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Digital control of dynamic systems / Gene F. Franklin, J. David Powell, Michael L. Workman. Article (PDF Available) with 2, Reads. Cite this publication. Digital Control of Dynamic Systems, 3rd Edition. Home · Digital Control of Dynamic Systems, 3rd Edition Author: Gene F. Franklin | soundofheaven.info Powell | Michael L. Library of Congress Cataloging-in-Publication Data. Franklin, Gene E. Digital control of dynamic systems/Gene F. Franklin, J. David. Powell. Michael L. Workman.

The transfer function in polynomial form is 2. C and we assume we can find values of "0 and R u as bounds on the of the complex variable for which the series Eq. That means that the magnitude plot must equal I at the same frequency that the phase plot equals '. Discrete Systems AnalysIs 4. A function fix is linear if!

Clearly, a balance between these two effects is required. It turns out that the optimal solution to this balance can be found as a function of the process noise intensity, R", and the sensor noise intensity. Since the only quantity affecting the result is the ratio R,J R,. An important advantage of using the optimal solution is that only one parameter.

This means that x t wtll converge to x t '. Errors in the model of the plant F, G, H cause additional errors to the state estllnate from those predicted by Eq. However, L can typically be chosen that the error IS kept small. It is important to emphasize that the nature of the plant and the estImator are quite different. The plant is a physical system such as a chemical process or servomechanism whereas the estimator is usually an elec.

The dynamics of the er- ror can be obtamed by subtractmg the estimate Eq. As In the control case, this is almost never done by hand. Rather, the' func- E lJons and place. The transpose of q. ThIS approach IS usuall supenor. The result is 2.

Possible locations for introducing the reference input: Combined Control and Estimation We now put all this together. If we take the control law Eq. The roots of this new closed-loop system can be shown to consist of the chosen roots of the controller plus the chosen roots of the estimator that have been designed in separate procedures in Sections 2. The poles and zeros of the compensator alone could be obtained by examining the system described by Eq. This scheme is shown schematically in Fig.

Using this approach. An alternative approach consists of entering the command r directly into the plant and estimator in an identical fashion as shown in Fig. Since the 1- Control Jaw Estimator: The solution is to incorporate an integral control tenn in the feedback similar to the integral control discussed in Section 2.

Integral control is accomplished using state-space design by augmenting the state vector with the desired integral x I" It obeys the differential equation 2. It is based on phase considerations which can easily be detennined graphically by hand, and are therefore very useful in checking computer based results. The open-loop transfer function can be determined experimentally or analytically.

Pole placement or optimal methods can also be used to arrive at the best estimator for this purpose. This equation is augmented to the state equations Eq. With this revised definition of the system. Summary 2. Ie Find the transfer function of the complete controller consisting of the control from pan a and the estimator from pan b. I;I' and modify the compensation so that the specifications are still met.

Ie Draw a block diagram of the system. With unity feedback. After completing the hand sketch. What is the minimum value of K to achieve a ,table system? However, most control systems today use digital computers usually microprocessors or microcontrollers with the necessary inpuUoutput hardware to implement the controllers. The intent of this chapter is to show the very basic ideas of design- ing control laws that will be implemented in a digital computer.

Unlike analog electronics, digital computers cannot integrate. These approxi- mation techniques are often referred to as numerical integration. This chapter shows a simple way to make these approximations as an introduction to digital control. Later chapters expand on various improvements to these approximations, show how to analyze them. In the final analysis, we will see that direct digital design provides the designer with the most accurate method and the most flexibility in selection of the sample rate.

From the material in this chapter. The system would be expected to give adequate performance if the sample rate is at least 30 times faster than the bandwidth of the system.

Chapter Overview In Section 3. Section 3. Even if 8t is not quite equal to zero. The sampled signal is y kT where k can take on any integer value. It is often written simply as v k. We make the assumption here that the 'sample period is fixed: There also may be a sampler and AID converter for the input command, r t , producing the discrete r kT from which the sensed y kT would be subtracted to arrive at the discrete error signal.

