BENJAMIN C. KUO. Automatic. Control Systems. THIRD EDITION. 2 Control Systems. Automatic EHER bo-. CO-O. EDITION. THIRD. PRENTICE. HALL. 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. Automatic Control Systems by Benjamin C. Kuo - Ebook download as PDF File . pdf), Text File .txt) or read book online.
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Automatic Control. Systems. FARID GOLNARAGHI. Simon Fraser University. BENJAMIN C. KUO. University of Illinois at Urbana-Champaign. WILEY. Download Automatic Control Systems By Benjamin C. Kuo, Farid Golnaraghi – Automatic Control Systems provides engineers with a fresh new controls book. PDF | On Jan 1, , B C Kuo and others published Automatic Control System.
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Root Locus 15 10 System: Angle of asymptotes: Then d and Poles: Angles of asymptotes: Root Locus 60 40 System: Root Locus 0. Root Locus 2. Root locus diagram, part a: Imaginary Axis locus diagram. Root Locus 5 4 3 Imaginary Axis 2 1 0 -1 -2 -3 -4 -5 -4 Breakaway points: Root Locus Imaginary Axis 0 Real Axis Root locus diagram, part b: Root Locus 10 8 6 Imaginary Axis 4 2 0 -2 -4 -6 -8 Answers to True and False Review Questions: Bode Diagram 20 Magnitude dB 0 System: To change the crossover frequency requires adding gain as: As , if GH is rearranged as: Bode diagram: L j Nyquist Diagram 1.
Nyquist Diagram 0. Nyquist diagram is added to the results of sisotool. Higher values of K resulted in unstable Nyquist diagram. Following is the Nyquist diagram at margin of stability. Part a , Nyquist at margin of stability: Nyquist Diagram 1 0. Using Routh criterion, the coefficient table is as follows: Location of poles in root locus diagram of the second figure will also verify that. Nyquist Diagram 20 15 10 Imaginary Axis 5 0 -5 -1 Inf Freq: Inf Stable loop -1 -1 The result and approach is similar to part a , a sample of Nyquist diagram is presented for his case as follows: Nyquist Diagram 15 10 Imaginary Axis 5 0 -5 -1 Using MATLAB sisotool, the transfer function gain can be iteratively changed in order to obtain different phase margins.
Following two figures illustrate the sisotool and Nyquist results at margin of stability for part a. Similar methodology applied as in part a. Following diagrams correspond to margin of stability: In this case, we need 0 CCW encirclements. See alternative solution to Root Locus 30 20 Imaginary Axis 10 0 0 10 20 Real Axis Part c , Gain and frequency that instability occurs: The Part e: Nyquist Diagram 8 6 4 Imaginary Axis 2 0 -2 -4 -6 -8 -1 Both of these values are consistent with the results of part a from sisotool.
CL Gain: Use G to obtain the gain-phase plots and Gm and Pm. Use the Bode plot to graphically obtain Mr. Sisotool Result shows that by changing K between 0 and inf. Therefore, the system is stable for all positive K. Inf 0 Stable loop 0 0 0 P.
PD controller design: The open-loop transfer function of a system is: The open loop transfer function of a system is: The transfer functions are generated and imported in sisotool as in According to the requirements the gain must be greater than must be less than or 0.
The Lead compensator T. To obtain a slightly higher PM, lead compensator zero was re-tuned, where the zero is pulled closer to imaginary axis from -5 to To include some integral action, Ki is set to 1. The open loop bode shows as PM of By try and error, 2 compensators a double lead compensator each with phase lead of 55 deg was found suitable.
Considering the change in cross over frequency after applying the lead filters, overall, a PM of 52 deg was obtained as seen in the bode diagram of compensated loop: Double Lead filter design: This was due to the shape of phase diagram affected by integral action i.
Bode diagram of compensated loop transfer function can be observed in the following figure, showing a PM pf The bode of the loop transfer function shows a PM of deg at 3. Considering the change in cross over frequency after applying the lead filters, overall, a PM of Bode diagram of compensated loop can be observed in the following figure, showing a PM pf In order to add some integral action, First, the bode plot of the Loop transfer function is obtained demonstrating a PM of By try and error, a double lead compensator, each with phase lead of 48 deg was found suitable.
The effect is similar to adding a Zero at K K.
At K results: The root locus diagram can be seen as: The zero and the gain of the PI controller needs to be designed. The place of the zero and the overall gain is iteratively changed in the MATLAB sisotool to achieve the crossover frequency of Now required to achieve PM 45 10 10,then PM 1 As a result, 91o phase lead is The crossover frequency is Therefore, This pole is usually placed at least 1 decade lower frequency wise than the slowest existing poles of the system.
