations from an initial displacement, forced vibrations caused by dynamic forces, self-excited vibrations from internal system characteristics rather than external forces, and parametrically excited vibrations where a system parameter varies over time.
3) Vibrations can also be classified based on how energy dissipates in the system, such as undamped
This document discusses fuzzy identification of systems and its applications to modeling and control. It begins by introducing fuzzy logic and fuzzy controllers. It then provides details on the format of fuzzy implications and reasoning algorithms using Takagi-Sugeno controllers. An identification algorithm is presented as a mathematical tool to build fuzzy models of systems. The document applies this fuzzy identification method to modeling a human operator's control of a water cleaning process and a converter in a steel-making process. Results show the fuzzy models accurately capture the operators' actions.
The document discusses the basic components of control systems including sensors, controllers, actuators, and feedback loops. It also covers different types of control systems such as open loop and closed loop systems. Finally, it describes different classifications of robots including industrial, medical, space, and domestic robots providing examples of robots in each category.
A control system uses sensors to monitor conditions, a microprocessor to process input data and compare it to preset values, and outputs to regulate the condition. Common examples are central heating, air conditioning, and refrigeration systems which use temperature sensors, microprocessors to turn heating/cooling on or off to maintain the set temperature, and fans or compressors as outputs. Control systems operate continuously and automatically but have disadvantages like cost, reliance on computer functioning, and inability to react flexibly like humans.
1. Entropy is defined as a quantitative measure of microscopic disorder for a system. It is a measure of energy that is no longer available to perform useful work.
2. The Clausius inequality, derived from the first law of thermodynamics, states that the net heat transfer of a system divided by the temperature can never be negative for a cyclic process. It is equal to zero only for reversible processes.
3. The definition of entropy is developed by recognizing that the quantity (net heat transfer/temperature) has a cyclic integral of zero, making it a property of the system. This property is called entropy. Entropy increases for irreversible processes and remains constant for reversible processes.
This document provides an introduction to state space representation of dynamic systems. It begins by stating that state space representation is an alternative to transfer function representation that allows analysis in the time domain rather than complex domain. It then uses the example of a mass-spring-damper system to illustrate the key steps: 1) Select state variables (here displacement and velocity), 2) Write first-order differential equations for each state variable based on the system model, 3) Collect the equations into matrix form allowing analysis with linear algebra tools. The representation allows rewriting a higher-order differential equation as a set of first-order equations.
Thermodynamics studies energy transfer as heat and work. The first law states that energy is conserved and can change forms but not be created or destroyed. The amount of heat absorbed by a system equals its change in internal energy plus work done by the system. The second law limits the direction of heat transfer and states that heat cannot be fully converted to work. Entropy is a measure of disorder in a system that always increases due to the second law. The Carnot cycle uses reversible, isothermal and adiabatic processes to achieve the maximum possible efficiency between two temperature reservoirs.
- The document discusses applying the Nyquist stability criterion using the mapping theorem to determine closed-loop stability.
- It establishes that for closed-loop stability, the contour mapped from the open-loop transfer function in the s-plane should encircle the origin P times in the counterclockwise direction in the closed-loop transfer function plane, where P is the number of open-loop poles within the contour.
- Subtracting 1 from the closed-loop transfer function shifts the corresponding contour in the open-loop transfer function plane, such that the number of encirclements of the origin in the closed-loop plane equals the number of encirclements of the point -1 in the open
This document discusses the Nyquist stability criterion and the mapping theorem.
[1] The mapping theorem states that a closed contour in the s-plane maps to a corresponding closed contour in the F(s)-plane, where F(s) is a ratio of polynomials. The number of clockwise encirclements of the origin in the F(s)-plane equals Z - P, where Z is the number of zeros and P is the number of poles of F(s) inside the s-plane contour.
[2] An example function F(s) with two zeros and two poles is analyzed. A closed contour in the s-plane maps to a closed contour in the F(s)-plane
- The document discusses frequency response analysis of control systems.
- It begins by reviewing state space representation and relating transfer functions to state matrices. Two homework problems are assigned involving checking these relationships for a mass-spring-damper system.
- The lecture then introduces frequency response, which analyzes the response of a system to a sinusoidal input. It considers providing a sinusoidal input u(t) to a stable linear time-invariant system and derives the output y(t). This lays the foundation for discussing how frequency response can be used for control design.
1) The document defines the transfer function of a system as the ratio of the Laplace transform of the output to the Laplace transform of the input, under zero initial conditions.
