# BENJAMIN KUO AUTOMATIC CONTROL SYSTEMS PDF

Automatic Control Systems by Benjamin C. Kuo - Ebook download as PDF File . pdf), Text File .txt) or read book online. Download Automatic Control Systems By Benjamin C. Kuo, Farid Golnaraghi – Automatic Control Systems provides engineers with a fresh new controls book. Farid Golnaraghi • Benjamin C. Kuo. Solutions Manual. Page 2., 9th Edition. AAutomatic Control Systems. Chapter 2 Automatic Control Systems, 9th Edition .

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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. 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.

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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. Also, if there is saturation and Kp value is too high, chattering phenomenon may appear. The torque generated by the motor is 0.

It takes 0. In order to get the same result as Problem , the Kp value has to increase by a factor of 5. It has less steady state error and a faster rise time than Problem , but has larger overshoot.

As the proportional gain gets higher, the motor has a faster response time and lower steady state error, but if it the gain is too high, the motor overshoot increases.

If the system allows for overshoot, the best proportional gain is dependant on how much overshoot the system can have. As the derivative gain increases, overshoot decreases, but rise time increases. Also, the amplitude of the output starts to decrease when the frequency increases above 0. Repeat the process for other frequency values and use the calculated gain and phase values to plot the frequency response of the system.

Enter the Step Input and Controller Gain values by double clicking on their respective blocks. Observe the steady state value change with K. Simulate the response and show the desire variables. Related Papers. A First Course in Li By Emmanuel Olvera. Nonlinear model predictive control using multiple shooting combined with collocation on finite elements.

By Jasem Tamimi. By waleed adel and waheed zahra. 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.

Signal flow single branch. Node as a summing point and as a transmitting point. Figure shows series In Section 3. In this section we shall give two simple illustrative examples. Signal flow graph of a feedback control system. For complex signal flow graphs we do not need to rely on algebraic manipu- lation to determine the input-output relation.

Then one set of independent equations representing cause-and-effect relation Ii. In this case it is more convenient to use the branch currents and node voltages designated as shown in Fig. It is noteworthy that in the case of network analysis.

I in order.

## Automatic Control Systems by Benjamin C. Kuo

Of course. The Laplace transform of the input voltage is denoted by Ein s and that of the output voltage is EJjs.

More elaborate cases will be discussed in Chapter where the modeling of systems is formally covered. Writing the voltage across the inductance and the the capacitor. One way is to solve for I s from Eq. Example current in network shown in Fig. Signal flow graph of the network in Fig. There are several possible ways of constructing the signal flow graph for these equations. When we have sl s take the Laplace transform. The signal flow graph portraying these equations is drawn as shown in Fig This exercise also illustrates that the signal flow graph of a system Now.

In order to arrive at algebraic equations. These four variables are related by the equations. As an The signal flow The signal flow alternative. Notice that in this signal flow graph the Laplace transform variable appears only in the form of s' 1 Therefore.

It must be emphasized that the gain formula can be applied only between an input node and an output node. Signal flow graphs Chapter 4 as the state diagrams. An error that is frequently regard to the gain formula is the condition under which it is valid. Sec 3. The signal flow graph is redrawn in Fig.

The forward-path M. There is one pair of nontouching loops. By use of Eq. There is only one forward path between gain is Eia and E. G s is There is only one loop. The following conclusions: Example Consider. The following conclusions flow graph 1. Since the system linear. The gain between one input and one output is detergain formula to the two variables while setting the rest of the is redrawn as shown in Fig. Ein graph of the passive network in Fig.

All the three feedback loops are in touch with the forward path. The following inputoutput relations are obtained by use of the general gain formula: Signal flow graph for Example To illustrate how the signal flow graph and the block diagram are related.

As described in Section 2. Figure a illustrates a linear system with transfer function G s whose input is the output of a finite-pulsewidth sampler. Since for a very small pulse duration p. At this point we can summarize our findings about the description of the discrete-data system of Fig. This may be obtained by or if r t is a unit impulse funcsampling a unit step function u. Although from a mathematical standpoint the meaning of sampling an impulse function questionable and difficult to define.

In the following There are several ways of deriving the transfer function representation of we shall show two different representa- tions of the transfer function of the system. When a unit impulse function is applied to the linear process. If a fictitious ideal sampler S2 which. Discrete-data system with an ideal sampler. Multiplying both sides of Eq. The z-transform relationship is obtained directly from the is definition of the z-transform.

The output of the ideal sampler is the impulse train. The pulse transfer relation of Eq. If c t is a description of the true output: The expression it in Eq. Figure illustrates two different situations of a discrete-data system which contains two cascaded elements. In the system of Fig. The Laplace transform of the output of the system C s in Fig. In discrete-data systems. Let us consider first the system of Fig. The two elements with transfer functions and G 2 s of the system in Fig.

For simplicity.

