# MODERN CONTROL DESIGN WITH MATLAB AND SIMULINK PDF

Modern Control Design With MATLAB and SIMULINK - Ebook download as PDF File .pdf), Text File .txt) or read book online. Download as PDF, TXT or read online from Scribd MATLAB, SIMULINK, and the Control System Toolbox 11 Advanced Topics in Modern Control MATLAB/SIMULINK combination has become the single most common - and industry-wide standard - software in the analysis and design of modern control.

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The idea of computer-aided design and analysis using MATLAB with the Ogata, K., Modern Control Engineering, 3rd ed., Prentice Hall, Englewood Cliffs. Request PDF on ResearchGate | Modern Control Systems Analysis and Design Using MATLAB and SIMULINK | From the Publisher:Modern Control Systems. Written for students and practicing engineers, this book presents the theory and applications of classical and state-space control system design. Topics covered.

Of particular importance are the protocol stacks that drive industry-standard interfaces like USB. The hardware blocks are put together using computer-aided design tools, specifically electronic design automation tools; the software modules are integrated using a software integrated development environment.

Once the architecture of the SoC has been defined, any new hardware elements are written in an abstract hardware description language termed register transfer level RTL which defines the circuit behavior, or synthesized into RTL from a high level language through high-level synthesis. These elements are connected together in a hardware description language to create the full SoC design. The logic specified to connect these components and convert between possibly different interfaces provided by different vendors is called glue logic.

Further information: Functional verification and Signoff electronic design automation Chips are verified for logical correctness before being sent to a semiconductor foundry. Bugs found in the verification stage are reported to the designer. Traditionally, engineers have employed simulation acceleration, emulation or prototyping on reprogrammable hardware to verify and debug hardware and software for SoC designs prior to the finalization of the design, known as tape-out.

Both technologies, however, operate slowly, on the order of MHz, which may be significantly slower — up to times slower — than the SoC's operating frequency. This is used to debug hardware, firmware and software interactions across multiple FPGAs with capabilities similar to a logic analyzer.

In parallel, the hardware elements are grouped and passed through a process of logic synthesis , during which performance constraints, such as operational frequency and expected signal delays, are applied.

This generates an output known as a netlist describing the design as a physical circuit and its interconnections. Kamaleshwar Sahai Tewari.

To my wife, Prachi, and daughter, Manya. Introduction 1 1. Linear Systems and Classical Control 13 2. State-Space Representation 3.

Why Do I Need It? The Canonical Forms 3. Solving the State-Equations 4. Control System Design in State-Space 5. Classical vs. Modern 5. Linear Optimal Control 6. Kalman Filters 7. Loop Transfer Recovery Exercises References Digital Control Systems 8. Regulators, Observers, and Compensators 8. Advanced Topics in Modern Control 9. Appendix A: At the end of each chapter. This book can be used as a textbook in an introductory course on control systems at the third.

Many textbooks are available on modern control. Many modern control applications are interdisciplinary in nature. Kalman filters. An effort is made to explain the underlying princi- ples behind many controls concepts. Bearing this in mind. Preface The motivation for writing this book can be ascribed chiefly to the usual struggle of an average reader to understand and utilize controls concepts.

As stated above. This book aims at introducing the reader to the basic concepts and applications of modern control theory in an easy to read manner.

Continuity in reading is preserved.

The numerical examples and exercises are chosen to represent practical problems in modern control. Since the book is intended to be of introduc- tory rather than exhaustive nature. Robert Hambrook. Dawn Booth and See Hanson for their encouragement and guidance in the preparation of the manuscript. This is to avoid over-dependence on a particular version of the Control System Toolbox.

I thank all. Readers are introduced to advanced topics such as HOC-robust optimal control. While the main focus of the material presented in the book is on the state-space methods applied to linear. After going through the book.