2) For a mass-spring-damper system, the transfer function is derived as F(s)/(Ms^2 + Cs + K), where F(s) is the input force and the denominator terms represent mass, damping and spring constants.
3) The transfer function allows predicting the system output for any input, or determining the needed input to achieve a desired output. The impulse response of a system is equal to its transfer function.
The document discusses the describing function, a tool for predicting oscillations in nonlinear systems. It introduces the concept through an example system with a nonlinear "bang-bang" controller driving a linear plant. By assuming the output oscillates sinusoidally, it shows how the describing function technique can determine the frequency and amplitude of limit cycles in the system. Specifically, it finds the frequency where the plant's transfer function induces a -180 degree phase shift, meeting the conditions for oscillation in the nonlinear closed loop.
1) The document discusses different variants of first order systems including: a) systems with non-unity steady state gain, and b) systems with time delay.
2) It then introduces second order systems, defined by a governing differential equation involving natural frequency, damping ratio, and input.
3) Based on the damping ratio value, second order systems can have complex conjugate poles (underdamped, ratio between 0-1), repeated real poles (critically damped, ratio equals 1), or distinct real poles (overdamped, ratio above 1). The implications of these cases will be explored next.
The document provides an introduction to control systems. It defines key terms like systems, control systems, open loop and closed loop systems. It explains that a system is a combination of components that work together, while a control system includes feedback to achieve a desired output. Open loop systems operate independently of feedback, while closed loop systems use feedback to adjust. Common examples of open and closed loop systems are also provided like electric hand driers and automatic washing machines. The basic elements of control systems like resistors, inductors, and capacitors are also introduced in the context of electrical systems.
1) The document discusses the mathematical preliminaries needed for studying control systems, including linear ordinary differential equations (ODEs) with constant coefficients.
2) It describes how the exponential function is the unique solution to linear ODEs, and presents the solutions to both homogeneous and inhomogeneous first order ODEs.
3) Laplace transforms are introduced as a tool to convert ODEs to algebraic equations, and standard inputs like impulse, step, ramp and sinusoidal functions are described that are used to analyze system responses.
This document provides an introduction to control engineering. It discusses several key points:
1) Control engineering deals with designing systems to control dynamic processes and improve response speed, accuracy, and stability. This includes analyzing both classical and modern control methods.
2) Modern control engineering uses state-space and eigenvector approaches to model multi-input multi-output systems as sets of first-order differential equations.
3) Automatic control systems are commonly used, where a controlled variable is measured and compared to a setpoint to generate an output that achieves the desired result. This reduces costs and improves quality and productivity over manual control.
This document provides an overview of the syllabus for a course on advanced electric drives. The course will begin with a generalized theory of electric machines to analyze different machine types within a common framework. It will then cover reference frame theory and modeling of machines like DC, induction, and synchronous motors. A significant portion of the course will focus on control strategies for induction motors like scalar, vector, and sensorless control. The course will also cover advanced control methods for synchronous motors and special drive systems like switched reluctance motors.
This document provides an overview of gas turbine power cycles. It begins with an animation showing the basic components and working of a gas turbine, including compression of air, combustion, expansion through a turbine, and heat recovery from exhaust gases. It then discusses the ideal Brayton cycle used to model gas turbines, showing pressure-volume and temperature-entropy diagrams. The document concludes by deriving an expression for the thermal efficiency of an ideal Brayton cycle in terms of pressure ratios.
This document provides definitions and concepts related to thermodynamics. It defines thermodynamics as the science relating heat, work, and properties of systems. A system is the part of the universe being studied, while the surroundings are what is outside the system boundaries. Systems can be closed, open, or isolated depending on if mass and energy cross the boundaries. Properties, state, path, process, and equilibrium are also defined. Temperature is a measure of how heat transfers between bodies in contact until thermal equilibrium is reached.
The document provides an introduction to automatic control systems. It discusses:
1. The objectives of understanding basic control concepts, mathematical modeling using block diagrams, and studying systems in time and frequency domains.
2. The differences between manual and automatic control systems, with examples of driverless cars versus manual driving.
3. A brief history of automatic control, including James Watt's flyball governor and Ivan Polzunov's water-level regulator.
4. An overview of control system components and their representation in block diagrams.
This document provides an overview and introduction to the study of machine dynamics and vibrations. It discusses:
1) Why the study of vibrations is important, as dynamic loading can cause higher stresses than static loading, and increasing thickness to strengthen a design may paradoxically increase stresses under dynamic loads.
