BENJAMIN C. KUO. Automatic. Control Systems. THIRD EDITION. 2 Control Systems. Automatic EHER bo-. CO-O. EDITION. THIRD. PRENTICE. HALL. Automatic Control. Systems. FARID GOLNARAGHI. Simon Fraser University . Golnaraghi also wishes to thank Professor Benjamin Kuo for sharing the. Automatic Control Systems by Benjamin C. Kuo - Ebook download as PDF File . pdf), Text File .txt) or read book online.
|Language:||English, Spanish, French|
|Distribution:||Free* [*Register to download]|
So lu t io ns M an ua l Automatic Control Systems, 9th Edition A Chapter 2 Solution ns Golnarraghi, Kuo C Chapter 2 2 2 1 (a) 10; Poless: s = 0, 0, 1, (b) Poles: s. Download Automatic Control Systems By Benjamin C. Kuo, Farid Golnaraghi – Automatic Control Systems provides engineers with a fresh new controls book. READ|Download [PDF] Automatic Control Systems Download by - Benjamin C. Kuo FULL ebook free trial Get now.
Because all physical systems have electrical and mechanical inertia. In this case the objective of control is the position of the rudder. Rudder control system. The position of the rudder as a function of time. In complex systems there may be a multitude of feedback loops and element blocks.
To regulate the tension. Figure l-7 a illustrates the elements of a tension control system of a windup process. Figure l-7 b shows a block diagram that illustrates the interconnections between the sensor is To elements of the system. During actual operation. The control system in this case is to maintain the tension of the material or web at a certain prescribed tension to avoid such problems as tearing. The unwind which is reel may contain a roll of material such as paper or cable to be sent into a processing unit.
When the system is in operation. The electric brake on the roller. The combination of the roller and the pivot arm is called the dancer. Basic elements of a feedback control system. The ideal position of the dancer is horizontal. Without feedback. The concept of feedback plays an important role in control systems.
The reduction of system now show that effects that feedback may bring upon a system. Web processing Windup reel increasing dia.
In reality. We demon- closed-loop strated in Section 1. We shall many. Let us consider the simple feedback system configuration shown in Fig. Without the necessary background and mathematical foundation of linear system theory. Feedback system. In a practical control system.
The reference of the feedback in the system of Fig. In general we can state that whenever a closed sequence of cause-and-effect relation exists among the variables of a system. The parameters G and ZTmay be considered as constant gains. When feedback is deliberately its existence is easily identified. We shall now investigate the effects of feedback on the various aspects of system performance. G and H are. This viewpoint will inevitably admit feedback number of systems that ordinarily would be identified as nonfeedback systems.
To understand the effects of feedback on a control system. The quantity GH may itself include a minus sign. It should be pointed out. Let us assume that the feedback —1. Since all environment and age. Effect of Feedback on Sensitivity Sensitivity considerations often play an important role in the design of control systems. It can be demonstrated that one of the advantages of incorporating feedback is that it can stabilize an unstable system.
In a nonrigorous manner. To investigate the effect of feedback on stability. Effect of Feedback on Stability whether the system will be able to follow the input command. If we introduce another feedsystem in Fig. The expression of the sensitivity function Sg can be derived by using Eq. The reader may derive the sensitivity of the system in Fig. It is apparent that in an open-loop system the gain of the system will respond in a one-to-one fashion to the variation in G.
Examples of these signals are thermal noise voltage in electronic amplifiers The effect and brush or commutator noise in electric motors. For instance.
With the presence of feedback. Referring to the system in Fig. Varying the magnitude of G would have 2 no effect whatsoever on the ratio.
We shall investigate what effect feedback has on the sensitivity to parameter variations. In the absence of feedback. In this case feedback is shown to have no direct effect on the output signal-to-noise ratio of the system in Fig. In other words. H Fig.
G 2 H n K ' Simply comparing Eq.. Let us assume that in the system of Fig. Feedback system with a noise signal. These one progresses into the ensuing material of this text. Linear feedback control systems are idealized models that are fabricated by the analyst purely for the simplicity of analysis and design.
Strictly speaking. When the magnitudes of the signals in a control system are limited to a range in which system components exhibit linear characteristics i. It is important that some of these more common is ways of classifying control systems are known so that proper perspective gained before embarking on the analysis and design of these systems. This type of control is found in many missile or spacecraft control systems.