I would like to specially thank Karen Mossman. Simon Plumtree. This is perhaps the only available controls textbook which gives ready computer programs to solve such a wide range of problems. The theoretical background and numerical techniques behind the software commands are explained in the text. Gemma Quilter. Older versions of this software can also be adapted to run the examples and models presented in the book. Answers to selected numerical exercises are provided near the end of the book.

Many of the examples contain instructions on programming. I would also like to thank my students and colleagues. I am grateful to the editorial and production staff at the Wiley college group.

I found working with Wiley. Writing this book would have been impossible without the constant support of my wife. Ashish Tewari. When referring to an isolated system. Let us take the example of the control system consisting of a car and its driver. The ultimate control system is the human body. The daily variation of temperature caused by the sun controls the metabolism of all living organisms. We have twice used the word system without defining it. We often do not realize how controlled the natural environment we live in is.

The composition. Imagine living in a world where the temperature is unbearably hot or cold. Even the simplest life form is sustained by unimaginably complex chemical processes.

The picture we see on the television is a result of a controlled beam of electrons made to scan the television screen in a selected pattern. A system is a set of self-contained processes under study. The temperature inside a refrigerator is controlled by a thermostat. You have to wonder who designed that control system! The controller is said to supply a signal to the plant. A compact-disc player focuses a fine laser beam at the desired spot on the rotating compact-disc in order to produce the desired music.

A controller could be either human. If we select the car to be the plant. A study of control involves developing a mathematical model for each component of the control system. By this broad definition. A control system by definition consists of the system to be controlled. Whether the control is automatic such as in the refrigerator.

The list is endless. When we use the word control in everyday life. While driving a car. The speed increase can then be the output from the plant. Say if the driver wants to make sure she is obeying the highway speed limit. Such a control system. By pressing the gas pedal control input she hopes that the car's speed will come up to the desired value. Now ask the driver to accelerate to a particular speed assuming that she continues driving in a straight line.

Note that in a control system. We have encountered the not so rare breed of drivers who generally boast of their driving skills with the following words: By now it must be clear that an open- loop controller is like a rifle shooter who gets only one shot at the target. Since the plant output is the same as the output of the control system. In other words. If the driver is overtaking a truck on the highway.

Figure 1. After understanding the basic terminology of the control system. If she wants to stop well before a stop sign. While driving in this fashion. On a block-diagram. If a sign is omitted. Now she can see her actual speed. In this situation. The actual car is not going to have such a simple transfer function. In the presence of uncertainty about the plant's behavior.

Suppose the driver knows from previous driving experience that. If one knows what output a system will produce when a known input is applied to it. Such an unknown and undesirable input to the plant. This circle is called a summing junction. The output of. Since in a closed-loop system the controller is constantly in touch with the actual output. The mechanism by which the information about the actual output is conveyed to the controller is called feedback.

This is a very simplified example. A block-diagram example of a possible closed-loop system is given in Figure 1. If the transfer function in the driver's mind was determined on smooth roads. Such a control system in which the control input is a function of the plant's output is called a closed-loop system. Note that now the driver. Suppose the driver decides to drive the car like a sane person i. In addition. Comparing Figures 1.

The transient response of a linear system depends largely upon the characteristics and the initial state of the system. Using the symbols u control input. In Chapter 4 we will see that the output of a linear system to an arbitrary input consists of a fluctuating sort of response called the transient response.

Whereas the desired output yd has been achieved after some time in Figure 1. In Figure 1. If the linear system is stable. The maximum overshoot is a property of the transient response. When the desired output. The controller transfer-function is the main design parameter in the design of a control system and determines how rapidly. We will see later how the controller transfer-function can be obtained.

In other closed-loop systems. Note that Figure 1. If the desired output is changing with time. A successful closed-loop controller design should achieve both a small maximum overshoot. In any case. In a feedforward control system.

While the most common closed-loop control system is the feedback control system. The feedforward controller incorporates some a priori knowledge of the plant's behavior. The reason why the output of a linear. To better understand this behavior of linear. Examples of energy dissipation processes are mechanical friction.

