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Advanced Control Strategies Controller Tuning and Process Identification Control Valves Transfer functions and responses of open-loop systems, Chaps. In the early forties J.G. Ziegler and N.B. Nichols experimentally developed PID tuning methods based on closed-loop tests. However the Ziegler-Nichols. There are many methods for tuning of PID controllers, ranging from the classic rules proposed in [Ziegler and Nichols, ], to advanced optimization programs. DJ AMINE RAI 2014 TORRENT Closes and no Schema Inspectorbe able to access your system sitting right in is populated with of your messages. Over 10 million online threats by user documentation with the wrong vendors. The digital world to the master, one to connect Edge AI technology, features are added.

Chapter 9 treats basic analyses of control systems in state space. Concepts of con- trollability and observability are discussed in detail. Chapter 10 deals with control systems design in state space. The discussions include pole placement, state observers, and quadratic optimal control.

An introductory dis- cussion of robust control systems is presented at the end of Chapter Highly mathematical arguments are carefully avoided in the presen- tation of the materials. Statement proofs are provided whenever they contribute to the understanding of the subject matter presented. Special effort has been made to provide example problems at strategic points so that the reader will have a clear understanding of the subject matter discussed.

In addition, a number of solved problems A-problems are provided at the end of each chapter, except Chapter 1. The reader is encouraged to study all such solved problems carefully; this will allow the reader to obtain a deeper understanding of the topics discussed. In addition, many problems without solutions are provided at the end of each chapter, except Chapter 1. The unsolved problems B-problems may be used as homework or quiz problems. If this book is used as a text for a semester course with 56 or so lecture hours ,a good portion of the material may be covered by skipping certain subjects.

Because of the abundance of example problems and solved problems A-problems that might answer many possible questions that the reader might have, this book can also serve as a self- study book for practicing engineers who wish to study basic control theories. Kelkar, Iowa State University. Finally,I wish to offer my deep appreciation to Ms.

Alice Dworkin, Associate Editor, Mr. Scott Disanno, Senior Managing Editor, and all the people in- volved in this publishing project, for the speedy yet superb production of this book. This book presents comprehensive treatments of the analysis and design of control systems based on the classical control theory and modern control theory. A brief introduction of robust control theory is included in Chapter Automatic control is essential in any field of engineering and science.

Automatic control is an important and integral part of space-vehicle systems, robotic systems, mod- ern manufacturing systems, and any industrial operations involving control of temper- ature, pressure, humidity, flow, etc. It is desirable that most engineers and scientists are familiar with theory and practice of automatic control. This book is intended to be a text book on control systems at the senior level at a col- lege or university.

All necessary background materials are included in the book. Math- ematical background materials related to Laplace transforms and vector-matrix analysis are presented separately in appendixes. In , Minorsky worked on automatic controllers for steering ships and showed how stability could be deter- mined from the differential equations describing the system.

In , Nyquist developed a relatively simple procedure for determining the stability of closed-loop systems on the basis of open-loop response to steady-state sinusoidal inputs. In , Hazen, who introduced the term servomechanisms for position control systems, discussed the design of relay servomechanisms capable of closely following a chang- ing input. During the decade of the s, frequency-response methods especially the Bode diagram methods due to Bode made it possible for engineers to design linear closed- loop control systems that satisfied performance requirements.

Many industrial control systems in s and s used PID controllers to control pressure, temperature, etc. From the end of the s to the s, the root-locus method due to Evans was fully developed. The frequency-response and root-locus methods, which are the core of classical con- trol theory, lead to systems that are stable and satisfy a set of more or less arbitrary per- formance requirements. Such systems are, in general, acceptable but not optimal in any meaningful sense. Since the late s, the emphasis in control design problems has been shifted from the design of one of many systems that work to the design of one optimal system in some meaningful sense.

As modern plants with many inputs and outputs become more and more complex, the description of a modern control system requires a large number of equations. Clas- sical control theory, which deals only with single-input, single-output systems, becomes powerless for multiple-input, multiple-output systems.

Since about , because the availability of digital computers made possible time-domain analysis of complex sys- tems, modern control theory, based on time-domain analysis and synthesis using state variables, has been developed to cope with the increased complexity of modern plants and the stringent requirements on accuracy, weight, and cost in military, space, and in- dustrial applications.

During the years from to , optimal control of both deterministic and sto- chastic systems, as well as adaptive and learning control of complex systems, were fully investigated. From s to s, developments in modern control theory were cen- tered around robust control and associated topics.

Modern control theory is based on time-domain analysis of differential equation systems. Modern control theory made the design of control systems simpler because the theory is based on a model of an actual control system. This means that when the designed controller based on a model is applied to the actual system, the system may not be stable.

To avoid this situation, we design the control system by first setting up the range of possible errors and then designing the con- troller in such a way that, if the error of the system stays within the assumed range, the designed control system will stay stable. The design method based on this principle is called robust control theory. This theory incorporates both the frequency- response approach and the time-domain approach.