2) Common types of vibrations classified by their source of energy - free vibrations from an initial displacement, forced vibrations caused by dynamic forces, self-excited vibrations from internal system characteristics rather than external forces, and parametrically excited vibrations where a system parameter varies over time.
3) Vibrations can also be classified based on how energy dissipates in the system, such as undamped
This document discusses fuzzy identification of systems and its applications to modeling and control. It begins by introducing fuzzy logic and fuzzy controllers. It then provides details on the format of fuzzy implications and reasoning algorithms using Takagi-Sugeno controllers. An identification algorithm is presented as a mathematical tool to build fuzzy models of systems. The document applies this fuzzy identification method to modeling a human operator's control of a water cleaning process and a converter in a steel-making process. Results show the fuzzy models accurately capture the operators' actions.
The document discusses the basic components of control systems including sensors, controllers, actuators, and feedback loops. It also covers different types of control systems such as open loop and closed loop systems. Finally, it describes different classifications of robots including industrial, medical, space, and domestic robots providing examples of robots in each category.
A control system uses sensors to monitor conditions, a microprocessor to process input data and compare it to preset values, and outputs to regulate the condition. Common examples are central heating, air conditioning, and refrigeration systems which use temperature sensors, microprocessors to turn heating/cooling on or off to maintain the set temperature, and fans or compressors as outputs. Control systems operate continuously and automatically but have disadvantages like cost, reliance on computer functioning, and inability to react flexibly like humans.
1. Entropy is defined as a quantitative measure of microscopic disorder for a system. It is a measure of energy that is no longer available to perform useful work.
2. The Clausius inequality, derived from the first law of thermodynamics, states that the net heat transfer of a system divided by the temperature can never be negative for a cyclic process. It is equal to zero only for reversible processes.
3. The definition of entropy is developed by recognizing that the quantity (net heat transfer/temperature) has a cyclic integral of zero, making it a property of the system. This property is called entropy. Entropy increases for irreversible processes and remains constant for reversible processes.
This document provides an introduction to state space representation of dynamic systems. It begins by stating that state space representation is an alternative to transfer function representation that allows analysis in the time domain rather than complex domain. It then uses the example of a mass-spring-damper system to illustrate the key steps: 1) Select state variables (here displacement and velocity), 2) Write first-order differential equations for each state variable based on the system model, 3) Collect the equations into matrix form allowing analysis with linear algebra tools. The representation allows rewriting a higher-order differential equation as a set of first-order equations.
Thermodynamics studies energy transfer as heat and work. The first law states that energy is conserved and can change forms but not be created or destroyed. The amount of heat absorbed by a system equals its change in internal energy plus work done by the system. The second law limits the direction of heat transfer and states that heat cannot be fully converted to work. Entropy is a measure of disorder in a system that always increases due to the second law. The Carnot cycle uses reversible, isothermal and adiabatic processes to achieve the maximum possible efficiency between two temperature reservoirs.
- The document discusses applying the Nyquist stability criterion using the mapping theorem to determine closed-loop stability.
- It establishes that for closed-loop stability, the contour mapped from the open-loop transfer function in the s-plane should encircle the origin P times in the counterclockwise direction in the closed-loop transfer function plane, where P is the number of open-loop poles within the contour.
- Subtracting 1 from the closed-loop transfer function shifts the corresponding contour in the open-loop transfer function plane, such that the number of encirclements of the origin in the closed-loop plane equals the number of encirclements of the point -1 in the open
This document discusses the Nyquist stability criterion and the mapping theorem.
[1] The mapping theorem states that a closed contour in the s-plane maps to a corresponding closed contour in the F(s)-plane, where F(s) is a ratio of polynomials. The number of clockwise encirclements of the origin in the F(s)-plane equals Z - P, where Z is the number of zeros and P is the number of poles of F(s) inside the s-plane contour.
[2] An example function F(s) with two zeros and two poles is analyzed. A closed contour in the s-plane maps to a closed contour in the F(s)-plane
1) The document discusses the Nyquist stability criteria, which uses the Nyquist plot of the open loop transfer function G(s)H(s) to determine the stability of the closed loop system.
2) The poles of the closed loop characteristic equation 1 + G(s)H(s) are called the open loop poles. The zeros are called the closed loop poles.
3) The key question is whether the Nyquist plot of the open loop transfer function G(s)H(s) can be used to comment on the stability of the closed loop system. The answer is yes, based on the mapping theorem.
