Root Locus Analysis

Root-locus analysis 5.3.1 System poles and zeros... 

Example 7.3B Repeat Example 7.3 (p. 7-5) with a root locus analysis. 

This is exactly what our root locus analysis showed. 

The entire curve is given in Fig. 19.10d. It crosses the negative real axis at = —0.087A,.. So the ultimate gain is = 1/0.087 = 11.6, which is the same result we obtained from a root locus analysis. 

The root-locus analysis cannot handle easily systems with dead time. Why Show how systems with dead time could be handled with root-locus analysis. 

Root-Locus Analysis 290 Things to Think About 295... 

Example 15.6 demonstrated that the root locus of a system not only provides information about the stability of a closed-loop system but also informs us about its general dynamic response characteristics as Kc changes. Therefore, the root locus analysis can be the basis of a feedback control loop design methodology, whereby the movement of the closed-loop poles (i.e., the roots of the characteristic equation) due to the change of the proportional controller gain can be clearly displayed. 

It is well known from classical root-locus analysis that as the feedback gain increases towards infinity, the closed-loop poles migrate to the positions of open-loop zeros. Thus, the presence of RHP-zeros implies high-gain instability. 

The utility of root locus diagrams has been illustrated by the third-order system of Examples 11.13 and 11.14. The major disadvantage of root locus analysis is that time delays cannot be handled conveniently, and they require iterative solution of the nonlinear and nonra-tional characteristic equation. Nor is it easy to display simultaneous changes in more than one parameter (e.g., controller parameters Kc and t/). For this reason, the root locus technique has not found much use as a design tool in process control. 

We have given up the pretense that we can cover controller design and still have time to do all the plots manually. We rely on MATLAB to construct the plots. For example, we take a unique approach to root locus plots. We do not ignore it like some texts do, but we also do not go into the hand sketching details. The same can be said with frequency response analysis. On the whole, we use root locus and Bode plots as computational and pedagogical tools in ways that can help to understand the choice of different controller designs. Exercises that may help such thinking are in the MATLAB tutorials and homework problems. 

In effect, we are adding a very large real pole to the derivative transfer function. Later, after learning root locus and frequency response analysis, we can make more rational explanations, including why the function is called a lead-lag element. We ll see that this is a nice strategy which is preferable to using the ideal PD controller. 

The system steady state gain is the same as that with proportional control in Example 5.1. We, of course, expect the same offset with PD control too. The system time constant depends on various parameters. Again, we defer this analysis to when we discuss root locus. 

One may question whether direct substitution is a better method. There is no clear-cut winner here. By and large, we are less prone to making algebraic errors when we apply the Routh-Hurwitz recipe, and the interpretation of the results is more straightforward. With direct substitution, we do not have to remember ary formulas, and we can find the ultimate frequency, which however, can be obtained with a root locus plot or frequency response analysis—techniques that we will cover later. 

The idea of a root locus plot is simple—if we have a computer. We pick one design parameter, say, the proportional gain Kc, and write a small program to calculate the roots of the characteristic polynomial for each chosen value of as in 0, 1, 2, 3,., 100,..., etc. The results (the values of the roots) can be tabulated or better yet, plotted on the complex plane. Even though the idea of plotting a root locus sounds so simple, it is one of the most powerful techniques in controller design and analysis when there is no time delay. 

Root locus is a graphical representation of the roots of the closed-loop characteristic polynomial (i.e., the closed-loop poles) as a chosen parameter is varied. Only the roots are plotted. The values of the parameter are not shown explicitly. The analysis most commonly uses the proportional gain as the parameter. The value of the proportional gain is varied from 0 to infinity, or in practice, just "large enough." Now, we need a simple example to get this idea across. 

To determine the shape of a root locus plot, we need other rules to determine the locations of the so-called breakaway and break-in points, the corresponding angles of departure and arrival, and the angle of the asymptotes if the loci approach infinity. They all arise from the analysis of the characteristic equation. These features, including item 4 above, are explained in our Web Support pages. With MATLAB, our need for them is minimal. 

In terms of controller design, the closed-loop poles (or now the root loci) also tell us about the system dynamics. We can extract much more information from a root locus plot than from a Routh criterion analysis or a s = jco substitution. In fact, it is common to impose, say, a time constant or a damping ratio specification on the system when we use root locus plots as a design tool. 

On the other hand, frequency response analysis cannot reveal information on dynamic response easily—something root locus does very well. Hence controller design is always an iterative procedure. There is no one-stop-shopping. There is never a unique answer. 

Root locus method gives us a good indication of the transient response of a system and the effect of varying the controller gain. However, we need a relatively accurate model for the analysis, not to mention that root locus does not handle dead time as well. 

With the Routh-Hurwitz analysis in Chapter 7, we should find that to have a stable system, we must keep Kc < 7.5. (You fill in the intermediate steps in the Review Problems. Other techniques such as root locus, direct substitution or frequency response in Chapter 8 should arrive at the same result.)... 

Do the root locus plots in Example 10-l(d). Confirm the stability analysis in Example 10-... 

In recent years a number of commercial programs have been developed that produce root locus plots (and provide other types of analysis tools). These software packages can speed up controller design. Some of the most popular include CC, CONSYD, and MATRIX-X. We will refer to these packages again later in the book since they are also useful in the frequency and z domains, as well as for handling multivariable systems. /... 

The linear model permits the use of all the linear analysis tools available to the process control engineer. For example, the poles and zeros of the openloop transfer function reveal the dynamics of the openloop system. A root locus plot shows the range of controller gains over which the system will be closedloop-stable. 

Frequency response or root-locus techniques for the analysis and synthesis of sampled-data control systems have not been included in this text. The procedure is analogous to that for continuous systems. For more information the reader can consult Chapter 15 in Luyben s text [Ref. 11]. 

Part IV (Chapters 13 through 18) covers the analysis and design of feedback control systems, which represent the control schemes encountered most often in a chemical plant. Emphasis has been placed on understanding the effects which various feedback controllers have on the response of controlled processes, and on the selection of the most appropriate among them. The subject of controller tuning has been deemphasized, and as a consequence, the traditional root-locus techniques and frequency response tuning methods have been scaled down. 

Let us close this chapter with one more example of the construction of the root locus for a reactor system and its use for the analysis of the system s dynamic response. 

Representing the roots of the characteristic equation in the complex domain offers a simple way to perform a stability analysis. The system is stable if and only if all the poles are located in the open left-half-plane (LHP). If there is at least one pole in the right-half-plane (RHP), the system is unstable. The representation is similar wiA the well-known root-locus plot used to evaluate the stability of a closed-loop system. 

The commercial software MATLAB makes it easy to generate root locus plots. The Control Toolbox contains programs that aid in this analysis. We illustrate in the following example the use of some simple MATLAB programs to generate root locus 1 plots. Similar programs will be used in Chapter 11 to compute frequency response results. 

In Chapter 14 we define mathematically the sampling process, derive the z transforms of common functions (learn our German vocabulary), develop transfer functions in the z domain, and discuss stability. Design of digital controllers is studied in Chapter 15 using root locus and frequency response methods in the z plane. We use practically all the stability analysis and controller design techniques that we introduced in the Laplace and frequency domains, now applying them in the z domain for sampled-data systems. 

For example, a Bode plot can be generated using the computer-generated transfer function or the A, B, C, D matrices in order to do a frequency response analysis. Root locus, pole placement, and other operations such as controllability and observability using the state space form are possible also using the model produced by the approach presented in this chapter. The result of the above matrix operations can be... 

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