Control systems are often classified according to the main purpose of the system. Quite often. These jets are often controlled in a full-on or full-off fashion. Linear Versus Nonlinear Control Systems This classification is made according to do not the methods of analysis and design. According to the types of signal found in is often made to continuous-data and discrete-data systems. But when the magnitudes of the signals are extended outside the range of the linear operation.
Other common nonlinear effects found in control systems are the backlash or dead play between coupled gear members. The schematic diagram of a closed-loop dc control system A is shown in Fig.
Most physical systems contain elements that drift or vary with time to some extent. A Unlike the general definitions of ac and dc signals used in electrical engineering. Error detector Controlled variable Fig. On the other hand. When one refers to an ac control system it usually means that the signals in the system are modulated by some kind of modulation scheme.
Among all continuous-data control systems. Schematic diagram of a typical dc closed-loop control system. Although a time-varying system without is still nonlinearity a linear system. Time-Invariant Versus Time-Varying Systems When the parameters of a control system are stationary with respect to time during the operation of the system.
Continuous-Data Control Systems continuous-data system is one in which the signals at various parts of the system are all functions of the continuous time variable t. In this case the modulated signals are demodulated by the low-pass characteristics of the control motor. Typical components of an ac control system are synchros. In this text the term "discrete-data control system" is used to describe both types of systems.
In this case the signals in the system are modulated. In general a sampled-data system receives data or information only inter- For instance. Sampled-Data and Digital Control Systems Sampled-data and digital control systems differ from the continuous-data systems in that the signals at one or more points of the system are in the form of either a pulse train or a digital code.
Schematic diagram of a typical ac closed-loop control system. Typical components of a dc control tem are potentiometers. Figure illustrates how a typical sampled-data system operates.
Notice that the output controlled variable still behaves similar to that of the dc system if the two systems have the same control objective. The schematic diagram of a typical ac control system is shown sys- in Fig.
The error signal e t control channels. Attitude of Digital-to- Digital coded input Digital missile computer. The sampling rate of the samthe output of the advantages of incorporating pler may or may not be uniform. There are many control system. Block diagram of a sampled-data control system. Because digital computers provide many advantages in size and Many aircomputer control has become increasingly popular in recent years.
Digital autopilot system for a guided missile. Figure basic elements of a digital autopilot for a guided missile. Laplace transform. Figure illustrates the complex j-plane. Modern control theory. In addition to the above-mentioned subjects.
Complex Variable complex variable j is considered to have two components: Since imaginary parts. Functions of a Complex Variable s if for said to be a function of the complex variable are corresponding value or there every value of s there is a corresponding imaginary parts.
Complex j-plane. G s is said to be points in the G s -plane is correspondence from points in the j-plane onto there are many functions for described as single valued Fig. If for value for G s [one corresponding point plane there is only one corresponding a single-valued function.
If a function G s is analytic and of s except at s it is said to have a pole of t. It can analytic in the j-plane except at these poles. As an example. Singularities and Poles of a Function are the points in the j-plane at which the funcpole is the most common type of singu- The singularities of a function tion or larity its derivatives does not exist. A and plays a very important definition of a pole role in the studies of the classical control theory.
At these two points the value of the function is infinite. In following two attractive features 1. The following examples serve as illustrations on how Eq. The The defining equation of Eq. This assumption does not place linear system problems. The Laplace transform converts the algebraic equation in It is differential equation into an then possible to manipulate the algebraic s equation by simple algebraic rules to obtain the solution in the solution is obtained by taking the inverse Laplace domain.
The final transform. These properties are presented in the following in the form of theorems. Important Theorems of the Laplace Transform The by the applications of the Laplace transform in many instances are simplified utilization of the properties of the transform. Equation represents a line integral that is to be evaluated in the j-plane.
Integration The Laplace transform of the respect to time is first integral of a function fit with is. J Shift in. Fraction Expansion 71 ' In a great majority of the problems in control systems. The following examples illustrate the care that one must take in applying the final-value theorem. When the function in s. Initial-Value Theorem is If the Laplace transform of fit lim f t if 7. Final-Value Theorem If the Laplace transform of fit is F s and ifsF s is analytic on the imaginary axis and in the right half of the s-plane.