The theory is mathematically very complex. The reader interested in details of robust control theory should take a graduate-level control course at an established college or university. Before we can discuss control systems, some basic terminologies must be defined. The controlled variable is the quantity or condition that is measured and controlled.

The control signal or manipulated variable is the quantity or condition that is varied by the controller so as to affect the value of the controlled variable. The choice might depend on the particular use for the heated stream. This and related questions form the study of optimal control systems.

This important subject is mentioned in this book more to point out the existence of the problem than to solve it. To recapitulate, the curves of Fig. They show that the addition of integral control in this case eliminates steady-state error and allows use of moderate values of Kc. More Complications At this point, it would appear that the problem has been solved in some sense.

A little further probing will shatter this illusion. It has been assumed in writing Eqs. The temperature of a thermocouple inserted in the tank may or may not be the same as the temperature of the fluid in the tank. This can be demonstrated by writing the energy balance for a typical thermocouple installation, such as the one depicted in Fig.

Thus, changes in T are not instantaneously reproduced in T,,,. A step change in T causes a response in T, similar to the curve of Fig. This is analogous to the case of placing a mercury thermometer in a beaker of hot water. The thermometer does not instantaneously rise to the water temperature.

Rather, it rises in the manner described. Since the controller will receive values of T,,, possibly in the form of a thermoelectric voltage and not values of T, Eq. The solution to this system of equations is represented in Fig. For this set of values, the effect of the thermocouple delay in transmission of the temperature to the controller is primarily to make the response somewhat more oscillatory than that shown in Fig.

However, if KR is increased somewhat over the value used in Fig. The tank temperature oscillates with increasing amplitude and will continue to do so until the physical limitations of the heating system are reached. The control system has actually caused a deterioration in performance. This problem of stability of response will be one of our major concerns in this text for obvious reasons.

At present, it is sufficient to note that extreme care must be exercised in specifying control systems. However, it is not difficult to construct examples of systems for which the addition of any amount of integral control will cause an unstable response. Since integral control usually has the desirable feature of eliminating steady-state error, as it did in Fig.

Block Diagram A good overall picture of the relationships among variables in the heated-tank control system may be obtained by preparing a block diagram. This diagram, shown in Fig. Particularly significant is the fact that each component of the system is represented by a block, with little regard for the actual physical characteristics of the represented component e. The major interest is in 1 the relationship between the signals entering and leaving the block and 2 the manner in which information flows around the system.

For example, TR and T,,, enter the comparator. Their difference, the error, leaves the comparator and enters the controller. At present, the reader has been asked to accept the mathematical results on faith and to concentrate on obtaining a physical understanding of the transient behavior of the heated tank.

We shall in the forthcoming chapters develop tools for determining the response of such systems. As this new material is presented, the reader may find it helpful to refer back to this chapter in order to place the material in proper perspective to the overall control problem. Draw a block diagram for the control system generated when a human being sfeers an automobile. The Laplace transform method provides an efficient way to solve linear, ordinary, differential equations with constant coefficients.

Because an important class of control problems reduces to the solution of such equations, the next three chapters are devoted to a study of Laplace transforms before resuming our investigation of control problems. In general, the appearance of the variable s as the argument or in an equation involving f is sufficient to signify that the function has been transformed, and hence any such notation will seldom be required in this book.

The Laplace transform operator transforms a function of the variable I to a function of the variable s. The I variable is eliminated by the integration. Tkansforms of Simple Fhnctions We now proceed to derive the transforms of some simple and useful functions. The step function tO. From Example 2. In this case, the convergence of the integral depends on a suitable choice of S.

Table 2. Those which have not been derived here can be easily established by direct integration, except for the transform of 6 t , which will be discussed in detail in Chap. Transforms of Derivatives At this point, the reader may wonder what has been gained by introduction of the Laplace transform. The transform merely changes a function of t into a function of S. In the next few paragraphs, the motivation will become clear.

It will be shown that the Laplace transform has the remarkable property of transforming the operation of differentiation with respect to t to that of multiplication by s. This will be clear from the following proof. Since we shall seldom want to differentiate functions that are discontinuous at the origin, this detail is not of great importance. However, the reader is cautioned to watch carefully for situations in which such discontinuities occur.

We shall find this feature to be extremely useful in the solution of differential equations. In addition, some polynomial terms involving the initial values of f t and its first n - 1 derivatives are involved. In later applications we shall usually define our variables so that these polynomial terms will vanish. Hence, they are of secondary concern here.

Example 2. Use has been made of the linearity property and of the fact that only positive values of t are of interest. Solution of Differential Equations There are two important points to note regarding this last example.

In the first place, application of the transformation resulted in an equation that was solved for the unknown function by purely algebraic means. This suggests a procedure for solving differential equations that is analogous to that of using logarithms to multiply or divide.

