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- [PDF] Automatic Control Systems Download by - Benjamin C. Kuo
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- Golnaraghi F., Kuo B. Automatic Control Systems

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.

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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.

The treatment of the analysis and design of control systems was all classical. The two major changes in the second edition, published in , were the inclusion of the state variable technique and the integration of the discrete-data systems with the continuous data system. The chapter on nonlinear systems was eliminated in the second edition to the disappointment of some users of that text. At the time of the revision the author felt that a comprehensive treatment on the subject of nonlinear systems could not be made effectively. The third edition is still written as an introductory text for a senior course on control systems. Although a great deal has happened in the area of modern control theory in the past ten years, preparing suitable material for a course on introductory control systems remains a difficult task. While some of the techniques in modern control theory are much more powerful and can solve more complex problems, there are often more restrictions when it comes to.

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-.

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. 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. 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. N constructing a signal flow graph. The signal-flow-graph representation of Eq. In A signal can transmit where j. It Fig. The branch that Finally. The signal flow The.

As another illustrative example. Nodes are used to represent variables. 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. Modification of a signal flow graph so that y 2 and y z satisfy the requirement as output nodes. Signals travel along branches only in the direction described by the arrows of the branches.

An input node branches. Output node sink. An output node is a node which has only incoming branches. The branch directing from node y k to j. Forward path. Erroneous way to make the node y 2 Fig. Rearranging Eq. 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.

Since the only proper way that a signal flow graph can be drawn is from a set of cause-and-effect equations. Path gain. Loop gain the loop gain of the loop is defined as the path gain of a loop. The product of the branch gains encountered in traversing a path is called the path gain. Forward-path gain. A loop is a path that originates and terminates on the same node is and along which no other node in Fig. Four loops in the signal flow graph of Fig.

Forward-path gain forward path. An example of this case is of the gains of illustrated in Fig. Signal flow single branch. 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.

In the signal flow graph of Fig. 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. More elaborate cases will be discussed in Chapter where the modeling of systems is formally covered.

Then one set of independent equations representing cause-and-effect relation Ii. Of course. It is noteworthy that in the case of network analysis.

I in order. In this case it is more convenient to use the branch currents and node voltages designated as shown in Fig. The Laplace transform of the input voltage is denoted by Ein s and that of the output voltage is EJjs. In order to arrive at algebraic equations.

There are several possible ways of constructing the signal flow graph for these equations. Writing the voltage across the inductance and the the capacitor. When we have sl s take the Laplace transform. Signal flow graph of the network in Fig. 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. Example current in network shown in Fig. One way is to solve for I s from Eq.

These four variables are related by the equations. As an The signal flow The signal flow alternative. It must be emphasized that the gain formula can be applied only between an input node and an output node. Notice that in this signal flow graph the Laplace transform variable appears only in the form of s' 1 Therefore.

Sec 3. 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. By use of Eq. The forward-path M. Example Consider. The following conclusions: The signal flow graph is redrawn in Fig. G s is There is only one loop. There is one pair of nontouching loops. The following conclusions flow graph 1. There is only one forward path between gain is Eia and E. 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. Since the system linear. 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.. When a unit impulse function is applied to the linear process. This may be obtained by or if r t is a unit impulse funcsampling a unit step function u. If a fictitious ideal sampler S2 which. At this point we can summarize our findings about the description of the discrete-data system of Fig.

Discrete-data system with an ideal sampler. 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.

The output of the ideal sampler is the impulse train. The z-transform relationship is obtained directly from the is definition of the z-transform.

Multiplying both sides of Eq. If c t is a description of the true output: The pulse transfer relation of Eq. In the system of Fig. The expression it in Eq. Figure illustrates two different situations of a discrete-data system which contains two cascaded elements. The Laplace transform of the output of the system C s in Fig. In discrete-data systems. For simplicity. The two elements with transfer functions and G 2 s of the system in Fig.

Let us consider first the system of Fig. Closed-loop discrete-data system. Consider the closed-loop system shown in Fig. 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. Substituting Eq. The output transforms. Let us consider the system shown in Fig. The z-transform of the output is determined directly from Eq. Flow Graphs. Diagram Network Transformation.

Data Systems 1. Discrete Prentice- B. The following where r t differential equations represent linear time-invariant systems. The transfer function matrices of the system are is shown in Fig. The block diagram of a multivariate feedback control system P Station A. Signal Flow Graphs and Englewood Cliffs. Signal Flow Graphs of Sampled. Box Linear Networks and Systems.

July Draw a signal flow graph for the following 3xi Xi 3 set of algebraic equations: 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. Find the transfer function C s jR s. Draw an equivalent signal flow graph for the block diagram in Fig. Find the P Figure P 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. Are the two systems shown in Fig. Given the signal flow graph of Fig. P a and b equivalent? 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 Construct an equivalent signal flow graph for the block diagram of Fig. The state-variable representation to linear systems and time-invariant systems. To begin with the state-variable approach. The state-variable method is often referred to as a modern approach. An important feature of this type of representation is that the system dynamics are described by the input-output relations.

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. The state variables of a system are defined as. Once the inputs of the system and the initial states defined t for t above are specified. An output of a system is a variable that can be measured. Definition set we may define the state variables as follows: Let us define these variables as these state variables must satisfy the following conditions 1.