- The document discusses constructing the Nyquist plot for a transfer function of the form s plus 1 divided by s plus 10.
- It is shown that the Nyquist plot for this transfer function is a semicircle in the first quadrant of the complex plane, with a center of 0.55 and radius of 0.45.
- The maximum phase angle of the transfer function occurs at the point where a tangent can be drawn to the semicircle, which is calculated to be 54.9 degrees using trigonometric relationships between the radius and tangent length.
1) The document discusses Nyquist plots for various transfer functions, including time delay and second order systems.
2) A time delay transfer function results in a unit circle Nyquist plot, where the phase shifts from 0 to -Tdω as ω goes from 0 to infinity.
3) For a second order system, the Nyquist plot takes the shape of an arc that starts at 1 and ends at 1 as ω goes from 0 to infinity, cutting the imaginary axis when ω equals the natural frequency ωn.
The document discusses an upcoming lecture on Nyquist plots. It begins by reviewing frequency response and bode plots. It then introduces Nyquist plots, explaining that they visualize the sinusoidal transfer function G(jω) by plotting its real and imaginary parts in the complex plane, known as the G(s) plane, as ω varies from 0 to infinity. Examples are given of plotting common transfer functions like 1/s, s, and 1/(Ts+1) on the Nyquist plane. Critical points like ω→0, ω=1/T, and ω→∞ are identified. The lecturer indicates they will demonstrate how to construct Nyquist plots and identify differences from bode plots.
- The professor discusses plotting the phase plot for a transfer function containing multiple factors. He calculates the phase contributions of each factor at different frequencies and plots the individual phase plots.
- To plot the overall phase plot of the transfer function, he calculates the total phase at two corner frequencies and connects them with a smooth curve. This results in a phase plot that transitions from 90 degrees to -90 degrees over the different frequency ranges.
- He explains that the bode plot can be used to experimentally determine an unknown transfer function by measuring the input-output magnitude and phase responses at different frequencies.
1. The document discusses constructing the Bode plot of a transfer function step-by-step using an example transfer function of s/(s+1)*(s+10).
2. The transfer function is rewritten as a product of four factors: 0.1s, s, 1/(s+1), and 1/(0.1s+1). The Bode plots of the individual factors are constructed and then combined.
3. The Bode plot consists of a line with a slope of +20 dB/decade from the s term, shifted down by -20 dB from the 0.1 term. Below the first corner frequency of 1 rad/s, this represents the combined low-frequency
This document is a transcript of a lecture on bode plots. The professor discusses drawing bode plots for second order transfer functions with different damping ratios. He draws the magnitude and phase plots, explaining how the resonant peak shifts left as damping increases. For undamped systems, he notes the magnitude would blow up to infinity at the natural frequency. He also discusses minimum phase and non-minimum phase systems, explaining how their phase plots differ at high frequencies. He leaves questions for students to think about generalizing these concepts to higher order systems and using bode plots to determine system properties.
- The document discusses the bode plot of a second-order transfer function.
- It shows that the high frequency asymptote has a slope of -40 dB/decade and intersects the low frequency asymptote at the natural frequency ωn, which is also the corner frequency.
- The magnitude of the transfer function reaches its maximum value at a frequency where the derivative of the denominator term is zero, which occurs at ω = ωn/√(1-2ζ^2).
This document discusses the magnitude and phase plots of first-order transfer functions. It begins by deriving the phase plot of 1/(Ts+1), showing that the phase approaches 0 degrees at low frequencies, -45 degrees at the corner frequency 1/T, and -90 degrees at high frequencies. It then derives the corresponding magnitude plot, showing it has the characteristics of a low-pass filter with a cutoff frequency of 1/T. Finally, it briefly discusses the magnitude and phase plots of the reciprocal transfer function T(s+1), showing the magnitude plot has a high-frequency asymptote of 20log(Tω) rather than -20log(Tω) as with 1/(Ts+1).
The document discusses Bode plots and frequency response analysis. It introduces the first-order transfer function 1/(Ts+1) as a new factor to analyze. The magnitude and phase expressions for this factor are derived. It is explained that the magnitude plot has low and high frequency asymptotes. The low frequency asymptote is 0 dB as the magnitude tends to 1 at low frequencies compared to 1/T. The high frequency asymptote is -20log(Tω) dB as the magnitude tends to 1/Tω at high frequencies. It is noted that the asymptotes intersect at the corner/break frequency of 1/T, where the responses change behavior.
- The document discusses bode plots and how to construct them for different transfer functions.
