Root Locus Design Methods

With continuous systems we made root locus plots in the s plane. Controller gain was varied from zero to infinity, and the roots of the closedloop characteristic equation were plotted. Time constants, damping coefficients, and stability could be easily determined from the positions of the roots in the s plane. The limit of stability was the imaginary axis. Lines of constant closedloop damping coefficient were radial straight lines from the origin. The closedloop time constant was the reciprocal of the distance from the origin. 

Sometimes other planes beside the z plane are used. The log z and ID planes offer some advantages to the z plane for some systems. We, will discuss these later in this chapter. 

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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. 

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. 

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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