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Questions and Answers
Who invented the Laplace transform?
Pierre-Simon Laplace
What is the Laplace transform of a function f(t)?
The Laplace transform allows us to find f(t) given F(s) through the _______ Laplace transform.
inverse
What property of Laplace transform helps in solving differential equations with discontinuous initial conditions?
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What is the distinguishing characteristic of an open-loop system?
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Closed-loop systems are less sensitive to noise, disturbances, and changes in the environment.
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Define 'Transient Response' in control systems.
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For a control system to be useful, the natural response must eventually approach zero, leaving only the _____ response.
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What are two major measures of performance in a control system?
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What is the purpose of a control system?
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Liquid-level control using a float valve is a system that helps to keep the ___ in the lower container constant.
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Match the control system development with its inventor:
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What are some consequences of an unstable system?
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Stability is a crucial requirement in control system design.
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What must the natural response of a system do for it to be stable?
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A time plot of an unstable system would show a transient response that grows without ____.
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Match the system with its time characteristics:
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Study Notes
Laplace Transform
- The Laplace transform was invented by French mathematician and astronomer Pierre-Simon Laplace in the late 1700s.
- Laplace's early work involved calculus and differential equations, while his later work focused on planetary movements, probability theory, and Bayesian inference.
Definition of Laplace Transform
- The Laplace transform of a function f(t) is defined as: 𝓛𝐟 𝐭 = ∫∞ 0 f(t)e^(-st)dt
- s = σ+jω, a complex variable, and u(t) is a unit step function.
- The unit step function u(t) ensures that the function f(t) is zero for t < 0.
Simplification of Laplace Transform
- Multiplying f(t) by u(t) yields a time function that is zero for t < 0, allowing the equation to be simplified to: 𝓛𝐟 𝐭 = ∫∞ 0 f(t)e^(-st)dt
- The notation for the lower limit allows for integration prior to discontinuities, enabling the Laplace transform of impulse functions.
Advantages of Laplace Transform
- The Laplace transform has distinct advantages when solving differential equations with discontinuous initial conditions at t = 0.
- It allows for the solution of initial conditions before the discontinuity, eliminating the need to solve for initial conditions after the discontinuity.
Inverse Laplace Transform
- The inverse Laplace transform is defined as: 𝓛⁻¹[F(s)] = (1/2πj) ∫∞ -∞ F(s)e^(st)ds
- It allows for the finding of f(t) given F(s).
Laplace Transform of Basic Functions
- Table 2.1 shows the Laplace transform of some basic functions.
- Examples of Laplace transforms include:
- 𝓛[𝐴𝑢 𝑡] = 𝐴/s
- 𝓛[𝐴𝑒^(𝑢 𝑡)] = 𝐴/(s-α)
- 𝓛[𝑡 𝑢 𝑡] = 1/s^2
- 𝓛[sin(5𝑡) 𝑢 𝑡] = 5/(s^2 + 25)
- 𝓛[𝑒^(𝑢 𝑡) sin(𝜔𝑡) 𝑢 𝑡] = 𝜔/(s^2 + 𝜔^2)
- 𝓛[𝑒^(5𝑡) cos(5𝑡) 𝑢 𝑡] = (s-5)/(s^2 - 10s + 50)
Inverse Laplace Transform Examples
- Examples of inverse Laplace transforms include:
- 𝓛⁻¹[F(s)] = 𝑓 𝑡 when F(s) = 𝐴/s
- 𝓛⁻¹[F(s)] = 𝑓 𝑡 when F(s) = 𝐴/(s-α)
- 𝓛⁻¹[F(s)] = 𝑓 𝑡 when F(s) = 1/(s^2 + 𝜔^2)
Introduction to Feedback Control Systems
- Control systems are an integral part of modern society, with numerous applications in various fields such as space-vehicle systems, robotic systems, and modern manufacturing systems.
- Automatic control is essential in any field of engineering and science, and it is desirable for most engineers and scientists to be familiar with the theory and practice of automatic control.
