It is a very interesting subject and a scoring one because most of the problems are based on numerical analysis and derivations and complex calculations and very less theory.

Even more interesting is when you get into the deep to the creation of algorithms for coding and decoding of information. There's a lot more if one is able to understand the concept of the study of this subject. The book is divided into simple chapters with illustrative figures and simpler formulations. The problems also are suitable for beginners and are not very extensive. Date uploaded Jan 12, Did you find this document useful?

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This cable was designed to be. New York University, Leonard N. Stern School of Business C You are now ready to enjoy the convenience. Soon after physicists began to understand electricity, the electrical engineering profession was born. Modern control methods became increasingly popular after with the invention of the computer, and the start of the space program.

Computers created the need for digital control methodologies, and the space program required the creation of some "advanced" control techniques, such as "optimal control", "robust control", and "nonlinear control".

These last subjects, and several more, are still active areas of study among research engineers. Branches of Control Engineering Here we are going to give a brief listing of the various different methodologies within the sphere of control engineering.

Oftentimes, the lines between these methodologies are blurred, or even erased completely. Classical Controls Control methodologies where the ODEs that describe a system are transformed using the Laplace, Fourier, or Z Transforms, and manipulated in the transform domain.

Modern Controls Methods where high-order differential equations are broken into a system of first-order equations. The input, output, and internal states of the system are described by vectors called "state variables".

Optimal Control In a system, performance metrics are identified, and arranged into a "cost function". The cost function is minimized to create an operational system with the lowest cost. Adaptive Control In adaptive control, the control changes it's response characteristics over time to better control the system. Nonlinear Control The youngest branch of control engineering, nonlinear control encompasses systems that cannot be described by linear equations or ODEs, and for which there is often very little supporting theory available.

Game Theory Game Theory is a close relative of control theory, and especially robust control and optimal control theories. In game theory, the external disturbances are not considered to be random noise processes, but instead are considered to be "opponents".

Each player has a cost function that they attempt to minimize, and that their opponents attempt to maximize. This book will definately cover the first two branches, and will hopefully be expanded to cover some of the later branches, if time allows. There are a number of other software tools that are useful in the analysis and design of control systems. Additional information can be added in the appendix of this book, depending on the experiance and prior knowledge of contributors. Equations that are labeled in such a manner are important, and should be taken special note of.

For instance, notice the label to the right of this equation: [Inverse Laplace Transform] Information which is tangent or auxiliary to the main text will be placed in these "sidebox" templates. Examples will appear in TextBox templates, which show up as large grey boxes filled with text and equations. Important Definitions Will appear in TextBox templates as well, except we will use this formatting to show that it is a definition.

Notes of interest will appear in "infobox" templates. These notes will often be used to explain some nuances of a mathematical derivation or proof. Warnings will appear in these "warning" boxes. These boxes will point out common mistakes, or other items to be careful of. Systems, in the barest sense, are devices that take input, and produce an output.

The output is related to the input by a certain relation known as the system response. The system response usually can be modeled with a mathematical relationship between the system input and the system output. There are many different types of systems, and the process of classifying systems in these ways is called system identification.

System Identification Physical Systems can be divided up into a number of different catagories, depending on particular properties that the system exhibits. Some of these system classifications are very easy to work with, and have a large theory base for studying. Some system classifications are very complex, and have still not been investigated with any degree of success.

This book will focus primarily on linear time-invariant LTI systems. LTI systems are the easiest class of system to work with, and have a number of properties that make them ideal to study. In this chapter, we will discuss some properties of systems, and we will define exactly what an LTI system is. Additivity A system satisfies the property of additivity, if a sum of inputs results in a sum of outputs. By definition: an input of results in an output of.

Homogeniety A system satisfies the condition of homogeniety if an input scaled by a certain factor produces an output scaled by that same factor.

Linearity A system is considered linear if it satisfies the conditions of Additivity and Homogeniety. In short, a system is linear if the following is true: We take two arbitrary inputs, and produce two arbitrary outputs: Now, a linear combination of the inputs should produce a linear combination of the outputs: This condition of additivity and homogeniety is called superposition.

A system is linear if it satisfies the condition of superposition. Causality Causality is a property that is very similar to memory. A system is called causal if it is only dependant on past or current inputs. A system is called non-causal if the output of the system is dependant on future inputs. This book will only consider causal systems, because they are easier to work with and understand, and since most practical systems are causal in nature.

Memory A system is said to have memory if the output from the system is dependant on past inputs or future inputs! A system is called memoryless if the output is only dependant on the current input. Memoryless systems are easier to work with, but systems with memory are more common in digital signal processing applications. Systems that have memory are called dynamic systems, and systems that do not have memory are instantaneous systems. Time-Invariance A system is called time-invariant if the system relationship between the input and output signals is not dependant on the passage of time.