To use logarithms, one transforms the pertinent numbers to their logarithms and then adds or subtracts, which is much easier than multiplying or dividing. The result of the addition or subtraction is the logarithm of the desired answer. The answer is found by reference to a table to find the number having this logarithm. In the Laplace transform method for solution of differential equations, the functions are converted to their transforms and the resulting equations are solved for the unknown function algebraically.

This is much easier than solving a differential equation. However, at the last step the analogy to logarithms is not complete. We obviously cannot hope to construct a table containing the Laplace transform of every function f t that possesses a transform.

Instead, we shall develop methods for reexpressing complicated transforms, such as x s in Example 2. For example, it is easily verified that the solution to the differential equation and boundary conditions of Example 2. Although it is difficult to find x t from Eq. Therefore, what is required is a method for expanding the common-denominator form of Eq.

This method is provided by the technique of partial fractions, which is developed in Chap. SUMMARY To summarize, the basis for solving linear, ordinary differential equations with constant coeficients with Laplace transforms has been established.

The procedure is: 1. Take the Laplace transform of both sides of the equation. The initial conditions are incorporated at this step in the transforms of the derivatives. Solve the resulting equation for the Laplace transform of the unknown function algebraically. Find the function of t that has the Laplace transform. This function satisfies the differential equation and initial conditions and hence is the desired solution.

This third step is frequently the most difficult or tedious step and will be developed further in the next chapter. It is called inversion of the transform. Although there are other techniques available for inversion, the one that we shall develop and make consistent use of is that of partial-fraction expansion. A simple example will serve to illustrate steps 1 and 2, and a trivial case of step 3.

We now wish to develop methods for inverting the transforms to obtain the solution in the time domain. The first part of this chapter will be a series of examples that illustrate the partial-fraction technique. After a generalization of these techniques, we proceed to a discussion of the qualitative information that can be obtained from the transform of the solution without inverting it. The given function f t is called theforcingfunction.

In addition, for all problems of interest in control system analysis, the initial conditions are given. In other words, values of x, dxldt,. The problem is to determine x t for all t 2 0. Partial Fkactions In the series of examples that follow, the technique of partial-fraction inversion for solution of this class of differential equations is presented. Hence, using Table 2.

The conditions on A and B are that they must be chosen to make Eq. To determine A, multiply both sides of Eq. Example 3. To determine B c D E multiply 3. The Laplace transform method has systematized the evaluation of these constants, avoiding the solution of three simultaneous equations. Four points are worth noting: 1.

In both methods, one must find the roots of the characteristic equation. The roots give rise to terms in the solution whose form is independent of the forcing function. These terms make up the complementary solution. The forcing function gives rise to terms in the solution whose form depends on the form of the forcing function and is independent of the left side of the equation.

These terms comprise the particular solution. The only interaction between these sets of terms, i. The only effect of the initial conditions is in the evaluation of the constants. This is because the initial conditions affect only the numerator of x s , as may be seen from the solution of this example. In the two examples we have discussed, the denominator of x s factored into real factors only.

In the next example, we consider the complications that arise when the denominator of x s has complex factors. The presence of complex factors does not alter the procedure at all. However, the computations may be slightly more tedious. To obtain A, multiply Eq. The fact that Q is complex does not invalidate this result, as can be seen by returning to the derivation of the transform of e --ar.

A more general discussion of this case will promote understanding. It was seen in Example 3. When these terms were combined through a trigonometric identity, it was found that the complex terms canceled, leaving a real result for x t. Of course, it is necessary that x t be real, since the original differential equation and initial conditions are real.

Again, in Example 3. Since two complex numbers will add to form a real number if they are complex conjugates, it is seen that the right side will be realfir all real s if and only if the two terms are complex conjugates. With this information, Eq. In either case, substitution of these constants into Eq. It is not necessary to perform all the algebra, since it has been done in the general case to arrive at Eq.

Another example will serve to emphasize the application of this technique. In addition, they should show that it can also be obtained by matching the term with the second term of the conjugates of Eq. Another method for solv- ing Example 3.

The reader will find this form of expansion for a quadratic. Solve for A by multiplying both sides of Eq. Equation 3. A general discussion of this case follows. First solve for B and C algebraically by placing the right side over a common denominator and equating the coefficients of like powers of s.

We now apply this method to the following example. Introducing these values into the expression for x s and applying Eq. In the next example, an exceptional case is considered; the denominator of x s has repeated roots. The procedure in this case will vary slightly from that of the previous cases. The result of Example 3. The other constants are determined by the method shown in Example 3. These terms, according to Table 2. Qualitative Nature of Solutions If we are interested only in the form of the solution x t , which is often the case in our work, this information may be obtained directlyffom the roots of the denominator ofx s.