Control System Definition
- A control system consists of subsystems and processes (or plants) assembled to obtain a desired output with desired performance, given a specified input.
- The system has three main components: input, process, and output.
Advantages of Control Systems
- Four primary reasons for building control systems:
- Power amplification: to produce the needed power amplification or power gain.
- Remote control: to control systems in remote or dangerous locations.
- Convenience of input form: to change the form of the input to provide convenience.
- Compensation for disturbances: to compensate for disturbances that affect the system's performance.
A History of Control Systems
- The Greeks began engineering feedback systems around 300 B.C. with the invention of a water clock.
- The concept of liquid-level control was applied to an oil lamp by Philon of Byzantium.
- Regulation of steam pressure began around 1681 with Denis Papin's invention of the safety valve.
- Speed control was applied to a windmill by Edmund Lee in 1745.
- James Watt invented the flyball speed governor to control the speed of steam engines in the 18th century.
- Control systems theory began to crystallize in the latter half of the 19th century with the work of James Clerk Maxwell and Edward John Routh.
Contemporary Applications
- Control systems are used in various applications, including:
- Guidance, navigation, and control of missiles and spacecraft.
- Process control industry, regulating liquid levels, chemical concentrations, and thickness of fabricated material.
- Home heating systems, home entertainment systems, and industrial robots.
System Configurations
- Open-loop systems:
- Do not correct for disturbances.
- Cannot compensate for any disturbances that add to the controller's driving signal.
- Examples: toasters, home heating systems.
- Closed-loop (feedback control) systems:
- Compensate for disturbances by measuring the output response and feeding it back to the controller.
- Can correct for disturbances and changes in the environment.
- Examples: temperature control systems, digital computer-controlled systems.
Analysis and Design Objectives
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Analysis: the process of determining a system's performance, including evaluating its transient response and steady-state error.
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Design: the process of creating or changing a system's performance to meet desired specifications.
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A control system is dynamic, responding to an input by undergoing a transient response before reaching a steady-state response.### Objectives of Systems Analysis and Design
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Three major objectives of systems analysis and design: producing the desired transient response, reducing steady-state error, and achieving stability
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These objectives are demonstrated using an elevator as an example
Transient Response
- Transient response is important for the comfort and safety of passengers in an elevator
- A slow transient response can make passengers impatient, while an excessively rapid response can make them uncomfortable
- In a computer, transient response affects the time required to read from or write to disk storage
- We establish quantitative definitions for transient response, analyze the system, and adjust parameters or design components to yield a desired transient response
Steady-State Response
- Steady-state response is the response that remains after transients have decayed to zero
- We are concerned about the accuracy of the steady-state response
- Examples of steady-state response include an elevator stopped near the fourth floor and a read/write head stopped at the correct track
- We define steady-state errors quantitatively, analyze a system's steady-state error, and then design corrective action to reduce the steady-state error
Stability
- A system's total response is the sum of its natural response and forced response
- Natural response describes how a system dissipates or acquires energy
- Forced response is dependent on the input
- For a system to be useful, its natural response must eventually approach zero or oscillate
- If the natural response grows without bound, the system is unstable and can lead to self-destruction
- Examples of instability include an elevator crashing through the floor, an aircraft going into an uncontrollable roll, or an antenna rotating out of control
Other Considerations
- Other important design considerations include factors affecting hardware selection, finances, and robust design
- Robust design aims to create a system that is not sensitive to parameter changes
- System parameters considered constant during design may change over time, affecting system performance
System Models
- System models can be classified as linear or non-linear, deterministic or stochastic, and time-invariant or time-varying
- Time-invariant systems have coefficients that do not change with time, while time-varying systems have dynamics that change over time
- Examples of time-invariant and time-varying systems include a bicycle, a car, and a rocket
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Description
This quiz reviews the Laplace transform, a mathematical concept developed by French mathematician and astronomer Pierre-Simon Laplace. Learn about its applications in calculus, differential equations, and probability theory.