In this case, since KI K 0. Use the transfer function for the open-loop system, and a series PID compensator in a unity feedback system. Assume a small electric time constant or small inductance and simplify to Equation The resulting zero in the right hand plane is troubling.
Looking at the TF poles, it seems prudent to design the controller by placing its zero farther to LHS of the s-plane. Now, we can choose pole p far enough from pole dominant of second order. Mathematical Model: Draw free body diagrams Assume both xc and xw are positive and are measured from equilibrium.
Refer to Chapter 4 problems for derivation details. For a better design, and to meet rise time criterion, use Example and Chapter 9 PD design examples. You get the next window. Enter the A,B,C, and D values. Note C must be entered here and must have the same number of columns as A. We us [1,1] arbitrarily as it will not affect the eigenvalues. Characteristic Polynomial: Transfer function: The A matrix is: If The solution s for x is , then or 0 , theerefore: The 2nd order desired characteristic equation of the 2 2 0 1 On the other hand: For overshoot of 4.
First convert the transfer function to a unity feedback system to make compatible to the format used in the Control toolbox. You can adjust K values to obtain alternative results by repeating this process. For observability, we define H as: Automatic Control Systems, 8th ed. 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. The schematic diagram of a closed-loop dc control system A is shown in Fig.
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. 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. Error detector Controlled variable Fig.
On the other hand. Schematic diagram of a typical dc closed-loop control system. Among all continuous-data control systems. 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. Although a time-varying system without is still nonlinearity a linear system. Typical components of an ac control system are synchros.
Automatic Control Systems, 9th Edition - Solutions Manual
Notice that the output controlled variable still behaves similar to that of the dc system if the two systems have the same control objective. In general a sampled-data system receives data or information only inter- For instance.
In this case the modulated signals are demodulated by the low-pass characteristics of the control motor. In this text the term "discrete-data control system" is used to describe both types of systems. 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.
Figure illustrates how a typical sampled-data system operates. The schematic diagram of a typical ac control system is shown sys- in Fig. In this case the signals in the system are modulated.
Schematic diagram of a typical ac closed-loop control system. Typical components of a dc control tem are potentiometers.
Digital autopilot system for a guided missile. Because digital computers provide many advantages in size and Many aircomputer control has become increasingly popular in recent years. Block diagram of a sampled-data control system. Attitude of Digital-to- Digital coded input Digital missile computer. Figure basic elements of a digital autopilot for a guided missile. The error signal e t control channels. There are many control system.
The sampling rate of the samthe output of the advantages of incorporating pler may or may not be uniform. Laplace transform. Complex Variable complex variable j is considered to have two components: In addition to the above-mentioned subjects. Modern control theory. Figure illustrates the complex j-plane. Since imaginary parts. Complex j-plane.
If for value for G s [one corresponding point plane there is only one corresponding a single-valued function. 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. 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.
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. 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.
If a function G s is analytic and of s except at s it is said to have a pole of t. As an example. It can analytic in the j-plane except at these poles. The following examples serve as illustrations on how Eq. The defining equation of Eq. 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.
In following two attractive features 1. The final transform. The This assumption does not place linear system problems. These properties are presented in the following in the form of theorems. Equation represents a line integral that is to be evaluated in the j-plane. 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.
J Shift in.. Integration The Laplace transform of the respect to time is first integral of a function fit with is. Since the function sFis has two poles on the imaginary axis. Initial-Value Theorem is If the Laplace transform of fit lim f t if 7. Fraction Expansion 71 ' In a great majority of the problems in control systems. The inverse Laplace transform operation involving rational functions can be carried out using a Laplace transform table and partial-fraction expansion.
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 following examples illustrate the care that one must take in applying the final-value theorem. When the function in s. Applying the partial-fraction expansion technique. It is assumed that the order of Q s in s is greater than that of P s..
The methods of partialmultiplefraction expansion will now be given for the cases of simple poles.. The zeros of Q s are either real or in a. X s The n - the coefficients that correspond to the multiple-order poles is described below.. The determination of r coefficients. C 2 s The coefficients in Eq. Let us illustrate the method by several illustrative examples. The advantages with the Laplace transform method are that. Example Consider the differential. Unlike the classical method..
To solve of Eq. Taking the Laplace transform on both sides of Eq. CO n with Eq. There- fore. In terms of matrix which will be discussed later. The three matrices involved here are defined to be algebra. The product of the matrices A andx is equal to the matrix y. Row vector. The matrix in Eq. Does not have a square matrix n minant. Matrix Determinant An with n rows array of numbers or elements columns. As a we always refer to the row first row and thejth column of the matrix.
