Root locus techniques

One of the basic limitations of root locus techniques is that deadtime cannot be handled conveniently, The rst-order Fade approximation of deadtime is frequently used, but it is often not very accurate. 

A related approach which has been used successfully in industrial applications occurs in discrete-time control. Both Dahlin (43) and Higham (44) have developed a digital control algorithm which in essence specifies the closed loop response to be first order plus dead time. The effective time constant of the closed loop response is a tuning parameter. If z-transforms are used in place of s-transforms in equation (11), we arrive at a digital feedback controller which includes dead time compensation. This dead time predictor, however, is sensitive to errors in the assumed dead time. Note that in the digital approach the closed loop response is explicitly specified, which removes some of the uncertainties occurring in the traditional root locus technique. 

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. 

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. 

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 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. 

The technique of using the damp ratio hne 0 = cos in Eq. (2-34) is apphed to higher order systems. When we do so, we are implicitly making the assumption that we have chosen the dominant closed-loop pole of a system and that this system can be approximated as a second order underdamped function at sufficiently large times. For this reason, root locus is also referred to as dominant pole design. 

For as instructive as root locus plots appear to be, this technique does have its limitations. The most important one is that it cannot handle dead time easily. When we have a system with dead time, we must make an approximation with the Pade formulas. This is the same limitation that applies to the Routh-Hurwitz criterion. 

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.)... 

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. 

Root locus diagrams can be quickly generated by using a hand calculator or a computer with root-finding techniques such as are provided in MATLAB. 

The locus taken by the roots of the characteristic differential equation of the load cell as the applied mass changes can be determined by automatic system identification techniques. Such a locus is illustrated in Fig. 7.5, and the roots of the compensating filter need to follow it. For each value of mass there is a corresponding final output of the compensating filter once oscillation has ceased. The trick is to make the parameters of the filter vary with its own output as dictated by the locus. 

---