Calculus Basic Concepts And Applications Pdf
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Calculus 2 Pdf
The fractional order calculus FOC is as old as the integer one although up to recently its application was exclusively in mathematics. Those characteristics have attracted the engineers' interest in the latter years, and now it is a tool used in almost every area of science.
This paper introduces the fundamentals of the FOC and some applications in systems' identification, control, mechatronics, and robotics, where it is a promissory research field.
The fractional order calculus FOC was unexplored in engineering, because of its inherent complexity, the apparent self-sufficiency of the integer order calculus IOC , and the fact that it does not have a fully acceptable geometrical or physical interpretation [ 1 , 2 ]. Notwithstanding it represents more accurately some natural behavior related to different areas of engineering, and now it is used as a promissory tool in bioengineering [ 3 , 4 ], viscoelasticity [ 5 , 6 ], electronics [ 7 , 8 ], robotics [ 9 — 11 ], control theory [ 12 , 13 ], and signal processing [ 14 , 15 ] between others.
Some instance are thermal diffusion phenomenon [ 19 ], botanical electrical impedances [ 20 ], model of love between humans [ 21 ], the relaxation of water on a porous dyke whose damping ratio is independent of the mass of moving water [ 22 ], and so forth.
On the other hand, direction the behavior of a process with fractional order controllers would be an advantage, because the responses are not restricted to a sum of exponential functions, therefore a wide range of responses neglected by integer order calculus would be approached [ 23 ]. Bearing these ideas in mind, this paper is organized as follows. Section 2 presents the fundamentals and analytical definitions.
Section 3 introduces several approaches to the solution of the years-old problem of the geometrical interpretation of the FOC. Then in Section 4 some applications in systems' identification, control, and robotics are presented. Finally Section 5 introduces the main conclusions and future applications of FOC.
It is a generalization of the IOC to a real or complex order [ 25 ]. Formally the real order generalization is introduced as follows: with. Its applications in engineering were delayed because FOC has multiple definitions [ 18 , 26 ], there is not a simple geometrical interpretation and the IOC seems, at first sight, to be enough to solve engineering problems.
However, many natural phenomena may be better described by a FOC formulation, because it takes into account the past behavior and it is compact when expressing high-order dynamics [ 27 , 28 ]. We can choose one definition or another, depending on the application and the preference of the designer. In [ 26 ] the authors compare these definitions in applications of control and signals processing, finding that the Cauchy definition preserves some important frequency properties, that also exist in IOC simplifying the data's interpretation.
Some other tools of interest for engineers are the classical transforms of Laplace and Fourier, that are valid and used in order to simplify operations like convolution and can be applied to solve FOC differential equations.
In FOC the Laplace transform is defined as [ 30 ]. As shown, this transform takes into account all initial conditions from the first to the derivative. In practice, the Fourier transform can be achieved by replacing by in 2. In addition to the problem for which definition must be chosen based on its properties or implementation complexity, the engineers may know the implications of using a mathematical tool.
An easy way to understand it, is by plotting it in a figure and seeing what is happening when it is applied. Pitifully for FOC it is a lack, but some approaches were proposed in the last decade, as will be presented in Section 3. In the case of integral order calculus, there is a well-accepted geometrical explanation which clearly relates some physical quantities, for example, instant rate of change of a function completely explains the relationship between concepts like position and speed of an object.
Unfortunately, until the last decade there was no geometrical interpretation of the fractional order derivatives. If is a value between 0 and 1, and is defined as. Then for we obtain , that is, the value of the function at evaluation time present appears with probability of.
For values of the value vanishes when the analysis point is far from the evaluation one. Therefore, the author suggests that the expression is the expected value of a random variable , where. Therefore the values near to the evaluation time present have more influence over the result than those that are far from it.