The differential equation of the continuous compensation is approximated by a difference equation which is the discrete approximation to the differential equation and can be made to duplicate the dynamic behavior of a D s if the sample period is short enough. The result of the difference equation is a discrete u kT at each sample instant. The resulting u t is then applied to the actuator in precisely the same manner as the continuous implementation.

One particularly simple way to make a digital computer approximate the real time solution of differential equations is to use Euler's method. This approximation1 can be used in place of all derivatives that appear in the controller differential equations to arrive at a set of equations that can be solved by a digital computer.

These equations are called difference equations and are solved repetitively with time steps of length T. The com- putatIOn of the error signaL e. D s , can all be accomplIshed In a digital computer as shown in Fig. The fundamental differences between the two implementations are that the digital system operates on samples of sensed plant output rather than on the continuous signal and that the dynamICs represented by D s are implemented by algebraic recursive equatIOns called difference equations.

We consider first the action of the analog-to-digital AID converter on a sIgnal. The conversion from the analog signal y f occurs repetitively at instants of time that are T seconds Digitization j Digital controller , T' I r kTj , , , , , ,: SectIOn 3. See Problem 3. Implement the control equations on an experimental laboratory facility like that depicted in Fig.

Compute the theoretical step response of the continuous system and compare that with the experimentally detennined step response of the digitally controlled system. Solution, Comparing the compensation transfer function in Eq. Discrete design methods described in later chapters will show how to achieve this performance and the consequences of sampling even slower if that is required for the computer being used.

First find the differential equation that corresponds to D s. After cross multiplying Eq. Rearranging Eq. For computalional effiCIency, it is convenient to re-arrange Eq. However, there is usually no requirement that values for all limes be saved In memory.

Therefore, the computer need only have variables defined for the current and past values for this first-order difference equation. The InstruCllons to the computer to implement the feedback loop in Fig. A delay in any feedback system degrades the stability and damping of the system.

Because each value of u kT in Fig. I b is held constant until the next value is available from the computer, the continuous value of 11 t consists of steps see Fig.

By incorporating a continuous approximation of this T12 delay in a continuous analysis of the system, an assessment can be made of the effect of the delay in the digitally controlled system.

The delay can be approximated by the method of Pade. This linear approximation of the sampling delay Eq. However, before those methods can be examined. Note that the 40 Hz sample rate about 30 x bandwidth behaves essentially like the contmuous case, ;-hereas the 20 Hz sample rate about 15 x bandwidth has a detectable mcreased overshoot slgmfymg some degradation in the damping.

The damping would degrade further lf the sample rate were made any slower. You willleam how to compute the response of a digital system in Chapter 4. For a sample rate of 40 Hz. Fur prior versions. Since the PM is approximately X.

Both analvsis methods a similar reduction in the damping of the syqem. One should. For the case with no Fig. The actual step p in Fig. The trend that deLTeasing. Figure 3. Use hoth linear analysis and the frequencY response method. The damping of the 'y,tem with the simple delay approximation added IEq, 3. For more precision. Equation t3. Reviewing again briefly. These three constants define the control. However, normally these terms are used together and, in this case, the combination needs to be done carefully.

The combined continuous transfer function Eq. The damping was degraded an increasing amount as the sample rate was reduced. Furthermore, it was possible to restore the damping with suitable adjustments to the control. Pick an appropriate sample rate. Compare the digital step response with the calculated response of a continuous system.

This implementation shows a considerably increased overshoot over the continuous case. The line with circles in the figure shows the improved performance obtained by increasing the sample rate to 10 kHz; i. It shows that the digital performance has improved to be essentially the same as the continuous case. Increasing the sample rate. A look at Fig. This suggests that the proportional gain. Some trial and error. The integral reset time. The sample rate needs to be selected first. But before we can do that.

The solid line in Fig. Based on Eq. Therefore, the sample rate would be about 3. Use of Eq. In order to analyze the system accurately for any sample rate. A zero-pole approximation for this delay is I I 'II. Compare the resulting difference equations with the fDrward reclangular Euler methDd. Assume the continuous values from Eq. Repeat the calculations using the backward retangular method see Problem 3. Use a sample rate of 6 kHz. Modify the MATLAB file fig32m so that you can evaluate the digital version of your lead compensation using Euler's forward retangular method.