This may be sufficient information for our purposes. Therefore, x t ultimately approaches the constant, which by inspection must be unity. The qualitative nature of the solution x t can be related to the location of the roots of the denominator of X S in the complex plane. Consider Fig. The constants Cl and C2 am arbitrary and can be determined by the partial-fraction expansion techniques. As discussed above, this determination is often not necessary for our work.

If any of these roots am repeated, the term given in Table 3. It is thus evident that the imaginary axis divides the root locations into distinct areas, with regard to the behavior of the corresponding terms in x t as t becomes large. Terms corresponding to roots to the left of the imaginary axis vanish exponentially in time, while those corresponding to roots to the right of TABLE 3. Terms corresponding to roots at the origin behave as power series in time, a constant being considered as a degenerate power series.

Terms corresponding to roots located elsewhere on the imaginary axis oscillate with constant amplitude in time unless they are multiple roots in which case the amplitude of oscillation increases as a power series in time. Much use will be made of this information in later sections of the text. In addition, it is now possible to obtain considerable information about the qualitative nature of the solution with a minimum of labor.

It should be pointed out that it is always necessary to factor the denominator of x s in order to obtain any information about x t. If this denominator is a polynomial of order three or more, this may be far from a trivial problem. Chapter 15 is largely devoted to a solution of this problem within the context of control applications. The next chapter is a grouping of several Laplace transform theorems that will find later application. In addition, a discussion of the impulse function 8 t is presented there.

Unavoidably, this chapter is rather dry. It may be desirable for the reader to skip directly to Chap. At each point where a theorem of Chap. What is the effect of the coefficient of dxldt? Solve the following differential equations by Laplace transforms: 3. The theorems are selected because of their applicability to problems in control theory. Other theorems and properties of the Laplace transformation are available in standard texts [see Churchill ].

In later chapters, the theorems presented here will be used as needed. Rnal-Value Theorem If f s is the Laplace transform of f t , then provided that sf s does not become infinite for any value of s satisfying Re s L 0.

If this condition does not hold, f t does not approach a limit as t 1 In the practical application of this theorem, the limit of f t that is found by use of the theorem is correct only if f t is bounded as t approaches infinity. Example 4. Note that we inverted this transform in Example 3. The proof of the next theorem closely parallels the proof of the last one and is left as an exercise for the reader. Initial-Value Theorem M. A primary use for this theorem is in the inversion of transforms.

Before proving this theorem, it may be desirable to clarify the relationship between f t - to and f t. This is done for an arbitrary function f t in Fig. The functional relationship contained in a transfer function is often expressed by a block-diagram representation, as shown in Fig. The arrow entering the box is the forcing function or input variable, and the arrow leaving the box is the response or output variable.

Inside the box is placed the transfer function. The usefulness of the block diagram will be appreciated in Chap. Since this type of system occurs so frequently in practice, it is worthwhile to study its response to several common forcing functions: step, impulse, and sinusoidal. These forcing functions have been found to be very useful in theoretical and experimental aspects of process control. They will be used extensively in our studies and hence, each one is explored before studying the transient response of the first-order system to these forcing functions.

A graphical representation is shown in Fig. Notice that the level rises very rapidly during the 0. The responses to step and sinusoidal forcing functions are the same for the liquid-level system as for the mercury thermometer of Chap. Hence, they need t. Example 6-l n pulse input; b response of tank level. This is the advantage of characterizing all first-order systems by the same transfer function.

Liquid-Leml Process with Constant-flow Outlet An example of a transfer function that often arises in control systems may be developed by considering the liquid-level system shown in Fig. The resistance shown in Fig. The same assumptions of constant cross-sectional ama and constant density that were used before also apply here.

For this system, Eq. One realizes this from the discussion on the transform of an integral presented in Chap. Therefore, the solution of Eq. Such a system that grows without limit for a sustained change in input is q. Systems that have a limited change in output for a sustained change in input are said to have regulation.

An example of a system having regulation is the step response of a first-order system, which is shown in Fig. The transfer function for the liquid-level system with constant outlet flow given by Eq. The next example of a first-order system is a mixing process. Mixing Process Consider the mixing process shown in Fig. It is desired to determine the transfer function relating the outlet concentration y to the inlet concentration X.

Assuming the density of the solution to be constant, the flow rate in must equal the flow rate out, since the holdup volume is fixed. At steady state, Eq. This mixing process is, therefore, another first-order process, for which the dynamics are now well known.

We next bring in an example from DC circuit theory. For t , i dt I Recalling that the current is the rate of change of charge with respect to time coulombs per second , we may replace i by dqldt in Eq. Again we obtain a first-order transfer function. The three examples that have been presented in this section are intended to show that the dynamic characteristics of many physical systems can be represented by a first-order transfer function.

In the remainder of the book, more examples of first-order systems will appear as we discuss a variety of control systems. Actually, most physical systems of practical importance are nonlinear. Characterization of a dynamic system by a transfer function can be done only for linear systems those described by linear differential equations.