Examples of energy storage devices are a spring in a mechanical system. Due to a transfer of energy from the applied input to the energy storage or dissipation elements. As the time passes. The input is the flow rate of water supplied to the bucket. This definition is a little confusing. This reduces the burden on the driver. It is clear from the present example that many practical control systems can benefit from the feedforward arrangement.

In this section. For example. In the car driver example. Now the feedback controller consists of the driver and the gas-pedal mechanism. We will return to the concept of state in Chapter 3. To understand such classifications. Note that if the feedback controller is removed from Figure 1. The state of a system is any set of physical quantities which need to be specified at a given time in order to completely determine the behavior of the system.

Since the characteristics of a deterministic system can be found merely by studying its response to initial conditions transient response. In this book. We should. An example of chaotic control systems is a double pendulum Figure 1. It consists of two masses.

A system is called chaotic if even a small change in the initial conditions produces an arbitrarily large change in the system's state at a later time. When we toss a perfect coin. Such a problem of unpredictability is highlighted by a special class of deterministic systems. Going back to the definition of state. A control system is said to be deterministic when the set of physical laws governing the system are such that if the state of the system at some time called the initial conditions and the input are specified.

A response to initial conditions when the applied input is zero depicts how the system's state evolves from some initial time to that at a later time. Then at a later time. Suppose we do not apply an input to the system. The laws governing a deterministic system are called deterministic laws.

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A system that is not deterministic is either stochastic. A stochastic also called probabilistic system has such governing laws that although the initial conditions i. Due to an uncertainty in our knowledge of the governing deterministic laws. Taking the car driver example of Section 1. Chaotic systems are so interesting that they have become the subject of specialization at many physics and engineering departments.

Another situation when a deterministic system may appear to be stochastic is exemplified by the toss of a coin deliberately loaded to fall every time on one particular side either head or tail. Any unpredictable system can be mistaken to be a stochastic system. Note that we know the governing laws of this deterministic system. There are other minor classifications of control systems based upon the systems' char- acteristics.

In modern applications. Depending upon whether the differential equations used to describe a control system are linear or nonlinear in nature. The mathematical nature of the governing differential equations provides another way of classifying control systems. Note that in the limit of very small time steps. A vibrating string. Chapter 8 is devoted to the study of digital systems.

Other classifications based upon the mathematical nature of governing differential equations will be discussed in Chapter 2. A more common nomenclature of distributed and lumped parameter systems is continuous and discrete systems. Yet another way of classifying control systems is whether their outputs are contin- uous or discontinuous in time. If the time steps chosen to sample the discontin- uous output are relatively large.

Here we depart from the realm of physics. Such control systems are called discrete in time or digital systems. A mass suspended by a spring is a lumped parameter system.. When we analyze and design control systems. A particular system can be treated as linear.

If one can express the system's state which is obtained by solving the system's differential equations as a continuous function of time.

The design of modern control systems using the state-space approach is introduced in Chapter 5. Chapter 6 also provides new computer programs for solving important optimal control problems. Chapter 2 also discusses important properties of a control system. This has been an age old method of analyzing a system. Chapter 7 introduces the treatment of random signals generated by stochastic systems. In classical control an object of Chapter 2 the distinction between single-input.

Two prominent members of this zoo are the unit step function and the unit impulse function. Chapter 6 introduces the procedure of designing an optimal control system.

## Navigation menu

Here we also learn how an optimal state estimation can be carried out. In this chapter. Some of the well known inputs applied to study a system are the singularity functions.

Chapter 8 presents the design and analysis of. Chapter 3 introduces the state-space modeling for linear control systems. We treat an unknown control system in a similar fashion. The solution of a linear system's governing equations using the state-space method is discussed in Chapter 4.