Try different sample rates. Approximate the effect of a digital implementa- tion to be 2iT G1,. The fundamental character of the digital computer is that it takes a finite time to compute answers, and it does so at discrete steps in time. The purpose of this chapter is to develop tools of analysis necessary to understand and to guide the design of programs for a computer sampling at discrete times and acting as a linear.

Needless to say, digital computers can do many things other than control linear dynamic systems; it is our purpose in this chapter to examine their characteristics when doing this elementary control task and to develop the basic analysis tools needed to write programs for real-time computer control. Chapter Overview Section 4. The tool for analyzing this sort of system, the: Use of the: Furthermore, state-space models of discrete systems are developed in this section.

Section 4. The last section. Assume that uK represems. Assume that no rabbits die and that a new pair begin reproduction after one period.

Thus at time k. We need a starting time k-value and some initial conditions to characterize the contents of the computer memory at this time.

For example, suppose we take the case 30 4. Here there are no input values. The first nine values are I. A plot of the values of u, versus k is shown in Fig. The results. However that may be, the output of the system represented by Eq. If the response of a dynamic system to any finite initial conditions can grow without bound.

We would like to be able to examine equations like Eq. Figure 4. The name "difference equation" derives from the fact that we could write Eq. Here we consider the treatment of the data inside the computer. Thus we write If we solve Eq. Although the two forms are equivalent, the recurrence form of Eq. We will continue, however, to refer to our equations as "difference equations: We plan to demonstrate later that with such equations the computer can control linear constant dynamic systems and approximate most of the other tasks of linear, constant.

To do so, it is necessary first to examine methods of obtaining solutions to Eq. For continuous. In the case oflinear, constant. Consider Eq. Now if we assume: The characteri,tic equation is I! This polynomial of second degree has two solutions,: Let's call these z, and: We can solve for the unknown constants by requiring that this general solution satisfy the specific initial conditions given.

These equations are easily solved to give. We can generalize this result. The equation in: If any solution of this equation is outside the unit circle has a magnitude greater than one , Figure 4.

Since both these roots are in,ide the unit circle. As an example of the origins of a difference equation with an extemal input. Suppose we have a continuous signal, e t. Three altematives are sketched in Fig.

In Eq. To take the one-s. Fmdthe is the unit function. I We will return to the analysis of signals and development of a table of useful in Section 4.

C and we assume we can find values of "0 and R u as bounds on the of the complex variable for which the series Eq. Applying Eq. It should be, because if e t is a straight line, the trapezoid is the exacf area. If we take the third choice, the area of the trapezoid is Each of these integration rules is a special case of our general difference equation Eq. We will examine the properties of these rules later, in Chapter 6. Thus we see that difference equations can be evaluated directly by a digital computer and that they can represent models of physical processes and approxi- mations to integration.

It turns out that if the difference equations are linear with coefficients that are constant. The Discrete Transfer Function We will obtain the transfer function of linear, constant, discrete systems by the method of z-transform analysis. A logical alternative viewpoint that requires a bit more mathematics but has some appeal is given in Section 4. The results are the same. We also show how these same results can be expressed in the state space form in Section 4. The numerator of Eq.

For example, the.: To find the relation, we proceed by direct substitution. We take the definition given by Eq. Solving it we obtain Now we multiply Eq. We get "". The discrete system is specified as' Although we have developed the transfer function with the z-transform.

Because H z is a rational function of a complex variable, we use the ter- minology of that subject. Suppose we call the numerator polynomial b z and the denominator a z.

The transfer function Eq. When completely factored. T where T is the sample period. The general input-output relation between transforms with linear, constant, difference equations is zeros poles 4. Thus, in this case. The transfer function of paths in parallel is the sum of the single-path transfer functions Fig.

The transfer function of paths in series is the product of the path transfer functions Fig. The transfer function of a single loop of paths is the transfer function of the forward path divided by one minus the loop transfer function Fig. The transfer function of an arbitrary multipath diagram i, given by combi- nations of these cases. Mason's rule" can also be used. It is interesting to connect this case with a block diagram way as for continuous system transfer functions.