The inverse Laplace transform operation involving rational functions can be carried out using a Laplace transform table and partial-fraction expansion.
Since the function sFis has two poles on the imaginary axis. Applying the partial-fraction expansion technique.
The zeros of Q s are either real or in a. The methods of partialmultiplefraction expansion will now be given for the cases of simple poles. It is assumed that the order of Q s in s is greater than that of P s. X s The determination of r coefficients. The n - the coefficients that correspond to the multiple-order poles is described below. C 2 s The coefficients in Eq. Let us illustrate the method by several illustrative examples. Unlike the classical method. The advantages with the Laplace transform method are that.
To solve of Eq. Example Consider the differential. Taking the Laplace transform on both sides of Eq.
CO n with Eq. There- fore. The three matrices involved here are defined to be algebra. The product of the matrices A andx is equal to the matrix y. In terms of matrix which will be discussed later. A one that has one row and more than one row matrix can also be referred to as a row Diagonal matrix. For matrix. When a matrix is written a. A row matrix is column. Row vector. Examples of a diagonal matrix are Tfln " "5 0" a 22 3 a As a we always refer to the row first row and thejth column of the matrix.
The matrix in Eq. A column matrix than one row. Matrix Determinant An with n rows array of numbers or elements columns. The order of a matrix refers to the total number example.
Does not have a square matrix n minant. A square matrix is one that has the same number of rows Column matrix. A null matrix is one whose elements are 0" all equal to zero. With each square matrix a determinant having same elements and order as the matrix may be defined.
In matrix form. When exam- ple. As an illustrative it is minant.
Two examples of symT6 5 1 metric matrices are 5 r 1 -4". When the matrix is used to represent a set of algebraic equations. A its changed with symmetric matrix has the property that if its rows are intercolumns. A square matrix is said to be singular if the value of its On the other hand.
The transpose of a matrix A is defined as the matrix that is obtained by interchanging the corresponding rows and columns in A. Transpose of a matrix. In this case the Therefore. Notice that the order of A n X m. Skew-symmetric matrix. The 1. Given a matrix A whose elements are represented by a tJ the conjugate of A. Example first As an example of determining a 2 x 2 matrix.
Let matrix of A. They are of the same order. It is important operations for scalar quantities. The corresponding elements a. Addition of Matrices Two order. This points out an important fact that the commutative law is not generally valid for matrix multiplication. This means that the number of columns of A must equal the number of rows of B.
It is may ing references are exist: The matrix C will have the same number of rows. In this case the products are not even of the. AB C if the product is conformable. For the associative law. A must be nortsingular.
In matrix algebra. Multiplication by a Scalar k Multiplying a matrix A by any scalar k is equivalent to multiplying each element of A by k. For the distributive law. A is a square matrix. A 1 denotes the "inverse of A. A" " In matrix algebra. Several examples maximum number of linearly independent in the largest nonsingular matrix contained is the order of of a matrix are as follows: A has an inverse matrix.
The reader can an example matrix. Properties 2 and 3 are useful in the determination of rank. Rank of AA'.
Given matrix A.. Rank of A'A. The matrix A n X n is negative semidefinite nonpositive and at least one of the eigenvalues is zero. A is A are of positive negative definite if all the leading principal minors A are positive negative. Equation is called the characteristic equation eigenvalues of A. Definiteness Positive definite.. Given the square matrix n matrix minors of an n X A "an a The leading are defined as follows.
These numbers are spaced T seconds apart. One way of describing the discrete nature of the signals is to consider that the input and the output of the system are sequences of numbers.
Figure illustrates a set of typical input and output signals of the sampler. This way. This is referred to as a sampler with a uniform sampling period T and a finite sampling duration p.
Let us first consider the analysis of a discrete-data system which is represented by the block diagram of Fig.? Another type of system that has discontinuous signals is the sampled-data A sampled-data system is characterized by having samplers in the system. A sampler is a device data.
To represent these input and output sequences by time-domain expressions. We may The quadratic form. With the notation of Figs. Block diagram of a discrete-data Fig. Input and output signals of a finite-pulsewidth sampler. Figure illustrates the typical input and output signals of an ideal sampler.