Later sections of Chapter 5 contains material about how this process of state estimation is carried out by an observer. In Chapter 2. Many people. Chapter 9 introduces various advanced topics in modern control. The repetitive linear algebraic operations required in modern control design and analysis are. Try to remember the last time you attempted to invert a 4 x 4 matrix by hand!

It can be a tedious process for any matrix whose size is greater than 3 x 3. At the end of the book. At the end of each chapter except Chapter 1. Some of the topics contained in Chapter 9. Although not required for doing the exercises at the end of each chapter. Modern control design and analysis requires a lot of linear algebra matrix multipli- cation. These exercises range from analytical to numerical. User's Guide. For solving many problems in control. Control System Toolbox 5.

References 1. Many solved examples presented in this book require the Control System Toolbox. The Math Works Inc. In the solved examples. With advanced features such as the Real Time Workshop for C-code generation. For this purpose. Suppose we are capable of solving Eq. To answer these questions. In a large class of engineering applications. SISO system Figure 2. Since n is the order of the highest time derivative of y f in Eq. This notation for derivatives of a function will be used throughout the book.

Recall that lumped parameter systems are those systems whose behavior can be described by ordinary differential equations.

For a general lumped parameter. In Eq. For simplicity. The concept of linearity is one of the most important assumptions often employed in studying control systems.

To determine the output y t. The superposition principle is also applicable for non-zero initial conditions. Since linearity is a mathematical property of the governing differential equations.

In short. In the present chapter. If the resulting output. It is possible to express Eq. A system with time-varying parameters is called a time-varying system. Example 2. When a control system is designed for maintaining the plant at an equilibrium point.

Under such circumstances. A large majority of control systems are designed for keeping a plant at one of its equilibrium points. Such a system is called an unforced system. On inspection of Eq. Upon the substitution of Eq.

## Description

Suppose we do not have an input. Such constant solutions for an unforced system are called its equilibrium points. Also included is an example which illustrates that such a linearization may not always be possible. The equation of motion of the simple pendulum in the absence of an externally applied torque about point O in terms of the angular displacement. Since the only nonlinear term in Eq. From our everyday experience with a simple pendulum. The following examples demonstrate how a nonlinear system can be linearized near its equilibrium points.

Let us examine the behavior of the system near each of these equilibrium points. Due to the presence of sin. We will discuss stability in detail later. Second order linear ordinary differential equations especially the homogeneous ones like Eqs. It is well known and you may verify that the solution to Eq. The comparison of the solutions to the linearized governing equations close to the equilibrium points Figure 2.

While Example 2. The distance of the satellite from the center of the planet is denoted r r. Assuming there are no gravitational anomalies that cause a departure from Newton's inverse-square law of gravitation. Equation 2. The following missile guidance example illustrates such a nonlinear system. In such a case.

This linearization is left as an exercise for you at the end of the chapter. Many practical orbit control applications consist of minimizing deviations from a given circular orbit using rocket thrusters to provide radial acceleration i. Note that we could also have linearized Eq. Such specifications are called boundary conditions. If a nonlinear system has to be moved from one equilibrium point to another such as changing the speed or altitude of a cruising airplane.

A guidance law provides the following normal acceleration command signal. Although the distance from the beam source to the target. The guidance strategy is such that a correcting command signal input is provided to the missile if its flight path deviates from the moving beam. The feedback guidance scheme of Eq.

Figure 2. The guidance law given by Eq. This example shows that the concept of linearity. We accelerate or decelerate until our velocity and acceleration become identical with our friend's car.

It can be seen in Figure 2. To understand this philosophy. Then the beam's normal acceleration can be determined from the following equation: In such a case.. The two distinct singularity functions commonly used for determining an unknown system's behavior are the unit impulse and unit step functions. The singularity functions are important because they can be used as building blocks to construct any arbitrary input function and.

From this description. Another interesting fact about the singularity functions is that they can be derived from each other by differentiation or integration in time.

A unit impulse function can be multiplied by a constant to give a general impulse function whose area under the curve is not unity. A common property of these functions is that they are continuous in time. The unit impulse function shown in Figure 2.