To use block-diagram analysis to manipulate these discrete-transfer-function relationships. See Franklin. We can now give a physical meaning to the variable:. Suppose we let all coefficients in Eq. Then H: But H ;: Uk' equals the input delayed by one period. Thus we see that a transfer function of Z-i is a delay of one time unit. We can picture the situation as in Fig. Since the relations of Eqs.

We can follow the operations of the discrete integrator by tracing the signals through Fig. This is the signal marked a, in Fig. After this. The discrete integration occurs in the loop with one delay. These can be solved by the methods of linear algebra or by the graphical methods of block diagrams in the same Figure 4.

The output equation is also immediate except that we must watch to catch all paths by which the state variables combine in the output.

The results we require and references to "rudy material are given in Appendix C. From Eq. To complete the representation of Eqs. The completed picture is shown in Fig. Having the block diagram shown in Fig. We can write Eq. The first one we will consider leads to the "control" canonical form. At this point we need to get specific; and rather than carry through with a system of arbitrary order. In the development that follows. With this convention which is simply using the property of z derived earlier.

In this equation. The terms with factors of z are time-shifted toward the future with respect to k and must be eliminated in some way. To do this, we assume at the start that we have the u k , and of course the e k. Now in this internal result there appear a,u and -b,e.

If we continue this process of subtracting out the terms at k and operating on the rest by Z-I, we finally arrive at the place where all that is left is u alone! But that is just what we assumed we had in the first place, so connecting this term back to the start finishes the block diagram. XI for example not only reaches the output through hi but also by the parallel path with gain -boG,.

The block diagrams of Figs. Another useful form is obtained if we realize a transfer function by placing several first- or second-order direct forms in series with each other. The cascade factors. Following the technique used for the control form. Discrete Systems AnalysIs 4.

We can also give a time-domain meaning to an arbitrary transfer function. Recall that the is defined by Eg. For example, let us look at the system of Fig. We can readily follow the pulse through the block and build Table 4. Thus the unit-pulse response is zero for negative k. A final point of view useful in the interpretation the dIscrete transfer function is obtained by multiplying the infinite polynorruals of E z and H ;: For purposes of illustration.

The dm? Table 4. A system with input e and output II is linear if superposition applies. A system is stationan-, or time invariant. If we repeat this experiment at any later time when the system is again at rest and we apply the shifted input. A constant coefficient difference equation is stationary and typically referred to as a constant system.

External Stability A very important qualitative property of a dynamic system is stability, and we can consider internal or external stability.

The general case is thus again 4. First we need more fonnal definitions of "linear" and "stationary: Finally, by linearity again, the total response at time k to a sequence of these pulses is the slim of the responses, namely. Values for j greater than k occur if the unit-pulse response is nonzero for negatIve By de6nition, such a system, which responds before the mput that causes It occurs, IS called noncausal.

This is the discrete convolution sum and IS the of the convolution integral that relates input and impulse response to output In lInear, constant, continuous systems. To venfy Eq. Listing these coefficients. Applying the test, we have x ex: Difference Equation Stability Consider the difference equation 4. The test gIven q IS ru ,. Otherwise we can be satisfied to consider only the external stabilitJ as given by the study of the input-Qutput relation described for the linear stationary case by the convolution Eq.

These differ in that some internal modes might not be connected to both the input and the output of a given system. For external stability, the most common definition of appropriate response is that for every Bounded Input, we should have a Bounded Output. If this is true we say the system is BlBO stable. A test for BIBO stability can be given directly in terms of the unit-pulse response. Suppose the input e, is bounded.

Let us call the particular output in response to the pulse shown in Fig. This response is the difference between the step response [to I t ] and the delayed step response [to 1 1 - n]. The Laplace transform of the step response is G s! Although it is possibly confusing at first, we follow convention and call the discrete transfer function G z when the continuous transfer function is G s.

Although G z and G s are entirely dIfferent functIOns, they do describe the same plant, and the use of s for the continuous transform and z for the discrete transform is always maintained, To is inside the unit circle, the corresponding pulse response decays with time and is stable, Thus, if all poles are inside the unit circle, the system with ratIonal transfer function is stable; if at least one pole is on or outside the unit circle, the corresponding system is not BlBO stable.