- It provides an example of constructing the bode plot for a transfer function G(s) = 1/s, showing that the magnitude plot is a straight line with a slope of -20 dB/decade and the phase plot is a horizontal line at -90 degrees.
- It then constructs the bode plot for a transfer function G(s) = s, finding that the magnitude plot has a slope of +20 dB/decade and the phase plot is a horizontal line at +90 degrees.
The document discusses Bode plots, which are used to visualize the frequency response of linear time-invariant systems. It begins by reviewing frequency response and how it characterizes the steady-state output of a system to a sinusoidal input.
It then introduces Bode plots, which have two components - a magnitude plot and a phase plot. The magnitude plot uses logarithmic scales for both frequency and magnitude in decibels. The phase plot uses a logarithmic scale for frequency and a linear scale for phase in degrees. Using logarithmic scales allows visualization of a wide range of frequencies and magnitudes on the same plot.
The document outlines common building blocks that comprise transfer functions, such as constants, integrals, derivatives, and
1) The professor derives that for a stable linear time-invariant system with a sinusoidal input, the steady state output will be a sinusoid of the same frequency as the input.
2) The steady state output magnitude is scaled by the magnitude of the system's transfer function evaluated at jω, and the phase is shifted by the phase of the transfer function evaluated at jω.
3) This property allows the experimental determination of a system's transfer function by analyzing the steady state response to various sinusoidal inputs.
The document discusses the state space representation of linear time-invariant (LTI) systems. It provides the following key points:
1. The state space representation of a single-input single-output (SISO) LTI system is given by x ̇(t) = Ax(t) + bu(t) and y(t) = cx(t) + du(t), where x(t) is the state vector, u(t) is the input, y(t) is the output, and A, b, c, d are matrices that characterize the system.
2. For multiple-input multiple-output systems, u(t) and y(t) become
This document summarizes a lecture on control system design using root locus analysis. The professor discusses plotting the root locus for a specific system and using it to determine the range of proportional gain (KP) that ensures both closed-loop stability and desired performance. By substituting values into the closed-loop characteristic equation, they determine that KP must be between 8 and 35.76 to satisfy both conditions. They also note that choosing KP between 12.5 and 35.76 would result in an underdamped second order system, as desired. As homework, students are asked to plot the root locus for negative values of KP to gain more experience with root locus analysis.
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1) The document describes deriving the governing equations of motion for a gear transmission system consisting of an input shaft connected to an output shaft via gears.
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- For positive feedback systems, one branch of the root locus always lies in the right half plane, making the system unstable for any positive K.
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Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
Null Bangalore | Pentesters Approach to AWS IAMDivyanshu
#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
- Basic understanding of AWS services and architecture
- Familiarity with cloud security concepts
- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
- Basics of IAM in AWS
- Implementing IAM Policies with Least Privilege to Manage S3 Bucket
- Objective: Create an S3 bucket with least privilege IAM policy and validate access.
- Steps:
- Create S3 bucket.
- Attach least privilege policy to IAM user.
- Validate access.
- Exploiting IAM PassRole Misconfiguration
-Allows a user to pass a specific IAM role to an AWS service (ec2), typically used for service access delegation. Then exploit PassRole Misconfiguration granting unauthorized access to sensitive resources.
- Objective: Demonstrate how a PassRole misconfiguration can grant unauthorized access.
- Steps:
- Allow user to pass IAM role to EC2.
- Exploit misconfiguration for unauthorized access.
- Access sensitive resources.
- Exploiting IAM AssumeRole Misconfiguration with Overly Permissive Role
- An overly permissive IAM role configuration can lead to privilege escalation by creating a role with administrative privileges and allow a user to assume this role.
- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
- Steps:
- Create role with administrative privileges.
- Allow user to assume the role.
- Perform administrative actions.
- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024Sinan KOZAK
Sinan from the Delivery Hero mobile infrastructure engineering team shares a deep dive into performance acceleration with Gradle build cache optimizations. Sinan shares their journey into solving complex build-cache problems that affect Gradle builds. By understanding the challenges and solutions found in our journey, we aim to demonstrate the possibilities for faster builds. The case study reveals how overlapping outputs and cache misconfigurations led to significant increases in build times, especially as the project scaled up with numerous modules using Paparazzi tests. The journey from diagnosing to defeating cache issues offers invaluable lessons on maintaining cache integrity without sacrificing functionality.
artificial intelligence and data science contents.pptxGauravCar
What is artificial intelligence? Artificial intelligence is the ability of a computer or computer-controlled robot to perform tasks that are commonly associated with the intellectual processes characteristic of humans, such as the ability to reason.
