Signal Processing and Linear Systems-B.P.Lathi copy

We observe t hat t he closer t he poles t o t he j w

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: t he r eady availability of computers, root locus can b e p roduced easily. Nevertheless, understanding these rules can b e a g reat help in developing the intuition needed for design. We shall present here the rules, b ut o mit t he proofs of some of t hem. We begin with a feedback system depicted in Fig. 6.42, which is i dentical to Fig. 6.18d, except for t he explicit representation of a variable gain K . T he s ystem in Fig. 6.36a is a special case w ith H (s) = 1. For t he s ystem in Fig. 6.42, T (s) _ K G(s) - 1 + K G(s)H(s) (6.89a) T he c haracteristic equation of t his s ystem is:j: tThis procedure was developed as early as 1868 in Maxwell's paper "On Governors". :j:This characteristi~ equation is also valid when the gain K is in the feedback path [I umped with H (s) ) rather than I n the forward path. The equatlOn applies as long as the gain K is in the loop at any point. Hence, the root locus rules discussed here apply to all such cases. 4 40 6 Continuous-Time System Analysis Using the Laplace Transform 1 + K G(s)H(s) = 0 6.7 Application t o Feedback and Control S tate Equations t he ready ~vailability o f computers and progran,s makes it much easier to draw actual locI. . T he first four rules are still very helpful for a quick sketching j t he root locI. 0 (6.89b) We shall consider the paths of the roots of 1 + K G(s)H(s) = 0 as K varies from 0 0. W hen t he loop is opened, the transfer function is K G(s)H(s). For this reason, we refer t o K G(s)H(s) as the o pen-loop t ransfer f unction. T he rules for sketching t he root locus are as follows. o to 1. R oot loci begin ( K = 0) a t t he open-loop poles and terminate on t he openloop zeros ( K = 0 0). T his fact means t hat t he number of loci is exactly n , t he order o f t he open-loop transfer function. Let G(s)H(s) = N (s)/D(s), where N (s) a nd D(s) a re polynomials of powers m and n, respectively. Hence, 1 + K G(s)H(s) = 0 implies D(s) + K N(s) = O. Therefore, D(s) = 0 when K = O. I n t his case, the roots are poles of G(s)H(s); t hat is, t he open-loop poles. Similarly, when K ~ 0 0, D(s) + K N(s) = 0 implies N (s) = O. Hence, the roots a re t he open-loop zeros. For t he system in Fig. 6.36a, t he open-loop transfer function is K / s(s + 8). T he open loop poles are 0 and - 8 a nd t he zeros (where K /s(s + 8) = 0 are b oth 0 0. We can verify from Fig. 6.41 t hat t he r oot loci do begin a t 0 a nd - 8 a nd terminate a t 0 0. Udnderstandin g these rul~s ~an be helpful in design of control systems as dem~ns(trate lat~r. T hey are an aId I II determining what modifications should be made or w hat ~md of ~ompen~ator t o add) t o t he open loop transfer function in order t o meet gIven deSIgn specIfications. • E xample 6 .21 Using t he fou~ rules of t he r oot loci, sketch t he r oot locus for a system w ith Io op transfer functIOn open K G(s)H(s) = K s (s + 2)(s + 4) 1. Rule 1: For this G (s)H(s) ' n - . Hence, t here are three root loci which begi t -3 t he poles of G (s)H(s); t hat is, a t 0, - 2 a nd - 4. ,na 2. Rule 2: T here are odd numbers of poles t o t he right of t he real axis segment s < -...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online