With modem computer programs available, finding the poles of a particular transfer function is no big deal. Sometimes, however, we wish to test for stability of an entire class of systems; or, as in an adaptive control system, the potential poles are constantly changmg and we wish to have a quick test for stability in terms of the literal polynomial coefficients. In the continuous case, such a test was provided by Routh; III the dtscrete case, the most convenient such test was worked out by Jury and Blanchard Most of the dynamic systems to be controlled, however, are continuous systems and, if linear, are described by continuous transfer functions in the Laplace variable s.

In this section we develop the analysis needed to compute the discrete transfer function between the samples that come from the digital computer to the DIA converter and the samples that are picked up by the NO convener. It is able to accept the system in any of the forms. We will apply the fonnula 4. The samples of this signal are I k T - e"Uu I k n, and the: We will next develop a fonnula using state descriptions that moves Ihe tedium to the computer. A continuous, linear, constant-coefficient system of differential equations was expressed in Eq.

For a scalar input. The output was expressed in Eq. The fluid mixer problem in Appendix. The term e';" represents the delay of Aseconds.

We assume that H 5 is a rational transfer function. To prepare this function for computation of the: To complete the transfer function. The first term is a unit step shifted left by In T seconds.

The corresponding: Consequently the final transfer function is Gl: Notice thatth: For specific values of the mixer. For these values. Also, we must often consider finite computation time in the digital controller, and this is exactly the same as if the process had a pure time delay. With the techniques we have developed here, it is possible to obtain the discrete transfer function of such processes exactly, as Example 4. We saw earlier how a single high-order difference equation could be represented by a state description in control or in observer canonical form.

Also, there is a very useful state description corresponding to the pat1ial- fraction expansion of a transfer function. State transformations can take a general description for either a continuous or a discrete system and, subject to some technical restrictions, convert it into a description in one or the other of these forms. We wish to use the state description to establish a general method for obtain- ing the difference equations that represent the behavior of the continuous plant.

Ultimately, the digital controller will take the samples Y k , operate on that sequence by means of a difference equation. The loop wilL therefore, be closed. To analyze the result. To do this. We will solve the general equation in two steps. We begin by solving the equa- tion with only initial conditions and no external input. This is the homogeneous equation Solution. The representations given by Eqs.

The satellite attitude-control example is shown in block diagram form in Fig. Therefore, the equations of motion Can be written as Often the sampled-data system being described is the plant of a control problem, and the parameter J in Eq. I" Hence, from Eq. Substituting Eq.

Hence we conclude that. It can be shown that the solution given by Eq. Select sampling period T and description matrices F and G. Go to step 5. We then find r from Eq. A discus,ion of the selection of N and a technique to compute V for comparatively large T is given by Kallstrom , and a review of various methods is found in a classic paper by Moler and Van Loan To compare this method of representing the plant with the discrete transfer functions, we can take the ;,-transform ofEq.

The r integral in Eq. G, and T for Simple cases, The left arrow A common and typically valid assumption is that of a zero-order hold ZOH with no delay. Random are treated in Chapter 9.

We wish to use this solution over one sample period to obtain a difference equation: If some other hold is implemented or if there is a delay between the application of the control from the ZOH and the sample point, this fact can be accounted for in the evaluation of the integral in Eq. The equations for a delayed ZOH will be given in the next subsection. To facilitate the solution of Eq. If w is a constant. If w is an impulse. To compute the poles numerically when the matrices are given, one would use an eigenvalue routine.

Combining the two parts of Eq. From matrix algebra the well-known requirement for this is that det Using the from the previous example. An interpretation of transfer-function poles from the perspective of the corresponding difference equation is that a pole is a value of z such that the equation has a nontrivial solution when the forcing input is zero.

Use Eqs. This determinant is the characteristic polynomial of the transfer function, and the zeros of the determinant are the poles of the plant. In Eg. The integral runs for I from 0 to T, which corresponds to t frum kT - I: Over this period, the control. J Using the discrete model sysD found in Example 4. The general solution to Eq. Now we presenlthe formulas for a time delay in the model and also a time prediction up to one penod whIch corresponds to the modified z-transform as defined by Jury.