If the input signal produces an output then any time shifted input, , results in a time-shifted output This property can be satisfied if the transfer function of the system is not a function of time except expressed by the input and output.

If a system is time-invariant then the system block is commutative with an arbitrary delay. We will discuss this facet of time-invariant systems later. LTI systems are one of the most important types of systems, and we will consider them almost exclusively in this book.

Lumpedness A system is said to be lumped if one of the two following conditions are satisfied: 1. There are a finite number of states 2. There are a finite number of state variables. Systems which are not lumped are called distributed. We will not discuss distributed systems much in this book, because the topic is very complex.

Relaxed A system is said to be relaxed if the system is causal, and at the initial time t0 the output of the system is zero. Stability Stability is a very important concept in systems, but it is also one of the hardest function properties to prove.

There are several Control Systems engineers will frequently say that an unstable system has different criteria for system stability, but the most common "exploded".

Some physical systems requirement is that the system must produce a finite output when actually can rupture or explode when they subjected to a finite input. For instance, if we apply 5 volts to the go unstable. Therefore, this book will spend a large amount of time discussing system stability. This book is going to consider both analog and digital topics, so it is worth taking some time to discuss the differences, and to display the different notations that will be used with each.

Continuous Time A signal is called continuous-time if it is defined at every time t. A system is a continuous-time system if it takes a continuous-time input signal, and outputs a continuous-time output signal. Discrete Time A signal is called discrete-time if it is only defined for particular points in time.

A digital system takes discretetime input signals, and produces discrete-time output signals. Quantized A signal is called Quantized if it can only be certain values, and cannot be other values.

Analog By definition: Analog A signal is considered analog if it is defined for all points in time, and if it can take any real magnitude value within it's range. An analog system is a system that represents data using a direct conversion from one form to another. Example: Motor If we have a given motor, we can show that the output of the motor rotation in units of radians per second, for instance is a function of the amount of voltage and current that are input to the motor.

For any value of v we can calculate out specifically what the rotational speed of the motor should be. Example: Analog Clock Consider a standard analog clock, which represents the passage of time though the angular position of the clock hands.

The positions of all the different hands of the clock are dependant on functions of time. Different positions on a clock face correspond directly to different times of the day. Digital Digital data is represented by discrete number values. By definition: Digital A signal or system that is discrete-time and quantized.

Digital data always have a certain granularity, and therefore there will almost always be an error associated with using such data, especially if we want to account for all real numbers. The tradeoff, of course, to using a digital system is that our powerful computers with our powerful, Moore's law microprocessor units, can be instructed to operate on digital data only. This benefit more then makes up for the shortcomings of a digital representation system. Discrete systems will be denoted inside square brackets, as is a common notation in texts that deal with discrete values.

Analog, or "non-discrete" values are denoted in regular expression syntax, using parenthesis. Example: Digital Clock As a common example, let's consider a digital clock: The digital clock represents time with binary electrical data signals of 1 and 0. Counting in binary, we can show that any given time can be represented by a base-2 numbering system: Minute Binary Representation 1 1 10 30 59 But what happens if we want to display a fraction of a minute, or a fraction of a second?

A typical digital clock has a certain amount of precision, and it cannot express fractional values smaller then that precision. Hybrid Systems Hybrid Systems are systems that have both analog and digital components. Devices called samplers are used to convert analog signals into digital signals, and Devices called reconstructors are used to convert digital signals into analog signals.

Because of the use of samplers, hybrid systems are frequently called sampled-data systems. Example: Car Computer Most modern automobiles today have integrated computer systems, that monitor certain aspects of the car, and actually help to control the performance of the car. The speed of the car, and the rotational speed of the transmission are analog values, but a sampler converts them into digital values so the car computer can monitor them.

The digital computer will then output control signals to other parts of the car, to alter analog systems such as the engine timing, the suspension, the brakes, and other parts.

Because the car has both digital and analog components, it is a hybrid system. Continuous and Discrete A system is considered continuous-time if the signal exists for all time. Frequently, the terms "analog" and "continuous" will be used interchangably, although they are not strictly the same. Discrete systems can come in three flavors: Note: We are not using the word "continuous" here in the sense of continuously differentiable, as is common in math texts.

Discrete time 2. Discrete magnitude quantized 3. Discrete time and magnitude digital Discrete magnitude systems are systems where the signal value can only have certain values. Discrete time systems are systems where signals are only available or valid at particular times.

A discrete-time system has as sampling time value associated with it, such that each discrete value occurs at multiples of the given sampling time.

We will denote the sampling time of a system as T.

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