The unit impulse function also called the Dime delta function. A set of such test functions is called singularity Junctions. For more information on missile guidance strategies. The height of the rectangular pulse in Figure 2. Like the unit impulse function. The unit step function. Recalling from Figure 2. It is aptly named. In fact. Comparing Eq. It can be expressed by multiplying the unit step function with f and a decaying exponential term.

The unit impulse function has a special place among the singularity functions. Recall from Section 1. As an alternative to singularity inputs which are often difficult to apply in practical cases. This fact is illustrated in Figure 2. While the singularity functions and their relatives are useful as test inputs for studying the behavior of control systems. An unstable system will have a transient response shooting to infinite magnitudes.

We shall study next how such a model can be obtained. We will see how this is done when we discuss the response to singularity functions in Section 2. The steady-state.

Of course. The complex space representation of the harmonic input given by Eq. Studying a linear system's characteristics based upon the steady-state response to harmonic inputs constitutes a range of classical control methods called the frequency response methods. Such methods formed the backbone of the classical control theory developed between 1 For these reasons.

Qlwt 2. For these powerful reasons. Modern control techniques still employ frequency response methods to shed light on some important characteristics of an unknown control system. This is an advantage. A simple choice of the harmonic input. If we choose to write the input and output of a linear system as complex functions. You will see that the equation is satisfied in each case. The particular solution is of the same form as the input.

We will shortly see the implications of a complex response amplitude. Consider a linear. While the transient response of a linear. By obtaining a steady-state response to the complex input given by Eq. In the complex space. We can also express the frequency response. You can easily show that if the harmonic input has a non-zero phase. From Eq. If we excite the system at various frequencies. Instead of the real and imaginary parts. The phasor representation of the steady-state response amplitude is depicted in Figure 2.

G ico. The length of the phasor in the complex space is called its magnitude. Equations 2. Representation of a complex quantity as a vector in the complex space is called a phasor. The magnitude of a phasor represents the amplitude of a harmonic function.

Bode plots can be plotted quite easily.

Polar plots have an advantage over the frequency plots of magnitude and phase in that both magnitude and phase can be seen in one rather than two plots. Bode plots are cumbersome to construct by hand.

In Bode plots. When talking about stability and robustness properties. With the availability of personal computers and software with mathematical functions and graphics capability. Such a plot of G ico in the complex space is called a polar plot since it represents G ico in terms of the polar coordinates.

Referring to Figure 2. Since the range of frequencies required to study a linear system is usually very large. In general. Despite this. G is the name given to the frequency response of the linear. If you don't specify w. The guitar player makes each string vibrate at a particular frequency.

Each string of the guitar is capable of being excited at many frequencies. Before we do that. These coefficients should be be specified as follows. The example given below will illustrate what Bode plots look like. Musical notes produced by a guitar are related to its frequency response.. Instead of plotting the Bode plot. In the bode command. When the switch. Just like the guitar. High pitched voice of many a diva has shattered the opera-house window panes while accidently singing at one of the natural frequencies of the window!

If a system contains energy dissipative processes called damping. A practical limitation of Bode plots is that they show only an inter- polation of the gain and phase through selected frequency points. The frequencies at which a system can be excited are called its natural or resonant frequencies. One could determine from the peaks the approximate values of the natural frequencies. The input to the system is the applied voltage. An undamped system. A natural frequency is indicated by a peak in the gain plot.

When we use the word excited. To verify whether this is the exact natural frequency. We are assuming. The Bode plots shown in Figure 2. The frequency response is used to define a linear system's property called bandwidth defined as the range of frequencies from zero up to the frequency. The plot is in polar coordinates. Linear systems with G ico having a higher degree denominator polynomial than the numerator polynomial in Eq.