We begin With a state-variable model that includes a delay in control action. From Eg. Because m T is restricted to be less than T, however. To do so we first convert 1'1 to a form similar to the integral for 1',. We need a test for deciding Oll the value of k. We propose to approximate the series for 'It. We will select k. A simpler method is to select k such that the size of FT divided by 2' is less than I. The rule is to select k such that which has been found effective by Moler and Van Loan The basic idea comes from Eq.

This final solution is easily visualized in terms of a block diagram. To do this we introduce e new variables such that In this case, we must eliminate u k - 1 from the right-hand side. We have thus an increased dimension of the state.

Kallstrom has analyzed a technique used by Kalman and Englar , The structure of the equations is 4. For that. In the Control Toolbox. Controls engineers commonly use numerical simulation of nonlinear models to evaluate the performance of control systems. To aid in the design synthesis of controllers and to gain insight into approximate behavior. We begin with the assumption thaI our plant dynamics are adequately de- scribed by a set of ordinary differential equations in state-variable form as Select F and T.

Now double 'IT k times. The program logic for computing 'IT is shown in Fig. This is equivalent to where the symbol rx means the smallest integer greater than x. The maximum of this integer and zero is taken because it is possible that II FT II is already so small that its log is negative. A Having selected k. To obtain the suitable formula for 'IT. In Section 4. A method for study of linear constant discrete systems is thereby indicated. Signal Analysis and Dynamic Response If the system description is available in difference-equation form.

The final step, however, is tedious if done by hand. Our approach to this problem is to present a 1. Compute the transfer function of the system H z. Compute the transform of the input signal, E z. Form the product. Invert the transform to obtain u k T. The accuracy of the approximation varies with the problem, but is generally useful in designing the control system. The final design of the control system should always be checked via numerical simulation of the nonlinear equations.

But now the notation is overly clumsy. If r IS a vector, we define its partial derivatives with respect to the vector x as the matrix called the Jacobean composed of rows of gradients. In the subscript notatiun, if we mean to take the partial of all components, we omit the specific subscript such as I or 2 but hold its place by the use of a comma aI, ax aI 2 ax The assumption of small signals can be reflected by taking x and u to be always close to their reference values x o ' u o ' and these values, furthermore, to be an equilibrium point of Eq, 4.

The Laplace transform of the unit step is I! We will explore this further later. In any event. To emphasize the connection between the time domain and the z-plane. Beside the z-plane.

The unit circle is shown for reference.

Thus, when given an unknown transform we will be by reference to these known to infer the major fea: To begm this process of attaching a connection between the time domain and the z-transform domain, we compute the transforms of a few elementary signals. This result is much like the continuous case, wherein the Laplace transform of the unit impulse is the constant 1.

The quantity E, z gives us an instantaneous method to relate sianals to sy: To characterize the system H z , consider the signalu k , is the umt pulse response: By exploiting the features of E4 z.

We collect these for later reference. The boundary of stability is the unit circle. If such a signal were the unit-pulse response of our system such as? We plot in Fig. We thus take first and compute 19 have not ,this formally. The demonstration. The corre- sponding system is BIBO stable. We can compute the settling time in samples, N, in terms of the pole radius, r.

A sketch of the unit circle with several points corresponding to various numbers of samples per cycle marked is drawn in Fig. The settling time of a transient, defined as the time required for the signal to decay to one percent of its maximum value, is set mainly by the value of the radius, r, of the poles. The number of samples per oscillation of a sinusoidal signal is determined bye. It demonstrates visually the features just summarized for the general sinusoid, which encompasses all possible signals.

It is useful to sketch several major features from the s-plane to the z-plane according to Eq. Such a sketch is shown in Fig. Each feature should be traced in the mind to obtain a good grasp of the relation. These features are given in Table 4. We note in passing that the map 20 Unless a pole of E: The result is readily shown to a polynomial in II.

For our first study we consider the effect of zero location. We let ;1 - PI and explore the effect of the remaining zero location. In every case. The situation in the ;-plane is sketched in Fig.

Tn addition to the two poles and one zero of H: There are many values of s for each value of ;. The great significance of this fact will be explored in Chapter 5. Lines of constant damping in the s-plane are mapped into the ;-plane ac- cording to Eq. We often refer to the damping of a pole in the in terms of this equivalent s-plane damping. Because of the usefulness of this mapping.