› ...
Artificial intelligence (AI) | Definitio
1. Control Systems
Prof. C. S. Shankar Ram
Department of Engineering Design
Indian Institute of Technology, Madras
Lecture – 02
Introduction to Control Systems
Part - 2
(Refer Slide Time: 00:17)
What do we mean by these models and how do we get them and what do we use them
for? So, that is something which we need to understand. So, once again I am revisiting
our idea of what we mean by a system. So, for us, a system is a mapping between )(tu
and )(ty . Now, suppose if, let me call this as problem 1 ok. The first problem I want to
solve is that, given )(tu and )(ty , we are asked to find the mapping S, that relates the
input )(tu and the output )(ty . This problem is what is called as the problem of
synthesis.
That is, suppose let us say, let me take an example of a room that is being air-
conditioned. So, let us say if I take the example of a room that is being air-conditioned
then suppose, if I want to regulate the temperature of the room, the temperature of air in
the room is the output of the system and the input to the system is the amount of cold air
that I blow into the room.
2. Let us say we have a room air conditioner and let us say the temperature of air in the
room is my output and let us say the amount of cold air provided right to the room
through the ac vents is the input of the system. So, what would this synthesis problem
entail? The problem of synthesis essentially means that I provide varying quantities of
the input to the system and then measure the output and try to get a mapping between the
two. So, that is what I do so that I get a mathematical model for the system.
So, in this case, if I provide various amounts of cold air to my air conditioner and I
measure the corresponding temperature and I get a relationship between the two, then I
will know how the temperature of air in this particular room would vary. So, once I do
this first problem of synthesis, I can go and do problem number 2, which is going to be
the following: Given the mapping S, once we find the mapping S, and given a particular
input )(tu , find )(ty .
Suppose, I have got a mathematical model for the temperature of air in this room
equipped with an air conditioner. So, tomorrow let us say, I want to essentially change
the air conditioner and let us say I have 5 choices. Without even purchasing them I can
use the model which I have developed to figure out how temperature would vary and
settle down to a desired value by doing this analysis right, once I have the mathematical
model for this particular process of air conditioning.
So, the advantage is that I can predict the temperature variation without even buying the
air conditioner, installing it and testing it so that I can make a well-informed choice as far
as which AC to buy. So, this problem is what is called the analysis problem or the
prediction problem.
Another problem, which we can do once we complete the synthesis problem is that,
given the mapping S and a desired output )(ty , find )(tu . That is the third problem
that we can solve. So, this problem is what is called the control problem.
So, what do I mean by this? Suppose, let us say I come into a room with an air
conditioner and I essentially want to regulate the temperature of air in this room. So, I set
the temperature of the air at 25 degree Celsius. The question is that, what is the amount
of cold air, which I should blow, which would get my temperature to 25 degree Celsius.
That is the problem of control.
3. So, I know what the mapping of the dynamic system is and I want the desired output,
what is the input which I have to provide? So, in a certain sense, it is the inverse problem
of what we do in problem number 2. So, this mapping helps us to find a )(ty given a
)(tu . In the problem of control, what we do is that, given )(ty I want to use this
mapping to find what )(tu will get me the output )(ty . So, that is the problem of
control.
So, what are a few examples of control that we have already look at? One is room
temperature control. So, that is, that is one example that we have discussed. Let us say,
we have also looked at an example, where we want to control the speed of a motor. And
our human body is a marvellous controller.
Say, for example, our human body maintains our body temperature in a very-very narrow
band. So, even if the temperature goes to 100 degrees Fahrenheit right, so we are in
trouble, correct. So, essentially our body maintains our internal body temperature, in a
very narrow band irrespective of what that environment temperature is.
So, it does not matter whether we are in winter or in summer you know the human
internal body temperature is maintained at almost a constant value and that is a great
control system. For example, the blood pressure that is maintained by your body. You
know it should be in a good range. Blood sugar level is another marvellous control
mechanism. And in fact, even our heart which is a pump which essentially pumps blood
throughout the body needs to work repeatedly again and again as far as we live. So, that
is extremely important to us. So, our human body is filled with marvellous controllers,
which regulate various variables around the desired values very accurately and for a long
time and that is something which is marvellous.
So, essentially we are going to look at various case studies, as we go along and then we
will see how to formulate practical problems as a control problem and solve them, ok.