## Automatic Control Systems by Benjamin C. Kuo | Control Theory | Control System

Transfer Functions of Closed-Loop Discrete-Data Systems In this section the transfer functions of simple closed-loop discrete-data systems are derived by algebraic means. Closed-loop discrete-data system. Consider the closed-loop system shown in Fig. Substituting Eq. Let us consider the system shown in Fig. The output transforms. The z-transform of the output is determined directly from Eq. The transfer function matrices of the system are is shown in Fig.

Data Systems 1. Station A. The block diagram of a multivariate feedback control system P Linear Networks and Systems. Box Signal Flow Graphs of Sampled. July Discrete Prentice- B. Flow Graphs. Signal Flow Graphs and Englewood Cliffs. The following where r t differential equations represent linear time-invariant systems. Diagram Network Transformation. G s H s Figure P A multivariable system with two inputs and two outputs Determine the following transfer function relationships Ci j is shown in Fig.

Draw a signal flow graph for the following 3xi Xi 3 set of algebraic equations: Draw an equivalent signal flow graph for the block diagram in Fig. Figure P Find the P Find the transfer function C s jR s. Are the two systems shown in Fig. Given the signal flow graph of Fig. P a and b equivalent?

P are all equivalent. Find the value of a so that the voltage e t is not affected by the source ed t. In the circuit of Fig. Construct an equivalent signal flow graph for the block diagram of Fig.

G 3 G4 u and 2 so that the output C is not affected by the disturbance signal N. A multivariate system is described by the following matrix transfer function relations C s S. H H Figure P Transfer function js valuable for frequency-domain analysis and The greatest advantage of transfer compactness and the ease that we can obtain qualitative information on the system from the poles and zeros of the transfer function.

As the word we should first begin by deimplies. To begin with the state-variable approach. The state-variable method is often referred to as a modern approach. The state-variable representation to linear systems and time-invariant systems. An important feature of this type of representation is that the system dynamics are described by the input-output relations. In this case. It is interesting to note that an easily understood example is the "State of the Union" speech given by the President of the United States every year.

Definition set we may define the state variables as follows: There are some basic ground rules regarding the definition of a state by a set variable and what constitutes a. An output of a system is a variable that can be measured. From a mathematical sense it is convenient to define a set of state variables and state equations to portray systems. Let us define these variables as these state variables must satisfy the following conditions 1. Once the inputs of the system and the initial states defined t for t above are specified.

Then system. Consider that the set of variables. The state variables of a system are defined as. The outputs of the system c k t.. The state equations and the output equations together form the set of equations which are often called the dynamic equations of the system. This is not surprising since an inductor is an electric element that stores kinetic energy.

RLC network.. One method of writing the state equations of the network. For a system with p inputs and q outputs. Using the convennetwork approach. An alternative approach is to start with the network and define the state variables according to the elements of the network. The objective. Rearranging the terms in Eq. Equation differs from Eq. We have demonstrated how the state equations of the RLC network may be written from the loop equations by defining the state variables in a specific way.

As stated in Section 4. We notice that using the two independent methods. For a linear system with time-varying parameters..

Then the state equations of Eq. Let x 2 t x r n X 1 where x t is defined as the state vector. We let x 0 the solution to Eq. One way of determining sides of Eq. RL network of Fig. As implies. Significance of the State Transition Matrix Since the state transition matrix satisfies the homogeneous state equation. In view of Eqs. This is left as an Example Consider the that is. Then Eq. Property of the state transition matrix. T Br r di last Now using the property of Eq. We start and assume that an input with Eq.

In the study of control systems. BR s ] 1 Using the definition of the state transition matrix of Eq. A -'BR s. Input voltage waveform is for the network in Fig. IE e t Fig. Let us consider that the input voltage to the RL network of Fig..

First for the time interval. The relationship between a high-order differential equation and the state equations is discussed in this section.. This simply involves the defining of the n state variables in terms of and its derivatives. The problem the output c 0 to represent Eq. We have shown earlier that the state vari- ables of a given system are not unique. The magnitude of the input for this interval t is 2E.

Theorem given by It is tem with single input shown in the following that any linear time-invariant and satisfying a certain condition of controllability see section 4. Anl B] nonsingular. If the n coefficient matrix Pll Pl2 Pin Pin P. Pi A"-'. Repeating the procedure leads to J'.

This is the the matrix condition of complete state controllability. Variable Characterization of Dynamic Systems Chap.. Pi obtained as a row matrix which be expressed in the phase-variable canonical form. Once P!

Since Pj is an 1 X n row matrix. Let us rewrite Eq.. To illustrate the point we consider the following example. Since the right side of the state equations cannot include any derivatives of the input r t. It is not ex- pected that one will always have these equations available for reference. The disadvantage with the method of Eqs.

It is interesting to investigate the relationship between these two representations. In Eq. Now we shall investigate the transfer function matrix relation using the dynamic equation notation. How- we shall later describe a more convenient method using the transfer function. Now equating the first term of each of the equations of Eqs. The resulting transformed. From the state-variable approach. E Sec. It can be defined an important part in the study of from the basis of the differential equation.

