Laplace transforms

Laplace transforms Laplace Transforms – Motivation convenience – differential eqns become algebraic eqns. – easy to handle time delays – frequency response analysis to determine how the system responds to oscillating inputs Block Diagram Algebra – doing math with pictures – arithmetic for manipulating dynamic components using boxes and arrows Laplace Transform – Review Given a function f(t) Notes f(t) defined for t from 0 to infinity f(t) suitably â€Å"well-behaved† – piecewise continuous, integrable Linearity of Laplace Transforms the Laplace transform is a linear operation we will use Laplace transforms to analyze linear dynamic systems if our models arent linear, then we will linearize Useful Laplace Transforms for Process Control We need a small library of Laplace transforms for – differentiation – step input – pulse/impulse functions – exponentials – oscillating functions because these are common functions that we will encounter in our equations Lets think about a simple linear differential equation example: with V and F as constants Library of Useful Transforms differentiation – initial conditions disappear if we use deviation variables that are zero at an in initial steady state unit step function (Heaviside fn.) Library of Transforms exponential – exponentials appear in solutions of differential equations  » a provides information about the speed of the response when the input changes. If a is a large negative number, the exponential decays to zero quickly  » What happens if a is positive? – After we have done some algebra to find a solution to our ODEs in the Laplace domain, we must invert the Laplace transform if we want to get a solution in the time domain. We sometimes use partial fraction expansion to express the Laplace expressions in a form that can be easily inverted. CSTR Example Transform Model (in deviation variables) using our library of transforms, the Laplace transform of the model is: For a step change in feed concentration at time zero starting from steady state. Tank Example Solution Solve for CA(s) If we like, we can rearrange to the form: This is the solution in the Laplace domain. To find the solution in the time domain, we must invert the Laplace transforms CSTR Example Solution inverse Laplace transform – Can be determined using a complex integral easiest approach is â€Å"table lookup† Use Table 4-1, entry 5 Maple is good at inverting Laplace transforms too The Impulse Function limit of the pulse function (with unit area) as the width goes to zero and height becomes infinite transform CSTR Impulse Response physically dump some pure A into reactor, all at once input function Transform time response Interpretation of Impulse Response dump a bag of reactant into the reactor in a very very short time we see an instantaneous jump to a new concentration due to the impulse input concentration then decays back to the original steady-state concentration Time-Shifted Functions Representation of Delays Laplace transform for function with time delay Just pre-multiply by an exponential. How could we prove this? – change of variables in integration in expression for Laplace Transform (see p. 103 of Marlin, p. 115 in first ed.) Reactor Example with Time Delay Suppose we add a long length of pipe to feed – assume plug flow – It will take a time period, q minutes, before the change in concentration reaches the tank, and begins to influence cA – delay differential equation  » difficult to solve directly in time domain  » easy to solve with Laplace transforms Tank Example with Time Delay – Solutionresponse to step input in cA0 time response Final Value Theorem An easy way to find out what happens to the output variable if we wait a long time. We dont have to invert the Laplace transform! Why is it true? – Consider the Laplace transform of a time derivative now let s approach zero provided dy/dt isnt infinite between t=0 and t ®Ã‚ ¥ (i.e y(t) is STABLE) This will be true if Y(s) is continuous for s ³0 Using the Final Value Theorem Step Response Reactor example final value after a step input What can we do with Laplace Transforms so far. Take Laplace transforms of linear ODEs (in deviation variables). Substitute Laplace transform expressions for different kinds of inputs we are interested in: – Steps, pulses, impulses (even with dead time) Solve for the output variable in terms of s. Invert the Laplace transform using Table 4.1 to get the solution in the time domain. Find the final steady state value of the output variable, for a particular input change, even without inverting the Laplace transform. Laplace transforms are mostly used by control engineers who want to determine and analyze transfer functions. compact way of expressing process dynamics relates input to output p(s), q(s) polynomials in s – q(s) will also contain exponentials if time delay is present Once we know the transfer function of the process, we can use it to find out how the process responds to different types of input changes:

Generations of Computer Essay

The development of the integrated circuit was the hallmark of the third generation of computers. Transistors were miniaturized and placed on silicon chips, called semiconductors, which drastically increased the speed and efficiency of computers. A chip is a small piece of semi conducting material(usually silicon) on which an integrated circuit is embedded. A typical chip is less than  ¼-square inches and can contain millions of electronic components(transistors). Computers consist of many chips placed on electronic boards called printed circuit boards. There are different types of chips. For example, CPU chips (also called microprocessors) contain an entire processing unit, whereas memory chips contain blank memory. Computer chips, both for CPU and memory, are composed of semiconductor materials. Semiconductors make it possible to miniaturize electronic components, such as transistors. Not only does miniaturization mean that the components take up less space, it also means that they are faster and require less energy. Instead of punched cards and printouts, users interacted with third generation computers through keyboards and monitors and interfaced with an operating system, which allowed the device to run many different applications at one time with a central program that monitored the memory. Computers for the first time became accessible to a mass audience because they were smaller and cheaper than their predecessors. Fourth Generation – 1971-Present: Microprocessors The microprocessor brought the fourth generation of computers, as thousands of integrated circuits we rebuilt onto a single silicon chip. A silicon chip that contains a CPU. In the world of personal computers, the terms microprocessor and CPU are used interchangeably. At the heart of all personal computers and most workstations sits a microprocessor. Microprocessors also control the logic of almost all digital devices, from clock radios to fuel-injection systems for automobiles. Three basic characteristics differentiate microprocessors: * Instruction Set: The set of instructions that the microprocessor can execute. * Bandwidth: The number of bits processed in a single instruction. * Clock Speed: Given in megahertz (MHz), the clock speed determines how many instructions per second the processor can execute. In both cases, the higher the value, the more powerful the CPU. For example, a 32-bit microprocessor that runs at 50MHz is more powerful than a 16-bitmicroprocessor that runs at 25MHz. The CPU is the brains of the computer. Sometimes referred to simply as the processor or central processor, the CPU is where most calculations take place. In terms of computing power, the CPU is the most important element of a computer system. On large machines, CPUs require one or more printed circuit boards. On personal computers and small workstations, the CPU is housed in a single chip called a microprocessor. Two typical components of a CPU are: * The arithmetic logic unit (ALU), which performs arithmetic and logical operations. * The control unit, which extracts instructions from memory and decodes and executes them, calling on the ALU when necessary. In 1981 IBM introduced its first computer for the home user, and in 1984 Apple introduced the Macintosh. Microprocessors also moved out of the realm of desktop computers and into many areas of life as more and more everyday products began to use microprocessors. As these small computers became more powerful, they could be linked together to form networks, which eventually led to the development of the Internet. Fourth generation computers also saw the development of GUI’s, the mouse and handheld devices. Fifth Generation – Present and Beyond: Artificial Intelligence Fifth generation computing devices, based on artificial intelligence, are still in development, though there are some applications, such as voice recognition, that are being used today. Artificial Intelligence is the branch of computer science concerned with making computers behave like humans. The term was coined in 1956 by John McCarthy at the Massachusetts Institute of Technology. Artificial intelligence includes: * Games Playing: programming computers to play games such as chess and checkers * Expert Systems: programming computers to make decisions in real-life situations (for example, some expert systems help doctors diagnose diseases based on symptoms) * Natural Language: programming computers to understand natural human languages * Neural Networks: Systems that simulate intelligence by attempting to reproduce the types of physical connections that occur in animal brains * Robotics: programming computers to see and hear and react to other sensory stimuli Currently, no computers exhibit full artificial intelligence (that is, are able to simulate human behavior). The greatest advances have occurred in the field of games playing. The best computer chess programs are now capable of beating humans. In May, 1997, an IBM super-computer called Deep Blue defeated world chess champion Gary Kasparov in a chess match. In the area of robotics, computers are now widely used in assembly plants, but they are capable only of very limited tasks. Robots have great difficulty identifying objects based on appearance or feel, and they still move and handle objects clumsily. Natural-language processing offers the greatest potential rewards because it would allow people to interact with computers without needing any specialized knowledge. You could simply walk up to a computer and talk to it. Unfortunately, programming computers to understand natural languages has proved to be more difficult than originally thought. Some rudimentary translation systems that translate from one human language to another are in existence, but they are not nearly as good as human translators. There are also voice recognition systems that can convert spoken sounds into written words, but they do not understand what they are writing; they simply take dictation. Even these systems are quite limited — you must speak slowly and distinctly. Today, the hottest area of artificial intelligence is neural networks, which are proving successful in a number of disciplines such as voice recognition and artificial intelligence.

