The root locus method

This is a controi system design technique deveioped by W.R. Evans (i948) that determines the roots of the characteristic equation (ciosed-ioop poies) when the open-ioop gain-constant K is increased from zero to infinity. 

The iocus of the roots, or ciosed-ioop poies are piotted in the. v-piane. This is a compiex piane, since. v = cr jw. It is important to remember that the reai part a is the index in the exponentiai term of the time response, and if positive wiii make the system unstabie. Hence, any iocus in the right-hand side of the piane represents an unstabie system. The imaginary part uj is the frequency of transient osciiiation. 

When a iocus crosses the imaginary axis, cr = 0. This is the condition of marginai stabiiity, i.e. the controi system is on the verge of instabiiity, where transient osciiia-tions neither increase, nor decay, but remain at a constant vaiue. 

The design method requires the ciosed-ioop poies to be piotted in the. v-piane as K is varied from zero to infinity, and then a vaiue of K seiected to provide the necessary transient response as required by the performance specification. The ioci aiways commence at open-ioop poies (denoted by x) and terminate at open-ioop zeros (denoted by o) when they exist. 

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The root locus method provides a very powerful tool for control system design. The objective is to shape the loci so that closed-loop poles can be placed in the. v-plane at positions that produce a transient response that meets a given performance specification. It should be noted that a root locus diagram does not provide information relating to steady-state response, so that steady-state errors may go undetected, unless checked by other means, i.e. time response. 

Eq. (2-34) is used in the root locus method in Chapter 7 when we design controllers. 

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. 

Thus the system is closedioop stable if K, > 1/Kp. This is exactly the conclusion we reached using root locus methods. So the Chinese frequency-domain conclusions are the same as the Russian Laplace-domain conclusions. 

For a more complex problem, the characteristic polynomial will not be as simple, and we need tools to help us. The two techniques that we will learn are the Routh-Hurwitz criterion and root locus. Root locus is, by far, the more important and useful method, especially when we can use a computer. Where circumstances allow (/.< ., the algebra is not too ferocious), we can also find the roots on the imaginary axis—the case of marginal stability. In the simple example above, this is where Kc = a/K. Of course, we have to be smart enough to pick Kc > a/K, and not Kc < a/K. 

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 state space state feedback gain (K2) related to the output variable C2 is the same as the proportional gain obtained with root locus. Given any set of closed-loop poles, we can find the state feedback gain of a controllable system using state-space pole placement methods. The use of root locus is not necessary, but it is a handy tool that we can take advantage of. 

Among other methods, root locus is the most instructive in this case. With a PI primary controller and numerical values, Eq. (10-3) becomes... 

First we will look at the question of stability in the z plane. Then root locus and frequency response methods will be used to analyze sampled-data systems. Various types of processes and controllers will be studied. 

We could make a root locus plot in the U) plane. Or we could use the direct-substitution method (let U) = iv) to find the maximum stable value of. Let us use the Routh stability criterion. This criterion cannot be applied in the z plane because it gives the number of positive roots, not the number of roots outside the unit circle. The Routh array is... 

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. 

Root locus plots are easy to generate for first- and second-order systems since the roots can be found analytically as explicit functions of controller gain. For higher-order systems things become more difficult. Both numerical and graphical methods are available. Root-solving subroutines can be easily used on any computer to do the job. The easiest way is to utilize some user-friendly software tools. We illustrate the use of MATLAB for making root locus plots. 

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. 

The first two sampled-data controller design methods use conventional root locus and frequency response methods, which are completely analogous to the techniques in continuous systems. Instead of looking at the s plane, however, we look at the z plane. The third sampled-data controller design method is similar to the direct synthesis method discussed in Chapter 9. 

The design of digital compensators was discussed in this chapter. The conventional root locus, frequency response, and direct synthesis methods used in continuous systems in the j plane can be directly extended to sampled-data systems in the z plane. 

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