A sampler whose output is a train of impulses with the strength of each impulse equal to the magnitude of the input at the corresponding sampling instant is called an ideal sampler.
For small p. Figure shows the block diagram of an ideal sampler connected in cascade with a constant factor p so that the combination is an approximation to the finite-pulsewidth sampler of Fig. Although it is conceptually simple to perform inverse Laplace transform on algebraic transfer relations.
One simple fact is The fact that Eq. This points to the fact that the signals of the system in Fig. This necessitates the use of the z-transform. Input and output signals of an ideal sampler. Our motivation here for the generation of the z-transform is simply to convert transcendental functions in s into algebraic ones in z.
The definition of z-transform is given with this objective in mind. The following examples illustrate some of the simple z-transform operations. If a time use of the same procedure as described in the of finding its z-transfunction r t is given as the starting point. Example In Example When the time signal r t is sampled by the ideal sampler.
The power-series method. The partial-fraction expansion method. With this in mind. A more extensive table may be found in the litera- ture. The inversion formula. In slight difference between carrying out the partial-fraction expansion. For example. Expanding R z lz by partial-fraction expansion. Inversion formula. The following example will first recommended procedure. Just as in the case of the Laplace transform. Now let us consider the same function used in Example Equation represents the ztransform of a time sequence that is shifted to the right by nT.
Some Important Theorems of the z-Transformation Some of the commonly used theorems of the z-transform are stated in the following without proof. The reason the right-hand side of Eq.
The function R z of Eq. Automatic Control. Data Control Systems. McGraw-Hill Book Company. Circuit Theory.. Englewood Cliffs. Introduction one. Legros and A. Box Station Illinois. Transform Calculus for Englewood Cliffs. Partial Fraction Expansion 7. Discrete Prentice- B C. Kuo Hall. Methods of Applied Mathematics. Book Company. Advanced Engineering Mathematics. UNIS Partial. Prentice-Hall Inc Cliffs. McGraw-Hill IEEE Trans.
New York. Coefficients of High Order pp. McGraw-Hill "dO.. Laplace Transforms. Analysis and Synthesis of Sampled. Solve the following differential equation by means of the Laplace transformation: Find the valid products. Carry out the following matrix sums and differences: Express the following of algebraic equations in matrix form: The following of sampled by an ideal sampler with a sampling period Determine the output of the sampler.
Determine the definiteness of the following matrices: One of the most important steps in the analysis of a physical system is the mathematical description and modeling of the system. A mathematical model ot a system is essential because it allows one to gain a clear understanding of the system in terms of cause-and-effect relationships among the system com-.
In this chapter we give the definition of transfer function of a linear system and demonstrate the power of the signal-flow-graph technique in the analysis of linear systems. From the mathematical standpoint, algebraic and differential or difference equations can be used to describe the dynamic behavior of a system In systems theory, the block diagram is often used to portray systems of all types.
For linear systems, transfer functions and. In general, a physical system can be represented that portrays the relationships and interconnections. Transfer function plays an important role in the characterization of linear time-invariant systems. Together with block diagram and signal flow graph transfer function forms the basis of representing the input-output relationships ot a linear time-invariant system in classical control theory.
The starting point of defining the transfer function is the differential. Consider that a linear time-invariant system described by the following nth-order differential equation tion of a. Once the input and the initial conditions of the system are specified, the output response may be. However, it is apparent that the differential equation method of describing a system is, although essential, a rather cumbersome one, and the higher-order differential equation of Eq.
More important is the fact that although efficient subroutines are available. To obtain the transfer function of the linear system that is represented by Eq. A transfer function between an input variable and an output variable defined as the ratio of the Laplace transform of the. All initial conditions of the system are assumed to be zero.
A transfer function is independent of input excitation. In a multivariate system, a differential equation of the form of Eq. When dealing with the relationship between one input and one output, it is assumed that all other inputs are set to zero.
Since the principle of superposition is valid for linear systems, the total effect on any output variable due to all the inputs acting simultaneously can be obtained by adding the. In this case the input variables are the fuel rate and the propeller blade angle. The output variables are the speed of rotation of the engine and the turbine-inlet temperature. In general, either one of the outputs is affected by the changes in both inputs. For instance, when the blade angle of the propeller is increased, the speed of rotation of the engine will decrease and the temperature usually increases.