Money For The Future College - 955 Words

Money For The Future Many say that education is a right, not a privilege. Others may argue that education, especially higher education like college, is a privilege. Everyone is entitled to an education, but what is holding a lot of students back is their financial situation. The cost of college education is getting out of hand; its prices are alarming, leaving students with little ways to pay for it, in the dust. The unfortunate reality of students dealing with debt is a fear-mongering tactic used so that students sike themselves out and never attend college. This harsh reality applies to many students who have to pay for their college expenses out of pocket, unlike the lucky few who attend accredited four-year colleges with the money of their financially stable parents. According to the essay Is College Tuition a Sound Investment, college is not only time consuming, but expensive. Its prices are rising higher and higher every single year, costing students roughly 15,000 dollars a ye ar for tuition and room and board. Some say that these prices are priced fairly and by doing so it creates competition between colleges around the nation to attract students to the commodities that these colleges offer. Though expensive, there is always financial aid and scholarships available to anyone who needs it. The author states that some colleges are expensive because of the different luxuries they offer which includes the high quality of education provided, which is inevitablyShow MoreRelatedCollege Education Essay1257 Words   |  6 PagesComparing between college education and the other secondary things we need in life, mostly education comes at the first. Many high school students know pretty much everything about how much college education is essential in their life and many students want to have college education. 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Essay on Accommodating Religion in the Workplace - 977 Words

The circumstances, images and fallout related to September 11, 2001 have caused many Americans to revisit their spiritual beliefs in attempt to process the horrific events of such an unimaginable day (Smith, 2003). In the years since the Fall of 2001, employees have developed an increasing desire to integrate their personal spirituality and religious beliefs with their professional lives (Cunningham, 2010). Managers are now faced with the difficult task of accommodating the varying spiritual beliefs of their workforce while tactfully mitigating religion-based issues in accordance with Title VII. Food Consumption When Maya failed to consume the majority of her dish after announcing her hunger at a recent business luncheon with†¦show more content†¦If Maya’s coworkers find her behavior so concerning that they want to peruse a conversation about it, they should look to have a brief discussion in private. If Maya elects not to delve into the rational behind her behavior, the coworkers will have no choice but to move on from the situation. Food Consumption in the Workplace From Ash Wednesday to Easter Sunday, the Catholic religion observes Lent, which is marked by many religious practices including self-denial. During this period management and coworkers may experience their peers choosing not to consume sweets, junk food, coffee or alcohol. Management should be prepared at this time to field questions or concerns related to the behavior of employees observing Lent if they choose not to participate, for example, in employee activities such as the morning coffee run or afterhours happy hour. Sunday Teambuilding After Jenny declined to be part of her new department’s softball team despite her colligate softball success, many coworkers characterized her as not being a team player without considering the rationale behind her decision. 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Moreover, the influx of people from different backgrounds bringing with them different religious practices has caused organizations to seek help from organizations such as the Equal Employment Opportunity Commission (EEOC), the courts, and legal firms on how to deal with this new reality. Thus, the complexities surrounding religion and spirituality