For a general system. In the present plot. Let us now draw a polar plot of G ico as follows note that we need more frequency points close to the natural frequency for a smooth polar plot. The resulting polar plot is shown in Figure 2. It can be shown rigourously that the Laplace integral converges only if u t is piecewise continuous i. The polar curve is seen in Figure 2.

Most of the commonly used input functions are Laplace transformable. Here we would like to consider the total response both transient and steady-state of a linear.

## Modern Control Design: With MATLAB and SIMULINK

The term bounded implies that a function's value lies between two finite limits. The direction of increasing CD is shown by arrows on the polar curve. The Laplace transform of a function u t is defined only if the infinite integral in Eq.

We saw how the representation of a harmonic input by a complex function transformed the governing differential equations into a complex algebraic expression for the frequency response.

For a general input. The convergence of the Laplace integral depends solely upon the shape of the function. Some important properties of the Laplace transform are stated below. If we apply the real differentiation property successively to the higher order time derivatives of f t assuming they are Laplace transformable.

For such unknown systems. By applying known inputs such as the singularity functions or harmonic signals and measuring the output. Fourier transform is widely used as a method of calculating the Input. The transfer function representation of a system is widely used in block diagrams.

It is easy to see that if the input. A special transform. The latter relationship is easily obtained by comparing Eq. To do so. G itw. U s Output. Note that inverse Laplace transform. The Laplace variable. We can grasp this fact by applying the inverse Laplace transform.

U ico as follows: The transfer function. However, use of G s involves interpreting system characteristics from complex rather than purely imaginary numbers. The roots of the numerator and denominator polynomials of the transfer function, G s , given by Eq.

The denominator polynomial of the transfer function, G s , equated to zero is called the characteristic equation of the system, given by. The roots of the characteristic equation are called the poles of the system.

In terms of its poles and zeros, a transfer function can be represented as a ratio of factorized numerator and denominator polynomials, given by the following rational expression:. As in Eq. Such systems are said to be proper. Also, note that some zeros, z,, and poles, PJ, may be repeated i.

Such a pole or zero is said to be multiple, and its degree of multiplicity is defined as the number of times it occurs. Finally, it may happen for some systems that a pole has the same value as a zero i. Then the transfer function representation of Eq. Pole-zero cancelations have a great impact on a system's controllabilty or observability which will be studied in Chapter 5. To get a better insight into the characteristics of a system, we can express each quadratic factor such as that on the left-hand side of Eq.

The damping ratio, g, governs how rapidly the magnitude of the response of an unforced system decays with time.

For a mechanical or electrical system, damping is the property which converts a part of the unforced system's energy to heat, thereby causing the system's energy - and consequently the output - to dissipate with time. Examples of damping are resis- tances in electrical circuits and friction in mechanical systems. From the discussion following Eq. For the present example, the poles are found by solving Eq. These numbers could also have been obtained by comparing Eq.

The natural frequency agrees with our calculation in Example 2. One can see the dependence of the response, y t , on the damping-ratio, g, in Figure 2. Clearly, the larger the value of the damping-ratio, g, the faster the response decays to zero. As soon as we see a linear system producing an unbounded response to a bounded input i. A further discussion of stability follows a little later. Locations of poles and zeros in the Laplace domain determine the characteristics of a linear, time-invariant system.

Some indication of the locations of a poles and zeros can be obtained from the frequency response, G icu. Let us go back to Figure 2.

Due to the presence of a zero at the origin see Eq. The difference between the number of zeros and poles in a system affects the phase and the slope of the Bode gain plot with frequency in units of dB per decade of frequency , when the frequency is very large i.

Note that the expressions in Eq. For example, the transfer function in Eq. Three different output variables in the Laplace domain are of interest when the aircraft is displaced from the equilibrium point defined by a constant angle of attack, O. Q, a constant longitudinal velocity, DO, and a constant pitch-angle, OQ: The input variable in the Laplace domain is the elevator angle, d s.