The following transfer relations may be written from steady-state tests performed on the system: The related to all the input transforms x. The impulse response of a linear system is defined as the output response of the system when the input is a unit impulse function.
Taking the inverse Laplace transform on both sides of Eq. Laplace transform of G s and is the impulse response sometimes also called the weighing function of a linear system.
Therefore, we can state that the Laplace transform of the impulse response is the transfer function. In practice, although a true impulse cannot be generated physically, a pulse with a very narrow pulsewidth usually provides a suitable approximation. For a multivariable system, an impulse response matrix must be defined and is.
Under such conditions, to analyze the system we would have to work with the time function r t and g t. Let us consider that the input signal r j shown in Fig. The output response c t is to be determined. In this case we have denoted the input signal as a function of r which is the time variable; this is necessary since t is reserved as a fixed time linear system.
Now consider that the input r r is approximated by a sequence of pulses of pulsewidth At, as shown in Fig. In the limit, as At approaches zero. We now compute the output response of the linear system, using the impulse-approxi-. Additional load inertia effect: As the overall inertia of the system is increased by 0. The above results are plotted for 5 V armature input. Study of the effect of disturbance: As seen, the effect of disturbance on the speed of open loop system is like the effect of higher viscous friction and caused to decrease the steady state value of speed.
Using speed response to estimate motor and load inertia: Using first order model we are able to identify system parameters based on unit step response of the system. The final value of the speed can be read from the curve and it is 8. Considering Eq. Based on this time and energy conservation principle and knowing the rest of parameters we are able to calculate B. However, this method of identification gives us limited information about the system parameters and we need to measure some parameters directly from motor such as Ra , K m , K b and so on.
So far, no current or voltage saturation limit is considered for all simulations using SIMLab software. Open loop speed response using Virtual Lab: Then the system time constant is obviously different and it can be identified from open loop response. Identifying the system based on open loop response: Open loop response of the motor to a unit step input voltage is plotted in above figure. Using the definition of time constant and final value of the system, a first order model can be found as: In both experiments 9 and 10, no saturation considered for voltage and current in SIMLab software.
If we use the calculation of phase and magnitude in both SIMLab and Virtual Lab we will find that as input frequency increases the magnitude of the output decreases and phase lag increases. Because of existing saturations this phenomenon is more sever in the Virtual Lab experiment In this experiments we observe that M 0. Apply step inputs SIMLab In this section no saturation is considered either for current or for voltage.
The same values selected for closed loop speed control but as seen in the figure the final value of speeds stayed the same for both cases. As seen, the effect of disturbance on the speed of closed loop system is not substantial like the one on the open loop system in part 5, and again it is shown the robustness of closed loop system against disturbance. Also, to study the effects of conversion factor see below figure, which is plotted for two different C.
Apply step inputs Virtual Lab a. The nonlinearities such as friction and saturation cause these differences. For example, the chattering phenomenon and flatness of the response at the beginning can be considered as some results of nonlinear elements in Virtual Lab software.
Comparing this plot with the previous one without integral gain, results in less steady state error for the case of controller with integral part. As seen in the figure, for higher proportional gains the effect of saturations appears by reducing the frequency and damping property of the system. Comments on Eq. In experiments 19 through 21 we observe an under damp response of a second order system. According to the equation, as the proportional gain increases, the damped frequency must be increased and this fact is verified in experiments 19 through Experiments16 through 18 exhibits an over damped second order system responses.
In following, we repeat parts 16 and 18 using Virtual Lab: Study the effect of integral gain of 5: In order to find the current of the motor, the motor constant has to be separated from the electrical component of the motor. The response of the motor when 5V of step input is applied is: This is the time constant of the motor. The current d When Jm is increased by a factor of 2, it takes 0. This means that the time constant has been doubled. The motor achieves this speed 0.
It does not change Part 2: It does not change. This is the same as problem It does not change d As TL increases in magnitude, the steady state velocity decreases and steady state current increases; however, the time constant does not change in all three cases. If there is saturation, the rise time does not decrease as much as it without saturation.