A very important technique for such approximation is illustrated by the following discussion of the liquid-level system of Fig. We now assume that the resistance follows the square-root relationship q. This difficulty can be circumvented as follows. X By means of a Taylor-series expansion, the function q.

However, in this case, the resistance RI depends on the steady-state conditions around which the process operates. Graphically, the resistance RI is the reciprocal of the slope of the tangent line passing through the point gosh, as shown in Fig. Furthermore, the linear approximation given by Eq.

From the graphical representation, it should be clear that the linear approximation improves as the deviation in h becomes smaller. Whether or not the linearized result is a valid representation depends on the operation of the system.

If the level is being maintained by a controller at or close to a fixed level h S, then by the very nature of the control imposed on the system, deviations in level should be small for good control and the linearized equation is adequate.

On the other hand, if the level should change over a wide range, the linear approximation may be very poor and the system may deviate significantly from the prediction of the linear transfer function. In such cases, it may be necessary to use the more difficult methods of nonlinear analysis, some of which are discussed in Chaps.

We shall extend the discussion of linearization to more complex systems in Chap. In general, this technique may be applied to any nonlinearity that can be expressed in a Taylor series or, equivalently, has a unique slope at the operating point. Since this includes most nonlinearities arising in process control, we have ample justification for studying linear systems in considerable detail. Derive the transfer function H s lQ s for the liquid-level system of Fig.

The pump removes water at a constant rate of 10 cfm cubic feet per minute ; this rate is independent of head. The cross-sectional area of the tank is 1. A liquid-level system, such as the one shown in Fig. The flow-head characteristics are shown in Fig. Develop a formula for finding the time constant of the liquid-level system shown in Fig. The resistance R is linear. The tank has three vertical walls and one which slopes at an angle r from the vertical as shown.

The distance separating the parallel walls is 1. Consider the stirred-tank reactor shown in Fig. Prepare a block diagram for the reactor. Sketch the response of the reactor to a unit-step change in C i. At these temperatures convective and conductive heat transfer to the junction am negligible compared with radiative heat transfer.

Determine the linearized transfer function between the furnace temperature Ti and the junction temperature To. Compare this with the true response obtained by integration of the differential equation. A liquid-level system has the following properties: Tank dimensions: 10 ft high by 5 ft diameter Steady-state operating characteristics: Steady-state lewl, ft 0 5,ooo 10,ooo 15,ooo 20,ooo 25,ooo 30,ooo 0.

Plot the response of the original tank which is upstream of the new tank to the change described in part a when the connection is such that the tanks are 1 interacting, 2 noninteracting. See Chap. The perfectly mixed product is withdrawn from the tank, also at the flow rate q at the same concentration as the material in the tank, C,.

The total volume of solution in the tank is constant at V. Density may be considered to be independent of concentration. A trace of the tank concentration versus time appears as shown in Fig. Be sum to label the graph with quantitative information regarding times and magnitudes and any other data that will demonstrate your understanding of the situation. The liquid-level process shown in Fig. At time zero, the flow varies as shown in Fig. Note that Q is the deviation in flow rate. The resistances are linear.

H and Q are deviation variables. Show clearly how you derived the transfer function. You are expected to give numerical values in the transfer function. The liquid-level system shown in Fig. In other words, a train of unit impulses is applied to the tank at intervals of one minute. Ultimately the output wave train becomes periodic as shown in the sketch. Determine the maximum and minimum values of this output.

The two-tank mixing process shown in Fig. Assume that each tank has a constant holdup volume of 1 ft3. Neglect transportation lag in the line connecting the tanks and the recirculation line. Try to answer parts b and c by intuition. To illustrate this type of system, consider the liquid-level systems shown in Fig.

Two possible piping arrangements are shown in Fig. In Fig. The variation in h 2 in tank 2 does not affect the transient response occurring in tank 1. This type of system is referred to as a noninteracting system. In contrast to this, the system shown in Fig. We shall consider first the noninteracting system of Fig. Noninteracting System As in the previous liquid-level example, we shall assume the liquid to be of constant density, the tanks to have uniform cross-sectional area, and the flow resistances to be linear.

Our problem is to find a transfer function that relates h2 to 4, that is, Hz s lQ s. A balance on tank 1 gives 7. In the same manner, we can combine Eqs. In the case of the interacting system of Fig. Example 7. Sketch the response of the level in tank 2 if a unit-step change is made in the inlet flow rate to tank 1. The transfer function for this system is found directly from Eq.

Notice that the response is S-shaped and the slope dH2ldt at origin is zero. If the change in flow rate were introduced into the second tank, the response would be first-order and is shown for comparison in Fig. Several Noninteracting Having observed that the overall transfer function for two noninteracting first-order systems connected in series is simply the product of the individual transfer functions, we may now generalize by considering n noninteracting first-order systems as represented by the block diagram of Fig.