You will see its use in the figure files of discrete root loci in Chapter 7. The generic dynamic test for controls is the step response. Our attention will be restricted to the step responses 'of the discrete system shown in Fig.

If a sinusoid at frequency 0" is applied to a stable, linear. If the transfer function is written in gain-phase form as H j: Ultimately, however, the test of a design is typically the actual time response, either by numerical simulation or an experimental evaluation. To summarize all these data, we plot the percent overshoot versus zero location in Fig.

The major feature of these plots is that the zero has very little influence when on the negative axis. Also included on the plots of Fig. For this case we again consider the system of Fig. In this case, the major influence of the moving singularity is on the rise time of the step response. In the figure we defined the rise time as the time required for the response to rise to 0.

Our conclusions from these plots are that the addition of a pole or zero to a given system has only a small effect if the added singularities are in the range from oto -I. The understanding of how poles and zeros affect the time response is very useful for the control system designer. The knowledge helps guide the iterative 21 Such systems are called nonminimum phase by Bode because the phase shift they impart 10 a sinu!

We can say almost exactly the same respecting the frequency response of a stable. If the system has a transfer function H z , 4. This is the phe- nomenon of aliasing, to which we will return in Chapter 5. If we substitute Eq. If we expand U z! If a unit-amplitude sinusoid is applied. It is worthwhile going through the calculations to fix ideas on this point.

And this is the case, which us t. Let the I1me tunctlOn In questIon be! For the analysis of real data, we need a transform defined over a finite data record, which can be computed quickly and accurately. The required formula is that of the Discrete Fourier Transform. Implementation of a version of the FFf algo- rithm is contained in all signal-processing software and in most computer-aided control-design software.

To understand the DFf, it is useful to consider two properties of a signal and its Fourier transform that are complements of each other: In ordinary Fourier analysis, we have a signal that is neither periodic nor discrete and its Fourier transform is also neither discrete nor periodic.

If, however, the time function f t is periodic with period To. In other words, if the function in time is periodic, the function in frequency is discrete.

The case where the properties are reversed is the z-transform we have just been studying. In this case, the time functions are discrete. We can summarize these results with the following table: A function fix is linear if! Applying this result to the definition of the z-transform, we find immediately that Dividing these results. We began a table ofz-transforms, and a more extensive table is given in Appendix B. H e1 l2: We will discuss in Chapter 12 the general problem of estimation of the total frequency response from experimental data using the DFTlFFT as well as other tools.

Some of these, such as linearity, we have already used without making a formal statement of it, and others, such as the transform of the convolution, we have previously derived. For reference, we will demonstrate a few properties here and collect them into Appendix B for future reference. In all the properties listed below, we assume that F', I X-I t: With this evaluation, we see that the sum we have been considering is ill! We apply this input to the system alld wait ulltil all transients hm'e died Glmy.

At this time.

QED As an illustration of this property. This is an example of radial projection whereby the roots of a polynomial can be projected radially simply by changing the coefficients of the polynomial.

The technique is sometimes used in pole-placement de- signs as described in Chapter 8. Final- Value Theorem: We should note here that the transform of the time shift is not the same for the one-sided transform because a shift can introduce terms with negative argument which are not included in the one-sided transform and must be treated separately. This effect causes initial conditions for the difference equation to be introduced when solution is done with the one-sided transform. See Problem 4.

Scaling ill the z-Plane: Thus the z-transform is a linear function. It is the linearity of the transform that makes the partial-fraction technique work. Convolution 01 Time Sequences: We have already developed this result in connection with Eq. It is this result with linearity that makes the transform so useful in linear-constant- system analysis because the analysis of a combination of such dynamic systems can be done by linear algebra on the transfer functions.

Time Shift: J 3,25z-' - 4. The process is identical to convening Ftz to the equivalent difference equation and solving for the unit-pulse response, and divide as follows Equation 4, 18 represents the transfonn of the system output. Such an expansion is especially easy if F Digital Design: Principles and Practices 3rd Edition. Understanding Digital Signal Processing, 3rd Edition.

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