Eigenvalues The roots of the characteristic equation are often referred to as the eigenIt is interesting values of the matrix A. X X2 located on the main diagonal. Another important property of the characteristic equation and the eigenvalues is that they are invariant under a nonsingular transformation. One of the motivations A matrix is all that if to A is a diagonal be matrix.

This is A are identical to those proved by writing si. A P is equal to the product of the determinants. We have to assume that all the eigenvalues of A are distinct. Ap 2 p2. J Therefore. The problem can be stated as. We show is. K state This transformation is also is equation of Eq. A cannot always be diagonalized if it has multiple-order eigenvalues.

The known as the canonical form. P can be formed by use of the eigenvectors of A. There are other reasons for wanting to diagonalize the A matrix.

If the matrix A is of the phase-variable canonical form.. Let Pn P. X n are the eigenvalues of A. The similarity transformation may be carried out by use of the Vandermonde matrix of Eq. Let the eigenvector associated with a. Since of the phase-variable canonical form. It A into a diagonal matrix 3 We shall follow the guideline that P contains the eigenvectors of A.

P2 3 -P All the elements below the main diagonal of are zero. When its is the nonsymmetrical matrix has multiple-order eigenvalues. The Is. A are the eigenvalues of the 2.

The number of Jordan blocks is equal to the number of independent eigenvectors. A2 A3 A4 generally has the following properties: The elements on the main diagonal of matrix. Typical Jordan canonical forms are shown in the following examples [A. There is one and only one linearly independent eigenvector associated with each Jordan block.

The number of 1 s above the main diagonal is equal ton — r. In Eqs. The matrix A is called the Jordan canonical form.

A Some of 4. Let us as- distinct eigenvalues among n eigenvalues Then the following transformation "A y 1 must hold: In the first place. The mined by eigenvectors associated with an mth-order Jordan block are deter- Jordan block being written as There- For the eigenvector associated with the second-order eigenvalue.

A - l 2 To P such A has a simple eigenvalue at A! In this section we introduce the methods of the state diagram. Multiplication of a machine variable by a constant tion is done by potentiometers and amplifiers. A state is constructed fol- lowing all the rules of the signal flow graph. Basic Analog Computer Elements Before taking up the subject of state diagrams.

## [PDF] Automatic Control Systems By Benjamin C. Kuo, Farid Golnaraghi Book Free Download

Multiplication by a constant. The computer block diagram symbols of the potentiometer and the operational amplifier are shown in Figs. These are discussed separately in the following. If a lies to realize the operation of Eq. First consider the multiplication of a variable by a constant.

We shall now show that these analog computer operations can be portrayed by signal flow graphs which are called state diagrams because the state vari- ables are involved. Amplification may be accompanied by algebraic sum. Operational amplifier used as a summer. Analog computer block diagram of an integrator.

Laplace transform on both sides of Eq. The integrator can also serve simultaneously as a summing and amplification device. The algebraic sum of two or more machine variables may be obtained by means of the operational amplifier. Signal-flow-graph rep- Fig.

In this case the transform operation is necessary. Equation flow graph as is is now algebraic An graph for Eq. For the integration operation. Since the branch gains are constants in these cases. The and the output equations can be determined from the diagram. The 4. A state diagram can be constructed directly from the system's This allows the determination of the state variables and the state equations once the differential equation of the system is given.

The state diagram can be used for the simulation of the system on a digital computer. From Differential Equation to the State Diagram a linear system is described by a high-order differential equation. The details of these techniques are given below. Thus we have established a correspondence between the simple analog computer operations and the signal-flow-graph representations.

Consider the following differential state When. Before embarking on several illustrative examples on state diagrams. A state function. State State diagram for Eq. Example Consider the following differential equation: We venient to obtain the transfer show later that.

From State Diagram It to Analog Computer Block Diagram was mentioned earlier that the state diagram is essentially a block diaprogramming of an analog computer. As the next step. Now as the variables r. In terms of Laplace transform.

The following examples illustrate same purpose as an analog practically eliminated. JT10 Example From the state diagram of Fig. The final practical version of the com- puter block diagram for programming from shown. From State Diagram to Digital Computer Simulation on the digital comThe solution of differential equations by puter has been well established. The typical CSMP statements for the mathematical equations listed: Analog-computer block diagram for the system described by Eq.

The outputs of the intediagram is redrawn as shown in Fig. Then the main program of the given as follows: Example grators are assigned as state variables and the state Consider the state diagram of Fig. With the state diagram. Some the state diagram.. From State Diagram The state to Transfer Function transfer function between an input and an output is obtained from the to zero.

The state transition equations in the time domain are subsequently obtained by taking the inverse Laplace transform. The state diagram of Fig.

This illustration will emphasize the importance of using the gain formula. Figure 8 b shows the state diagram with the initial states or a Fig. State diagram of Fig. The right side of the equation contains the state t variables. Full Name Comment goes here. Are you sure you want to Yes No. Be the first to like this. No Downloads. Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. Kuo 1. Kuo 2. Book details Author: Benjamin C.

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