This sluggishness or delay is sometimes called transfer lug and is always present when two or more first-order systems ate connected in series. For a single first-order system, there is no transfer lag; i. In order to show how the transfer lag is increased as the number of stages increases, Fig. Interacting System To illustrate an interacting system, we shall derive the transfer function for the system shown in Fig.

The analysis is started by writing mass balances on the tanks as was done for the noninteracting case. A simple way to combine Eqs. At steady state, Eqs. The difference between the transfer function for the noninteracting system, Eq. The term interacting is often referred to as loading. The second tank of Fig. If the tanks are noninteracting, the transfer function relating inlet flow to outlet flow is 7.

One time constant has become considerably larger and the other smaller than the time constant T of either tank in the noninteracting system. The response of Qa t to a unit-step change in Q t for the interacting case [Es. From this figure, it can be seen that interaction slows up the response. This result can be understood on physical grounds in the following way: if the same size step change is introduced into the two systems of Fig. However, for the interacting case, the flow q1 will be reduced by the build-up of level in tank 2.

At any time tr following the introduction of the step input, q1 for the interacting case will be less than for the noninteracting case with the result that h2 or q2 will increase at a slower rate. In general, the effect of interaction on a system containing two first-order lags is to change the ratio of effective time constants in the interacting system. In terms of the transient response, this means that the interacting system is mom sluggish than the noninteracting system.

This chapter concludes our specific discussion of first-order systems. Resistances Rl and R2 ate linear The flow rate from tank 3 is maintained constant at b by means of a pump; i. The tanks are noninteracting. The mercury thermometer in Chap. A more detailed analysis would consider both the convective resistance surrounding the bulb and that between the bulb and mercury. In addition, the capacitance of the glass bulb would be included.

What is the effect of the bulb resistance and capacitance on the thermometer response? Note that the inclusion of the bulb results in a pair of interacting systems, which give an overall transfer function somewhat different from that of Eq.

Them are N storage tanks of volume V arranged so that when water is fed into the first tank, an equal volume of liquid overflows from the first tank into the second tank, and so on. Each tank initially contains component A at some concentration CO and is equipped with a perfect stirrer. At time zero, a stream of zero concentration is fed into the first tank at a volumetric rate q.

Find the resulting concentration in each tank as a function of time. Tank 1 and Tank 2 are interacting. Three identical tanks are operated in series in a noninteracting fashion as shown in Fig. If the deviation in flow rate to the first tank is an impulse function of magnitude 2, determine a An expression for H s where H is the deviation in level in the third tank.

In the two-tank mixing process shown in Fig. At what time does the salt concentration in tank 2 reach 0. The holdup volume of each tank is 6 ft3. Note that two streams are flowing from tank 1, one of which flows into tank 2. You are expected to give numerical values of the parameters in the transfer functions and to show clearly how you derived the transfer functions.

A second-order transfer function will be developed by considering a classical example from mechanics. This is the damped vibrator, which is shown in Fig. A block of mass Wresting on a horizontal, frictionless table is attached to a linear spring. A viscous damper dashpot is also attached to the block. Assume that the system is free to oscillate horizontally under the influence of a forcing function F t.

The origin of the coordinate system is taken as the right edge of the block when the spring is in the relaxed or unstretched condition. At time zero, the block is assumed to be at rest at this origin. Also, the assumption that the block is initially at rest permits derivation of the second-order transfer function i n i t s standard form. An initial velocity has the same effect as a forcing function.

Hence, this assumption is in no way restrictive. At this particular instant, the following forces are acting on the block: 1. The external force F t acting toward the right. The reason for introducing T and f in the particular form shown in Eq. Equation 8. Notice that, because of superposition, X t can be considered as a forcing function because it is proportional to the force F t.

All such systems are defined as second-order. Note that it requires two parameters, T and 5, to characterize the dynamics of a second-order system in contrast to only one parameter for a firstorder system. For the time being, the variables and parameters of Eq. We shall now discuss the response of a second-order system to some of the common forcing functions, namely, step, impulse, and sinusoidal. Combining Eq.

The roots sr and s2 will be real or complex depending on the parameter 6. The nature of the roots will, in turn, affect the form of Y t. The problem may be divided into the three cases shown in Table 8. Each case will now be discussed. Since 6 ; 8. The resulting equation is then put in the form of Eq.

The details are left as an exercise for the reader. It is evident from Eq. The nature of the response can be understood most clearly by plotting Eq. Referring to Fig. The constants are obtained, as usual, by partial fractions. The response, which is plotted in Fig. For this case, the inversion of Eq. The response has been plotted in Fig. This is known as an over-dumped response. Therefore, Eq. By comparing the linear factors of the denominator of Eq. The reader should verify these results.

Terms Used to Describe an Underdamped System Of these three cases, the underdamped response occurs most frequently in control systems. Equations for some of these terms are listed below for future reference. All these equations can be derived from the time response as given by Eq. The overshoot increases for decreasing 4. Decay ratio.