A column matrix than one row. Examples of a diagonal matrix are Tfln " "5 0" a 22 3 a A row matrix is column. The order of a matrix refers to the total number example. For matrix. A one that has one row and more than one row matrix can also be referred to as a row Diagonal matrix. When a matrix is written a. A square matrix is one that has the same number of rows Column matrix. A its changed with symmetric matrix has the property that if its rows are intercolumns.
With each square matrix a determinant having same elements and order as the matrix may be defined. In matrix form. When the matrix is used to represent a set of algebraic equations. A null matrix is one whose elements are 0" all equal to zero.
When exam- ple. As an illustrative it is minant. A square matrix is said to be singular if the value of its On the other hand. Two examples of symT6 5 1 metric matrices are 5 r 1 -4". Notice that the order of A n X m. In this case the Therefore. Transpose of a matrix. Skew-symmetric matrix. The transpose of a matrix A is defined as the matrix that is obtained by interchanging the corresponding rows and columns in A. Let matrix of A. Example first As an example of determining a 2 x 2 matrix. The 1.
Given a matrix A whose elements are represented by a tJ the conjugate of A. It is important operations for scalar quantities. The corresponding elements a. They are of the same order. Addition of Matrices Two order.. It is may ing references are exist: The matrix C will have the same number of rows. 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. In other words.. In this case the products are not even of the. In matrix algebra. A is a square matrix. AB C if the product is conformable. For the distributive law.
For the associative law. A 1 denotes the "inverse of A.
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A must be nortsingular. Multiplication by a Scalar k Multiplying a matrix A by any scalar k is equivalent to multiplying each element of A by k. A has an inverse matrix. The reader can an example matrix. Several examples maximum number of linearly independent in the largest nonsingular matrix contained is the order of of a matrix are as follows: A" " In matrix algebra.
Rank of AA'. Given matrix A. Rank of A'A. Properties 2 and 3 are useful in the determination of rank. The leading are defined as follows.
Automatic Control Systems by Benjamin C. Kuo
Given the square matrix n matrix minors of an n X A "an a The matrix A n X n is negative semidefinite nonpositive and at least one of the eigenvalues is zero. Definiteness Positive definite. 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. Let us first consider the analysis of a discrete-data system which is represented by the block diagram of Fig.
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. Block diagram of a discrete-data Fig. We may The quadratic form. Figure illustrates a set of typical input and output signals of the sampler. This is referred to as a sampler with a uniform sampling period T and a finite sampling duration p.
To represent these input and output sequences by time-domain expressions. This way. Another type of system that has discontinuous signals is the sampled-data A sampled-data system is characterized by having samplers in the system..
These numbers are spaced T seconds apart. A sampler is a device data. With the notation of Figs.. Input and output signals of a finite-pulsewidth sampler. For small p. Figure illustrates the typical input and output signals of an ideal sampler.
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. 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.
One simple fact is The fact that Eq. This points to the fact that the signals of the system in Fig. Although it is conceptually simple to perform inverse Laplace transform on algebraic transfer relations.
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. This necessitates the use of the z-transform. The definition of z-transform is given with this objective in mind.
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.. The following examples illustrate some of the simple z-transform operations. Example In Example A more extensive table may be found in the litera- ture. The inversion formula. The power-series method. When the time signal r t is sampled by the ideal sampler.
With this in mind. The partial-fraction expansion method. In slight difference between carrying out the partial-fraction expansion.
Expanding R z lz by partial-fraction expansion. The following example will first recommended procedure. For example. Inversion formula.. The reason the right-hand side of Eq. The function R z of Eq. Some Important Theorems of the z-Transformation Some of the commonly used theorems of the z-transform are stated in the following without proof..
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. Just as in the case of the Laplace transform. Box Station Illinois. Coefficients of High Order pp. Kuo Hall. McGraw-Hill IEEE Trans. Laplace Transforms. Transform Calculus for Englewood Cliffs. Prentice-Hall Inc Cliffs. McGraw-Hill Book Company. Methods of Applied Mathematics.
Partial Fraction Expansion 7. Circuit Theory. Discrete Prentice- B C.
Automatic Control. Englewood Cliffs. Book Company.. New York. Advanced Engineering Mathematics. Analysis and Synthesis of Sampled. Legros and A. UNIS Partial.. Data Control Systems. Introduction one.. McGraw-Hill "dO. 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: Determine the definiteness of the following matrices: The following of sampled by an ideal sampler with a sampling period Determine the output of the sampler.
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-.
Some systems simply have parameters that vary with time a predictable or unpredictable fashion. For instance, the transfer characteristic of a guided missile in flight will vary in time because of the change of mass of the nmsile and the change of atmospheric conditions. On the other hand, for a simple mechanical system with mass and friction, the latter may be subject to unpredictable variation either due to "aging" or surface conditions thus the control system designed under the assumption of known and fixed parameters may fail to yield satisfactory response should the system parameters vary.