This interpretation is shown in Figure 1. A geometric interpretation based on Riemann-Louville-definition 2. This definition can be written as with. With this information a tridimensional graph is drawn with axes , and as shown in Figure 2. The projection of the area below , over the plane , is , the same as the integer integral definition. The projection of the area below the curve, over the plane , is. Note that it is the same definition as 3.
Another geometrical interpretation, this time based fractal dimension was proposed in [ 32 ]. Here the author argues that the Riemman-Lioville 2. For , the function is undetermined. By increasing fractional integral case , the kernel takes into account the effect of the past values, weighting them Figure 3.
If integer integral case , then has perfect memory and all the past is equally weighted. In the derivative case , the interpretation cannot be explicit obtained from 2. Using the Leibniz rule on 2. Note that the kernel of 3. The derivative value is the sum of the effect of the initial condition and the value of the integer derivative, both regulated by the kernel. The kernel behavior is similar to the Cantor set [ 33 ], that is, an iterative function that removes the middle section of a line Figure 4.
The fractal dimension of the Cantor set is defined by. In this case the dimension represents the density of the remaining bars and it is analogous to the kernel , but in discrete time.
Fractional order can represent systems with high-order dynamics and complex nonlinear phenomena using few coefficients [ 6 , 34 , 35 ], since the arbitrary order of the derivatives gives an additional degree of freedom to fit an specific behavior. Another important characteristic is that the fractional order derivatives depend not only on local conditions of the evaluated time, but also on all the history of the function. However, it is also a problem when fractional derivative functions are implemented in logical circuits, because they require a huge quantity of physical memory.
The strategies to simulate fractional order systems are classified in three groups [ 36 , 37 ]. These methods present multiple parameters and are complicated to analyze, as it is necessary to evaluate every single point in the function and its history; moreover, the explicit equation if often difficult to obtain.
The analytical system is replaced by its discrete equivalent in frequency space. On the other hand, if the series has a lot of coefficients, it limits simulation in real time, as it requires more processing cycles. This is approximated by rational continuous approach, but the series must be truncated; therefore, it must be limited to a specific frequency range of operation.
Another way to obtain the response of a fractional order system is by using analogical circuits with fractional order behavior as shown in Figure 5 or systems with fractal configuration as shown in Figure 6 a. Here three methods are introduced. The approximation of the transfer function is done by the recursive circuit shown in Figure 5. The gain between and in Laplace transform is the continuous fraction approximation to the original system [ 38 ], that is where and.
This circuit has two principal disadvantages: it has a limited frequency band of work, and this is an approximation, therefore it requires a lot of low tolerance components, depending on the accuracy required by the designer.
The designer implements the circuit component by component into a FPAA. It allows changing of the dynamical behavior of the fractional order system with a few simple modifications and each element has custom tolerance.
It is a capacitor with fractional order behavior introduced in [ 40 ]. In general it consists in a capacitor of parallel plates, where one of them presents fractal dimension Figure 6 a. Anyone of these approaches could be used in engineering applications, in this paper we introduce its use in systems' identification, control theory and robotics.
Fractional order dynamical systems can be modeled using the Laplace transform-like transfer functions [ 41 ] as with , , and. Some high-order systems would be approximated with a compact fractional order expression, it is useful in cases where an approach between holistic and detailed description of the process is required.
As an instance the model of the 5th order [ 7 ]. Many real systems are better identified as fractional order equations [ 16 , 42 ] than integer ones. In fact, some responses cannot be approximated just as a linear combination of exponential functions [ 43 ], and the arbitrary order is an additional degree of freedom that yields a better approximation to the real system while describing it in a compact way [ 44 ]. In [ 45 ] it was used this fact to identify a fractal system, typically modeled in frequency as:.
Adjusting the model is accomplished by finding the parameters that minimize the mean error with the real data. Another instance of the fractional order formulation is presented in [ 46 ], the authors approximated a complex system, a flexible structure with five vibration modes, modeling it with few parameters, being still valid for a wide range of frequencies.