So, before we get into the mathematics I just want to give another physical perspective,
as far as how we look at control and so on.
4. (Refer Slide Time: 08:41)
So, essentially in a very broad sense, people classify control as what is called an open
loop control and closed loop control. What are these terms? Let us say we consider the
example of a ceiling fan. So, I have a ceiling fan, I have a regulator and I have a switch,
right. I can come, switch on the fan and I can vary the speed setting, maybe in discrete
quantities by using the fan speed regulator and that is about it, right.
So, I just get some output, which is a fan RPM. So, that is an open loop controller in the
sense that, it cannot account for any disturbances, which come during the operation of
the fan. For example, if there is a voltage fluctuation, then, the output of the fan or the
speed of rotation of the fan is going to fall down or go up depending on whether the
voltage is falling down or going up. So, then, you know like, the question is that like I
lose certain performance right, but the question we need to ask ourselves is that, is it
important, for a ceiling fan? So, typically ceiling fans are open loop control systems,
where the speed settings are calibrated in the factory and the regulator essentially, adjusts
the fan RPM in a discrete set of values.
Essentially in the presence of disturbances, obviously, the system response gets affected.
So, that is the characteristic of open-loop control. So, what are the various characteristics
of open loop control: There is no feedback (we will shortly define what is called as
feedback). So, essentially, it cannot tolerate disturbances; it is not robust to disturbances
and so on, ok. But on the flip side, the cost and complexity are lower.
5. On the other hand, if we want a fan, whose RPM needs to be maintained at a baseline
value, let us say 100 RPM and I cannot afford a huge variation in the RPM, right, for
some industrial application, what would I do? What I would do is that, I would measure
the actual RPM using a speed sensor and then I would take the actual RPM at each and
every instant of time, compare it with what is the desired value and then take the
difference, quantify the difference between what I desire and what is actually being
obtained, take that error and pass it through what is called as a controller and the
controller will then adjust the electrical input to the fan. Then, when that happens we
have what is called as closed-loop control and closed loop control systems have
feedback. Feedback is the process of measuring variables that need to be regulated or
controlled so that corrective action can be taken ok, that is the process of feedback.
So, consequently closed loop control is more tolerant or robust to uncertainties that come
in disturbances and so on (what we call as un-modelled dynamics). So, but anyway let
me group them under uncertainties, ok. But the flipside of closed-loop control is that the
cost and complexity are higher.
In this particular course, we are going to deal with closed-loop control. So, that is what
we are going to deal with in this particular course. So, let me just construct an example to
just explain what we did. Suppose, let us say, we have, let us say, a DC motor right. So,
the DC motor is our system or plant, ok. And to this DC motor, let us say, I provide a
voltage )(tV as my input and )(t is my output.
Suppose, let us say I want a desired output (desired RPM of rotation from the motor) and
this is what we call as a reference input and what we do then is the following. Suppose if
I measure the actual output, then what I do is that I go and compare, then I take the
difference between what I desire and what I measure, which is what is called as the error
and then, let us say, we pass it through an element called as a controller and the
controller then calculates what should be the input that should be provided to the DC
motor, ok.
When the controller calculates the input that should be provided to the system, the
adjective control is added to the term ‘input’. So, the input to the system becomes what is
called as the control input. So, this is feedback, ok. So, this is a typical layout of a closed
loop control system with what is called as negative feedback, ok.
6. So, this is essentially closed loop control system with negative feedback. Why is it
negative feedback? Because at the summing junction you can see that, we are subtracting
the feedback signal from the desired reference input. So, that is why it is called as
negative feedback.
So, we are taking the difference between the reference input and the actual output and
calculating the error. This is what is called a closed loop control system with negative
feedback. So, we can see that this is what we are going to essentially do in this particular
course. So, some more terms that I want to introduce here is that, like, this process of
essentially taking this output measurement and feeding it back is what is called as
feedback. This is what is called a feedback path.
(Refer Slide Time: 17:47)
Typically, we can have a mapping in the feedback path, say we can have some sensor
dynamics coming into play here. So, let us say I use a sensor for measuring speed, the
sensor may have its own dynamic characteristics. I may need to use a filter to filter out
noise, then a mapping comes in the feedback path ok.
So, if a mapping comes in the feedback path (when the mapping is not one ) we call it as
a non-unity feedback. So, when this mapping in the feedback path is 1, we call it as unity
feedback. We will see how this affects our analysis later on right and many times when
the controller calculates and provides a control signal to the system, it is typically
realized by what is called an actuator.