The three transfer functions separately defining the relationship between the input, 5 5 , and the three respective outputs, v s , a s , and 0 s , are as follows:.

Since we know that the denomi- nator polynomial equated to zero denotes the characteristic equation of the system, we can write the characteristic equation for the aircraft's longitudinal dynamics as. Comparing the result with that of Example 2.

These values are the following: Note that the CST command damp also lists the eigenvalues, which are nothing but the roots of the characteristic polynomial same as the poles of the system. We will discuss the eigenvalues in Chapter 3. Alternatively, we could have used the intrinsic MATLAB function roots to get the pole locations as the roots of each quadratic factor. As expected, the poles for each quadratic factor in the characteristic equation are complex conjugates.

Instead of calculating the roots of each quadratic factor separately, we can multiply the two quadratic factors of Eq.

The first mode is highly damped, with a larger natural frequency 1. The second characteristic mode is very lightly damped with a smaller natural frequency 0. While an arbitrary input will excite a response containing both of these modes, it is sometimes instructive to study the two modes separately.

There are special elevator inputs, 8 s , which largely excite either one or the other mode at a time. You may refer to Blakelock [3] for details of longitudinal dynamics and control of aircraft and missiles. Figures 2. From the Bode plots Figures 2. The peaks due to complex poles sometimes disappear due to the presence of zeros in the vicinity of the poles. As expected, the natural frequencies agree with the values already. However, Figure 2. Hence, both modes essentially consist of oscillations in the pitch angle, 9 ia.

The present example shows how one can obtain an insight into a system's behavior just by analyzing the frequency response of its transfer function s. Note from Figures 2. Such a decay in the gain at high frequencies is a desirable feature, called roll-off, and provides attenuation of high frequency noise arising due to unmodeled dynamics in the system. We will define sensitivity or robustness of a system to transfer function variations later in this chapter, and formally study the effects of noise in Chapter 7.

Using Eq. A system with transfer function having poles or zeros in the right-half s-plane is called a non-minimum phase system, while a system with all the poles and zeros in the left-half s-plane, or on the imaginary axis is called a minimum phase system. We will see below that systems which have poles in the right-half s-plane are unstable. This usually results in an unacceptable transient response. Popular examples of such systems are aircraft or missiles controlled by forces applied aft of the center of mass.

For this reason, a right-half plane zero in an aircraft or missile transfer function is called 'tail-wags- the-dog zero'. Control of non-minimum phase systems requires special attention. Before we can apply the transfer function approach to a general system, we must know how to derive Laplace transform and inverse Laplace transform of some frequently encountered functions. This information is tabulated in Table 2. Note that Table 2.

It is interesting to see in Table 2. Let us assume that the system has the. The output. Let us first derive the system's governing differential equation by applying inverse Laplace transform to the transfer function with zero initial conditions. To understand why this is so. Here we will apply a similar approach to find out a linear system's response to singularity functions.

Then the residue command is used as follows to give the terms of the partial fraction expansion: Laplace transform. All one has to do is to specify the numerator and denominator polynomials of the rational function in s. For a system with complex poles such as Example 2. If a pole. In terms of the elements of p and k.

## Modern Control Design With MATLAB and SIMULINK

We had ended Section 2. One can thus obtain G s from g t by applying the Laplace transform. Then from Eqs. Since the transfer function contains information about a linear system's characteristics. We know from Table 2.

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In a manner similar to the impulse response. Using the partial fractions ex- pansion of G s. Since the poles can also be represented in terms of their natural frequency. The same computation in a low-level language. Note that both t and g are vectors of the same size.What is the magnitude of the maximum deviation from the circular orbit?

The transfer function representation of a system is widely used in block diagrams. Ashish Tewari. The CST command step is a quick way of calculating the step response. A study of control involves developing a mathematical model for each component of the control system. The first mode is highly damped, with a larger natural frequency 1. GJJ S. Such a compensator is called a lead-lag compensator. While discussing steady-state error. Note the ease with which response.