They propose the transfer function: with , and. A real value of models the damper behavior without increasing the order of the system, and maintaining a compact expression too, valid for the frequency range. As the author demonstrate by the experiment shown in Figure 8 , by applying a sine voltage and analyzing the current over the object.
A nonparametric method introduced in [ 47 ] uses a continuous neural network CNN in order to identify nonlinear systems. This type of networks uses integral blocks instead of time delays.
From this kind of network is possible to separate the static nonlinear system neural network from the dynamical one integral blocks. If the integral blocks are fractional order blocks, then the captures the fractional behavior too. In order to train the network, the authors used the square mean error between the system output and the neural network output Figure 9.
Just as an example, we propose an experiment with synthetic data, simulating the vibration present in a gearbox. These kinds of systems are highly complex as several frequencies and their harmonics are exited by the rotation of the axes, unbalanced pieces, meshing between gears, bearing balls interaction, backslash between pieces among others.
When the system has a failure, harmonics and side-bands are added to the frequency spectrum and the dynamical model of the system may change. If these models were known a predictive maintenance strategy would be proposed based on comparison between them.
Unfortunately as there are many components interacting and some have nonlinear behavior, a dynamical model of integer order is frequently difficult to obtain and involve several parameters that are hardly comparable. When a failure is introduced, the model of the system change. In this case the failure was identified with just one parameter, the order of the equation. Dynamic systems are typically fractional order, but often just the controller is designed as that, as the plant is modeled with integer order differintegral operators.
A robust fractional order controller requires less coefficients than the integer one [ 48 ]. Grouping by type of plant and controller, the systems are classified in four sets [ 49 ], as shown in Table 1. In [ 49 ] it is proved that fractional order controllers are more robust than integer order. The authors proposed two dynamic systems with three coefficients, an integer system of second order and a system of fractional order with three coefficient.
They optimized those controllers and found that fractional algorithms were more stable taking into account stationary error and the overshoot percentage.
It is a controller similar to the of integer order in its architecture, but replacing the proportional component by a function like , with.
Calculus 1 Pdf
Calculus 1 Pdf. Unit 1 - Limits and Continuity. Free math lessons and math homework help from basic math to algebra, geometry and beyond. Remember the expectation isn't that they finish all the problems the expectation is that they give their personal best for 1 minute. Math teachers love to factor things.
Summary: Calculus and Its Applications, Thirteenth Edition is a comprehensive, yet flexible, text for students majoring in business, economics, life science, or social sciences. The authors delve into greater mathematical depth than other texts while motivating students through relevant, up-to-date, applications drawn from students' major fields of study. The authors motivate key ideas geometrically and intuitively, providing a solid foundation for the more abstract treatments that follow. Every chapter includes a large quantity of exceptional exercises-a hallmark of this text--that address skills, applications, concepts, and technology. The Thirteenth Edition includes updated applications, exercises, and technology coverage. The authors have also added more study tools, including a prerequisite skills diagnostic test and a greatly improved end-of-chapter summary, and made content improvements based on user reviews. Specifically written for undergraduate students, this guide provides an accessible introduction to computer hardware and architecture.
Calculus 2 Pdf In Chapter 6, basic concepts and applications of integration are discussed. Swokowski Calculus 6th Edition Pdf. One is a symbolic expression such as sin x or x 2. Given a differential equation and an initial value: a. Active Calculus - single variable is designed to support an active learning approach in the standard first two semesters of calculus, including approximately activities and exercises. You signed out in another tab or window. Integral calculus, sequences and series, parametric curves, polar coordinates, optional: vectors.
Chapter R. This chapter covers basic concepts related to functions, graphing, and modeling. It is an optional chapter based on the prerequisite skills students.
Fundamental Theories and Their Applications of the Calculus of Variations
It seems that you're in Germany. We have a dedicated site for Germany. Authors: Lao , Dazhong, Zhao , Shanshan. This book focuses on the calculus of variations, including fundamental theories and applications.
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