7. Say, for example, let us say you know I have to essentially move, say, design a motion
control system that essentially displaces a work piece along one axis right. Let us say,
simple translation of a work piece in a machining system. So, let us say, I want to
regulate the position of the work piece. I give a, a voltage signal to, let us say, an electric
motor drive system, which essentially provides translational motion to the work piece
right. Suppose, if the controller calculates that, at this point of time, provide 5 volts of
input right to the electric motor or if it calculates and tells me that look, you know like,
provide, you know like, 10 Newton’s of force to the work piece, in order to essentially
move by some distance x , right. So, the controller may say, provide 10 Newton’s at this
instant of time, but then there may be a small, what to say, response time before the 10
Newton’s is actually realized in practice through the electric motor and drive system,
right.
So, that, if that is important then you know, we need to figure out what is called as
actuator dynamics in the design process, ok and sometimes we can have disturbances
coming into the system. Let us say, I have a sudden load that may come on a DC motor,
right, for example, you know that I mean to model as a disturbance and then like see how
we can overcome such disturbances and so on, right.
So, one can see that, you know, we can add varying levels of complexity to this,
feedback system, ok. We are going to study the basic feedback loop in this particular
course and as we go along maybe when we go to some case studies, we will add some
more blocks and then see how the design varies.
To summarize, what we are going to do in this course is the following. So, as a summary
of all this discussion that we did, the title of our course is control systems, right. So, what
we are going to do, what we are going to learn is the following: Closed Loop Feedback
Control of SISO LTI Causal Dynamic Systems. So, that is what we are going to do in
this particular course, ok.
So, that is the class of systems that we are going to do, SISO LTI causal dynamic system
and we are going to do what is called as closed loop feedback control, right, of this class
of systems and it so turns out that this class of systems, SISO LTI causal dynamic
systems, the mathematical models that are typically used to characterize this class of
8. systems usually take the form of a Linear Ordinary Differential Equations, what are
abbreviated as ODEs, with constant coefficients.
So, typically the mathematical models that are used to characterize this class of systems
that we are going to study in this particular course take the form of linear ordinary
differential equations with constant coefficients, ok. So, that is the class of equations that
we would be focusing on.
And, another important aspect is that the kind of mathematical models we are going to
essentially look at are what are called as spatially homogeneous. That means that we do
not consider the variations of variables with space. So, we only consider temporal
variations of variables. For example, let us say, I want to analyze the variation of the
temperature of air in this room, right. So obviously, the temperature of the air in this
room can be different depending on the point where I am measuring and also the time at
which I measure, right. So, but then what we do is that we lump all the points in this
room i.e. the temperature of all the points in this room into a single entity, which
essentially can be represented as a function of time, ok.
So, in a certain sense, what we are doing is that we are lumping the effects as far as
spatial variation is concerned and we are going to assume that the entire room can be
characterized by a single temperature which is only a function of time, ok. So, spatially
homogeneous means we are essentially following sort of a lumped parameter approach
towards our modelling.
Then we are going to have what are called continuous time, dynamic, deterministic,
mathematical models, ok. That is the class of models we are going to use, right. So,
essentially in a certain sense, continuous time means that we treat time as a continuous
variable. So from a pragmatic perspective, it just means that we are going to get ODEs
right, Ordinary Differential Equations. Dynamic means we are going to have derivatives
in the equations.
So, what is a dynamic model? It is something which explicitly considers future states of
the system, right, so, and it essentially incorporates what is going to happen to the system
in the future by incorporating derivatives of the variables. So, that is a dynamic model.
Deterministic means we essentially neglect any stochastic effects, ok. We do not consider
9. variables as random variables. We consider all variables as deterministic variables and
we are going to have deterministic models, ok.
So, this is the class of models that we are going to use. So, in summary, what we are
going to do in this course is closed loop feedback control of SISO LTI causal dynamic
systems using spatially homogeneous, continuous time, dynamic, deterministic
mathematical models, ok. So, that is what we are going to essentially learn how to do in
this course, ok. So, what we would subsequently do is essentially have a brief recap of
what is the mathematical background that is required to do this analysis, ok.
So, that is something which we are going to recap. It is assumed that some mathematical
courses have already been completed before one comes to this particular course,
particularly courses on complex variables, ordinary differential equations and Laplace
transform. So, we would quickly recap some of these tools and then we will move
forward, ok. So, that is going to be the plan of action. Fine.
Thank you.