In order that the system may have the ability of self-correction or selfadjustment in accordance with varying parameters and environment it is necessary that the system's transfer characteristics be identified continuously or at appropriate intervals during the operation of the system.
One of the methods of identification is to measure the impulse response of the system so that design parameters may be adjusted accordingly to attain optimal control at all times In the two preceding sections, definitions of transfer function and impulse response of a linear system have been presented. The two functions are directly related through the Laplace transformation, and they represent essentially the same information about the system. However, it must be reiterated that.
The evaluation of the impulse response of linear a system is sometimes an important step in the analysis and design of a class of systems known as the adaptive control systems.
In real life the dynamic characteristics of most systems vary to some extent over an extended period of time. For instance, the block diagram of Fig.
Automatic Control Systems by Benjamin C. Kuo Ebook PDF
The main components of. If the mathematical and functional relationships of all the system elements known, the block diagram can be used as a reference for the analytical or the computer solution of the system. Furthermore, if all the system elements are assumed to be linear, the transfer function for the overall system can be obtained by means of block-diagram algebra. The essential point is that block diagram can be used to portray nonlinear as well as linear systems.
For example, Fig. In the. Block diagram of a simple control system, a Amplifier shown with a nonlinear gain characteristic, b Amplifier shown with a linear gain Fig. The motor is assumed to be linear and its dynamics are represented by a transfer function between the input voltage and the output displacement. Figure b illustrates the same system but with the amplifier characteristic approximated by a constant gain. In this case the overall system is linear, and it is now possible to write the transfer function for the overall.
One of the important components. The block-diagram elements of these operations are illustrated as shown in Fig.
It should be pointed out that the signals shown in the diagram of Fig. In Fig. It simply shows that the block-diagram notation can be used to represent practically any input-output relation as long as the relation is defined.
For instance, the block diagram of tiplication,. The physical components involved are the potenti-. As another example, shows a block diagram which represents the transfer function of a. The following terminology often used in control systems is defined with reference to the block.
In principle at least, the block diagram of a system with one input and one output can always be reduced to the basic single-loop form of Fig. However, the steps involved in the reduction process. Block Diagram and Transfer Function of Multivariable Systems is defined as one that has a multiple number of block-diagram representations of a multiple-variable system with p inputs and q outputs are shown in Fig.
The case of Fig. Figure shows the block diagram of a multivariable feedback control The transfer function relationship between the input and the output of is. I G 5 H j is nonsingular. However, it is still possible to define the closed-loop transfer it is.
Consider that the forward-path transfer function matrix and the feedback-path transfer function matrix of the system shown in Fig.
N constructing a signal flow graph. In the case when a system is represented by a equations. A signal flow graph may be defined as a graphical means of portraying the input-output relationships between the variables of a set of linear algebraic equations. It Fig. As another illustrative example. In A signal can transmit where j.
The signal flow The. The signal-flow-graph representation of Eq. The branch that Finally. The equations based on which a signal flow graph is drawn must be algebraic equations in the form of effects as functions of causes. Step-by-step construction of the signal flow A signal flow graph applies only to linear systems. Nodes are used to represent variables. Modification of a signal flow graph so that y 2 and y z satisfy the requirement as output nodes. An input node branches. Output node sink.
The branch directing from node y k to j. An output node is a node which has only incoming branches. Signals travel along branches only in the direction described by the arrows of the branches.. Rearranging Eq. Erroneous way to make the node y 2 Fig. Since the only proper way that a signal flow graph can be drawn is from a set of cause-and-effect equations.
A path is direction. Signal flow graph with y 2 as an input an input node. A forward path is a path that starts at an input node and ends at an output node and along which no node is traversed more than once.
Forward path. Path gain. Forward-path gain. Four loops in the signal flow graph of Fig. Loop gain the loop gain of the loop is defined as the path gain of a loop.
A loop is a path that originates and terminates on the same node is and along which no other node in Fig. The product of the branch gains encountered in traversing a path is called the path gain. Forward-path gain forward path. Fi a xl y x tfisJi Parallel branches in the same a single direction connected between two branch with gain equal to the sum nodes can be replaced by the parallel branches.
An example of this case is of the gains of illustrated in Fig. In the signal flow graph of Fig.Let us consider first the system of Fig.
Now let us consider the same function used in Example Published on Jul 3, Four loops in the signal flow graph of Fig. In the absence of feedback. Comparing this plot with the previous one without integral gain, results in less steady state error for the case of controller with integral part.
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