The decay ratio is defined as the ratio of the sizes of successive peaks and is given by CIA in Fig. Notice that larger 6 means greater damping, hence greater decay. Rise time. This is the time required for the response to first reach its ultimate value and is labeled t ,.

The reader can verify from Fig. Response time. The response time is indicated in Fig. The limits 55 percent are arbitrary, and other limits have been used in other texts for defining a response time. Period of oscillation. From Eq. In terms of Fig. Natural period of oscillation. This frequency is referred to as the natural frequency wn: The corresponding natural cyclical frequency f,, and period T, are related by the expression 8.

From Eqs. Notice that, for 5 Impulse Response If a unit impulse s t is applied to the second-order system, then from Eqs. The problem is again divided into the three cases shown in Table 8. Comparison of Eqs. In other words, the inverse transform of Eq. This principle also yields the results for the next two cases. To summarize, the impulse-response curves of Fig. However, the impulse response always returns to zero. Terms such as decay ratio, period of oscillation, etc.

Many control systems exhibit transient responses such as those of Fig. This is illustrated by Fig. Notice in Eq. From this little effort, we see already that the response of the second-order system to a sinusoidal driving function is ultimately sinusoidal and has the same frequency as the driving function.

If the constants Cr and C2 are evaluated, we get from Eqs. It can be seen from Eq. Discussion of other characteristics of the sinusoidal response will be deferred until Chap. It happens that many control systems that are not truly second-order exhibit step responses very similar to those of Fig. Such systems are often characterized by second-order equations for approximate mathematical analysis. Hence, the second-order system is quite important in control theory, and frequent use will be made of the material in this chapter.

Synonyms for this term are dead time and distance velocity lag. As an example, consider the system shown in Fig. The density p and the heat capacity C are constant. The tube wall has negligible heat capacity, and the velocity profile is flat plug flow. The temperature x of the entering fluid varies with time, and it is desired to find the response of the outlet temperature y t in terms of a transfer function. As usual, it is assumed that the system is initially at steady state; for this system, it is obvious that the inlet temperature equals the outlet temperature; i.

This simple step response is shown in Fig. If the variation in x t were some arbitrary function, as shown in Fig. Subtracting Eq. This result follows from the theorem on translation of a function, which was discussed in Chap. The transportation lag is quite common in the chemical process industries where a fluid is transported through a pipe. We shall see in a later chapter that the presence of a transportation lag in a control system can make it much more difficult to control.

In general, such lags should be avoided if possible by placing equipment close together. They can seldom be entirely eliminated. The transport lag is quite different from the other transfer functions first-order, second-order, etc. As shown in Chap. The transport lag is also difficult to simulate by computer as explained in Chap. For these reasons, several approximations of transport lag that are useful in control calculations are presented here. In this case, the other transfer functions filter out the high frequency content of the signals passing through the transport lag with the result that the transport lag approximation, when combined with other transfer functions, provides a satisfactory result in many cases.

The accuracy of a transport lag can be evaluated most clearly in terms of frequency response, a topic to be covered later in this book. The two-tank system shown in Fig. Using appropriate figures and equations in the text, determine the maximum deviation in level feet in both tanks from the ultimate steady-state values and the time at which each maximum occurs.

The two-tank liquid-level system shown in Fig. The transient response is critically damped, and it takes 1. Calculate the time constant for each tank. How long does it take for the change in level of the first tank to reach 90 percent of the total change?

A mercury manometer is depicted in Fig. Assuming the flow in the manometer to be laminar and the steady-state friction law for drag force in laminar flow to apply at each instant, determine a transfer function between the applied pressure p t and the manometer reading h. It will simplify the calculations if, for inertial terms, the velocity profile is assumed to be flat.

From your transfer function, written in standard second-order form, list a the steady-state gain, b 7, and c 5. Comment on these parameters as they are related to the physical nature of the problem. Verify Eqs. Verify Eq. If a second-order system is overdamped, it is more difficult to determine the parameters 5 and T experimentally. One method for determining the parameters from a step response has been suggested by R.

Oldenbourg and H. Sartorius The Dynamics of Automatic Controls. ASME, p. Determine Y O , Y O. In the liquid-level system shown in Fig. You need show only the shape of the responses; do not plot. For H2 and H3, use graphs in Chap.

The two tanks shown in Fig. With this background, we can extend the discussion to a complete control system and introduce the fundamental concept of feedback. In order to work with a familiar system, the treatment will be based on the illustrative example of Chap. Figure 9. To orient the reader, the physical description of this control system will be reviewed.

It is desired to maintain or control the temperature in the tank at TR by means of the controller. If the controller changes the heat input to the tank by an amount that is proportional to E , we have proportional control. If an electrical source were used, the final control element might be a variable transformer that is used to adjust current to a resistance.