From a mathematical sense it is convenient to define a set of state variables and state equations to portray systems. 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. There are some basic ground rules regarding the definition of a state by a set variable and what constitutes a.

Consider that the set of variables. Then system. The outputs of the system c k t. For a system with p inputs and q outputs. RLC network. One method of writing the state equations of the network. The state equations and the output equations together form the set of equations which are often called the dynamic equations of the system.. Using the convennetwork approach. This is not surprising since an inductor is an electric element that stores kinetic energy. As stated in Section 4.

Rearranging the terms in Eq. An alternative approach is to start with the network and define the state variables according to the elements of the network. 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. Equation differs from Eq. The objective. We notice that using the two independent methods.

Let x 2 t x r n X 1 where x t is defined as the state vector.. Then the state equations of Eq. For a linear system with time-varying parameters. We let x 0 the solution to Eq. One way of determining sides of Eq. Significance of the State Transition Matrix Since the state transition matrix satisfies the homogeneous state equation. As implies. In view of Eqs. RL network of Fig. This is left as an Example Consider the that is.

Then Eq. Property of the state transition matrix. In the study of control systems. We start and assume that an input with Eq.. BR s ] 1 Using the definition of the state transition matrix of Eq. T Br r di last Now using the property of Eq. A -'BR s. Input voltage waveform is for the network in Fig. Let us consider that the input voltage to the RL network of Fig.

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

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. Theorem given by Anl B] nonsingular.. If the n coefficient matrix. Pi A"-'. Pll Pl2 Pin Pin P. Repeating the procedure leads to J'. Variable Characterization of Dynamic Systems Chap.

Once P! Pi obtained as a row matrix which be expressed in the phase-variable canonical form This is the the matrix condition of complete state controllability.. Since Pj is an 1 X n row matrix.. Let us rewrite Eq. Since the right side of the state equations cannot include any derivatives of the input r t. To illustrate the point we consider the following example.. How- we shall later describe a more convenient method using the transfer function. It is interesting to investigate the relationship between these two representations.

In Eq. It is not ex- pected that one will always have these equations available for reference. The disadvantage with the method of Eqs. Now we shall investigate the transfer function matrix relation using the dynamic equation notation. Now equating the first term of each of the equations of Eqs. The resulting transformed. E Sec. It can be defined an important part in the study of from the basis of the differential equation.

From the state-variable approach. This is A are identical to those proved by writing si. One of the motivations A matrix is all that if to A is a diagonal be matrix. A P is equal to the product of the determinants. 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. Eigenvalues The roots of the characteristic equation are often referred to as the eigenIt is interesting values of the matrix A.

The problem can be stated as.

Ap 2 p2. A cannot always be diagonalized if it has multiple-order eigenvalues. We show is. There are other reasons for wanting to diagonalize the A matrix. K state This transformation is also is equation of Eq. The known as the canonical form. We have to assume that all the eigenvalues of A are distinct. P can be formed by use of the eigenvectors of A.

J Therefore. If the matrix A is of the phase-variable canonical form.. X n are the eigenvalues of A.

Let Pn P. It A into a diagonal matrix 3 We shall follow the guideline that P contains the eigenvectors of A. The similarity transformation may be carried out by use of the Vandermonde matrix of Eq. Since of the phase-variable canonical form. Let the eigenvector associated with a. P2 3 -P A are the eigenvalues of the 2. The displacement of actuator is changing proportionally with the temperature differences: The heat flow through the sides is: Substituting equation 3 , 4 , 5 and 6 into equation 7 and 8 gives the model of the system.

The changes if the temperature of heat sink is supposed to be zero, then: The state model of the system is given by substituting equations 2 , 3 , and 6 into these equations give. The behaviour of the valve in this system can be written as: The state equations can be rewritten by substituting P2, Pv, Ps and Q2-v from other equations. The dynamic for the well can be written as two pipes separating by mass m: For the walking beam: According to the equation of angular motion: Thus, N 9.

See Chapter 6. Thus e ss f. Since the system is linear, then the effect of X s is the summation of effect of each individual input. Thus Z n Thus, Z n Thus Kt 0. Z n The coefficients are ordered in descending powers. Other parts are the same. Therefore, the equation of motion is rewritten as: For description refer to Chapter 9. Therefore the natural frequency range in the region shown is around 2.

Overshoot increases with K. No need to adjust parameters. For the sake simplicity, this problem we assume the control force f t is applied in parallel to the spring K and damper B.

We will not concern the details of what actuator or sensors are used. Lets look at Figure and equations and This is now an underdamped system.

The process is the same for parts b, c and d. Use Example as a guide. Use Acsys to do that as demonstrated in this chapter problems. Also Chapter 2 has many examples. You may look at the root locus of the forward path transfer function to get a better perspective. For a better design, and to meet rise time criterion, use Example 5- Open loop speed response using SIMLab: The form of response is like the one that we expected; a second order system response with overshoot and oscillation.

Considering an amplifier gain of 2 and K b 0. To find the above response the systems parameters are extracted from: Study of the effect of viscous friction: The above figure is plotted for three different friction coefficients 0, 0. As seen in figure, two important effects are observed as the viscous coefficient is increased.

First, the final steady state velocity is decreased and second the response has less oscillation. Both of these effects could be predicted from Eq. 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.