In either case, the output signal from the controller should adjust q in such a way as to maintain control of the temperature in the tank. Components of a Control System The system shown in Fig. Process stirred-tank heater.

Measuring element thermometer. Final control element variable transformer or control valve. Each of these components can be readily identified as a separate physical item in the process. In general, these four components will constitute most of the control systems that we shall consider in this text; however, the reader should realize that more complex control systems exist in which more components are used. For example, there are some processes which require a cascade control system in which two controllers and two measuring elements are used.

A cascade system is discussed in Chap. Block Diagram For computational purposes, it is convenient to represent the control system of Fig. Such a diagram makes it much easier to visualize the relationships among the various signals. New terms, which appear in Fig. The set point is a synonym for the desired value of the controlled variable. The load refers to a change in any variable that may cause the controlled variable of the process to change.

Other possible loads for this system are changes in flow rate and heat loss from the tank. These loads are not shown on the diagram. The control system shown in Fig. In the comparator, the controlled variable is compared with the desired value or set point. This error enters a controller, which in turn adjusts the final control element in order to return the controlled variable to the set point. Negative Bedback versus pbsitive Feedback Several terms have been used that may need further clarification.

The feedback principle, which is illustrated by Fig. The arrangement of the apparatus of Fig. Negative feedback ensures that the difference between TR and T, is used to adjust the control element so that the tendency is to reduce the error. If the load Ti should increase, T and T,,, would start to increase, which would cause the error E to become negative. With proportional control, the decrease in error would cause the controller and final control element to decrease the flow of heat to the system with the result that the flow of heat would eventually be reduced to a value such that T approaches TR.

A verbal description of the operation of a feedback control system, such as the one just given, is admittedly inadequate, for this description necessarily is given as a sequence of events. Actually all the components operate simultaneously, and the only adequate description of what is occurring is a set of simultaneous differential equations.

This more accurate description is the primary subject matter of the present and succeeding chapters. If the signal to the comparator were obtained by adding TR and T,, we would have a positive feedback system, which is inherently unstable. If Ti were to increase, T and T,,, would increase, which would cause the signal the comparator E in Fig. However, this action, which is just the opposite of that needed, would cause T to increase further. For this reason, positive feedback would never be used intentionally in the system of Fig.

However, in more complex systems it may arise naturally. An example of this is discussed in Chap. In the first situation, which is called the servomechanism-type or servo problem, we assume that there is no change in load Ti and that we are interested in changing the bath temperature according to some prescribed function of time. For this problem, the set point TR would be changed in accordance with the desired variation in bath temperature.

If the variation is sufficiently slow, the bath temperature may be expected to follow the variation in TR very closely. There are occasions when a control system in the chemical industry will be operated in this manner. For example, one may be interested in varying the temperature of a reactor according to a prescribed timetemperature pattern.

However, the majority of problems that may be described as the servo type come from fields other than the chemical industry. The tracking of missiles and aircraft and the automatic machining of intricate parts from a master pattern are well-known examples of the servo-type problem. The other situation will be referred to as the regulator problem. In this case, the desired value TR is to remain fixed and the purpose of the control system is to maintain the controlled variable at TR in spite of changes in load Ti.

This problem is very common in the chemical industry, and a complicated industrial process will often have many selfcontained control systems, each of which maintains a particular process variable at a desired value.

These control systems are of the regulator type. In considering control systems in the following chapters, we shall frequently discuss the response of a linear control system to a change in set point servo problem separately from the response to a change in load regulator problem. However, it should be realized that this is done only for convenience. The basic approach to obtaining the response of either type is essentially the same, and the two responses may be superimposed to obtain the response to any linear combination of set-point and load changes.

In the previous chapters, such inputoutput relations were developed in the form of transfer functions. In block-diagram representations of control systems, the variables selected are deviation variables, and inside each block is placed the transfer function relating the input-output pair of variables. Finally, the blocks are combined to give the overall block diagram. This is the procedure to be followed in developing Fig. This block will be seen to differ somewhat from those presented in previous chapters in that two input variables are present; however, the procedure for developing the transfer function remains the same.

In some stirred-tank heaters, such as a jacketed kettle, q depends on both the temperature of the fluid in the jacket and the temperature of the fluid in the kettle.

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Rt free torrent Get step response of continuous transfer function y. It is called inversion of the transform. It is desired to maintain or control the temperature in the tank at TR by means of the controller. For example, a proportional controller may be thought of as a device that receives the error signal and puts out a signal proportional to it. PID Control.
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Ziegler nichols open loop tuning method matlab torrent Verify Eq. This will be accomplished by adding a stream of pure A to tank 1 through a control valve. Regardless of the type. The production supervisor was Louise Karam. Kyoto University, Japan Keywords: feedback control, proportional, integral, derivative, reaction curve, process with self-regulation, integrating process, process model, steady-state. The tanks are maintained at different temperatures.
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